The Rules of the Flock: Self-Organization and Swarm Structure in Animal Societies 2019947625, 9780198853398

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THE RULES OF THE FLOCK

THE RULES OF THE FLOCK Self-Organization and Swarm Structure in Animal Societies

Helmut Satz

1

1 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Helmut Satz 2020 The moral rights of the author have been asserted First Edition published in 2020 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2019947625 ISBN 978–0–19–885339–8 DOI: 10.1093/oso/9780198853398.001.0001 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY

Preface The past fifty years have witnessed the emergence of a new field of scientific research: the study of swarm behavior. The question which led to these studies is immediately evident: how is it pos­ sible that large numbers of simple individuals, each interacting only with a few near-by neighbors, produce dramatic large-scale collective behavior? Flocks of birds execute striking swarm maneuvers above us in the sky; schools of fish do the same in the depths of the sea. Glow worms in the Asian jungle perform light displays in which thousands of individual bugs radiate in perfect synchronization. These and various other, similar phenomena have led mathematicians and physicists to join forces with their colleagues from biology in the attempt to show that the under­ lying structure of animal swarms is in fact quite universal and many ways similar to that studied in the physics of many interacting particles. It is found that the formation and structure of a bird swarm is very much like the magnetization pattern of a piece of iron, in which the spins of most of the atoms suddenly point in the same direction. And the synchronization of the glow worm radiation is shown to use mechanisms quite similar to those leading to the radiation of light by a laser. On the other hand, ants, bees and other social insects have developed very efficient collective schemes for the solution of various problems, such as finding the shortest path between two points—schemes which since some decades have found their use in human logistics as well. And in addition, their swarm structure gives a new meaning to what survival of the fittest means. Evolution has here led to swarm constructs which completely modify the individual members: they are no longer autonomous animals, but have instead become parts of a superorganism, with specific functions and specific benefits. While an independent existence for members of mammal herds or bird flocks may have

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Preface

its difficulties, it is completely impossible for the members of social insect states, where collective efforts are necessary not only for the existence of the state, but for that of each member as well. What about human societies? Although the behavior of these is much more complex, there exist features which are evidently also the result of self-organization. In particular, evolution led to the creation of language as a tool to describe abstract as well as concrete aspects; this tool allowed the planning and organization which resulted in the human dominance over the entire Earth. The aim of this book is to describe the swarm behavior of ­animal societies and then confront it with the counterparts in physics and informatics. It is meant to address a general readership and it will therefore use very little and only very simple math­em­at­ics; also the physics and biology involved will be on a level accessible to non-specialists. We humans experience a deep sense of wonder when we see a whirling swarm of birds or a ­glistening school of fish, and a feeling of great amazement at the achievements of ant or bee states. The book wants to show that such feelings become still enhanced when we learn how the striking performances come about, when we see that in fact they follow general patterns which nature uses in the inanimate world as well. It is a pleasure to thank Irene Giardina for helpful comments on the STARFLAG project, to Johannes Fritz for providing photos of the Waldrapp project and to Susette von Reder for help in preparing the manuscript. Helmut Satz Bielefeld, March 2019

Contents 1. Introduction

1

2. The Eighth Plague

8

3. The Onset of Connectivity

15

4. The Birds of Rome

19

5. Spins and Magnets

27

6. The Rules of the Flock

34

7. Complexity and Criticality

42

8. Fiery Clouds in the Jungle

51

9. The Audible Silence

56

10. Coupled Oscillators

60

11. Laser Beams

65

12. The Way to Go

71

13. How to Fly

79

14. Avian Aerodynamics

89

15. The Ant State

95

16. Kin Selection: My Sister’s Keeper

102

17. Of Bees and Flowers

110

18. Communication and Language

115

19. Epilogue

122

Bibliography Person Index Subject Index

127 129 130

1 Introduction The locusts have no king, yet all of them go forth in ranks. The Bible, Proverbs 30:27

Moses led the people of Israel from Egypt to the Promised Land. Caesar’s legions conquered Europe. Genghis Khan’s hordes threatened the Occident. Napoleon’s army reached Moscow. The history of mankind always records that many were led by a few who ruled and decided. This has also had a formative effect on our view of the collective behavior in nature. When many “unimportant” individual objects combine to form a unified larger body, we are always ready to look for a king, a leader, a central initiator. But much of nature does not operate that way: locusts, bees, ants, fish, starlings, antelopes and many other animals form functioning societies, swarms, schools, flocks, herds and more, and yet these societies have no king, no ruler. The whole is more than the sum of its parts—that is well known at least since Aristotle. And that great builders can combine many small bricks to construct magnificent cathedrals has also been shown often enough. But that many small identical entities can come together completely on their own, without any initiator, without any plan, to create a new form, endowed with its own, special properties, that is a rather novel discovery in natural science. The key words are emergence and self-organization. Locusts, up to a certain point solitary individuals, suddenly come together to form immense swarms that darken the sky and devour all that is edible. Thousands of fireflies in the Asian jungle send out absolutely synchronized light signals, all flash at the same instant, without

The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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The Rules of the Flock

any coordinator. Ants establish complicated road systems, without any planning authority. Birds and fish form extensive flocks or swarms, consisting of thousands of animals, swarms which expand, contract and execute complex maneuvers in space—again without any leader or organizer. In all these cases it is of little or no help to study an individual animal in arbitrary detail, to determine all its ways of functioning; from such knowledge one can never derive or predict the observed collective behavior. It is the combination of the many simple and similar components that results quite on its own in a totally unexpected behavior of the whole, which as such acquires its own properties, its own existence. It came as some surprise to physicists as well as to biologists that such phenomena are in fact found in both areas, that selforganization is a much more general, transdisciplinary concept. The Nobel laureate Ilya Prigogine and his colleague Gregoire Nicolis in Brussels wrote in 1977 that “complexity is no longer confined to biology, but is invading physical sciences as well.” At the same time, their colleague Jean-Louis Deneubourg, working in Brussels as well, first introduced mathematical models to study the detailed behavior of ant colonies: mathematics was invading biology. The investigation of self-organization as a general scientific research field, reaching beyond statistical physics, thus entered the stage in the last quarter of the past century, less than fifty years ago. It actually started with mathematical models of complexity, in which many identical simple objects moved according to very simple rules: one found that these mindless “robots” indeed showed patterns of behavior similar to those observed in many animal and sometimes also in human congregations. Less than a hundred years ago there were still proposals that had flocks of birds communicate by telepathy, or that attributed the synchronized flashing of the fireflies to a blinking of the eyes of the observer. Today one can show that already very simple ­models necessarily lead to such behavior. Their robots are inanimate objects, devoid of any reasoning; they only follow strictly a few

Introduction

3

very simple rules, and “voilà” the unexpected collective behavior appears. It is unexpected only because we still don’t have much experience with self-organization. In physics and chemistry one has an excellent understanding of the atoms which form the different elements. They are bound states of positive nuclei and negative electrons; one can calculate the orbits of the electrons around the nuclei, determine the size of the atoms and the details of their structure. But all that helps little when we are faced by a system of very many such atoms. One iron atom is just that; but many iron atoms can form a magnet (at low temperature) or not (at high temperature). Helium is the only element first observed in the sun, hence the name. It never becomes a solid, it never freezes; at very low temperatures it forms a perfect liquid flowing without any resistance. Neither for iron nor for helium did the knowledge of their atomic structure lead to any predictions of the striking properties observed when they form matter: these concern collective behavior and were found only in separate, specific studies. Even in the inanimate nature then, collective properties can in general not be derived from an understanding of the individual components, however perfect that understanding may be. Not only is the whole more than the sum of its parts: the parts on their own can combine to form a completely new entity with completely unexpected properties. These often make us stand back and wonder. How can the thousands of birds in a flock manage to coordinate their dance in the sky? As mentioned, serious biologists went as far as to think of telepathy. And how can thousands of fireflies in the Asian jungle flash their lights simultaneously with a precision of thousands of a second? Swarms must have secret codes of their own, by which they achieve these striking performances. Our challenge then is to decode their behavior, to figure out how they achieve the order we see. One way to address this challenge, as we had already indicated, is to study what happens if we consider mindless machines, robots,

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The Rules of the Flock

programmed to just obey simple rules: follow your neighbor, move as fast as it does, but don’t collide with it. And today this does not even have to take place in the real world: we can do it through computer simulation, by letting points on a computer move according to the proposed rules. If such rules suffice to obtain a set of virtual robots living on the hard disk of a computer to behave just like an animal swarm, then we feel rightly justified to conclude that the behavior of the swarm is not based on causal decisions of the individuals which form it. In the past three decades, the theoretical investigation of selforganization and of collective swarm behavior has experienced a dramatic growth, through the advent of computer simulation as well as more generally through the study of mathematical m ­ odels of swarms. Every year hundreds of scientific works appear, with the aim to explain in mathematical terms the collective behavior of many individual beings, from microorganisms to herds of antelopes. At the same time, biologists have provided more detailed, quantitative experimental studies of the behavior of animal so­ci­ eties, of birds, ants, crickets and more. As an outcome, a new science has appeared, the science of swarm behavior. It is obviously an interdisciplinary field, in which biology, physics and math­em­ at­ics overlap. In this book, I want to explain to the general reader what problems and questions the investigations of this science address and how one may hope to answer them. There are four main areas to be considered: swarm formation, swarm structure, swarm synchronization and swarm roads, and they are to be addressed both in experiment—field and laboratory studies—and in theory—through mathematical models and computer simulation. As mentioned, I will keep the tech­nical level of the book very general and qualitative. For the reader interested in more details, I will address one essential aspect, critical behavior of complex systems, somewhat more formally in Chapter 7. The basic issue in swarm formation is how many separate identical components can be combined to create one connected entity. The way we are familiar with uses a construction plan, telling us

Introduction

5

how to build a house out of many building blocks. In living beings, such a plan is built-in, genetic, determining how a plant or an animal will grow out of a few starting cells. A different, complimentary way would be to simply combine the components at random—placing stones without any plan on the squares of a chess board, or watching how falling rain drops will create a puddle of water. The study of the onset of such randomly created connectivity, percolation, is also a field which has witnessed a striking growth in the past fifty years. For our topic, this field is evidently of great interest: the locusts in a swarm, the birds in a flock are not put into their position according to some plan—they just assemble somehow. The next question is how the components of the swarm co­ord­ in­ate their motion. Feeding birds in a field move randomly, some here, others there; but when they are frightened and rise up to flee, they choose to adopt a common direction of motion. How do they agree on that? In the investigation of such swarm structure, two research efforts have played a decisive role, one the­or­­ etic­al, the other experimental. In 1995, Tamás Vicsek (Budapest) and collaborators have shown that bird swarms and magnetic iron actually follow rather similar laws of alignment. Since then, an impressive number of biologists, physicists and math­emat­icians have corroborated and extended these considerations, leading by now to a proper area of science on its own. On the experimental side, high statistics studies of starling swarms, carried out by a team of biologists and physicists from the University of Rome, have provided a firm empirical basis of the structure of bird swarms. The issue of swarm synchronization has been around for a long time, ever since large congregations of fireflies were seen to blink in rhythm; subsequently, specific species of crickets were found to form synchronized chirping communities. This synchronization must be the consequence of some dedicated timing device each animal has built-in. It is not possible that the animal blinks or chirps when it sees its neighbor doing that: the degree of precision of the synchronization, less than a thousandth of a second,

6

The Rules of the Flock

precludes that. It was only in 1990 that Steven Strogatz and Renato Mirollo showed that here the physics of coupled os­cil­lators provides a mechanistic framework leading to such syn­chron­ized behavior. The mechanism causing the fireflies to flash together is quite similar to that causing the turn signal on your car to flash, and the ultimate form of this type of synchronization is achieved in laser beams. The establishment of animal roads, particularly those of ants, has fascinated scientists for quite some time. Through collective pattern formation, ants can in fact very effectively determine the shortest route between nest and food source. The methods they employ can today also be simulated by robots, and this has in fact led to algorithms used in the solution of transport logistics problems. The crucial feature here is to have a number of individuals carry out trials and then let the swarm choose the most effective outcome of the different attempts: a scheme that only works through a combined effort of many, and one which does not need a leader. The paths taken by animals on the ground are quite readily studied. For birds the problem becomes more complex, and given the wonders of long-range biannual migrations, it is indeed quite a problem (as it is in fact also for fish migrations). Besides the issue of orientation, we would like to understand how birds can use collective efforts to minimize their energy expenditure enough to carry out their travel over truly immense distances. Only very recently has the advent of miniaturized electronic devices, together with the availability of ultra-light flying equipment, made it ­possible to accompany birds closely on their migration voyages. A joint Austrian–British team has just published pi­on­eer­ing results on this, showing that the birds indeed use optimal aerodynamic flock patterns. In the final parts of the book we consider the states of social insects, to show that many, if not most, of the activities we consider today as achievements of our own society do in fact exist also in insect societies of far greater number of members than

Introduction

7

ours. There are societies that have agriculture, stock raising, building construction and specialized armies. And there are so­ci­ eties which also show many of the negative sides of human life: they attack others, kill them or enslave them. But all these different activities are, in contrast to the human world, carried out in a collective, self-organized way. There never is a commander or a leader. In the case of social insects, however, the formation of a swarm has in the course of evolution basically modified the individuals that make it up. A swarm of birds or antelopes still consists of animals having an independent life, males and females, feeding, mating, raising offspring. In insect states, in contrast, the members fall into well-defined castes with well-defined functions. There is one queen who produces all the offspring, with the help of males who have only on sole function, to mate with the queen and then die. They cannot even find their own food, but have to be fed by sterile female worker bees. The Darwinian principal of evolution, the survival of the fittest, seems to be canceled: no matter how good the worker bees fulfill their tasks—building the hive, collecting food, feeding the larvae—they never have children to which they can pass on these capabilities. And neither the queen nor the drones which mate with her have ever shown any such ability. How can such a swarm structure ever arise? The answer, as we shall show, lies in the modified genetic structure of the swarm members. In summary, studying the organization and operation of swarms has become an immensely interesting field of investigation, a field where biology, physics and mathematics meet in order to provide us with an understanding of phenomena which at first sight seem almost miraculous. I hope to show that this sense of w ­ onder remains or even grows when one understands how these h ­ appenings arise and proceed.

2 The Eighth Plague The cloud was hailing grasshoppers. The cloud was grass­ hoppers. Their bodies hid the sun and made darkness. laura ingalls wilder, On the Banks of Plum Creek

Since times immemorial, locust swarms have been a threat to mankind. In the Near East, in Africa and in America, billions of locusts have periodically darkened the sky and devoured all that was edible. In the Bible, they became the Eighth Plague in Egypt; when the Pharaoh refused to let the people of Israel go, God warned I will bring the locusts into your land, and they shall cover the face of the earth, so that one cannot see the face of the earth, and they shall eat all which there is (Exodus 10, 4/5). And that is indeed what happened, there remained not any green thing in the trees, or in the herbs of the field, throughout all the land of Egypt. Many later happenings of this kind are recorded in con­sid­er­ able detail. A record-setting locust invasion took place in the United States in the latter half of the nineteenth century. At that time, huge swarms of the so-called Rocky Mountain locust attacked the prairie states of the USA, and this event was extensively docu­ mented by the local agricultural administrations. According to these records, a swarm of some 12.5 trillion locusts invaded the Midwest in the year 1875. It was the largest mass of animals ever recorded; the number of locusts in this swarm is more than a thou­ sand times the entire present human population of the earth. The swarm extended over a length of 2000 km and a width of 175 km. The animals moved through the country like a  snowstorm and devoured all that could be eaten, including blankets and clothing.

The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

The Eighth Plague

9

Rocky Mountain locust (Melanoplus spretus).

The American writer Laura Ingalls Wilder, quoted at the begin­ ning of this chapter, witnessed the invasion on the farm of her parents in Minnesota; her family, like many others, lost every­ thing and had to start new on land outside of the locust path. Nevertheless, the particular species of locusts involved in these prairie swarms did in fact not survive the subsequent invasion by humans: the onset of large-scale agriculture destroyed the condi­ tions for their reproduction, and the last such grasshopper was recorded in Canada early in 1900. The locust threat as such has remained, however; since then, locusts have devastated the fields of all continents. Since 2000, huge invasions have taken place in Argentina, Brazil, Israel and in Africa. As a result, many countries in Africa and in South America have to devote an ever-growing fraction of their national income to locust defense. In recent years, swarms of up to 500 billion locusts have invaded Madagascar. Since each animal devours per day an amount of about twice its body weight, these swarms destroy some 100,000 tons of plants daily, and that can endanger the food supply for entire populations. What kind of animals are these locusts? Initially, a locust is a solitary creature which avoids contact with others and peacefully goes its own way. Males and females meet briefly, copulate and then separate again; the female lays eggs, and out of these little grass­ hoppers emerge (the biologists call them nymphs), looking much like the adults, except that their wings are not yet ­developed.

10

The Rules of the Flock

Development of a grasshopper from nymph to fully grown adult.

In the course of the next few weeks, they grow and shed their rigid outer skins several times (they molt), before they are finally grown up, have fully developed wings and can fly. The adolescent nymphs already wander around in little groups and look for food, grass and other plants. The situation becomes problematic only through a peri­od­ic­ al­ly arising chain of climatic events. Sudden abundant rain falls in the home ranges of the locusts, such as the arid regions of Northern Africa or the Middle East, can lead in a short time to luscious vegetation. This in turn then results in an explosion of the locust population. The locust eggs were initially deposited in dry sand; now the warm ground has become humid—an en­vir­on­ment which in a very short period leads to the birth of huge numbers of little locusts. These wander around and devour every­thing edible

The Eighth Plague

11

in their vicinity, and so sooner or later food becomes scarce. The search now brings together more and more animals in an ever smaller area. At first, this leads to mutual repulsion, but eventu­ ally, when the grasshopper density surpasses some 75 insects per square meter, a sudden and dramatic transition occurs: the strong repulsion becomes an intensive attraction. The solitary loner becomes a social crowd-seeker, all locusts stream together and if an animal is accidentally separated from the crowd, it becomes frightened and quickly returns. The swarm is born—born, so it seems, simply through a critical dens­ity of insects. We shall return shortly to the determination of the critical dens­ ity; first we want to note that of course there also has to be some physiological trigger for the sudden change of character, and sci­ entists have indeed identified that. The close proximity and the contact with many next neighbors induce in the insect a sudden production of the hormone serotonin. The press has sometimes labeled serotonin the happiness hormone—here it indeed results in relaxation and mutual attraction. That was checked in the la­bora­tory: serotonin injections lead even for isolated locusts to an attraction to others, and removing individuals from the swarm caused their serotonin levels to drop. From a biological point of view, the relation between serotonin levels and swarm coherence is of course of great importance; however, the fact remains that in nature just a sufficient density of insects leads to swarm formation. Swarm formation begins already in the nymph stages of the locusts’ life cycle. Already before they are able to fly, the adoles­ cent locusts form swarms, “go forth in ranks” and devour every­ thing edible in their path. After the final molt they are then prepared to fly, so that locust clouds now darken the skies and attack the land before and below them: the biblical plague has started. And it does not stop: adult locusts with high serotonin levels also prod­uce offspring with high levels, which then again form swarms. For this reason one had initially thought that there were two distinct species: solitary grasshoppers and gregarious locusts forming swarms. That was not unreasonable, since the

12

The Rules of the Flock

Locust in solitary (top) and in gregarious state (bottom).

change of character even changes the appearance of the animals, in particular their color. Further studies then showed that there was indeed only one species which exists in two different states, depending on the population density, and that transitions from one state to the other were possible. By now, numerous species of locusts have been identified, on all continents, and all show the inherited possibility to exist in two distinct states. The population density determines the state and thereby also the appearance of the individual. To determine the critical density for swarm formation, a group of scientists (J. Buhl et al.) has carried out an extensive experiment with locust nymphs, that is, animals which could not yet fly. Different num­ bers of these were placed in a circular arena and tracked with video cameras and computers over long periods of time. It was found that for densities up to some 10–15 animals per square meter, the insects just moved around randomly and ignored each other. When the density was increased further, small groups of insects formed, with the members of each group marching together in little circles pointing in the same direction; however, the different groups moved in arbitrary ways, independent of each other. This

The Eighth Plague

13

pattern continued until the density reached some 75 insects per square meter: at that point and from then on, the different groups joined forces and eventually all marched in a common circular path around the arena for the entire duration of the experiment. To study the onset of swarm formation, it is of course helpful to quantify this behavior a bit more. The locusts are marching around in a circular arena, and so much of the time they are on circular paths. We can therefore form two groups: given N insects per square meter, N c travel clockwise, while N cc move counter­ clockwise, with N c + N cc = N. The difference N c - N cc then tells us if there is an overall preference or order, and if we divide this differ­ ence by the total number N, we obtain a measure for such order, D= N c N cc / N. The vertical bars in |x| here mean that we only consider the size of the difference, ignoring whether it is positive or negative: using mathematical terminology, we denote by |x| the abso­ lute value of x. In the case of randomly moving locusts or locust groups, as many move clockwise as anticlockwise, so that Δ = 0; for the other extreme, a unified swarm of all members moving in the same sense, clockwise or counterclockwise, we get Δ = 1. Hence overall disorder is indicated by Δ = 0, complete order by Δ = 1. Plotting this behavior as a function of the number of locusts, one thus obtains the pattern shown in the figure, with a rather sudden onset of swarm formation around N = 75 animals per square meter. This type of behavior is quite well known in phys­ ics. The density of water suddenly changes at 100 degrees centi­ grade, where evaporation sets in and turns the liquid into gas, to quote a familiar example. Such phenomena are generally called phase transitions; they indicate that the state of the system changes from one form to another. From this point of view, the locusts undergo a phase transition from the solitary to the gregarious state once their density reaches some 75 animals per square meter. The behavior of the animals thus shows a “tipping point,” a critical density at which previously separate animals join together

The Rules of the Flock

14 1.0

∆

0.5

N 50

100

Swarm order measure Δ as function of the number N of insects per square meter.

to form one connected swarm. How can that happen? The transi­ tion from solitary to gregarious behavior of locusts is evidently a biological process, a change in the behavior of living beings. But the formation of a connected whole out of many separate identi­ cal components, whatever their nature, appears to be something more general. It is therefore reasonable to ask if the onset of con­ nectivity cannot be studied abstractly, detached from the specific case on hand, invoking mathematical models. The simplest form of such an approach will be addressed in the next chapter.

3 The Onset of Connectivity E pluribus unum (Out of many, one) Great Seal of the United States of America

Great structures are usually created by combining many small components. Following a building plan, houses are made of building blocks. Matter consists of atoms, and these in turn have electrons circling around a central nucleus. In crystals, such atoms are then combined in well-defined geometric ways, such as in a cubic structure. In these and numerous similar cases, the com­ bin­ation of the constituents follows definite rules. Here we now want to consider a basically different form of connection: how can a connected structure appear out of randomly distributed identical components, without any construction plan or manual? Can order be created in a random way? Let us randomly throw small circular disks, like beer coasters, onto a table having a size of one square meter. We assume each disk to have a size of 100 square centimeters, implying a radius of a little less than 6 cm. When we throw the disks, some will land partially on top of each other—but that doesn’t matter. In the following picture, we show three successive configurations obtained in this way. Besides randomly distributed single disks, we get islands of two or more overlapping disks. As we keep throwing, the islands continue to increase in size; they consist of more and more overlapping individual disks. And then, suddenly, throwing one more disk, we find an island extending from one side of the table to the other. Besides this big island, there still remain other, smaller ones, but the big one now bridges The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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The Rules of the Flock

Successive distribution of disks on a table.

the table, connects opposite sides. How many disks do we have to throw to have that happen? The result will differ a little from game to game, but given the sizes of table and of disks, the relevant mathematical theory tells us that we need some 110 disks. The sum of the required disks thus corresponds to an area somewhat larger than that of the table: that is the effect of the partial overlap of disks, so that there still remains some open space. If we continue throwing, that space will become ever smaller, until finally the entire table is covered. Mathematicians and physicists call such a sudden onset of connectivity percolation; to percolate means to flow through something. In percolation theory one studies how a random combination of separate identical components can abruptly produce a connected whole. One consequence of percolation is that now even components far removed from each other become correlated. Percolation is today a very active and timely area of research, with applications ranging from primordial matter shortly after the big bang to the formation of galaxies in the early universe. When atomic nuclei are compressed to high density, a new state of matter is formed, the quark plasma, which made up the very early universe. And when gas clouds in the later universe were attracted to each other and merged, that led to the appearance of galaxies. In today’s world, numerous technological problems, from the construction of networks to the detection of oil fields and the prevention of forest fires, require applied percolation theory for their solution.

The Onset of Connectivity

17

In the USA, machines for making coffee are referred to as percolators: they inject boiling water into a container filled with ground coffee, and when the amount of water has reached a certain limit, it suddenly flows out as coffee. Something similar happens when we water potted flowers: the water seeps in, is absorbed, and then suddenly flows out at the bottom of the pot. The surprising thing in these phenomena is their sudden onset: it’s not that first a few drops, a little water or coffee comes out, and then more and more—rather we get first nothing and then abruptly full flow. This is just what we had found in our game with the beer coasters. First there was a disk here, another there, even some overlapping disks, but still much of the table remained empty. Only when we got near the critical density the covered area suddenly increased very much and reached almost the same dimension as that of the table. If we keep throwing, eventually the entire table will be ­covered. This behavior is illustrated in the figure below. We obtain a variation of our beer coaster game if the disks are metallic and if there is a voltage difference between the opposite sides of the table. In that case, when the final disk thrown establishes the connection between the two sides, current will start to flow. Such a sudden onset of electrical conductivity is therefore also an example of percolation. Still another one, beloved by the percolation theorists, has the table become a pond and the disks

% covered area

100

0 percolation point

disk density

Throwing disks onto a table: increase of area coverage with disk ­density.

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The Rules of the Flock

water lilies on the pond. Percolation then means that an ant can now cross the pond without getting its feet wet. The formation of a connected whole by identical independent pieces, as described by percolation theory, is thus in a sense the simplest and most general form of swarm formation. The individual pieces don’t interact in any way—the onset of connectivity occurs simply when the density reaches a certain value; but at that point, we find critical behavior. We thus conclude from these considerations that combination of many identical individual components to a connected whole does not occur gradually, but it rather happens quite abruptly. And the similarity of the above figure with that of the swarm formation of locusts discussed in the previous chapter is certainly not accidental. If we assume that in its solitary state each locust defines an area of about 100 square centimeters as its proper territory, a region in which it does not tolerate neighbors, then with some 110 locusts per square meter a dense connected system would be created. A serotonin current could now flow and the gregarious swarm would be born. The actual numbers are of course a bit arbitrary—but if we put a locust on each beer coaster, disk percolation would correspond to swarm formation. Nevertheless, many open questions remain to be solved concerning the understanding of swarms—questions which simple percolation theory cannot answer. After all, the swarm is not just a mass of animals somehow squeezed together: they also all move together in the same direction. How does the mass of randomly crawling animals, one going here, the other there, manage to form an orderly marching group, with all moving at the same speed in the same direction? That will be our next theoretical issue; but first we want to illustrate the problem by a very similar case, the flocking of birds.

4 The Birds of Rome

Photo: Tommy Hansen

The Palazzo Massimo lies in the center of Rome, across from the Central Station Roma Termini, and it contains one of the largest collections of antique Roman and Greek art. In three winters, from 2004 to 2006, rather unusual events took place there. A group of scientists from the Roman University La Sapienza, biologists and physicists, had installed high speed cameras on the roof of the Palazzo, in order to carry out a remarkable pioneering project. The object of their study was a well-known European bird, the common starling (Sturnus vulgaris). These birds spend the summer in central Europe and then migrate in the winter to the milder The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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The Rules of the Flock

Common starling (Sturnus vulgaris).

climate of the region in central Italy. Starlings from many different parts of Europe come together here, forming congregations of millions of birds. During the day they look for food in the fields of the surrounding countryside, and at night they come to sleep in the trees of parks around the railroad station. However, for reasons still completely unknown even today, every evening the birds form huge swarms which carry out large aerial maneuvers before they finally settle to rest. For up to half an hour, the swarms circle, expand, contract, rise and descend again, thus creating large mobile geometric formations in the sky above Rome. How these swarms form and how they are co­ord­in­ated had always been enigmatic—there certainly was no choreographer who planned this dance of the birds. The swarms must somehow organize that and their motions by themselves, through some kind of information exchange between individual birds. How this actually takes place had puzzled biologists for a long time, and up to the Roman study only sparse and rather inconclusive data dealing

The Birds of Rome

21

with some tens of birds was available. On the other hand, the computer theorist Craig Reynolds in California had developed a program in which he let individual agents (he called them boids, abbreviating birdoids) move according to a simple local behavioral rule: • Stay with your neighbor boids, steer in the same direction as they do, but don’t crowd them. The result of such a program was the formation of boid aggregates moving around in a collective, swarm-like fashion. It was clearly necessary to obtain more detailed information on the structure and behavior of actual bird flocks. The aim of the mentioned research project—it was called STARFLAG and supported by the European Union—was to provide the lacking empirical basis for a quantitative study of the phenomenon. When my colleague Giorgio Parisi, the co­ord­in­ ator of the project, told me about their results, I noted that it should be possible to understand their findings in terms of similar structures in statistical physics. Giorgio replied that with such an idea I would unfortunately be too late—it had already been proposed some ten years ago, even before the Roman data had become available. The Hungarian physicist Tamás Vicsek and his collaborators had developed a mathematical model similar to those used in spin physics, with the aim to understand the swarm behavior of different animal congregations, of birds, of fish and others. We shall return to this model, published in 1995, in more detail in a subsequent chapter. The communication within a flock of birds has, as mentioned, puzzled scientists for quite some time. How is it possible that so many birds simultaneously start a turn or simultaneously land? How is the information passed on between members of the flock? A hundred years ago, the British ornithologist Edmond Selous even proposed that some kind of telepathy must be at work, that there is some collective intelligence connecting all the birds with each other. Today, however, statistical physics has shown us that short-range interactions between adjacent particles can lead to

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The Rules of the Flock

striking long-range consequences between distant partners. The intrinsic local range of interaction between two neighbors and the resulting global range between distant birds in the flock are indeed quite different issues. But let us return to the starlings in Rome. They were photographed in flight, from the top of the Palazzo Massimo, 30 m above ground, simultaneously by two cameras some 25 m apart, with ten frames a second in sequences of 8 seconds each. The swarms consisted of 100–4000 birds and were typically about 100 m away. The aim was to obtain a running three-dimensional record of all the birds in the space of their flight. As we had mentioned, this had previously been tried with much smaller groups, ten birds or so, but the evaluation of even these attempts soon hit measure-technical limits. The Roman group here had decisive advantages. They had access to the analysis methods used by nuclear physicists to study the production of many particles in high energy collisions. In these processes, the collisions of two protons results in the production of hundreds, even thousands of additional particles, whose tracks were electronically recorded. The analysis of such phenomena was carried out on large highperformance computers, using algorithms specifically designed for such analyses. This form of investigation was applied to the pictures taken of the bird swarms, and it indeed turned out to be successful: the positions and directions of flight of the birds within the large swarms could be completely specified. That meant a true break-through in swarm studies. First of all, one could now determine the size and form of the starling flocks. They were not, as it had looked from the ground, bubbles, spherical or egg-shaped. Instead, they were more like thin pancakes, moving largely parallel to the surface of the Earth. Any expansion upward required work against the force of gravity, and that the birds tried to avoid as much as possible. In the plane parallel to the Earth, there were wide and as well as long flocks, often changing from one form to the other. In contrast to right turns in the military, in which the inner soldiers almost stop, while the

The Birds of Rome

23

A turning flock of starlings.

outer turn faster, bird swarms maintain their form in the turn, only the individual birds change their direction of flight. The same pattern applies in motions upward or downward. So, one began to understand fairly well the motion of the swarm as a whole. But how is that related to the motion of the individual members? It turned out that the crucial feature for each bird was the behavior of its immediate neighbors. Neighbors were, for this purpose, the closest birds flying in a layer next to each other. The birds could not see what was behind them, and also their view to the front was limited, since their eyes are on the sides of their heads. Toward the front they moreover they kept a certain distance simply to avoid collisions. For the distribution of birds within the flock one thus got the following picture: around any given bird there arose an oval pattern, determined by the closest neighbors. The more neighbors one included, the more this form became spherical. The deviation from such a spherical distribution thus specified with how many other birds our given one, let’s call it A, was in direct connection. And that, according to Andrea Cavagna and Irene Giardina, STARFLAG spokespersons, was just six or seven. Seven starlings form the direct interaction nuclei within each the flock. Why not more? One possible answer would be that starlings can only count to seven . . . But in addition, things

24

The Rules of the Flock

were a bit more complicated. The six neighbors of bird A were not always a certain distance away; in fact, distance did not matter, the relevant neighbors were the six closest birds, no matter how far away they were. This definition of neighbor was of course crucial as the swarm expanded or contracted. We can thus picture the swarm as knots (birds) connected by elastic strings. Such constructs have been a research topic in statistical physics for many years, and we turn to this in detail in the next chapter. Here we briefly return to a question which, as we had already mentioned, so far remains without an answer. Why do the birds perform these evening maneuvers every day? We just can’t find a reason. Are they just happy to be there, do they celebrate a day well-done? For fish, the formation of swarms reduces the danger to fall prey to a predator, so the swarm makes sense. The same holds true for herds of antelopes, since more eyes see more, or for swarms of insects searching for food. Even for birds traveling across the countryside, a swarm would increase the safety of all. So in general, swarms mean advantages of some kind. Only the starlings on their goodnight flights just seem to enjoy the party. The second interesting quantity was the correlation range within the swarm. When the swarm expanded, there would be large groups of birds moving to the right on the right side of the swarm and another group moving to the left on the left side. In a contraction, the opposite would take place. One could thus draw an im­agin­ary line through the swarm, separating left and right and thus defining correlated regions. It turned out that these regions grew without limit with the size of the swarm—the larger it was, the larger was the set of birds correlated to each other, even though each bird interacted directly with only its seven nearest neighbors. The third important observable of the swarm was what the researchers called its polarization. It specifies the overall direction in which the swarm is moving and can be obtained by averaging over the directions of all birds. When the swarm is on the ground, each bird randomly moving around and picking up food here and there, the polarization is zero—there is no preferred direction;

The Birds of Rome

25

Correlation regions within a swarm.

a physicist would denote this as rotational invariance. If the birds are scared, say by a loud noise, they all rise up and fly in a certain direction: the polarization now attains a specific value, ideally unity, if all birds fly precisely in parallel. Which direction the birds chose is a priori arbitrary; the crucial thing is that they all fly in one and the same direction. The physicist would now speak of a spontaneous breaking of the rotational symmetry—spontaneous because there is no-one telling them to line up and in which direction to go. One essential aspect of swarm structure has already been mentioned—let us emphasize it again. When the swarm turns, the operation involves birds much further away from each other than their direct interaction distance. That is just what makes swarms so interesting: short-range interactions between neighboring birds give rise to much longer range effects—the entire swarm turns. The creation of global effects through local connections is in a way the basis of swarm theory. Something quite similar has been observed for mammals in Africa. In the Serengeti in Tanzania, large herds of wildebeest antelopes migrate annually between summer and winter pastures, forming herds of more than 100,000 animals. These herds start their march in a wide front, which in time acquires a wave-like form.

26

The Rules of the Flock

Migrating wildebeest herd in the Serengeti, East Africa (Sinclair, 1977).

Here as well only a small number of animals are in direct contact with each other, and yet the wavelength, the distance between peak and valley of the resulting formation, is much greater, as seen in the following photo. One has found that such a behavior is obtained if the members of the herd just follow three simple rules: • Speed up or slow down when your neighbor does, • If you fall back more than a certain distance, catch up, • If you move ahead more than a certain distance, slow down. These rules suffice to create the wave structure of the wildebeest herds. Some disturbance causes an animal to briefly slow down or speed up. And if all animals follow the mentioned rules, this leads to the observed pattern—with variations which are not of the size of the local interaction distance between individuals, but much larger. The wave length of these variations is determined by the density of the herd and is much greater than the distance any specific animal can see. So we find also here that short-distance interactions lead to long-range consequences. In the next chapter we shall see that this is not only a crucial feature of swarm formation, but that it is in fact a fundamental result of statistical physics.

5 Spins and Magnets Natura non facit saltus (Nature does not make leaps) Carl von Linné, Philosophia Botanica (1751)

Since antiquity, natural philosophy assured us that nature does not make sudden jumps in the evolution of processes. Before Carl von Linné wrote down the above statement, Gottfried Wilhelm von Leibniz and Sir Isaac Newton had developed infinitesimal calculus, a mathematical formalism which is based on a sequence of infinitesimally small steps, so that any progress is continuous. Ever since then, one denotes in mathematics any deviation from such a continuity as singular, and the points at which nature dared to jump after all are called singularities. Physicists call such a behavior critical; it happens at critical points. In our everyday world there are many such breaking points. A  rope stretches until it suddenly tears, a beam bends until it breaks, and one can think of many more such instances. In physics, the best-known critical point, with the greatest influence on the development of the science, is the Curie point: it is the temperature at which iron turns magnetic. Magnets were known already long ago in China as well as in ancient Greece and Rome—pieces of rock which pointed to the North if one allowed them to swing freely. For the Romans these stones came from the Greek city Magnesia and hence they called them magnets. It was also found that this mysterious behavior could be transferred to other pieces of iron by touch. The explanation of this miraculous phenomenon, which through the invention of the compass soon became crucial for navigation,

The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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28

was found only many years later. Iron consists of atoms, which can be pictured as small spinning tops—each atom has an inherent spin which points in a certain direction. At high tem­per­at­ures, the spins of neighboring atoms point in random directions, so that the average over a region of many spins becomes zero, there is no preferred overall orientation. One therefore speaks here of  disorder; the state is called paramagnetic, anticipating what is to come. With decreasing temperature, the spins are more and more attracted to each other. Specifically, neighboring spins like to point in the same direction; however, the fluctuations induced by temperature counteract this. But below a certain temperature, now called the Curie temperature, the attraction wins and the majority of the spins do point in the same direction. In general, that can be any direction, except on Earth, where they point to the North, since the Earth as a whole has a magnetic field orient­ed in the direction of its axis of rotation, and it therefore forces the atoms to do the same. Below the Curie point, the average over the spins is thus no longer zero, but reaches a finite value: we have a ferromagnet, and that is the beginning of order. For iron, that transition occurs at 768 degrees centigrade. And we know today that at the lower end of the scale, when the temperature reaches its value of absolute zero (−273 degrees on the centigrade scale), all spins point in the same direction, the order has become perfect. It  was shown both experimentally and in many mathematical

(a)

(b)

(c)

Transition from a paramagnetic state above the Curie temperature (a) to a ferromagnetic state below that temperature (b) and a state of perfect order at absolute zero (c).

Spins and Magnets

29

formulations that any spin interacts only with its nearby ­neighbors, it knows nothing about the orientation of distant spins. The order which arises below the Curie point thus shows that in many-body systems local, spatially restricted interactions lead to global, ­far-reaching effects. The transition between the disordered paramagnetic and ordered ferromagnetic states was first described by the French physicist Pierre Curie in his doctoral dissertation of 1895. It was only the first in a number of discoveries associated with the name of Curie. In the same year 1895 Pierre married his Polish colleague Maria Sklodowska, and shortly afterward the couple determined the basis of radioactive radiation. For this they and Henri Becquerel received the 1903 Nobel Prize in physics. In 1911, Marie Curie was also awarded the Nobel Prize in chemistry, making her the first (and so far only) female scientist holding two such prizes. Her

Pierre Currie (1859–1905).

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The Rules of the Flock

husband Pierre had already died before this, in 1905. And then, in 1935, their daughter Irene and her husband Frederic Joliot were honored with the Nobel Prize in physics, for the discovery of artificial radioactivity. All in all, the family Curie thus holds five Nobel Prizes. The phenomenon observed in iron and other metals at the Curie point, the transition from paramagnetic to ferromagnetic behavior, is in physics generally called spontaneous symmetry breaking. The state of iron at high temperatures is symmetric, since the spins of its atoms are randomly oriented, with as many pointing up or down, left or right, and so on. The average over all spins thus is zero. Below the Curie point that is no longer the case; there now exists a preferred direction, in which most spins point: the symmetry is broken, the average is no longer zero. It is more­over ­broken spontaneously, since no one was there to force the spins into a given direction. And calculations show that although spins only interact with nearby neighbors, even far distant spins now point in the same direction—the symmetry breaking is due to selforganization, to a long-range collective effect in the many-body system, caused by local few-body interactions. Here we should make a small detour, in order to avoid objections from experts. Why is iron at room temperature, well below the Curie temperature of 768 degrees centigrade, not always magnetic? Why does it become magnetic only when subjected to an external magnetic field? This puzzle was solved in 1906 by the Alsatian physicist Pierre-Ernest Weiss. Decreasing the temperature below the Curie point does not generally lead to one single large magnetic region, in which all spins point in the same direction. Instead numerous smaller areas form, smaller than the whole but much larger than atomic distances. Within each such domain, the spins point in the same direction, but the orientation of the different domains is random, disordered, causing their overall spin of the whole system to vanish. Within each Weiss domain, the symmetry is thus spontaneously broken, while for the sum of all domains it persists: the whole does not behave magnetic.

Spins and Magnets

(a)

(b)

31

(c)

Iron above (a) and below (b) the Curie point, with Weiss domains: in (c), an external field (long arrow) produces the overall magnetization.

It becomes so only through application of an external magnetic field, aligning all domains. The reason for this fragmentation is of energetic nature. The field of a magnet contains a certain amount of energy, and if that is not provided from the outside, a splitting into many smaller, randomly oriented domains is energetically favorable, since the field of the whole magnet is not required. But if we now apply an external field, the necessary energy is delivered, all domains align themselves to that field, and we have a true magnet. And that remains so also when the external field is removed. The twofold character of metals, paramagnetic and ferromagnetic, with a transition between the two states, quite naturally makes us wonder why this is so. What changes with temperature to produce the transition and the different states? Over the past years, statistical physics has clarified this issue: there are ef­fect­ ive­ly two opposing effects at work. We had already mentioned this; let us now come to it in some more detail. Temperature as heat energy causes the spins to fluctuate, to flip around, so that they sometimes point here, then there. The interaction between neighboring spins, on the other hand, wants spins to align, to point in the same direction: that makes the interaction energy as small as possible. Temperature and spin–spin interactions thus

The Rules of the Flock

32

pursue opposite aims. It requires energy to disturb the alignment of adjacent spins, and the temperature, if it is high enough, can provide this and thus produce a disordered system. With decreasing temperature, however, the influence of the spin interactions becomes ever stronger, islands of aligned spins form and grow and at the right temperature, the Curie point, the spin inter­actions win. There still are regions where the spins don’t yet follow the majority, but these regions become smaller and fewer. At absolute zero, there are no more thermal fluctuations and all spins are parallel. The Curie point thus separates a thermal region from an interaction region. So we have two different but equivalent descriptions of what happens at the Curie temperature. From the point of view of symmetry, we have above the transition temperature a state invariant under rotations, while below that temperature the rotational symmetry is spontaneously broken by the formation of an overall spin, there is a preferred direction. An alternative view is based on order, with the magnetization m(T), that is, the spin average (over a domain when applicable) as the relevant order parameter: in the paramagnetic state it vanishes, there is disorder, in the ferromagnetic state it is finite and there is (at least partial) order. The temperature variation of the magnetization is illustrated in the figure: it vanishes above the Curie point and then becomes finite. The

1

m(T )

ferromagnetic

0

paramagnetic

Tc

T

The order parameter for magnetism.

Spins and Magnets

33

r­ eason for the transition between the two states was outlined just above: above the Curie point, the temperature dominates, resulting in disorder, below the spin interactions win, leading to order. At this point we should emphasize that order and correlation are not the same thing. By order, we mean that in a given region, the spins point in the same direction, thus defining a domain. This is a necessary, but not a sufficient condition for correlation. If two spins are correlated, they fluctuate together around their equilibrium position. Well below the Curie point, that is not the case; now most spins point in the same direction, there is order, since the system is ferromagnetic, but the spins oscillate slightly around that direction in an independent manner. It is a bit like people sitting on a moving train: seen by a stationary observer, they all move at the same speed in the same direction, but their respective activities—reading, eating, looking at their laptops—are completely uncorrelated. For spin systems, there is only one point at which order and correlation coincide: at the critical point. At this point, the spins not only form a domain by pointing in the same direction, they also fluctuate together. We can thus determine whether a system is critical or not by studying it fluctuations. With this situation in mind, it seems not unreasonable to ask if bird swarms are not perhaps something like biological ferromagnets, systems with long-range order created by short-range inter­ actions. That will be the topic of the next chapter.

6 The Rules of the Flock Yet all evidence indicates that flock motion must be merely the aggregate result of the actions of individual animals, each acting solely on the basis of its own local perception of the world. Craig Reynolds, Flocks, Herds and Schools, Computer Science 21 (1987) 25

We had already noted that when a flock of pigeons in the square of a town, with birds moving here and there on the ground, is scared to fly up and away in some specific direction, this implies a change of symmetry of the system. Initially, the system of pigeons is invariant under rotations: it looks more or less the same from all angles. Once the pigeons rise up and fly away as a swarm in some given direction, that symmetry is spontaneously broken. It seems tempt­ ing to consider that happening as something like a transition from a paramagnetic to a ferromagnetic state, but now in the world of birds. The view of a bird swarm as an analog of a ferromagnet is based on the work of the Hungarian physicist Tamás Vicsek and his col­ laborators. They considered the direction of flight of a given bird as the counterpart of the spin in a metal, and they replaced the spin rule “point in the same direction as your neighbor” by the swarm rule “fly in the same direction as your neighbor.” Craig Reynolds had already required that in his simpler computer simu­la­tions. The pointing spin, a vector in mathematics ter­min­ ology, thus was replaced by another vector, the flight direction of the bird. But if we continue that line of thought, we encounter a difficulty. In metals, a high temperature can make the spin flip The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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35

Tamás Vicsek.

around and hinder alignment. The problem thus is to find also the counterpart of temperature in bird swarms. What had to be sufficiently low to allow bird swarms to remain coherent? Vicsek based his considerations on a set of objects which he called “self-propelled particles;” we will nevertheless often refer to them as birds. They are defined as points on a computer and move around in a plane, all having equal speeds, but flying ini­ tially in random directions. These velocity vectors were to play the role of the spin vectors in magnetic systems. Next, he assumed a time evolution: in the course of time, his birds moved from on spot to another in the plane, obeying certain rules. To simplify things, time passed in discrete steps, and after each step, each bird found itself in a new position. To reach that position, every bird adjusted his flight orientation to coincide with the average orien­ tation of all birds within a certain fixed distance around itself;

36

The Rules of the Flock

it adjusted its flight to become aligned with that of its neighbors. Actually, the rule was not quite that strict: it stated only that the bird should try to adjust its orientation in that way. The formalism allowed the birds to do so with a certain margin of error: its direc­ tion could differ from the average of its neighbors by some hope­ fully small angle ϑ. And this angle, this uncertainty would play the role of temperature. For small angles, in the course of time, the birds would form a coherent swarm, while for sufficiently large errors, that would never be attained. So the direction of flight in swarms corresponded to the spin orientation in metals, while the uncertainty in alignment between neighbors was the counterpart of the temperature for magnetic systems. For swarms yet another factor became relevant: their density ρ. If the density of Vicsek’s self-propelled particles was too low, a given particle would have no neighbors within a certain range, so that the swarm formation mechanism could not become operative. Let us here deviate briefly from our main line of argument. The observers of the starlings in Rome had found that their birds considered the six closest birds as their neighbors, no matter how far away these were. In this point, the Vicsek model thus departs from what we know today about bird swarms: it considers as neighbors all the birds within a fixed (“metric”) distance around a given one, so that in a dilute system, most birds don’t have any neighbors. It turns out that this difference is in fact not completely trivial—we shall return to this point later on. Vicsek started with randomly distributed birds and then let the computer determine time-step by time-step their further spatial distribution. To begin, at time 0, all birds have random velocity distributions. For a given bird A, one now determines the average orientation of all birds within a certain fixed range, including A itself. This will then become the new orientation of bird A, but with a given uncertainty ϑ. When all particles are dealt with in this way, they move a fixed distance according to

The Rules of the Flock

37

The temporal evolution in the Vicsek model: at the start (left), the ­average orientation of all birds in the dashed circle is determined (dashed arrow). This defines the new orientation (right) of the center bird (open arrow). This procedure is repeated for all other birds.

their new velocity orientation, arriving at their new position at time 1. With the new distribution given, we now repeat the pro­ ced­ure for the whole set of birds and thus come to time 2. After many iterations, we will in this way reach a distribution which no longer changes in structure with further iterations: we have reached an equilibrium, a distribution of birds whose form does not change with time. That is the pattern of the set of birds for the given parameters of alignment uncertainty ϑ and density ρ. In the figure, we illustrate the updating procedure. One can play this game for different values of ϑ and ρ. At low density and for large fluctuations, nothing happens: the birds continue their initial pattern, flying around randomly. When the density increases and the fluctuations decrease, the birds begin to form small groups, which move around randomly in different directions, independent of each other. And finally there is a crit­ ic­al point: for sufficiently large density and small fluctuations, the groups merge, the set of birds becomes a coherent swarm, all moving in the same direction, with only small deviations bird by  bird. At this moment, the previous rotational symmetry is ­broken, the swarm flies in a certain, albeit arbitrary direction. Let us assume that all birds fly with the same speed; then the average over all velocity angles becomes a measure of order. Up to a cer­ tain point of density and fluctuation it is zero, and then it becomes

38

The Rules of the Flock

suddenly finite: the swarm is born, in the form of what physicists call a dynamical phase transition. So far, we have said nothing about the boundaries in Vicsek’s model. In the course of the simulation, we obviously don’t want the birds to fly into a wall arising from the end of the computer storage space. To avoid that, one uses a well-known trick from computer simulation practice: periodic boundary conditions— the end is the new beginning. The simplest form of such a scen­ ario is letting the birds fly on the surface of a sphere, then they never encounter a limit.

The evolution of structure in the Vicsek model: (a) random start, (b) individual group formation for intermediate fluctuation and density, (c) random pattern for high fluctuation and low density, and (d) swarm formation for low fluctuation and high density (figure courtesy T. Vicsek).

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39

Let’s look at the velocity average in a little more detail. We form the sum over all individual velocities, for simplicity assum­ ing the speed v (the magnitude of the velocity) to be the same for all birds. Only the orientations differ, so that the sum of a bird flying to the right and one flying to the left gives zero. The sum over all N birds is then divided by Nv. It’s easy to see that then the average v becomes one when all birds fly in the same direction and zero if their directions are randomly oriented. The following picture shows what actually happens in the model. If the density is high enough and the directional uncertainty is sufficiently small, the requirement to align yourself with your neighbors forces all birds to fly in the same direction. With increasing fluc­ tuation that requirement becomes less and less effective, there are more deviations, and finally no swarm is formed any more. The behavior found here is once more the familiar pattern of a phase transition: we have a transition from a rotationally invariant ­system to one whose overall motion is oriented in an arbitrary but specific direction. We can thus conclude: the formation of a swarm of starlings is essentially the result of the same set of rules as the magnetization of iron. The swarm thus arises by self-organization in a system with short-range interaction: there is neither a commander tell­ ing all birds what to do, nor do the birds exert immense effort to 1.0 0.8

v

0.6 0.4 0.2 0

1

2

η

Distribution of the average velocity v at fixed density ρ as function of the fluctuation η.

40

The Rules of the Flock

keep even the far-away members of the swarm in sight. They sim­ ply follow the basic swarm code: follow your neighbor as well as you can. As Vicsek and collaborators have shown, this code suf­ fices to make even robots in a computer program form swarms quite analogous to those of the starlings in Rome. Before passing onto the next topic, we return to the men­ tioned feature of the starling swarms which is not taken into account in the Vicsek model. In the actual bird swarms, neigh­ bors are defined in terms of the seven nearest birds, no matter how far away they are; their metric distance is not relevant. As we had noted, that is necessary in order to keep the swarm structure operational as the swarm expands or contracts. In the Vicsek model, the neighbors were those self-propelled particles which were found within a fixed metric radius of the given particle, so in a sufficiently dilute system, many particles would have no neighbors. This model thus contains two param­ eters which determine the state of the system, the density ρ and the alignment error η. If the density of birds is too low, most birds lead a solitary existence without neighbors; they have no chance to align with others, no matter how precise their alignment cap­ abil­ities are. If the density is high enough to provide a particle with neighbors, determined by a fixed neighborhood radius, it is then the error margin which decides whether or not there is alignment. Detailed studies have shown that in this model the transition from disorder to order for decreasing error does not occur directly. In a first step, density fluctuations will result in the formation of clusters, which move around randomly in the plane. The clusters themselves are connected entities, but differ­ ent clusters are disjoint. A second step then leads to the fusion of the different clusters into a connected whole. The situation is thus similar to the condensation of water, where in the water vapor first single droplets form, which with increasing density then fuse into a connected liquid. In statistical physics, such pro­ cesses are denoted as first-order phase transitions: the disordered state proceeds to the ordered one through an intermediate coexistence

The Rules of the Flock

41

regime, in which both ordered and disordered regions exist ­simultaneously. With increasing density, the ordered sections are “squeezed” together and eventually form an ordered fluid. This behavior is evidently very similar to that observed for the marching locust nymphs, so that we can consider the formation of locust swarms as occurring in the form of a first-order phase transition, from a state of individual animals first to an intermedi­ ate state of disjoint little groups and then, at the end of this mixed state, to one ordered swarm. In the actual case of the starlings, where the nearest neighbors are just that, no matter how far away they are in meters, density evidently does not matter. In a dilute flock, the group of a bird and its nearest neighbors simply occupy a larger area. In this situ­ ation, the alignment parameter η remains the only variable: once it becomes small enough, the birds form a flock. There now is no intermediate cluster phase; the transition is continuous, from dis­ order to order, as found in the case of magnetization. From the point of view of the starling flocks, the behavior seen in the upper right of the picture of the Vicsek model results thus is an artifact of the model, due to a different definition of neighborhood than that used by the birds. There is yet a further point to note. Vicsek’s birds, when they are close to each other, like to fly in the same direction. But there is no in-built attraction between the birds—something obviously needed and present in all flocks as well as herds of mammals or schools of fish. It is this feature which keeps the members bound together. Neither is there a repulsion, to prevent birds from collid­ ing. In a biologically more realistic model, one would thus include both an attraction between the members of the swarm and a repul­ sion once they get too close to each other. An attraction in particu­ lar can result in swarm formation even if the overall density is quite low. In this sense, the Vicsek model is a simplified description to understand the crucial feature, swarm formation. Much subse­ quent work has included both the mentioned missing features; we will comment on some details in the following chapter.

7 Complexity and Criticality The laws of physics are simple, but nature is complex. Per Bak, How Nature Works, Springer, New York, 1996

The basic topic here is the behavior of a macroscopic system made up of many microscopic constituents, a central theme, as we saw, in both statistical physics and swarm behavior. We can consider such a medium either from a geometric or from a dynamical point of view (or include both aspects, of course). In the geometric case, we attribute to the constituents an intrinsic size, such as the disks in Chapter 2. For a very dilute system, this leaves two scales: its overall size of the system and the intrinsic size of the isolated disks. With increasing density, a third scale emerges between these two: connected clusters appear of ever increasing size, and their size provides us with a new, further scale, larger than the disk size and smaller than that of the overall system. In dynamical systems, we initially have many non-interacting point particles, all moving around randomly without taking note of each other. Such a system is usually called a free or ideal gas. So far, there is only one scale, the overall size of the system, and for its behavior, that does not matter: if we cut it in two, or double it in size, the new system is in all aspects (except size) identical to the original one. A second scale enters if we introduce two-body interactions of a fixed range between neighbors, as we had seen in spin systems, so that we now again have two intrinsic scales. Here the third scale emerges because through a chain of successive interactions, the correlation of more distant particles becomes The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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possible. The thermal motion of the particles counteracts such an information transfer, so that its range is limited: the new emergent scale is the collectively produced correlation length. Both the isolated disks and the non-interacting particles constitute simple systems. We shall speak of complex systems, whenever the intrinsic simple features give rise to a new, emergent scale. In other words, complexity is an emergent feature, created in a selforganized fashion by a many-body system, once its constituents no longer ignore each other. The surprising feature of complex systems is that something striking happens when the new emergent scale becomes as large (or larger) than the overall size of the system. We now have two limiting cases: all constituents are independent (for molecules, the ideal gas) or all constituents are connected (for molecules, a  liquid). Complexity thus quite naturally leads to a two-state structure of matter, paramagnets vs. ferromagnets, insulators vs. conductors, gases vs. liquids, and many more. Quite often the transition from one state to the other is gradual; but there are also many cases where the transition is abrupt, where nature does make jumps. The system shows criticality, and the points where the jumps occur are called critical points. Let us look at the case of a molecular gas in a little more detail. In such a gas, there is interaction between two adjacent, freely moving molecules, and through successive interactions, the effect can be extended to more distant constituents. The thermal motion of the molecules in the gas modifies the interaction more and more, so that after a distance ξ any further correlation ceases. Hence ξ is denoted as the correlation length, which is generally larger than the intermolecular distance, but shorter than the size of the container. As the temperature is lowered, the thermal agitation of the molecules is reduced, and at a certain critical point, it is no longer strong enough to prevent the molecules to join forces and form a connected net, the liquid: condensation sets in. Coming from the low temperature side, we have oscillations of the mol­ ecules in the liquid, around their positions in the net, but these

44

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oscillations are not strong enough to fully break the intermolecular binding. Only at the evaporation/condensation point, the thermal motion of the individual molecules becomes sufficiently strong to break the liquid bonds, and the system evaporates, turns into a gas. So above the evaporation temperature, a given molecule can move freely throughout the entire system, whereas below this point, it is constrained to remain in its given neighborhood. In more mathematical terms, the direct interaction between two isolated molecules separated by a spatial distance r is generally described by the function

1 g(r ) = , r

that is, it decreases with increasing separation r. If we now ­consider a pair A and B of constituents inside a medium at a ­sep­ar­ation such  that between the two there are many other constituents, the communication between A and B has to proceed successively. The intermediate transfer agents are subject to thermal oscillation, which hinders the transfer and thus limits the range which the interaction information can travel. This effect is known as damping or screening, and the distance the interaction is still felt is, as mentioned above, called the cor­re­la­tion or screening length ξ. It depends on the temperature, and the correlation function between A and B now becomes

G(r,T ) =

e - r /x . r

With increasing temperature, the growing thermal agitation destroys the correlation more effectively, so that in the limit of infinite temperature, ξ goes to zero, and the correlation function G(r,T) does so likewise. The constituents no longer experience any interaction, the high temperature limit leads to an ideal gas of free particles. On the other hand, as we approach the critical condensation point from above, the correlation becomes stronger and stronger, and at the critical temperature Tc, it becomes in­fi­ nite: all molecules are now unified in the liquid. This means that

Complexity and Criticality

45

directly at the critical point, the correlation function becomes scale-invariant, there is no more emergent intermediate scale ξ: 1 G(r,Tc ) = . r More explicitly: the correlation function for two constituents at separation r is by a constant factor two larger than it is for sep­ar­ ation 2r, no matter what r is: it is scale-free. In contrast, above the critical point, when ξ is still finite, the correlation is suppressed exponentially with r, so that its relative decrease is stronger for larger r: the system is not scale-free. These concepts are evidently quite relevant for swarm formation and structure. The connectivity of a flock of birds shows that even the most distant members can be correlated, no matter how large the flock is. The empirical studies mentioned in Chapter 4 had in fact shown that the correlation range does increase, without any limit, with the flock size. The state of the flock thus seems to correspond to a critical point in terms of statistical physics. The theory of critical behavior had remained quite enigmatic for many years, since the success of physics was for centuries based on “divide and conquer,” study the behavior of a small subsystem and then combine many such systems into one large one. At the critical point, this procedure breaks down. The correlation domains become as large as the entire system, which now realizes “how big it is” and refuses to be divided into independent small subsystems. It is now more than the sum of its parts. The ul­tim­ ate solution came with the renormalization theory of the American theorist Kenneth Wilson; it brought him the 1982 Nobel Prize in physics. It is based on the existence of domains of all sizes in the critical region; this implies that a change of scales (“renormalization”) will only rescale a given quantity. A stick of one meter length is equivalent to one-hundred times a stick of one centimeter length: L(100 cm) = 100L(1 cm). The general form for an observable L(a) becomes

L( xa ) = x b L( a ),

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The Rules of the Flock

where the intrinsic scale a is rescaled by the factor x and where the critical exponent β is a quantity characteristic of the system (it was one in the above example). Different observables, such as magnetization, energy, specific heat and more, lead to a set of such exponents. Systems, or mathematical models, sharing the same set of such critical exponents, are said to belong to the same universality class of critical behavior. How can such a scale-invariant critical state be attained? In the physics of equilibrium systems, in which the state of the system is time-independent, there is a well-defined procedure: adiabatic ­tuning. We lower the temperature a little bit, give the system time to adjust, then lower it some more, and so on. At each point, the system is allowed to reach equilibrium, before the temperature is tuned down further. With each step, the average domain size increases a little more, and as we approach the critical temperature, it diverges. We have reached criticality, keeping the system in equilibrium all the time: that is the procedure of adiabatic tuning. Such an approach to reach criticality is in fact quite an ideal­ iza­tion. Most processes in nature are non-equilibrium reactions; they change in time and need an energy input to keep going. The locusts and the birds move around and need to feed to be able to do so, and they are certainly not put into order by some im­agin­ ary tuning operator: they somehow manage to do that on their own. How such an order can be achieved was quite mysterious until the seminal work of the Danish theorist Per Bak, who in 1986 introduced the concept of self-organized criticality. The issue still remains mysterious enough, but on a higher level . . . Bak illustrated his ideas in term of sand piles. Imagine pouring fine sand onto a flat surface. This will create an ever growing conical pile of sand, whose slopes become steeper and steeper. When they reach a certain critical value, the addition of further sand will lead to avalanches sliding down the pile, and continued sand addition will lead to more and more avalanches. It runs out that the number of avalanches thus produced will depend on their size: the larger the avalanche, the fewer times it is produced. One thus

Complexity and Criticality

47

finds that the number N(s) of avalanches of size s, specified through its surface area, behaves as

N ( s ) ~ 1/ s.

The scale-free nature of the process here means that avalanches of all sizes are specified by the same law. Avalanches of size s occur twice as often as those of size 2s, no matter what the actual size s of the avalanche is. The process is scale-free. The underlying physics of what is happening in this experiment is determined by a non-equilibrium process, with a constant input—the pouring to the sand—leading more and more toward what the physicists call “a critical attractor,” the goal of the evolution, so to speak. In a landscape of hills and valleys, it is the lowest point, to which all water will eventually flow. Here it is the critical slope resulting in avalanche formation. When that is reached, we have criticality, and the system produces an output of avalanches of scale-free size. In the Vicsek model for the flocks of birds discussed above, there are two intrinsic param­ eters, the degree of alignment and the metric distance defining the neighbors for a given bird. Hence the relevant parameters for the model are the flock density and the intrinsic alignment fluctuation, and the critical attractor is a line in the plane of these variables. However, the data of the STARFLAG project showed that in fact the real birds don’t use a metric scale: their neighbors are the closest birds around, no matter how far away these are in meters. Hence the more realistic version of such a model defines neighbors topologically, and now the alignment fluctuation remains the only relevant parameter, corresponding to the temperature of a spin system. The model, as well as the flocks of starlings, becomes scale-free: the flock converges on its own to the critical point. Here we can return to an observation made above: at a critical point, the birds should not only fly in the same direction—their oscillations around that direction should also be in step. And that is in fact what happens, according to the STARFLAG observations.

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The Rules of the Flock

In one of the correlation regions which they form, all the birds deviate a bit to the left, in the other region all to the right. And in general, a similar effect was observed for the speeds: in a given correlation region, they all were a little above (or below) the average. This, together with the fact that the domain size grows with system size, is a good indication that the flock is indeed poised at the critical point. The surprising result thus is that not only do flocks of birds behave quite similar to the interacting spins or particles of statistical physics, but that moreover their state seems to correspond just to the critical point. In a rather idealized case, one can follow this line of argument even further: we assume the flock to be strictly two-dimensional, the birds fly in a plane. We can then consider a model consisting of a set of N particles randomly distributed on a two-dimensional surface. If we now define the neighborhood of a given particle as all those points of the plane closer to it than to any other particle, we obtain the picture shown in the figure. It shows the so-called Voronoi portioning, a “tessellation” of the plane into neighborhoods, named after the Russian mathematician Georgy Voronoi. It is inherently scale-invariant, topological: the only thing that matters is the relative orientation of the particles, not their ­metric separation. If the entire configuration is blown up uniformly, the scale is increased, then the neighborhoods grow correspondingly, and the overall picture remains unchanged. In this scen­ario, one can calculate the average number of neighbors which a given particle has, and it is found to be six (try it out in the picture), quite close to what the starlings showed. The flocks observed in the STARFLAG project, though relatively flat, were not really two-dimensional. It is thus not clear if there is a connection between the data and the two-dimensional topological results of the Voronoi approach. It does, however, show new possible perspectives for a scale-free treatment of swarm structure, as it arises from self-organized criticality. We had already indicated that one essential feature missing in the Vicsek model is an attractive force between the members of

Complexity and Criticality

49

A random set of points in a plane, with their Voronoi neighborhoods.

the flock. If we start from a fixed number of randomly distributed birds in a given very large area (remaining in two space dimensions for simplicity), then most birds will have no neighbors in the original metric version of the model. But even if we define neighbors in the topological way, we end up with totally unrealistic flocks, in which the closest members are kilometers apart— something we would hardly call collective. A cohesive force between the members of a flock is therefore an essential ingredient, as already noted by Craig Reynolds, when he made his boids stay near each other. This force brings the members together and thus allows a flock moving in an otherwise empty space. On the other hand, we don’t want this force to be so strong that it makes the flock members collide in their drive

50

The Rules of the Flock

to be close to each other. The force generally assumed today is therefore repulsive up to a certain exclusion range around each bird, and beyond this it is attractive for the nearest neighbors. Combining such a force with the alignment mechanism (“fly in the same direction as your nearest neighbors”) then provides an operational scheme for the construction of swarms very similar to those observed in nature.

8 Fiery Clouds in the Jungle

Engelbert Kaempfer in Japan.

In the year 1690, the German physician and explorer Engelbert Kaempfer passed through the Kingdom of Siam, today’s Thailand. He was on his way to Japan, which he did reach as one of the first Europeans, and where he spent quite some time. After his return he wrote an extensive History of Japan, which became quite wellknown. But on his way, when passing from Bangkok on a river through the mangrove jungles of Siam, he made an observation which puzzled biologists for centuries and which only some thirty years ago found its final explanation. In the Siamese jungle he observed something truly extraordinary: “A fiery cloud of insects takes possession of a tree and spreads itself over the branches, then shows an intense The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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light and suddenly extinguishes it again, to show it again and again, with utmost regularity.” Kaempfer’s observation was the discovery of synchronized light emission, by now observed for a number of firefly species. With the help of bioluminescence, light production through reactions of body substances, these creatures emit a sequence of light flashes, about two per second, then remain dark for a while, and subsequently repeat the performance. We are all familiar with fireflies, glowworms, which sparkle here and there in the darkness of the night. But their Siamese relatives do that in an absolutely synchronous fashion: thousands of animals flash in exactly the same rhythm, all simultaneously. Years later a species was observed in America which had the light travel on in the form of waves, starting at a certain place and then propagating on and on, like the Ola-wave in a soccer stadium. The phenomenon provided two puzzles to biologists. How could such a simple animal define an exact rhythm (twice a second), and how could thousands of such animals light up sim­ul­ tan­eous­ly? There were many explanations, ranging from the supposition that the flash was caused by the observer blinking his eyes to the idea that there was a king of the fireflies giving signals. In 1935, the American biologist Hugh Smith concluded in resignation that some of the explanations were indeed stranger than the phenomenon itself. Then there were those who thought that the origin of the synchronization was empathy: the animals preferred to flash when their neighbors did. That in a way came closest to today’s interpretation; it required, however, that these creatures had a sense of feeling and were capable of making very fast decisions. The work of the American biologist John Buck and his wife Elizabeth provided an essential boost to the investigation of the phenomenon. Already in 1938 Buck, still a young scientist at the time, had published a comprehensive report about the syn­chron­ ous flashing of the Asiatic bugs, in which he also collected the many different explanations provided at time. More than thirty years

Fiery Clouds in the Jungle

53

Malaysian firefly (Pteroptyx). Photo: Ben Pfeiffer-Firefly.org.

later, during a stay in Thailand, he and his wife captured a large number of male fireflies and liberated them in groups of fifty in their darkened hotel room. Initially, the bugs flew around randomly and blinked now and then. Eventually, they settled on the walls of the room in smaller groups, with each group blinking in time. And finally the groups coordinated their act and the whole set flashed at the same time. Today one knows that the syn­chron­ ous flashing only persists when enough animals come together, and one also knows that the aim of the procedure is to signal to the females, which normally remain on the ground, that the males above are ready to mate. In fact, one can also analyze the chemical process leading to the production of the emitted light. There remain the mentioned questions. How can an animal emit signals of a precisely specified frequency? That concerns the individual animal, and it is in fact also understood today—we shall return to it in a moment. The crucial issue is how thousands of animals can coordinate their signals so perfectly, that they all appear simultaneously. That is evidently a question of selforganization of many animals, and it is this aspect that has kept the scientists guessing for so long. But let us first address the individual timing.

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The Rules of the Flock

The illumination display of the bugs in fact implies that each animal has an inherent timing device. Even an isolated firefly flashes with the same frequency as shown eventually by the whole swarm, about two times per second. Each bug must therefore have a built-in timer, like a metronome, which allows it to flash in precisely defined intervals. That may seem surprising at first sight, but it does occur quite often in nature—for example, in the heartbeat of all warm-blooded animals. Buck and his wife tested to see if they could modify the blinking frequency by shining light signals, and they found that that was indeed possible: a given bug adjusted his radiation such that his series of flashes started in coincidence with that of the light of the Bucks’. The animals thus not only had a timer, but they could even regulate it. This observation became the start for the subsequent understanding of synchronous radiation of insects. To obtain a basis as precise as possible, the Bucks carried out quite exact measurements. The frequency of the emitted flashes was one per 560 ± 6 milliseconds—that corresponds to the mentioned two flashes per second. The signals of different bugs coincided up to 20 milliseconds with each other. That meant that the synchronization could not arise through a simple coordination of neighbors: if a given bug saw its neighbor flash, it could not itself flash that quickly. In other words, the sequence of flashes must have somehow managed to synchronize directly all the timekeeping devices of the different bugs. How is that possible? The great Dutch physicist Christiaan Huygens was the inventor of the pendulum clock. In 1665 an illness had forced him to remain in bed for some days, and that allowed him to observe that two such pendulum clocks in his room, when started randomly, would after some time syn­chron­ ize their beats. Certainly these clocks did not possess any feeling of empathy? The later explanation was simpler than one had expected: through their common support, the clocks transmitted vibrations and thereby brought themselves into unison. We can easily reproduce the effect. Put two metronomes, set to the

Fiery Clouds in the Jungle

55

Two metronomes ticking randomly (left), in common beat (right).

same frequency, onto a rigid support such as a board, supported by two freely movable cylinders. At first, each metronome beats in its own rhythm, but soon, as the cylinders roll slowly back and forth, the two metronomes get together and tick in a common beat. That looks somehow miraculous: why should these mechanical devices like to join in a common beat? This can be made even more dramatic: there are videos of 50 and more metronomes on a movable platform; they start in a completely incoherent way, but in the course of a few minutes, they are all in phase, their ticks are synchronized. How can that happen? Are there phenomena in the material world, in a world without any mind or intelligence, which almost completely coincide with the behavior of animal swarms? Before we turn to this question, however, we want to consider yet another case of syn­ chron­ized insects.

9 The Audible Silence If moonlight could be heard, it would sound just like that. Nathaniel Hawthorne (1804–1864)

Surprisingly enough, there exists an acoustic counterpart to the synchronized flashing of the fireflies in Asia and America. Grasshoppers and crickets as we know them indeed chirp, but they do so sporadically, in an uncoordinated way. In North America, however, one has found a particular species, the snowy tree cricket, for which the individual animals synchronize their chirping, singing in time, as a chorus. Their song is quite well-known there, and the writer Nathaniel Hawthorne, quoted above, has called it the audible silence. In this case, the miraculous aspects extend even further: the frequency, the beat of the chirping, depends on the temperature of the environment, and it does so with such precision that it serves rather well as a therm­om­eter. Crickets are closely related to grasshoppers, even though they look a little more like beetles. Both species “sing,” although the sounds are not made by vocal cords; they are made by rubbing the extended wings on each other (for crickets) or by rubbing leg and wing together (for grasshoppers). And in both cases, it is only the males that sing, with the aim of attracting females for copulation. That is also why they join in a chorus: the more males chirp together, the more emphatic is the invitation. So the reason for the performance is here the same as it was for the fireflies: the concert is staged to attract females, and the synchronization enhances the signal. The chirping of the crickets consists of a long sequence of trills; each of these is about a tenth of a second long and consists in turn

The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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57

Snowy tree cricket (Oecanthus fultoni).

of five or eight pulses. That’s why a given trill sounds almost like a single tone. Per minute, depending on the temperature, some 40–120 trills are emitted. If we plot the emission as a function of the time, it looks like the pattern shown in the figure. Acoustically, that then becomes the well-known “chirp-chirpchirp . . . ,” with some two to four or more chirps per second. Let’s concentrate on a specific cricket and count how often per minute it chirps. Most people probably couldn’t really care less, but somehow one had noticed in America that crickets chirp faster when it gets warmer. The American physicist and inventor Amos Emerson Dolbear (in spite of a number of lost court processes versus Bell and Marconi he is considered as one of the inventors of the telephone) had in the year 1897 established a rule determining the temperature on the basis of cricket chirping. Count the number of chirps per fifteen seconds, that is, for a quarter of a minute, and then add 40. That gives you the temperature in degrees Fahrenheit. Dolbear’s law thus reads

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=IIIII

0

0.5

1.0 time (seconds)

1.5

2.0

The time structure of cricket chirps.

TF = 40 + N (15 seconds). If the crickets chirp three times a second, as illustrated above, or 45 times per quarter minute, the temperature is about 85 degrees Fahrenheit (corresponding to about 29 degrees Centigrade). A ­little later, Dolbear’s law was refined a little: TF = 40 + N (13 seconds), and in the range 60–90 degrees Fahrenheit it provides a temperature determination with a precision of one to two degrees. It’s not surprising then that in America the snowy tree cricket is known as the temperature cricket. That explains the chirping of a single cricket. What happens if two or more crickets get together? This was investigated around 1969 by the American biologist Thomas J. Walker of the University of Florida, and as a result of his work, we are today as well informed about the synchronous chirping of crickets as we are about the joint flashing of fireflies. Walker studied the behavior of a cricket when it was confronted by a recorded row of chirps. If the chirp of the observed cricket occurred in the latter part of the recording, then it held back its next chirp in order to be in time with the next recorded beat. If its chirp fell into a break between two recorded chirps, it reduced its own interval such that it again was in time the next round. In other words, the tested cricket behaved such as to achieve synchronization as quickly as possible. Next, Walker noticed that if he placed two crickets in separate containers near each other, they behaved in the same way; sometimes one,

The Audible Silence

59

sometimes the other would take the lead role. The result was a remarkably precise synchronization; the starting times of the two chirps agreed to within some thousandth of a second. Since such a rapid coordination between neighbors was definitely impossible, it meant that the crickets had calibrated their timing to agree with the previous chirps and then retained this calibration. Today biologists are quite sure that this behavior is not the result of a conscious decision of the animals, but that it rather is a reflex reaction. If you throw a ball at someone, that person will “instinctively” try to shield itself. He or she does not consciously have to decide “now I have to hold a hand in front of my face”—that happens as a reflex. In the same way, one extends one’s arms when stumbling; that is also not the result of a thinking process. And in this way, thousands of crickets achieve unison in their song of audible silence. As we had already mentioned, the song is meant to attract females; but it also seems to be useful to defend the singers against cricket-eating predators, since it becomes difficult to localize a specific animal. The chorus of the crickets, just as the joint flashing of the fireflies, is considered as a form of self-organization. One cricket may define the beat, but this role is unimportant and the lead changes many times in the course of the song. In this way, the swarm of the crickets defines its rhythm. Here as well it is therefore of great interest to see if it is possible to construct a mechanical system which can be programmed to show exactly such behavior. That will be the topic of the next chapter.

10 Coupled Oscillators

Christiaan Huygens (1629–1695).

A flock of birds may fly in formation, but when the birds settle down to rest in their trees, their chirping becomes quite uncoor­ dinated. Perhaps two birds may “talk” to each other, like the black­ birds on our roof tops do to define their territory, but the whole swarm certainly does not form an orderly chorus. In the case of humans, that can be different: the applause after a concert can be turned into a rhythmic clapping event, if a few listeners, The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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independent of each other, happen to produce a dominant beat that has many of the others join. Does the appearance of such a phenomenon require a sense of feeling, empathy or even think­ ing? What we know about fireflies or crickets seems to indicate that that cannot be the case. Could it then be that there exists a synchronization mechanism shared by the living and the ma­ter­ ial worlds? Could it be a mechanism similar to the one that brings the metronomes into phase? The Dutch physicist Christiaan Huygens was, as already men­ tioned, the inventor of the pendulum clock. To measure time, we need a process which repeats itself periodically, and the swinging pendulum was exactly that, just like the metronome. In this way, sunrise and sunset quite naturally define a day as a unit of time, the period between two full moons gives us a month, and the time between two summers or winters makes a year. These were therefore the earliest measures of time. “It happened many win­ ters ago,” in the words of the American Indians. An oscillator, in physics terminology, is a device which undergoes periodic motion, flipping back and forth between two extreme points—it oscillates, like the metronome or the pendulum clock. If we take two or more oscillators and establish a connection between them, such as the common movable basis of several metronomes or the not so rigid support of Huygens’s pendulum clocks, we have coupled oscillators: they disturb each other in their motion—unless they somehow manage to get into step. And just as Tamás Vicsek and collaborators brought physics into the study of bird swarms by considering the velocity of a bird as the analog of spin in a metal, so the mathematicians Steven Strogatz and Renato Mirollo provided a physical basis for the synchroniza­ tion of insect swarms, by considering them as systems of many coupled oscillators swinging in step. Let us see what that means. We can follow the motion of a metronome by considering the speed with which the arm moves: at one of the two extreme points, it comes momentarily to rest, then begins its return trip, speeds up more and more, passes through the vertical position at

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maximum speed, and on the other side slows down gradually, coming to rest again at the other extreme. If we now couple two such oscillators by connecting their arms with an elastic string, they will interfere with each other’s motion: if at a given point in time, A is moving faster than B, it will pull to speed B up, while B is dragging to slow A down. Both actions require energy, and nature tries to arrange its behavior such as to minimize energy expenditure. Here there is only one way to reach that goal: get in step. If A and B swing in perfect time, no-one pulls and no-one drags. And that is the reason why Huygens’ pendulums as well as our metronomes eventually ended up swinging in perfect harmony. In the example we have just considered, the coupling was done mechanically, with a rubber band. Another form, essential for the existence of all warm-blooded animals, drives as well as c­ ouples oscillators through electricity: one of its many applications is the heart-beat. To construct an electric oscillator, we need a device in which the voltage—the difference between positively and nega­ tively charged poles—increases with time. When this difference reaches a certain limit, there is a sudden discharge, just as a bolt of lightning ends the voltage gap between clouds and ground. If, however, the source of the voltage difference persists and the voltage buildup starts again as before, then such a setup results in a regular emission pattern of discharge signals, as illustrated in the figure. A familiar device of this type is the turn signal, the “blinker,” on cars, flashing in a regular rhythm. discharge

voltage

discharge

time

Electric oscillator.

discharge

Coupled Oscillators

63

A

B time

Two coupled oscillators.

Evidently the emission of the signal, the discharge, requires a certain amount of energy, and only when the voltage has reached a sufficient level, this energy is available and the discharge occurs— for a single such oscillator. If we have two or more c­ oupled oscil­ lators of this type, the signal emitted by A will give a small but finite kick to oscillator B, increasing its voltage and thereby bring­ ing it to discharge a little sooner than it would have otherwise. Its earlier flash will then similarly affect oscillator A, and the final result of such mutual interference is that the two will become syn­ chronized, will flash simultaneously. We thus expect that a system of arbitrarily many such os­cil­ lators will eventually become synchronized, with all discharging simultaneously. That had in fact been suggested in 1975 by the mathematician Charles Peskin, and subsequent computer simu­ lations by Steven Strogatz had confirmed it. However, in spite of the conceptual simplicity of the problem, the underlying math­ ematics turned out to be extremely complicated, and only in 1990 Strogatz and his fellow mathematician Renato Mirollo were finally able to prove that coupled oscillators indeed always con­ verge to synchronization. So we know today how the fireflies do it: they don’t do it at all, it happens on its own: a “mindless” phys­ ical system is the source of the phenomenon. The proof of Mirollo and Strogatz provided the explanation for a great variety of bio­ logical synchronization phenomena, from fireflies and crickets to the pacemaker function of the human heart.

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The human heart consists of thousands of cells, each acting like one of the oscillators we had just considered. Such a cell can be isolated and placed in the right voltage environment; it will continue to beat on its own, with the same rhythm as the whole heart. Through their synchronization, these cells then determine the rhythm of our heartbeat, sixty to eighty times a minute, bil­ lions of beats during our life. The cells set the pace collectively, so that if one of them should malfunction or die, the heart will sur­ vive and carry on. And if there should be a major malfunction of the whole system, today a sufficiently rapid reboot with the help of an external electric shock device, a defibrillator, can often still restore the regular function. The mechanism for the synchronized flashing of the Malaysian fireflies as well as that for the chorus of the snowy tree crickets is thus the same as natural pace-setter of the human heart. Each of the insects contains oscillator cells which, when stimulated through a biologically initiated voltage difference, provide the periodic beat eventually maintained by the entire swarm. This beat is to some extent controllable, just as we can speed up our heart-beat through physical or mental actions. The twofold puz­ zle we had faced when we observed the concerted actions of fire­ flies and crickets—how can an individual oscillate at a given rate, and how can a multitude of such individuals get into a uniform step—that puzzle was answered when we found that the same functions are carried out by the heart cells of our own body.

11 Laser Beams

In the inanimate world, synchronization turned out to play another major role. Light, as we know it in our daily life, is radiated by many atoms (such as in light bulbs) or by nuclei (in the sun). Each atom, each nucleus radiates independently of all the others, and the radiation is emitted in all directions. Their light in the visible range is distributed over different wavelengths, and the position of the peaks and dips in each wave are completely unrelated to those of the other waves. In physics terminology: the light is incoherent. Since about a hundred years, we know that it does not have to be that way, as first shown by a less well-known yet extremely important work of Albert Einstein. In 1916, he explained the process of emission and absorption of light by atoms. The consequences

The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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of this work were manifold: bar-code scanners in the supermarket, CD players, GPS systems for navigation and last, not least, the blade of the eye surgeon for retina operations, the laser. The word laser is an acronym, “Light Amplification by Stimulated Emission of Radiation.” In a laser, millions of atoms radiate in full synchronization, sending out light of the same wavelength and hence the same color, all waves moving in the same direction, in phase, peak on peak, valley on valley. How can that be achieved? For the explanation we first have to recall a few details about atoms. A single atom, left alone, is in what is called its ground state, in which its energy is as low as it can be. This energy is of kinetic origin: it arises from the motion of the electrons orbiting around the atomic nucleus, a little like planets around the sun, but with one important difference. Quantum theory tells us that an electron cannot circle around the nucleus on any arbitrary orbit, but that only specific concentric orbits are allowed. That means that the energy of the electrons and hence also those of the atoms are quantized: it can only have specific discrete values. Let’s look at the case of the simplest atom, that of hydrogen. The nucleus of this atom is a single positively charged proton, around which a single negatively charged electron is orbiting. The smallest orbit corres­ ponds to the ground state we had mentioned, with an energy E0 . The next orbit then gives the first excited state E1, followed by further excitation states, E2 , E3 and on. If we now add to an atom in the ground state E0 a sufficient amount of energy, for example the amount E1 - E0, by sending in a photon of that energy, then the atom is excited to the higher

E 0 E1 E2

E0 E1 E2

Hydrogen atom in the ground state (left) and in the first excited state (right).

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67 Photon

Photon E0

E1

E0

E1

Absorption and emission of photons creates transitions between energy levels.

state E1. But if we then leave the atom on its own, it will not remain excited forever. At some later time, according to the “lifetime” of the excited state, it will fall back into the ground state, “decay,” and in doing so emit a corresponding quantum of light of that energy. The analogous process happens in any transition from a higher to a lower excitation state, and this radiation, emitted by atoms in such transitions, is the light we get from a light bulb. The atoms in the wire of the bulb are through electricity excited to higher energy states, decay and thereby emit light, are then re-excited, and so on. In this context, one has to take into account that there are easy and more difficult transitions. The dynamics of the binding process can cause the threshold between E0 and E1 to be much larger than that between E1 and E2 . We can therefore consider the whole situation as a slope formed by steps of different heights. Each step consists of a trough, in which the electron rolls around, back and forth, until it finally overcomes the edge of the trough and falls into the next lower state, and in doing so, it emits a photon whose energy is just the difference between the two energy levels. The time the electron spends in each such trough is the lifetime of that state. If it so happens that the lifetime of E1 is much greater than that of the higher excited levels, then it is possible that as a result of a general excitation an unusually large number of atoms end up in that state and remain there for quite some time. Normally most

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E3

E3 Photon

E2

E2

E1

E1 E0

E0

An electron in state E 2 (left) and the transition from E 2 to E 1, ac­com­ pan­ied by the radiation of a photon (right). E1 E0 normal occupation

occupation inversion

Occupation inversion: more atoms are in the excited state than in the ground state.

atoms are in the ground state, but now we have for some period of time more in E1 than in E0 . This situation is referred to as occupation inversion, and something like that is needed for the construction of a laser: you need a material which when heated—in other words, when its electrons are excited—leads to such a con­fig­ur­ ation, with a population excess in an excited state. We need such a situation, because now something unexpected takes place. A single atom would just fall back into its ground state at the end of the lifetime of the excited state. This lifetime is not a fixed number; instead it is given by a distribution curve, some atoms decay very quickly, a little later more, then still more, then again fewer, finally very few, until all are in the ground state. The listed lifetime for a state of an individual atom is just the average of all these times. But if we have a medium of densely packed atoms, the photon emitted in the first decay will hit another

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The induced photon avalanche: a laser beam.

atom. This atom is normally not yet ready for decay; its electron would still remain for a while in the excited state. But if it is hit by the photon from the first decay, it also decays immediately, emitting a photon which is absolutely identical to the one by which it was hit. The two photons now continue as a coherent system, one from the first “natural” decay, the second from the “induced” decay. Both these continue and the process is repeated: each of the two hits an undecided electron and causes it to descend to the ground state, again with emission of a coherent photon. After this step, we already have four, then eight, then sixteen photons and so on. The first photon has triggered an avalanche, consisting of exactly identical photons, all formed in induced transitions. The resulting light is fully coherent, the same color, the same direction, the same phase: we have a laser beam. The physical process forming the basis of the whole phenomenon was, as we had indicated, first explained in 1916 by Albert Einstein: an incoming photon induces the decay of an excited atom, leading to the radiation of a further photon. That this can lead to a coherent beam of light was shown much later, by the American physicists Charles Townes and Arthur Schawlow. It brought Townes the 1964 Nobel Prize in physics, his brother-in-law Schawlow got nothing. Even the Nobel committee was apparently not at ease with this decision, and so Schawlow was awarded the 1981 Nobel Prize for an application of the laser. When Townes and Schawlow had proposed the possibility of a laser beam, applications were far from evident. Critics even talked about an answer

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in search of a question. Today our life is hardly conceivable without lasers, from medicine to different aspects of technology and on to space research: everywhere lasers are absolutely crucial. In the case of the laser, the first and “naturally created” decay photon induces the other atoms in the medium to emit successive identical photons and thus starts a chain reaction leading to the formation of a coherent beam. It appears that a corresponding process took place to create the synchronous light emission of the fireflies or the precise chorus of the crickets.

12 The Way to Go

Caminante, son tus huellas el camino, y nada mas; caminante, no hay camino, se hace camino al andar. (Wanderer, your footsteps are the road, nothing else; ­wanderer, there is no road, you make it as you go.) Antonio Machado (1875–1939)

Lapland is one of the few remaining wilderness areas in Europe, a vast country without roads and with hardly any human beings. A few marked hiking trails do exist, such as kungsleden, the royal trail in Swedish Lapland. In most parts, however, you need a map, a compass and good rubber boots to get from A to B. On the map The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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you can then connect A and B with a straight line and try, with the help of the compass, to follow this line in nature. If you do that, you soon discover that it was not a good idea. Between A and B you encounter mountains which you don’t really want to cross at the highest ridges, and there are deep valleys into which you don’t want to descend, since a little later you have to get back out of them. There are swamps which stop you from continuing, and there are rivers which can only be crossed in a few shallow places. In other words, the straight line is not what you should aim for. Instead you should try to find a path which is as level as possible, avoids swamps and lakes and hits rivers at the passable points. And when you finally succeed, you find to your surprise that there are tracks on that path, that the trail is already marked. The tracks are made by hoofs, not by feet: you now follow the trail of the reindeer. Over the years, many generations of animals have traveled here to create this trail, and the more it was used, the more it was accepted by the next generation. The trail was chosen by a vote with the feet. It is easily seen how that is achieved. In the following picture we note that there is an impassable range of mountains between the feeding grounds of the reindeer and the watering hole. Looking for water, some animals follow trail (a), others trail (b). Since it is much faster to reach the water via (b), this path is used more and more and eventually becomes the accepted reindeer trail. mountains

(a)

pasture

water (b)

Determination of the road through successive use.

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We shall see that this method of voting with the feet to determine the way to go is used by a multitude of creatures, from ants to humans—who through enough common use finally also authorize the path made across the well-kept lawn. The determination of the trail through hoof marks on the ground is, however, not a possibility for many animal species. How do ants recognize their path—they leave no footprints as they pass. It becomes even more difficult for bees: how can they tell their fellow bees where the source of food is that they just found? Surprisingly enough these examples confirm what we just concluded from the reindeer behavior: the road is that connection between two points which is used by the most members of the group. Let’s look at that in a little more detail, starting with the ants. The methods used by ants to specify their roads has fascinated many, and the solution of the ants—today we call it the ant algorithm—has led to applications also in various areas of human activity. We will come back to ants later on in different contexts, in particular concerning the organization of the ant state. Here we concentrate for the moment on a specific ant, leaving its anthill in search of food (see the figure). Once it has found a source of food, it returns to its home along a similar path. On both trips, going and returning, it marks its track with an aromatic substance, the so-called pheromone.

Ant in search of food, from anthill (black) to food source (gray); outward trip dark arrows, return light arrows.

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Back in its anthill, it regurgitates some of the food. Some fellow ants consider this as a report of success and now set out on their own for the source. They proceed with equal probability on one of the two marked tracks, and in going they mark these again themselves. The four possible outcomes are illustrated in the next figure. Those ants which managed to hit the shortest roundtrip (path (c) in the figure) are the first to be back, so that the next group out finds path (c) more strongly marked than the others, whose users have not yet made it back. The next therefore chose this path, and after one or two more iterations, the shortest path to the food source is determined and marked. A single ant could never achieve this; therefore this method is considered as one of the prime examples of what we today call swarm intelligence. Here again the road is the path used by the largest number of group members—a pheromone democracy, the ant algorithm. At this point, we have to add a small reservation. Actually, the ants do not chose the shortest path, but rather the fastest. If nasty biologists place a pad with a very rough surface in their way, so that a slight detour would mean a shorter trip, then the ants indeed take the longer but faster route. And our presentation is altogether only meant to be rather schematic—there are in fact

(a)

(b)

(c)

(d)

Four different path combinations following the announcement of the first ant; the shortest path (c) is determined by earliest return and hence most subsequent use.

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differently flavored pheromones, for example, one to indicate “don’t go, the source is empty.” Besides the determination method of the ant roads, the rules for the use of these roads have been subject of numerous studies, since the ants move around quickly and transport very effectively different object, leaves, insect bodies and more. It seems that the efficiency of ant traffic is the result of three basic rules rigorously adhered to by all: 1 . Everyone proceeds with the same speed. 2. Everyone keeps in their lane—no overtaking. 3. At a bottleneck, travelers wait in small groups to allow alternating passage of the traffic in each direction (“zipper system”). Ant traffic was studied by introducing narrow as well as wide bridges on the ant roads; it was found that even for narrow bridges, the adherence to these rules kept an undiminished passage rate. The rules as such were enough—no intelligent decisions were required by individual ants. And we have in fact seen that similar rules make the automobile traffic in large American cities such as Los Angeles move much more effectively than it does in most old European towns. Could it be that in the millions of years of their existence, insect states have in a Darwinian sense eliminated ­cultures based on individual decisions? Let us now turn to bees. Here we don’t have roads in the proper sense of the word, but as before there remains the problem of how a single bee, having found a source of food, can communicate this information, the location of the source, to its fellow insects. The method the bees use seems surprisingly intellectual—or should we say human? And it brought the Austrian scientist Karl von Frisch, who was the first to decipher it, the Nobel Prize. To begin with, we have to point out that bees—in contrast to humans—can identify the position of the sun even in case of a cloudy sky. Our daylight, coming from the sun, is polarized. The light waves, which in principle can oscillate three directions, do this for sunlight only in a plane orthogonal to the direction of

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= k. Se 1

α

1

km

the sun rays. This polarization persists also for the ultraviolet light passing through the clouds. We can’t see this light, but the bees can, and they moreover also “see” the polarization. So they always know where the sun is: they always have a reference point. Now once a bee has found a source of food, it returns to the hive and attracts the attention of its sister bees by doing a dance: the waggle dance made famous by the work of Karl von Frisch. In this dance, it roughly executes a figure eight, except that the connection of the two circles is a straight line (see the following figure). Traversing this line, it waggles its rear end, while it passes through the circular parts quietly. It sounds incredible, but the direction of the waggle line defines the position of the food source relative to the orientation of the sun, while the frequency of the waggles specifies the distance to the source, with one second roughly ­corresponding to one kilometer. And the more often it repeats the dance, the more attractive is that source. But the transfer of information is in fact even more subtle. Inside the hive it is often too dark to use the sun as reference point. The bee then uses the direction provided by the gravity, up and down, as substitute, with up meaning sun, and it executes its waggle dance with respect to this axis. The other bees understand the switch and when they are outside the hive to depart for the food source,

The waggle dance: positioning food source (left) and specifying its ­distance (right).

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they convert the gravity orientation to the sun reference and chose their flight direction accordingly. From time to time the bee colony divides itself into several smaller groups and thus starts new colonies. These need independent new housing, and so the search for a new nesting place is added to the search for food. The housing search and the final decision for the right place poses another interesting problem for the understanding of the bee world. Its solution, so the experts, was in fact the origin of the waggle dance of the bees, and it is indeed a highly social enterprise. Some of the members of the old colony leave the hive where they lived so far and then congregate at some specific place, the branch of a tree, for example—the bees “swarm.” Numerous individual members of this swarm now start out in search of the new home, such as a hollow in a tree. The new site has to satisfy a number of conditions, location, size, entrance and more. Since the bees will produce quite a bit of honey each year, the size has to be adequate for the storage—the scout bees have to “keep this in mind.” The search thus takes typically several days, and when it is done, the scouts return one by one, and perform a waggle dance in front of the swarm; each one indicates the direction and the distance of its proposed new site. Subsequently it returns to that site, accompanied by those fellow scouts and swarm members it has convinced that the proposed site would warrant a look. So now a number of evaluation parties check out a number of different sites. If a given party is convinced of the qualities of the new site, they fly together back to the swarm, signal their conclusion by a common dance and return again to the site in question. This may trigger further bees to join them, and if now at the new site some fifteen or more bees come together (a “quorum”), they jointly return to the swarm, announce the success of the search, and the entire swarm moves to the proposed place. In this case, the number of bees plays the role which the intensity of the pheromone tracks did for the ants. However, the transfer of information as carried out by the bees is considerably more subtle, and it is suspected that such an abstract and yet

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precise form of passing on information exists only for bees and for humans. Humans have to learn it, while for bees it seems inherited. We thus found three different methods to specify the way to go, using hoof tracks, pheromone trails or optical signals. In add­ition, there are the long-range determinations employed by migrating birds or wandering fish schools in order to reach des­ tin­ations thousands of kilometers away. The orientation needed to get there, however, is largely based on individual capabilities, not on collective actions. The flight patterns of certain bird species, however, do require collective action; we turn to this in the next chapter.

13 How to Fly

Photo: Frank Liebig

The biannual migrations of many bird species are among the most amazing phenomena on Earth. Birds ranging in sizes from hummingbirds and swallows to geese and cranes manage to travel many thousand miles to avoid the dangers of winter and then again to return to the exact locality where they were born, had lived and had raised their young. The two most challenging aspects of these migrations are how the birds determine and stay on the right route and how they can efficiently use the energy of their bodies to sustain such long flights. To underline these challenges: a bird as small as a swallow flies in the fall from northern Europe to South Africa and returns the next spring; each trip covers up to 10,000 km. For both problems, evolution has led to various solutions, and in both cases we are only slowly beginning to understand the almost miraculous ways by which they work. The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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The orientation is by and large not a collective achievement. Juvenile storks, which have never made the trip, nevertheless leave before their parents do and still reach the wintering grounds. Therefore they must have an inherited genetic code, and indeed a young stork from Eastern Europe, where the defined migration route to Africa passes through Turkey and the Near East, will travel that way all alone if humans displace it to France, where the local storks travel via Gibraltar. For other birds, geese or cranes, the young need adult leaders who know the route and can show it to them; that way, the knowledge is passed on from generation to generation. For the orientation itself, various means come into play: the position of the sun or the stars, the magnetic field of the Earth, rivers and mountain ranges, and more. For more details on this fascinating topic, we refer to some excellent books listed in the Bibliography; since they are not really issues related to collective behavior, they are not a topic for this book. Interactions between the members of the flock do come into play in establishing the formation pattern of the flock in flight. We had seen aspects of this in detail in the structure of the flock, as described in Chapters 3 and 4. A further feature concerns the positioning of the birds in the formation of the flight—a welldefined wedge-shaped V or a long line echelon, for example. Why do they fly the way fighter pilots fly? Do the birds understand aerodynamics? So the topic of this chapter will be the formation strategy of migrating bird flocks. To begin, we recall that birds have two ways of flying: active flight, carried out by flapping the wings, and soaring-gliding, performed by clever use of the motion of the air. Some species of birds use only one or the other, many both. The latter is a matter of bird and air—there is nothing collective about it. Thermal updrafts, caused by an irregular heating of the ground by the sun, provide a natural elevator to bring birds high up into the sky; they can then cover large distances by gliding back down, only to take the next thermal lift back up. This way, birds of prey, eagles, falcons and also vultures, circle endlessly without flapping a wing. On the

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Bird flight formations: wedge or V (left) and echelon (right).

ocean, the winds differ with the height directly above the water, getting stronger as one goes up. Sea birds use this differential to lift them up from sea level, rise high and then glide back down. This pattern requires essentially no effort, and so an albatross can spend days and weeks flying above the open ocean, without ever resting on land. Birds using this technique can fly alone, as do birds of prey, or in groups, such as storks; the efficiency of the flight is not affected by the other members of the group. Active flight is a different matter: the birds now have to work and therefore expend energy. This is a crucial issue—about a third of all birds migrating for the first time don’t make it; they die of exhaustion. To avoid this and to reduce the expenditure of energy, the birds make use of a remarkable form of collective action: they fly in formation. That is perhaps noteworthy in itself; but in such formations, there are more and there are less favorable positions. We know this from the peloton of bicycle racers, who rest a little while traveling in the “wind shadow” of other racers and then reciprocate the help by moving up to the front after some time. And so also the birds take turns—after some time, the one with the better position switches and gives it to one who had a worse one. This seems to contradict Darwin’s original view of evolution; his survival of the fittest assumed that maximizing the benefit for an individual would increase the chances for the whole species.

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Today we know that there are many cases where an individual makes a choice not optimal for itself, but better for the group as a whole—something Darwin had in fact also considered in the case of social insects. And so letting a fellow bird rest a while enhances the chances for the whole flock to arrive safely at its destination. Let us here just note just what happens when a bird flies; in the following chapter we shall turn to the underlying aerodynamics to explain why that happens. Air is a fluid, just like water, and when a bird moves through it, it causes the fluid to move, just as a swimmer or a boat disturbs the water by moving through it. The passing bird pushes the air below its wings downward (the so-called “downwash”); the reaction to this action, as we shall see shortly, provides the desired lift. Beyond its wingtips, on the other hand, the air moves rapidly upward (the “upwash”), as illustrated in the figure. The overall situation is indeed quite similar to a boat moving rapidly through water. The water directly below the boat is pushed down, and as reaction the boat is lifted up out of the water. Next to the boat and behind it, this leads to a bow wave, the wake, rising up and slightly spreading out behind the boat. A water skier can use the bow wave to be momentarily propelled into the air, “jumping the wave.” This gives us a clue for why the flight in V-formation is so useful: a bird flying behind another bird, but on a trajectory displaced upwash

upwash lift

downwash

downwash

Schematic view of a bird in flight.

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sideways, will travel in the upwash of the one in front of it, and thus it will get an additional lift free of charge. As a result, all birds in such a favorable position will save energy—estimates vary from 10 to 70 percent compared to a solo trip. The leader in the front, however, does not have that advantage, and so the birds frequently change positions, to spread evenly effort and advantage. But how do we know that the birds actually do save energy by flying in formation, and how can we really be sure that the do take turns in position? Getting reliable data here was as difficult as it was to study interactions within a flock of starlings. Only the last twenty years have brought real progress in such studies, largely through the advent of miniaturized electronic devices to record and broadcast information about the individual birds, rate of wing flapping, heartbeat and more. The expected relation between formation flying and energy saving for birds was introduced in 1970 by Peter Lissaman and Carl Shollenberger from the California Institute of Technology in Pasadena. Based on models and specific flight configurations, they predicted savings of as much as 70 percent. And they concluded, not surprisingly, that what was now needed were good data of actual bird formation flying. First such data came in 2001 from the group of Henri Weimerskirch and collaborators in France. They had gone to a national park in Senegal, Africa, and trained eight great white pelicans to fly behind a boat or a light plane, traveling over a lake. The pelicans were filmed in flight and carried heartbeat loggers attached to their backs. It was found that birds flying in formation had a rate of heartbeat some 10–15 percent lower than that of a bird flying alone, and they flapped their wings at almost half the rate of a solitary bird. Both observations supported the idea of Lissaman and Shollenberger, although the saving in energy was less than predicted. A cross-check was carried out by Dietrich Hummel and Markus Beukenberg of the Technical University of Braunschweig in Germany. They had two fixed-wing twin engine propeller planes (type DO-28) fly in formation, measuring the fuel consumption

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x

Formation flight of propeller planes at wing tip distance x.

for different flight positions. It was found that the plane in the rear by about one plane length (see figure) needed less fuel the smaller the wing tip separation x to the front plane became. In particular, when the plane tips were aligned (x = 0), the fuel consumption of the rear plane was some 15 percent lower in such a formation flight than it was if the plane flew alone, thus confirming the energy saving scenario. Nevertheless, for the actual problem of long range flight of bird flocks, there remained open questions. What about wing flapping and the relative flapping phases? And how do the birds decide who flies where? Here a breakthrough came less than five years ago, and it is indeed a story worth telling. Until three hundred years ago, Europe was the home for a remarkable bird, the so-called Northern bald ibis, “Waldrapp” in German, a European relative of the African ibis. It has a shiny, deep black plumage, is some two to two-and-a-half feet tall and has a long, curved red beak on a featherless head. In past centuries, its meat was considered a delicacy, and through intensive hunting it became extinguished in central Europe around 1630. Small colonies survived in Morocco and Turkey. It ranks high on the list of endangered species, with some 500 free-living individuals (in 2005) and about 2000 animals in ­captivity. Around the year 2000, attempts were started in Austria to reintroduce the bird into central Europe, and in 2014 a long-range European Union project with that goal was approved. By the end

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Northern bald ibis (Geronticus eremita).

of 2019, it should lead to the establishment of three Waldrapp col­ onies in Austria and Germany, with a total of 120+ birds, having a common wintering area in Tuscany in Italy. The latter was needed since the birds are migratory, spending the winter in warmer southern regions. However, their migratory patterns are not in­herit­ed; young birds are led by experienced adults to learn the way. That clearly posed a problem: there were no experienced birds which could play that role. It was solved in an ingenious way. The Nobel laureate Konrad Lorenz, often considered the founding father of ethology (the scientific study of animal behavior), had shown that freshly hatched birds would consider as parents or at least as reference authorities the first living beings they would have contact with. So a number of fertilized eggs of the bald ibis were taken to Austria, where they were hatched. The newly born chicks were then tended for by two human foster mothers, and these were indeed readily accepted by the ibis chicks. When the birds were old enough, the foster mothers started to practice flying with them, with an ultra-light flying device (paraglider) for the humans. The action was fully accepted, and in the fall of their first year, the birds were led from Austria to the projected wintering area, a nature preserve in Tuscany, by their foster mothers in the paraglider. Such joint flights of humans in ultra-light flyers and freely flying birds had been carried out before, notably with

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geese. Thus Christian Moullec from France led in 1990 a team of geese from central Sweden to a wintering area in Germany, from which they eventually returned to Sweden the following summer. The great advance of the Austrian–British project was that their birds were fully wired, so to speak, allowing precision measurements of absolute and relative wing flap rates, flock positions and more. The project was later extended to a colony in Bavaria/Germany, and until 2010, four groups of ibises were led to Tuscany. In 2011, the first birds came back to Bavaria on their own, and since then every year in the fall, the young birds follow their experienced elders to Italy, and every spring they all return to Germany. The project has thus established the first autonomously migrating colonies of Northern bald ibis in almost 400 years. It is assumed that from now on, no further human-led migrations will be ­necessary, that the birds now can take over on their own. Besides showing the young ibis how to reach their winter residence, the trip provided the team with excellent opportunities

Human-guided migration of Northern bald ibis crossing the Alps (photo: C. Esterrer, Waldrapp team).

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to  study in-flight migration. The birds carried miniature GPS devices, attached to their backs, and were in contact with the plane most of the time. This allowed numerous measurements to be carried out and led to a unique wealth of new data. In particular, it was now clearly established that the birds coordinated position and wing flapping such as to minimize energy expenditure. Consider a given bird in level active flight. Its wing tips trace an imaginary up-and-down line through the air, with peaks and ­valleys as in a regular wave. This is also the line of the updrafts created in the air by the passage, trailing behind the bird. If a following bird, flying over to the side and back, adjusts its wing flaps to trace a line coincident with the one produced by the advance bird, then it will receive upward pushes by the draft left behind by that bird. If the birds fly such that their wing beats are in phase, they will, so to speak, ride the wake of the birds before them. And the Waldrapp team found that that was exactly what the birds did. In fact, they did even more: if for some reason the birds momentarily ended up flying in a straight line one directly behind the other, then the upwash was replaced by a downwash. The birds now adjusted their wing-flapping to be out of phase, their line had peaks where the predecessor had valleys. This did not provide an additional enhancement, but it avoided the negative effect of the downwash. In short, the birds successfully passed both problems of their aerodynamics test: upwash gain and downwash avoidance. As mentioned, such formation flying is advantageous for all but the leading bird. This could lead to a cooperation dilemma: who is going to fly in front? The Waldrapp project provided first clear evidence that the birds indeed solve this problem by frequently taking turns. They did not wait until the front bird was exhausted and fell back, but instead changed so often that exhaustion never arose. The favorite grouping was to have two or three birds get together inside the wedge formation, and within such a group, the lead bird would swap places with another once a minute or less. This reciprocity—first I, then you, then I again and so

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Northern bald ibis in in-phase flight (photo: M. Unsoeld, Waldrapp team).

on—assured an optimal distribution of work over the entire flock. This type of formation pattern and the resulting aerodynamic gain is evidently not restricted to just the studied northern bald ibis. Model studies have shown that a crucial aspect is that the speed of the bird in flight is sufficiently much greater than the wing flapping rate; otherwise, the positive updraft cannot build up. For this reason, small birds, such as starlings, fly in dense and rather unstructured swarms. There may be small local benefits, but the large overall synchronization seen in the above picture of the migrating Northern bald ibis never takes place. Even though there are bird migrations, such as those of the Northern bald ibis, in which experienced adults show the way, the formation of the flock is again reached by collective decisions; there is no commanding bird to fix the marching order. In the Waldrapp project, it was tested whether the birds preferred relatives (siblings or parents) in the group formation, but no clear evidence to this end was found. The flight pattern was effectively determined simply in a way to minimize the collective cost.

14 Avian Aerodynamics

Mus´ee Antoine Vivenel

How can birds fly? This has been a subject of wonder and envy to humans as far back in history as we know. In Greek mythology, Icarus constructed wings out of wax and feathers to escape from his labyrinth prison on the island of Crete. And he would have succeeded, had he followed the subsequently classical advice of his father Daedalus, “medio tutissimus ibis” (in the middle you’ll go safest). Ignoring this, he flew too high and came close to the sun, the feathers of his wings melted and that was the end of his flight. In a way, the flight of birds through the air has many simi­lar­ ities to a swimmer in water. Both are subject to four basic forces: lift, weight, thrust and drag. For a bird, they are illustrated in the figure. The force of gravity of the Earth attracts the bird with a

The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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strength determined by its mass: that is the weight of the bird, pulling it down. To counteract this, the bird flaps its wings, push­ ing the air down. The reaction to this action is the lift, pushing the bird up. The simplest form of swimming, treading water, uses the same principle: by forcing the water down, we push our body up against the downward force of gravity. If the bird flaps its wings not just up and down, but at the same time also tilts them and moves them backward, the reaction now pushes it forward: it has thrust. The forward motion is hindered by the friction of the body moving through the air: that is the drag. Competition swimmers have tried to reduce the drag of the water by wearing body suits of a material having as little friction as possible—but now this is generally forbidden, in order to test the capabilities of  the swimmers, not that of the textile chemists in designing frictionless materials. In any case, both birds and swimmers have to make sure that the drag resistance of the return motion of the wings or the arms is more than overcome by the thrust of the forward propulsion. So all in all, there are two external forces acting on the bird, weight and drag, to be compensated by two forces created through efforts of the bird itself, lift and thrust. lift

thrust

drag

weight

The four basic forces exerted on a flying bird.

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What we have described here corresponds to a most rudimen­ tary form of active avian flying. In the real world, the two forces arising from the work of the bird itself, lift and thrust, are in­tim­ ate­ly connected; they arise in one smooth motion of the wings through the air flowing by. In particular, thrust alone is already sufficient to cause lift, without any wing flapping, provided the angle of the wing relative to the direction of motion is chosen accordingly. A water skier, with a slight tilt of the skis, stays afloat simply because of the force of the pulling boat; children tilt their kites into the wind and then get them to rise up in the air by just pulling and running fast enough. And after all, none of our air­ planes flap their wings—the combination of the motion pro­ vided by propellers or jets with the wing flap (aileron) angle is enough to lift them up into the air. Helicopters are an exception and their way of flying, in contrast to that of fixed wing planes, is  more similar to that of birds. In vertical position, the rotors provide only lift; they have to be tilted to get thrust and hence forward motion as well. Let us therefore have a look at how wings cause lift. One form, the simplest, we have already mentioned: the action of the wing pushing the air down leads to the reaction of pushing the wing up. This form is used when a bird hovers, flies on the spot, like a hummingbird “standing” in the air in front of a flower to extract nectar. Drag and thrust are now unimportant. The other, more generally used, combines lift and thrust to overcome weight and drag. We look at that by first considering a stationary wing in an airstream—that is equivalent to a flying wing moving through stationary air. In both cases, we have air molecules being deflected by hitting a hard object, the wing. What that does to an airstream is the subject of a well-known theorem by the famous Swiss physi­cist and mathematician Daniel Bernoulli (1700–1782). Consider air flowing through a tube of a fixed diameter; what happens when that diameter is suddenly decreased? To avoid congestion, the air now has to flow faster, which means that the momentum of the air molecules has to be redistributed. Before

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the constriction, they had a certain large momentum in the flow direction and a smaller fraction orthogonal to it. After the con­ striction, in the faster flow, the orthogonal momentum was further reduced in order to allow the faster flow along the pipe direction. Since there is no additional source of energy, the over­ all kinetic energy per molecule (the sum of longitudinal and orthog­ onal momentum fractions) remains the same, and so the increase in the longitudinal direction has to be paid for by a decrease in the orthogonal momentum. Since this orthogonal momentum deter­ mines the pressure on the walls of the tube, that pressure is less in the constricted part than it was before. So faster flow means lower pressure—and that is what Bernoulli’s theorem says. The so-called Venturi tube provides a classical illustration. Consider now an idealized cross-section of a bird’s wing, placed into a wind tunnel. Above the wing, the passage of air is restricted, squeezing layers of moving air closer together: the flow becomes faster and the pressure is correspondingly reduced. Below the wing, we have an opposite effect, and the pressure increases. The overall result of increased pressure from below and decreased pressure above means that there is lift. This lift is evidently not connected to any wing flapping—it is there as well for an air­ plane with fixed wings and a source of thrust to propel it through the air. A bird flapping its wings thus gains lift from two sources: the reaction of the air to the downward push by the flap of the

Pressure

Fluid velocity

The Venturi tube, showing decreasing pressure within increasing flow velocity.

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wing, and the dynamic effect arising from faster flow above and slower flow below the wing moving through the air. The motion crucial for the latter effect is produced by tilting the wing when flapping, so that it pushes backward as well as downward. A third phenomenon, which is in fact the essential one for the collective aspects for the formation flying, comes into play at the wing tips and beyond: the counterpart of the bow wave. The pressure below the wing is higher, above the wing lower. Since the two regions are connected, the air flows such as to equalize things, creating a form of turbulent motion: directly beyond the wing tip there is an upward draft, due to the air from the high pressure region below flowing into the upper region of lower pressure. As we move further out in the direction of the wing, this upwash eventually decreases; it is strongest close to the wing tip. The upwash persists in the air for a while after the bird or the plane has passed, until it is finally damped out by air friction. Its  presence is, as we had already noted, the reason why it is energy-wise favorable to fly behind another bird along the line traced by the wing-tip of the front bird: the upwash provides part of the needed lift and thus saves energy. And if the front bird flaps its wings, the line of upwash oscillates up and down, with a ­certain wavelength determined by the flap rate. To best use the upwash, the following bird has to adjust to the oscillation: if it is just exactly one wavelength λ behind, it has to flap up-down in time with its leader’s up-down. On the other λ

time

Periodic wing flapping, with a wavelength λ.

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λ

λ/2

Synchronized wing flapping.

hand, if it is only half a wavelength back (or one and a half), it has to flap down-up in time with its leader’s up-down. The mentioned Austrian–British project has shown that that is exactly how the birds fly—in a phase determined formation, to make full use of the upwash energy gain. This does not preclude other advantages of such a formation; in particular, an echelon line rather than one behind the other allows the birds to see those in front of it. That, in addition to the energy gain, shows the advantages of collective flight patterns for bird flocks—flight patterns which the birds adopt by self-organization, not by any command of some squadron leader . . . We had mentioned the simi­ larities between swimming and flying, and so an obvious question appears: do fish also swim in schools in order to save energy? Does the slipstream of the front fish play the role of the upwash of the birds? This issue has been addressed in a number of studies, but so far they remain inconclusive . . . waiting for electronic devices sufficiently small to attach to fish and for under­water paragliders in which humans can accompany the course of the school.

15 The Ant State

The ants look as if they know what they’re doing. Richard Feynman, Surely You’re Joking, Mr. Feynman

For every human being alive today, there are on Earth approximately one million ants, so that our planet presently is the home of some ten thousand trillion ants. All known species of ants are organized in states; we therefore share the Earth with numerous other organized societies. In the course of their evolution, however, insect states have developed a form whose structure is quite different from that adopted by mammals, birds or fish. Not only is the state of ants or bees more than the sum of its members; the members, the ants or bees, can only survive as integral parts of the whole; a solitary existence is for them no longer possible. Here the self-organization created through evolution determines the parts as well as the whole. Biologists call such a structure a superorganism: a system whose components cooperate and thereby assure simultaneously the existence of the whole and their own individual survival. In a flock of birds, every bird finds its own food and every pair raises its own young. In contrast, in an insect society there is one animal, the queen, which gives birth to all the young of the state; all members of the state thus are brothers and

The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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sisters. Certain members provide the food for all; others are responsible for construction and maintenance of the housing structure, still others for nursery duties or for the defense of the inhabitants. The state thus consists of different departments, each of which is responsible for specific functions necessary for the survival of the whole as well as that of its individual members. Such an insect community thus has more in common with a large modern industrial enterprise than with a flock of starlings. As mentioned, an isolated single ant normally cannot survive, unless it is a fertilized queen capable of laying eggs; through their further development it can then eventually start a new state and thereby end its solitary existence. All ant states are hierarchically structured: their populations are divided into castes with specific functions. There is one queen (sometimes also several), males and other females. The males only play a short role in the reproduction, after which they die and at best serve as food for the females. The females, in turn, are subdivided into sections with different functions: queen, nursery attendants, nest builders, food col­lec­ tors or soldiers. The lone queen is by no means a ruler: its only function is to lay eggs, and the caste of the other females is determined at birth. That means that all workers in the ant state are sisters; all have the same mother, but, as we shall see, there are many different fathers. From a human point of view, the ant state lacks certain at­trib­ utes of a state: there is no president or leader of any kind; the state has no ministers, no executive, no parliament. And if we come back to the comparison with an industrial enterprise: there is no board of directors, no council, no administration. On the other hand, there definitely is a state police, an army and some other organizational and social entities. Moreover, ants do have some form of language to communicate with each other. Since in general everything functions very well in an ant state, one is tempted to ask how necessary the additional organs of our human states or enterprises really are—and how it came to be that today insect states surpass human states by so many orders of magnitude.

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Let us therefore look at an ant state in a little more detail, starting from its creation. There are ants of many different sizes, forms and behavioral patterns; we therefore imagine some kind of standard model, in which the prevalent features of different species are combined. An existing state produces young virgin queens and males (drones), which leave the anthill at some given time and depart on a wedding flight. Each of the young queens mates with several males and preserves the sperms it receives in these acts in a specific bag it has for this purpose. These millions of sperms are the potential basis for a future ant state. The fertilized queens therefore now proceed to look for a suitable location for a future anthill. The males die shortly afterward and are, as mentioned, often eaten by the workers of the home state of the young queens. But the queens themselves also very rarely achieve their goals. The world is full of ant eaters of all kinds; so on average only one queen in a hundred thousand succeeds in finding the desired protected place for the beginning of a new state, a place where it can now start to lay eggs. For the egg production, there are two possibilities. The queen may or may not add sperms to the egg it is laying; in other words, it can decide whether or not to fertilize the egg. If it does, the egg will give rise to a female, if not, to a male. In the starting stage of the new state, the queen fertilizes most eggs, so that practically only females are produced. The queen so far does not have access to any food whatsoever, and the newly hatched ants, the larvae, also have no source of energy. For this reason, they become largely infertile, their ovaries remain undeveloped. In the new state, the queen is thus surrounded by a multitude of infertile daughters, the new workers. These now begin to build the new community. They collect pine needles, leaves and more, to construct the anthill. They swarm out looking for food, to be collected for the queen as well as for the continuously produced brood of larvae, which they clean, house in nurseries and feed. The basis for the new state is now provided; it is a state consisting only of infertile sisters, daughters of one and the same mother, born out of the sperms it had received on its wedding flight.

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In the more advanced stages of the state, the queen is more amply provided for, and also some of the larvae get a little more food. As a result, these favorite daughters develop functioning ovaries; they are fertile and later become the next generation of virgin queens. It is not known how larvae are selected for this track, or who does the selecting. The queen now off and on also lays eggs which it has not fertilized and which therefore become males. The ant state is now complete: it has a queen, some young virgin queens, some males and a vast number of infertile female workers. As long as these workers are young, they act mainly as nursery assistants within the anthill. When they get older, they then become delegated to outside activities, collecting food for the queen and the workers inside, as well as material needed for the maintenance of the anthill. That provides an idea of the overall structure shared by most ants. In more detail one finds developments which are incredibly similar to what is found in the evolution of human societies. There are nomadic ants, which travel around in search of food and therefore have to build new nests again and again. There are resident ant communities with extensive anthill structures, in which they practice agriculture as well as stock farming. Ant states attack other states and enslave their populations. And in most ant states, there are soldiers to protect the roaming members of their community as well as to destroy or subjugate competing ant states. One feature, however, is common to absolutely all: there never is a leader, a ruler or a commander. Let us look at three specific examples. The driver ants found in Africa and Asia are nomadic hunters of insects and other small animals; their states are very numerous and so they continuously have to look for new hunting grounds. Their forays generally start as a trail, a long band of ants marching side by side, with soldiers protecting the troops on both sides. Once this army has penetrated sufficiently far into new lands, it splits up into more and more branches, so that eventually a broad, fan-like search machine is rolling over the land. Everything in its

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path which is not too big or not able to flee in time is killed, cut up and carried back to the nest: insects, reptiles and small mammals. We emphasize again: the whole enterprise, which sounds a bit like the conquering hordes of Genghis Khan, has no leader. It is completely self-organized. The individual ants run forward, in the general direction of the whole army. Sometimes a given ant will stray a little to the side, even turn around for a moment— but the troop as a whole remains in motion in the specified direction. The march continues up to a certain time, probably fixed by the position of the sun; the army then comes to a halt and returns to the nest, following back the odor track laid on the way out. A particularly interesting form of ant is the leafcutter, inhabiting tropical jungles in Central and South America: they live on fungi which they grow themselves. To raise these fungi, they need the leaves of living plants. Using the scissors formed by their jaws, they cut pieces out of leaves of the local vegetation. These are then carried back to dark, humid earth chambers specifically built for this purpose, and there they are used to cultivate the

Leafcutter ants transporting leaves to their nest.

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fungi. The fungi grow nutritious tips, and only these are consumed by the ants, without destroying the fungal cultures as such. The whole activity is thus indeed a form of well-defined agriculture on quite a large scale: the chambers can extend to a diameter of 30 m and more, and a given ant community can contain millions of individuals. The fungi cultivation is moreover of mutual benefit: the ants need the fungus to survive, but without the ant cultivation, the fungus itself can also not exist. Certain ant states also carry out a form of stock-farming. In many forest areas one finds the so-called honeydew, a form of excrement produced by plant lice (aphids) and scale insects. These small insects live on the sap they extract from plants; they are, however, not able to digest the sugar contained in the sap, and so they excrete this immediately in the form of what we call honeydew, a sweet, sticky liquid. Bees love this honeydew and use it to produce the so-called forest or pine honey, a dark and very sweet form of honey. Certain species of ants keep large herds of such plant lice: they house them, feed them with leaves, protect them and raise their offspring. As a reward, the ants get the desired drops of honeydew, and if the lice don’t produce quickly enough, the ants carefully tap them—they “milk” them.

Ant milking a plant louse (aphid).

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Finally we briefly want to mention how ants communicate. In contrast to mammals and birds, vision does not play a significant role. Ants communicate by giving off or registering chemical odor substances, pheromones, by touching each other and through acoustic signals in form of earth vibrations. Pheromones are secreted to tell fellow ants where a source of food is to be found, whether that source is still rewarding enough, whether enemy forces are threatening, whether nursery activities are needed, if the queen requires feeding and much more. Touching a fellow ant can cause it to share stored food or to follow along a certain path. By drumming on the ground, an ant can warn its co-habitants of a possible danger. So there is a form of ant language, quite multifaceted and very different from ours, yet quite suitable for the transfer of information. We thus see that in ant states, a great variety of organized activities can take place. There are orderly forays to rob and plunder; there is fungus cultivation and the raising of lice; there is road building and traffic regulation. There are even very complex nest structures, which weaver ants build by pulling leaves together and then gluing them fixed. All these activities are carried out by communities of animals consisting of up to millions of in­di­vid­ uals, without any kind of leadership or direction. Already the number of participants clearly that rules out. Everything occurs as a result of a self-organization formed and perfected in millions of years of evolution. Not only the body structure of the individual ants is inherited; the pattern of behavior is as well programmed by inheritance: every animals knows from birth on what its function is, just as in a human every cell of the lung knows its function. That is why biologists call insect states superorganisms.

16 Kin Selection: My Sister’s Keeper I will confine myself to one special difficulty, which first seemed to me insuperable, and actually fatal to my whole theory. Charles Darwin, On the Origin of Species (1859)

We had noted that in insect societies, ants, bees and more, the individual members can only survive as part of the whole, as cells in a superorganism, in which they fulfill specific functions and in return receive certain benefits. In this chapter, we want to look in more detail at the evolutionary pattern through which such a society structure could arise. In particular, we will see that Darwin’s survival of the fittest can have unexpected and surprising aspects. It initially implied that those best adapted to their environment would through their descendants pass on their genes to the coming generations. In insect societies, the workers, though perfectly adapted, pass on nothing, while the queen, un­able to live on its own, is the source of all future generations. How can such a structure evolve? In the sexual reproduction of humans, just as of other mammals, reproductive cells (gametes) of the mother (the ovum) and the father (the sperm) form by cell splitting. Each gamete contains one-half of the forty-six chromosomes of a normal cell, and when ovum and sperm merge to form the new cell of the future child, that again has the full 46 chromosomes. The child thus receives twenty-three chromosomes randomly taken from those of its mother, twenty-three from its father. Such a reproduction mechanism is referred to as diploid, from the Greek diploos = double.

The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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All cells of the resulting being thus are diploid, with the exception of the reproductive cells, the gametes. These, as mentioned, are formed by a splitting of the parent cells and thus contain only one set of chromosomes; hence they are denoted as haploid, from haploos = single. Going back in our family tree, we have two hereditary sources (our parents) for our own generation, four from the previous one, the grandparents, eight from the great-grandparents, and so on, leading to the generation sequence 1, 2, 4, 8, 16... ® 2 n starting with one for ourselves. Our degree of relation to each parent as well as to each child thus is 0.5, to each grandparent 0.25, and so on; similarly, full siblings (brothers and sisters) are related by 0.5, half-siblings, nieces and nephews by 0.25, cousins by 0.125 and so on. In insect societies, as already indicated in the last chapter, the situation is quite different. The case of honey bees is quite similar to that of ants: there is typically one queen, whose sole function is to lay eggs. During her wedding dance, she had mated with a number of males and in a special body compartment for that purpose (spermatheca) retained an immense number of their sperms for future use. If she does use them to fertilize an egg, a female bee is produced, while unfertilized eggs lead to males (drones). The females thus receive an equal number of chromosomes from the mother and from one of the fathers. In contrast, the males have cells based on one-half of the chromosomes of the mother only—they are haploid, while the females, having two sets of chromosomes, are diploid. This asymmetric situation, haplodiploidity, in which females have a cell structure based on com­bin­ation of two sets of chromosomes, as is the general case for mammals and most other animals, while males receive one set from their mother only, leads to a rather different genealogical pattern. A given male bee has no paternal chromosomes: it only has the one set inherited from its mother, and it can only pass this on.

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All the sperms produced by this male are thus identical clones of each other. As a result, two daughters with the same father are now more closely related than they would be in the mammalian world: they have obtained all of their father’s and half of their mother’s chromosomes, so that the degree of relation between sisters is now 0.75 instead of the 0.5 of the mammalian world. On the other hand, if they were to receive a child of their own by mating with some unrelated male, their relation to that child would only be 0.5. In other words, in the world of the honey bees, sister–sister relations are closer than mother–daughter relations. This feature, as we shall see shortly, is most likely a major reason for the queen–worker structure in the bee society, in which the workers prefer to take care of the larvae of the queen, their sisters, rather than having children of their own. The genealogical pattern for the male bees is also quite interesting. Denoting it as generation one, the preceding generation source is also one, since it consisted only of the maternal half set. This set was produced in the mating of its grandmother with a male grandfather, providing the single set of the male and half the set of the grandmother, two sources in total. The grandfather again had only a single source, while the grandmother had two, making a total of three. Continuing this pattern leads to the sequence 1, 1, 2, 3, 5, 8, 13, 21... ® fn = fn-1 + fn-2 . for the number of sources at each stage. Some further reflection will show that in fact it holds equally for the number of male ancestors at each stage as well as for the number of females. The scheme is illustrated in the figure, where one can also carry out the counting. This sequence, in which each term beyond the first is the sum of the two previous ones, is in fact well-known in mathematics: it is the so-called Fibonacci sequence. Fibonacci (short for Figlio di Bonacci), or Leonardo da Pisa, was a math­em­at­ician in Pisa in the twelfth century and is considered one of the greatest in the Middle Ages. He used the sequence to describe the propagation of rabbits, but in different contexts it was already known in old Indian

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3 2 1

male female

The genealogical pattern for male and female bees.

and Greek writings. Moreover, Fibonacci had good contacts to Northern Africa, where bee keeping was abundant, so that it may even have originated from the study of bees. As a sideline we note here that the Fibonacci sequence makes its mysterious appearance also in various other natural phenomena. The seeds of many flowers, such as sunflowers, are arranged in the form of spirals; the number of seeds in the spiral arms is generally one of the Fibonacci numbers, such as 21, 34 or 55. And since antiquity, mathematicians have studied what they call the golden ratio: it is determined by the requirement that the ratio φ of two numbers a > b is the same as the ratio of their sum and the larger one: a a+b j= = . b a Its solution is an irrational number, j = (1 + 5) / 2 = 1.61803 …, and it has often been considered as a divine proportion, having inspired artists and scientists alike. The ratios of increasing successive Fibonacci numbers converge to this value, that is, 34/21, 55/34, 89/55 and so on approach in the limit the golden ratio φ. The origin of this result, as well as that of the patterns of bees, rabbits and flowers, remains a mystery. The best answer is perhaps the statement of a German mathematician who said that mathematics is the language God uses when he speaks to humans. Summarizing the role of insect genealogy, we note that bee and ant colonies show a unique reproductive structure, in which

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the queen can determine the sex of its descendants by adding or withholding sperm. As a result, the children are diploid (females) or haploid (males), having two or only one set of chromosomes. This in turn leads to a different relationship structure, in which sisters can be closer related than mother and daughter, and it may well be the evolutionary origin of the formation of a sterile worker cast, in which females tend for the children of the queen, their sisters, rather than having children of their own, which would be less closely related to them. Let us then now turn to the guidelines operating in the evolution of animal species. Charles Darwin, in his momentous Origin of Species, had the fittest survive, and that left the structure of a bee or ant colony as a puzzle. Why did fit females eventually become sterile and devoted to care for the children of one single queen? The survival of the fittest seemed to rule out altruistic behavior, in which individuals acted such as to support another individual at the expense of their own survival. Darwin tried to generalize his proposal to a survival of the fittest families, rather than individuals, but even that did not fully clarify the situation. Eventually, however, it turned out that things were not quite as simple as our human-oriented thinking had suggested. Let me suppose I am fit in the mentioned biological sense, I carry a fitness gene. My brother will carry the same gene with a probability of 0.5, and his children with one of 0.25. Nature does not care about passing me personally on to posterity, it only wants to pass on the fitness gene I have. If my brother has six children, that chance is 6 × 0.25 = 1.5. If I have only one child, my contribution to passing on the desired gene is 0.5. So if I don’t have any children myself, but instead help him to assure that his six survive, I am doing three times more to contribute to the survival of the fittest. From this, we learn that the survival of the fittest can proceed through two distinct avenues: I pass my fitness genes on to my direct descendants, or I help my close relatives to pass theirs on to a sufficiently large number of their descendants. The latter is known as inclusive fitness, in contrast to the former direct

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fitness, and they both serve the same purpose, the survival of the fitness gene. This more general view of the meaning of survival of the fittest was initiated by the British biologists R.A. Fisher and J.B.S. Haldane and formulated in the present form by W.D.  Hamilton, late research professor at Oxford; it is now generally referred to as Hamilton’s rule. For the precise formulation, we define the cost C to the giver (it would be C = 2 if I renounce having two children) and the benefit B to the receiver (this would be B = 6 if my brother must fully rely on my help to raise all six of his children). These quantities are compared to the degree rBC of relation between giver and descendants of the receiver, and the degree rC between the giver and his own descendants. Hamilton’s rule then requires that the ratio B to C be larger than that of rC to rBC, B /C > rC /rBC , in order to make the inclusive fitness more effective than the direct fitness. In our example, B/C = 6/2 = 3, while rBC = 0.25 and rC =0.5, so that rC/rBC = 2. Hence the condition requires 3 > 2 and thus is in fact satisfied. In general, it means that the less closely the descendants of the receiver are related to the giver, the more he receiver has to produce in order to make inclusive fitness more effective than the direct approach. However, one has to be careful with the application of the rule: if my brother could in fact raise two children even without my help, his benefit would be only the additional 4, so that then the two approaches would have equal weights, and if on his own he could take care of three, it would be better for gene survival if he proceeded without my help and I have the desired two children of my own. This leads to an interesting side issue: are bees and ants in fact capable of distinguishing relatives of different degrees? If a bee encounters another bee it had never seen before, can it distinguish a full sister from a half-sister or a cousin? It caused considerable amazement in the scientific world when the answer was shown to be affirmative. This result was obtained in 1979 by Les Greenberg,

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then a student at the University of Kansas, USA, in a study devoted to a specific bee species, the so-called sweat bee (Lasioglossum zephyrum). These bees live in underground nests, and their life and behavior can be readily studied in the laboratory by constructing earth ­layers between two glass panes. The entrance to a nest is generally guarded by a female bee that wards off intruder bees and other insects. Greenberg now studied how the genealogical relation between doorkeeper and intruder affected the latter’s chance of getting in. To obtain a wide range of relationship variables, he first used in-breeding to produce bees of fourteen different degrees of relation, and then had these attempt to enter a door guarded by a known given bee. While full sisters were almost always readily admitted, cousins had less than a fifty–fifty chance and unrelated bees were generally refused entrance. The pattern of his study, which is shown in the figure, illustrates strikingly that a bee, in  contrast to humans, can rather readily tell if and how an unknown bee is related to it. How it does so is only partially resolved. Greenberg proposed that it is largely decided by smell. The bee would carry in its brain an odor template and then check to what extent the odor pattern of the intruder matched this

acceptance probability

1.0 0.8 sisters

0.6

cousins aunts/nieces

0.4

unrelated

0.2

0

0.2 0.4 0.6 0.8 genetic relatedness

1.0

Kin recognition of sweat bees (Lasioglossum zephyrum), from Greenberg (1979).

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template. There may, however, be still more basic mechanisms at work. A human body accepts transplantation of tissue from other parts of the same body, but rejects tissue from foreign bodies. In general, however, tissue from an identical twin is accepted. Keeping this in mind, we now begin to understand the origin of the structure of insect societies, with their given reproduction pattern. If a strong and healthy female bee has ten children of her own, the factor for gene-passing is 10 × 0.5 = 5. On the other hand, if she instead takes care of ten young full sisters, daughters of her mother queen who is unable to do this herself and of the same father, the factor becomes 10 × 0.75 = 7.5. If a hundred bees of a hive behave in this way, the gene survival measure is by a factor 250 higher than it would be if the bees would all have children of their own. In societies of such reproduction biology, inclusive fitness thus by far surpasses direct fitness. And even in an overall comparison, we must remember that there are about one million ants for every human on Earth. We have thus seen that evolution has produced forms of swarms quite different from the ones we started from. These swarms, locusts, starlings or wildebeests, consisted of similar or identical individual members which led their own individual lives—feeding, finding a partner, breeding—apart from their swarm activities. In the course of evolution, however, nature through trial and error discovered another way of dealing with the issue of having the fittest survive. It found that in sufficiently large animal societies, a division of labor and a dedication to specific tasks could result in a more efficient survival of fitness. It made insect colonies early predecessors of production line technology, with one big difference: in the case of ants and bees, there is no foreman, no boss, no director. The entire structure has evolved in a self-organized fashion, such that each member knows from birth what is function is.

17 Of Bees and Flowers I can understand how a flower and a bee might slowly become modified and adapted in the most perfect manner to each other. Charles Darwin, On the Origin of Species, 1859

In the previous chapter we had seen that the structure of insect societies is very different from that of humans, mammals or birds. Instead of autonomous individuals coming together as a swarm, in which each member still has its own existence, insects form superorganisms, in which each part has a specific functions; each can only exist as a part of the whole, and the whole can only survive by a precise interplay of the parts. Here we now want to point out that collective cooperation exists on an even larger scale, in form of the so-called mutualistic interaction between plants and pollinators, where the existence of both partner groups crucially depends on their cooperation. In this case as well, bees—in their interaction with flowers—provide the best known example. Let us briefly recall the reproductive mechanism of plants. The earliest form in their evolution had open (non-encased) seed and pollen stands, and the male sperm, the pollen, were abundantly produced and transferred to the female ovules mainly by wind. Such gymnosperms (gymno = naked) persist today in the form of many trees (pines, birches, etc.) as well as in most grain species (wheat, rye, barley, etc.). The fertilized ovules then become seeds in the form of pine cones or grain ears. The evolutionary later form of the so-called angiosperms (encased seed stands) constitutes by far the largest part of the plant kingdom, the flower plants. The female

The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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part, the ovule, is encased in the ovary, and the male sperms are produced in separate pollen bags, both contained in the same structure, the flower. To best fertilize the seed, the pollen has to be transported from the pollen bag of one given flower to the ovary of another, more specifically to a surface of that ovary, from which it is then passed on to the ovule. This transport generally requires a sperm transport agent: enter the insects, and to a much lesser degree birds (Colibris) and mammals (bats). With this construct, evolution effectively produced the first, primeval form of commerce. The plants need the transport agents for their reproduction, and so they have to offer some bene­fit to attract them: nectar and/or pollen. In return, the construction of the flower assures that the agents in the process of acquiring the reward carry out the pollination. In addition, several other aspects come into play, showing some of the subtleties of evolution. To attract the pollinators, the plants have to advertise by sight and smell: they have to produce colorful flowers with attractive scents, and these flowers must provide the reward in the form of nectar as well as pollen. The flowers have to be grouped into identifiable sets: a bee carrying the pollen of an apple tree flower should deposit it on another apple tree, not on a rose, where it would do no good. To prevent inbreeding, the flower construction should moreover assure that the pollen is not deposited on the same flower—inbred reproduction turned out to produce less suitable descendants. And to assure the distribution of the final seeds, a further, additional attraction is generally offered: the seed stand forms a fruit, an apple, a berry or such, which other animals consume and thus distribute. In the course of ­evolution, all these requirements were eventually satisfied. The crucial relation, however, is the mutualistic one between flower and bee: neither can continue to exist without the other, the botanists speak of obligate mutualism. Evidently all animals rely on a source of food to survive, but in general the search for food is an individual act: a squirrel consumes or even stores nuts, a horse feeds on grass, a lion kills a zebra. In the case of bees (and ants or

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wasps as well), it is a fully structured collective enterprise. Search teams go out to find a source of food and then communicate that to the fellow bees. Worker bees then fly to the source, collect the nectar and pollen, bring it back to the hive and store it there. When needed, it is then fed by caretaker bees to the larvae or the queen. So the different components of the swarm each play their specific role in finding, collecting, storing and using the food provided by plant flowers. On the other hand, the plants can only survive as species through the pollination carried out by the bees. Eliminating bees totally would be the end of the growth of apples, almonds, blueberries and many more. We really have a complete interdependence of bee societies and fruit plants. Curiously enough, mankind has already in prehistoric times chosen to enter the scene, to make use of the efforts of bee ­so­ci­eties. Ten thousand year old cave paintings in Spain, from the

Honey collector, stone age painting in the Cuevas de la Araña (~10,000 bc).

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Stone Age, show honey hunters in action, and beekeeping started in Anatolia some 7000 years ago. In the course of history, man made use of many animals, but it always was a relation in which man interacted with individual animals as such. A horse allowed faster travel, a dog could guard the house, a cow could provide milk. In the case of bees, however, man wanted products prod­uced by the bee society, honey and wax. We took (and take) from them what they produce through complex collective efforts for their own use. An individual bee is of no interest to us—we need the col­lect­ ive, self-organized state, with queen, drones and different types of

Bee hives in the fourteenth century (Taccuinum Sanitatis, Casanatense 4182).

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workers interacting in just the right way, to produce what that need and what we want. For many centuries, until sugar cane and sugar beets became available, honey was essentially the only source of sugar. And the wax of the bees provided the material needed for the production of candles. For this reason, many medieval monasteries practiced beekeeping. Today, beekeeping is practiced both for the production of honey and to assure a source of pollinators for fruit trees and plants. In regions of large-scale fruit production, such as California or Northern Italy, an intensive commercial beekeeping industry has developed, with beekeepers traveling with their hives to wherever they are needed. In the USA, there are several hundred thousand such beekeepers, with over two million hives; in Germany, around one hundred thousand keepers hold about a million hives. Beekeeping thus always was and has remained a significant agricultural enterprise, producing more than 10 percent of all human food. Without the efforts of the bee swarms, human life would indeed be much poorer.

18 Communication and Language In the beginning there was the Word. The New Testament, John 1:1

The swarm behavior found in insect states is undoubtedly the most rigidly structured form in the realm of self-organized so­ci­ eties. Nevertheless, different versions of collective behavior have developed more or less in parallel in the course of evolution, and it is probably premature to consider one or another form most suitable for survival. During the past 50,000 years, however, one species has spread over the entire globe and become dominant over all others: the humans. While rule and leadership clearly played and continues to play an important role in human society, there are still many aspects in the social behavior pattern of humans which show definite collective features. So let us then consider human communities as specific forms of swarms. If we do so, we are immediately confronted by that one crucial question: what has made the human swarms so effective? The bands of prehistoric humans roaming the Earth some fifty thousand years ago were presumably not so different from the groups of chimpanzees or gorillas living today in the African jungle. Yet in a comparably short span of time, these humans, who were not stronger, faster or tougher than their competitors for food and space, occupied and dominated all corners of the Earth. Why and how did this happen? One feature which we have so far addressed only off and on is the communication between the members of the swarm. Sight, touch, smell all play a role; different pheromones provide different

The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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messages in ant states, and the waggle dance of a bee can localize food sources quite precisely to the hive members. The sudden turn of a bird is transmitted to its immediate neighbors and t­ ravels on to the entire flock; the whistle of a guard marmot alerts the entire colony. All animal groups have some way of communication, to warn members of an imminent danger, to advertise food sources, to attract partners and much more. But all these messages deal with the physical reality in which the animals find themselves; it is the actual reality at that given time. As far as we know, no animal can say to its fellow animal “last winter was tough and our beloved grandmother died” or “in three weeks we will migrate to Africa.” Animal languages do not address abstract issues. A specific ant or bee can tell whether another individual does or does not belong to its community; but it presumably does not have an abstract concept of the community as a whole. The evolution of human thinking has managed to create a language allowing abstraction: there is the actual apple, and then there is the word apple. By saying “apple” to a fellow human, I create in his mind the idea, the concept of an apple, even though no actual fruit is present. This was followed by creating words for concepts which have no touchable reality at all: wisdom, fear, love and many more. And once one could speak of such abstract concepts, one could also travel in time, to happenings that took place long ago or to events expected in the future. It was the language which provided the human communities with a scheme of organized thinking, with a mental framework, which no other species had at its command. In human history, we have come to denote numerous transitions as “revolutions.” There was a point at which some nomadic bands of prehistoric humans stopped foraging and hunting to obtain food; they settled down in fixed locations, where they raised crops and cattle for survival. Then there came a time when some human groups began to congregate in cities, in which agriculture was no longer possible. They performed certain tasks, making clothes and shoes, knives and plow shares, which they

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traded for food with village inhabitants. And within historical memory, the industrial revolution created a framework in which humans no longer produced complete products; the production process was subdivided into many steps, each of which was carried out by single humans: a production line. All these transitions were undoubtedly crucial in the creation of the human society as we know it today. But more fundamental than all of them was the evolutionary appearance of language in the human sense, creating the word “tree” independently of the object “tree” and thus providing humans with an abstract reality, independent of the concrete one in which they lived. It was this transition that separated humans and animals. So indeed it is the word which marked the beginning, the creation of humans. The rudimentary forms of language, which must have appeared in the course of evolution tens or hundreds of thousands of years ago, provided the roaming bands of humans with a tool no other animal society had. It was now possible to plan, organize, recall past events and discuss future possibilities. And the appearance of language was evidently a self-organized phenomenon—there was no primordial teacher. We have, of course, no way to go back and reconstruct the beginnings of language; what we can do is

Painting from the Cave of Lascaux, France, dated at 17,000 to 15,000 bc.

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look at the earliest records left by humans of what they saw in their world. Cave paintings dating back more than 15 thousand years show bulls, horses and more, providing us with a glimpse of the world of prehistoric man. So at that point, man was clearly already on a different level than animals. We don’t know of any animal which makes pictures of its world. It seems reasonable to assume that at that time there was, besides the real horse, also the word “horse,” which finally led to a painted horse. The creation of such paintings required considerable effort, and hence the appearance of a simpler form of record is only natural: some five thousand years ago, the written language first appeared. We define this today as the transition point between prehistoric and historic events. The first written records deal with accounting: how many cows, how many barrels of food, and records of similar concrete items. Sumerian cuneiforms, Egyptian hieroglyphs and Chinese characters all started as pictures of real objects, birds, cows, houses, rivers and more. One had invented a sound, a word, to symbolize an object, and to record the symbol, one reverted to a much simplified picture of this object. And to specify the number of the objects in question, it was obviously useful to have symbols for numerals, generalizing the counting with fingers. Given a language to describe concrete objects, the extension to abstract concepts was only a next step, but it was an absolutely crucial one. Mankind now had at its command a tool which allowed it plan and organize in ways hitherto unknown. The human mind, the “cogito ergo sum” (I think, therefore I exist), requires as foundation the existence of a language. In the beginning, there was the word, and in the end, it led humans to conquer the entire planet. Although leaders played a decisive role in human history, there thus are collective mechanisms operating at a perhaps more basic level, mechanisms which are a pre­requis­ite to all that followed. Since language is such an essential element of all human so­ci­ eties, and since it is presumably the outcome of collective developments, it seems natural to look for common structures present in

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all languages, however different they may be. We will close this chapter with one such structure, which is very striking but certainly not yet fully understood. It is known among linguists as Zipf’s law, named after the American language expert George Kingsley Zipf, who popularized it (around 1935) but did not claim to have invented it; there had been some previous indications in that direction. To obtain the law, one considers extensive literary texts, the bible, lexica and the like, and determines the frequency of words used. The words are then ranked according to the frequency observed: rank 1 for the word used most often, rank 2 for the next, and so on. The rank k thus specifies how many words exist that are used as much or more often than the word ranked k; for a word of rank 3, we thus know that there are three such words, the word itself, that of rank 2 and that of rank 1. In such a study, one classifies N words, with N as large as possible; present evalu­ ations go up to millions of words. In its simplest form, Zipf’s law then states that the frequency of use, F(k), of a word of rank k is inversely proportional to k, 1 F (k ) = a ´ , k where a is some constant, dependent on how many words were studied altogether. It is readily seen what this means: the most often used word occurs twice as often as the next frequent, three times as often as the third most frequent, and so on. The higher the rank of words, the closer their frequencies get to each other; the frequencies for rank 100 and rank 101 become almost identical. In other words, the distribution of F(k) falls steeply at the beginning, from 1 to 1/2 to 1/3 and so on, and then flattens more and more. The amazing thing is that this apparently simple game works. In the English language, the leaders are the, of, a, and and to. Zipf’s law then predicts that if we multiply the number of times of is used by two, we get the number of times the was used. Five times the number for to should also do it. And with some 10–15 percent

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fluctuations, particularly at the beginning, the agreement is quite good. In the figure we show an illustration, with the, of, and, a and to as leaders. To show the effect most clearly, one generally adopts a logarithmic scale of the frequency, log F, and compares this to the logarithmic value of the rank, log k. For the above formula, this gives us log F = log k + log a, a straight line. The law holds in fact over an immense range, up to millions of words; in the following picture, we show an ana­lysis of up to 10,000 words, based on the story Moby Dick by the American writer Herman Melville. What makes this law even more striking is that it apparently holds for all languages, with some fluctuations at the beginning the

10000 of

whale

1000 Frequency (log scale)

and to a in that it his i

100

10

1 1

10

100 Rank (log scale)

1000

Zipf’s law applied to the English language.

10,000

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and the end; even the form of the deviations is universal. And it holds also for ancient languages which are not yet deciphered: we don’t know what the characters mean, but their frequency satisfies Zipf’s law. This regularity thus seems to be a structure common to all human languages, however, wherever and whenever they developed. Its origin has been much discussed and disputed, but it seems fair to say that so far there is no generally accepted ex­plan­ ation. Zipf himself proposed that it arose because in all societies both speaker and listener wanted to exert a minimum effort to communicate, and this eventually led to the observed pattern. From the point of view of physics, it is most remarkable that the law has exactly the form obtained from self-organized crit­ic­ al­ity (see Chapter 7): the scale-invariant inverse power law. The relative rank of a word plays the role of the size of the avalanche in the sand-pile model. Doubling the size of an avalanche decreases it occurrence frequency by a factor two, no matter what the actual size is. Here doubling the rank of a word decreases its frequency of use by the same factor two, no matter what the rank is. Perhaps the evolution of the language has through self-organization driven it to the critical point specified by Zipf’s law, just as the addition of sand drives the pile to the critical slope for avalanche production.

19 Epilogue In the evolution of animals on Earth, swarm formation has shown to have great advantages, compared to an existence as isolated individuals or small groups. Estimates indicate that some 1016 ants live on Earth, and a single ant colony can consist of hundreds of millions of ants. The largest recorded locust swarm, mentioned in Chapter  2, had some 1013 members. To get some feeling for these numbers, we recall that the present human popu­la­tion of Earth is less than 1010 persons, leaving us with more than a million ants for every human being. And moreover human life itself is also becoming increasingly swarm-like. So as far as numbers are concerned, social animals outnumber by far those preferring a more solitary existence. And it is perhaps not surprising that in the course of time quite different swarm patterns have developed. The formation of swarms, their structure and their co­ord­in­ ation can occur, as we have seen, in a great variety of ways. There are simple assemblies of many individuals, such as locusts, who move together in the same direction once their density is great enough. The coherence of a flock of birds is found to arise through the interaction of each bird with its six or seven nearest neighbors. And the synchronization of the signals emitted by fireflies and crickets requires the tuning of many internal oscillators. All these swarms consist of identical members with an autonomous existence—they all contribute equally to the swarm, and the swarm as such is fully self-organized, there is no leader or dir­ect­or. Hence such swarms resemble in many ways the inanimate many-particle systems of physics. There are clear parallels between the birds in a flock and the spins in magnetic materials, between signal-emitting The Rules of the Flock: Self-Organization and Swarm Structures in Animal Socities. Helmut Satz, Oxford University Press (2020). © Helmut Satz. DOI: 10.1093/oso/9780198853398.001.0001

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insects and the light emission by excited atoms in a laser. The methods used by ants to find the shortest way between home and food source can be simulated by mechanical robots and are in fact used to obtain algorithms for optimizing human logistics problems. Such parallels between mathematical models of physical systems and similar model for biological systems have greatly advanced our understanding of the swarm behavior of animals. Nevertheless, a sense of miracle remains—the universality which had initially amazed physicists when they found that the same laws govern very different phys­ic­al systems: magnets, liquids, galaxies and much more—this universality is now found to include even the self-organized swarm behavior of animal societies. Nature seems to stick to a method once it has been found to work. In the collective behavior of social insects, on the other hand, a new structure has come into play; in the course of evolution, the differentiating pattern of insect states must have shown itself as superior to a simple massing of identical individuals. Even though these states also remain self-organized, have no leader of any kind, the members are no longer all the same. There is a queen, there are female workers and there are drones, with specific functions and capabilities in each case. The queen has only one function, to produce offspring. It cannot feed itself, it has to rely on the workers for that, and it contributes in no way to the work that has to be carried out in the state, such as construction of the hive or maintenance of the young. The workers are sterile, presumably as a consequence of the food (or lack of food) they received as larvae. They carry out different tasks according to their age: the very young take care of the larvae, the young adults construct the hive or nest and the older ones go out to collect food. The males also rely on food from the workers, and their only function is to mate with young virgin queens at one specific time, when these fly out on their mating excursion. Each queen mates with a number of males and retains for the rest of its life the sperm it receives. The males then die or are killed by the workers, and in fact most of the queens also perish soon afterward, with

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only very few managing to eventually create their own state. This swarm pattern, in which the different members fall into different castes, must have somehow evolved from the massing of equal members in the course of evolution: it seems to present a new and more effective way to assure the passing of the most prominent genes on to the next generations. The biological basis for such structured societies lies in their reproduction scheme, in which the queen can decide to produce diploid females or haploid males. The latter then pass on their complete gene pattern to their descendants, and as a result the relation between sisters is closer than between parents and children. To assure the survival of specific genes, it is thus more effective to have females maintain their sisters rather than to have children of their own. It may thus seem that self-organized animal societies will dominate life on Earth in the future. Nevertheless, there are ­limits of such projections, and so we shall close with two classical counterexamples, in which swarm formation proved to be quite counterproductive. In the year 1800, some fifty million buffaloes populated the prairies of North America; they formed gigantic herds, which moved collectively between summer and winter pastures. They were hunted over centuries by the native Americans in a balanced way, leaving their numbers effectively constant. The arrival of the Europeans and their indiscriminate hunting, effectively butchering the buffaloes with firearms, left only some 300 animals surviving a hundred years later. Only last moment rescue efforts prevented complete extinction. The existence of huge herds was certainly crucial for such rapid mass extermination; in this case, swarm formation was evidently detrimental to survival. To show this, consider the coyote, which lives in small packs or family groups. In pre-Columbian time, its range covered what is now the Southwest of the US and northern Mexico. In the course of changing living conditions, it proved itself a most adaptive animal, and its range today covers the entire North American con­tin­ent as well as Central America down to Panama, and it is

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still increasing. Another gregarious but non-swarming master of adaption is the rat—most large cities today have more rats than human inhabitants; for illustration, present counts give two rats per human in Paris. A second example is provided by the passenger pigeon (Ectopistes migratorius), which lived in the prairie states of the USA in the 1800s. With some 50 billion animals, it is today considered as the most populous bird species ever. Here however, unrestrained human killing led to a definite end; the shooting was particularly ef­fect­ ive because the densely packed flocks provided such simple targets. The last passenger pigeon, a bird called Martha, died in captivity in 1914. Swarm formation is thus not always the optimal survival strategy, to use the phrase coined by Raghavendra Gadagkar, the leading expert on wasp sociology. Under adverse circumstances, it can even produce the opposite effect, as we just saw in the cases of the buffalo and the passenger pigeon. On the whole, however, the social animal species that developed swarm structures were

Shooting passenger pigeons in Louisiana, USA (1875).

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immensely successful. As mentioned, there are today an estimated one million ants on Earth for every human, and adding the other social insects brings this figure to more than a billion per human. All these societies have developed intricate rules to determine their lives, their behavior and their interactions with each other and with the environment. And while we humans admire the miraculous sights provided by whirling flocks of birds, glistening schools of fish or the multitudes of organized insect societies, we continue with our attempts to understand how the patterns of such behavior come about—universal patterns shared by both the animate and the inanimate nature.

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Person Index Bak, Per  42, 46 Bernoulli, Daniel  91, 92 Beukenberg, Markus  83, 128 Buck, John  52, 54, 127 Buck, Elizabeth  52, 54 Buhl, J.  12, 127 Cavagna, Andrea  23, 127 Curie, Pierre  29 Curie, Marie  29 Darwin, Charles  81, 82, 102, 106, 110 Deneubourg, Jean-Louis  2 Dolbear, Amos Emerson  57, 58

Leibniz, Gottfried Wilhelm von  27 Linné, Carl von  27 Lissaman, Peter  83, 128 Lorenz, Konrad  85 Mirollo, Renato  6, 61, 63 Moullec, Christian  86, 128 Newton, Sir Isaac  27 Nicolis, Gregoire  2, 128 Parisi, Giorgio  21 Peskin, Charles  63 Prigogine, Ilya  2, 128 Reynolds, Craig  21, 34, 49

Einstein, Albert  65, 69 Feynman, Richard  95, 127 Fibonacci  104, 105 Fisher, R. A.  107 Frisch, Karl von  75, 76 Gadagkar, Raghavendra  125, 127 Giardina, Irene  23, 127 Greenberg, Les  107, 108, 127 Haldane, J. B. S.  107 Hamilton, W. D.  107, 127 Hawthorne, Nathaniel  56 Hummel, Dietrich  83, 128 Huygens, Christiaan  54, 60, 61, 62 Kaempfer, Engelbert  51, 52

Schawlow, Arthur  69 Seeley, Thomas  128 Selous, Edmond  21 Shollenberger, Carl  83, 128 Smith, Hugh  52, 128 Strogatz, Steven  6, 61, 63, 128 Townes, Charles  69 Vicsek, Tamás  5, 21, 34, 35, 36, 37, 38, 40, 41, 47, 48, 61, 128 Voronoi, Gregory  48, 49 Walker, Thomas J.  58 Weiss, Pierre-Ernest  30, 31 Wilder, Laura Ingalls  8, 9 Zipf, George Kingsley  119, 120, 121, 128

Subject Index adiabatic tuning  46 angiosperm 110 ant  1, 2, 4, 6, 18, 73, 74, 75, 77, 95, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 109, 111. 116, 122, 123, 126 ant-algorithm  73, 74 attraction  11, 28, 41, 111 bee  1, 7, 73, 75, 76, 77, 78, 95, 100, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 116, 127 boid  21, 49 buffalo  124, 125, 128 chirping  5, 56, 57, 58, 60 connectivity  5, 14, 15, 16, 17, 18, 45 continuous transition  41 correlation  33, 42, 43, 44, 45, 48 correlation function  44, 45 cricket  4, 5, 56, 57, 58, 59, 61, 63, 64, 70, 122 critical point  27, 33, 37, 43, 45, 47, 48, 121 Curie temperature  28, 30, 32 decay  67, 68, 69, 70 diploid  102, 103, 106, 124 disorder  13, 28, 32, 33, 40, 41, 127 domain  30, 31, 32, 33, 45, 46, 48 downwash  82, 87, 128 drag  62, 89, 90, 91 driver ant  98 drone  7, 97, 103, 113, 123 electron  3, 15, 66, 67, 68, 69 excitation  66, 67 ferromagnetic  28, 29, 30, 31, 32, 33, 34 Fibonacci sequence  104, 105 firefly  52, 53, 54 flight formation  81, 128 fluid velocity  92 formation flight  84, 128

generation sequence  103 GPS device  87 gravity  22, 76, 77, 89, 90 gymnosperm 110 haploid  103, 106, 124 honey  77, 100, 103, 104, 112, 113, 114 honeydew 100 Lapland 71 laser  5, 6, 65, 66, 68, 69, 70, 123 leafcutter ant  99 lift  80, 81, 82, 83, 89, 90, 91, 92, 93 locust  1, 5, 8, 9, 10, 11, 12, 13, 14, 18, 41, 46, 109, 122, 127 metronome  54, 55, 61, 62 migration  6, 79, 80, 86, 87, 88, 127 mutualism 111 Northern bald ibis  84, 85, 86, 88 nymph  9, 10, 11, 12, 41 occupation inversion  68 order  3, 7, 13, 14, 15, 19, 28, 29, 30, 32, 33, 37, 40, 41, 46, 58, 78, 88, 90, 92, 94, 127 order parameter  32 oscillator  6, 60, 61, 62, 63, 64, 122 paraglider  85, 94 paramagnetic  28, 29, 30, 31, 32, 34 passenger pigeon  125 pendulum  54, 61, 62 percolation  5, 16, 17, 18, 128 photon  66, 67, 68, 69, 70 plant louse  100 polarization  24, 25, 76 pressure  92, 93 queen  7, 95, 96, 97, 98, 101, 102, 103, 104, 106, 109, 112, 113, 123, 124

Subject Index repulsion  11, 41 rotational invariance  25 self-organization  1, 2, 3, 39, 59, 94, 95, 101, 121, 127, 128 self-organized criticality  46, 48, 121, 127 spontaneous symmetry breaking 30 Starflag  21, 23, 47, 48, 127 starling  1, 5, 19, 20, 22, 23, 24, 36, 39, 40, 41, 47, 48, 83, 88, 96, 109, 127 superorganism  5, 95, 101, 102, 110 sweat bee  108 symmetry  25, 30, 32, 34, 37 synchronization  4, 5, 6, 52, 54, 56, 58, 59, 61, 63, 64, 65, 66, 88, 122

131

tessellation 48 thrust  89, 90, 91, 92 transition  11, 12, 13, 14, 28, 29, 30, 31, 32, 33, 34, 38, 39, 40, 41, 43, 67, 68, 69, 116, 117, 118, 128 upwash  82, 83, 87, 93, 94, 128 Venturi tube  92 waggle dance  76, 77, 116 Waldrapp  84, 85, 86, 87, 88 weaver ant  101 weight  9, 89, 90, 91, 107 wildebeest  25, 26, 109 Zipf’s law  119, 120, 121, 128