Lateral Asymmetry in Animals: Predator-Prey Interactions, Dynamics, and Evolution (Ecological Research Monographs) 9811913404, 9789811913402

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Ecological Research Monographs

Michio Hori Satoshi Takahashi Editors

Lateral Asymmetry in Animals Predator-Prey Interactions, Dynamics, and Evolution

Ecological Research Monographs Series Editors Yoh Iwasa, Kyushu University, Fukuoka, Japan

The book series Ecological Research Monographs publishes refereed volumes on all aspects of ecology, including Animal ecology Population ecology Theoretical ecology Plant ecology Community ecology Statistical ecology Marine ecology Ecosystems Biodiversity Microbial ecology Landscape ecology Conservation Molecular ecology Behavioral ecology Urban ecology Physiological ecology Evolutionary ecology The series comprise books and edited collections by international experts in their fields.

Michio Hori • Satoshi Takahashi Editors

Lateral Asymmetry in Animals Predator-Prey Interactions, Dynamics, and Evolution

Michio Hori Kyoto University Kyoto, Japan

Satoshi Takahashi Faculty Division of Natural Sciences Research Group of Environmental Sciences Nara Women’s University Nara, Japan

ISSN 2191-0707 ISSN 2191-0715 (electronic) Ecological Research Monographs ISBN 978-981-19-1340-2 ISBN 978-981-19-1342-6 (eBook) https://doi.org/10.1007/978-981-19-1342-6 © Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: Attack of Lake Tanganyikan scale eater, Perissodus microlepis (Illustrated by Megumi Katayama) This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This book reviews recent studies, mainly of ours, on the laterality of fish and other animals. The studies started from a discovery of asymmetric mouth opening of scale-eating cichlids in an African lake, and extended to lateral dimorphism and corresponding behaviors of all the fishes, and now eventually stretches to some aquatic invertebrates. During the studies, we often received queries, such as “Why the fish developed such a character?” and “Could they act more efficiently when they have a symmetric body and unbiased behaviors?” We always answered such as “We don’t know the exact reason at present, but we can only say that they seemed to develop and retain the character as it has been more useful for their life. Incidentally, we suppose that you too have a clear handedness, as either righty or lefty, haven’t you?” The remarkable nature of laterality is such that most people make little account of it, despite the facts that everyone are bound by and also that this dimorphism has been maintained with a certain ratio in every population in every era. Historically, the laterality had been studied mainly on the handedness of human being, as its unique property, and had been related to the cerebral lateralization. A large number of papers, books, and even an international journal specific to the lateral asymmetry had been published. After that, studies on animals have been started on mammals, especially on non-human primates and experimental animals (the rat and mouse), and then spread over companion and/or domestic animals, birds, and then fishes in aquariums. In these studies, however, the laterality has been studied mainly from the viewpoint of the behavioral asymmetry in relation to the brain lateralization, but the morphological asymmetry was rather disregarded except the asymmetry in central nerve structure. These studies have issued mainly from a viewpoint of a characteristic of species, or a property of individual. Whereas, this book, focusing on fish and aquatic invertebrates, addresses the relation of morphological asymmetry to the behavioral laterality clarified mainly by our recent studies. Our studies put emphasis on interspecific interaction, especially predator–prey interaction in community, and pay attention to frequency-dependent selection. We will discuss also the phylogenetic relation of the laterality of fish, as well as its origin and inheritance, to the tetrapod animals, and the relation to that v

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of invertebrates. Although an extensive adjustment of our works to the result of conventional studies remains as a subject in future, we hope that readers can enjoy a new field of the laterality of animals. * “The ratio of scale-eater’s laterality seems to oscillate, but the difference from 0.5 is not significant.” My study on lateral asymmetry was elicited by these words. I (S.T.) first met Prof. Hori at the seminar of Animal Sociology Lab. in Osaka City University, where he talked about scale-eater’s lateral asymmetry. At that time, I got my first job position at the Department of Mathematics in O. C. U. and was studying fractals. Hori’s data of scale-eaters laterality ratio between 1980 and 1990 seemed to oscillate with 5-year period, but the difference from 1:1 ratio is not significant for each year. To show the oscillation, I approached from two directions, statistics and modelling. Though I was not familiar with many statistical methods, I could calculate probabilities. I accumulated laterality ratio data of different years by Fourier transform and showed its significance. This statistical approach is summarized in Chap. 7 of this book. Another approach to prove the oscillation was modelling oscillation mechanism. Incorporating minority advantage of lateral dimorphism in predation and time delay of 2-year growth period into the model of scale-eater replicated its 5-year periodic oscillation of laterality ratio. This population dynamics modelling approach is summarized in Chap. 5. I enjoyed collaboration with Prof. Hori, observing scale-eater’s attack on a goldfish in his laboratory tanks, etc. From the viewpoint of theoretician, the system of lateral asymmetry dimorphism is just beautiful. Usually, physical environment conditions affect different morphs in a species in different ways. In case of color polymorphism, different color morphs suffer different predation risks depending on surrounding backgrounds. In case of lateral asymmetry dimorphism, however, effects of physical environment operate equally to both of the morphs. Only the different frequencies of two lateral asymmetry morphs of same or other species affect differently to two morphs. The system also has the symmetry: exchanging lefty morphs with righty ones in all the species reproduces same dynamics. Systems of lateral asymmetry dimorphisms are ideal to study frequency dependence in itself. Series editor Prof. Yoh Iwasa recommended me writing a book in the series of Ecological Research Monograph. I together with Prof. Hori planned this book as collaboration work of field ecologists and theoretical biologists. Among contributors, four are field ecologists and three are theoretical biologists. This book treats variety of topics about lateral asymmetry, but they consistently related to predation. Chapters and related research fields or object organisms are summarized below. Behavioral ecology (Chaps. 1–4, 9), population dynamics (Chaps. 1–6, 10), physiology (Chaps. 1–4, 9), evolution (Chaps. 2, 8–10), genetics (Chaps. 8 and 10), development, learning and neuroscience (Chap. 4), statistics (Chaps. 1, 2, 8–9), fish (Chaps. 1–8), shrimp, prawn, and crayfish (Chaps. 2 and 4), crab (Chap. 2), cuttlefish

Preface

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(Chap. 2), and snake and snail (Chaps. 9 and 10). I hope this book interests the readers, impresses them by rich fruits of collaboration between field ecologists and theoretical biologists, and inspires new direction of their research. Kyoto, Japan Nara, Japan

Michio Hori Satoshi Takahashi

Acknowledgments

We are grateful to the following persons and teams: T. Sota, K. Watanabe, and other members of the Laboratory of Animal Ecology, Department of Zoology, Graduate School of Science, Kyoto University, for their stimulating discussions and support for our field works; M. Kohda and other members of the Laboratory of Animal Sociology, Osaka City University, and members of the Laboratory of Mathematical Biology, Nara Women’s University, for their valuable discussions. We also wish to thank Y. Iwasa for providing us the opportunity to editing this book in the series Ecological Research Monographs. We show our gratitude to all the contributors to create this book together. We thank all the institutes, museums, aquariums, researchers, and fishermen who helped us for correcting the fish materials, and M. Kataoka for illustrating the cover of book. The research in Lake Tanganyika was conducted with permission from the Zambian Ministry of Agriculture, Food, and Fisheries. We appreciate staff of Lake Tanganyika Research Unit of Fishery Department, Zambia, and members of Japanese Tanganyika Research Team. This study was supported by grants from the Ministry of Education, Culture, Science and Technology, Japan; 21st Century COE Program (A14), Global COE Program (A06), Priority Area (14087203), and JSPS KAKENHI Grant Numbers 21370010, 19570020, 15H05230, 16H05773, 17K14934, 17H01673. Michio Hori Satoshi Takahashi

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Contents

1

2

Laterality of Fish: Antisymmetry in Fish Populations Maintained by the Interspecific Interaction . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Michio Hori

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Universality of Laterality Among Fish and Invertebrates in Aquatic Communities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Michio Hori

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Predominance of Cross-predation or Parallel-predation in Fish . . . . . . Masaki Yasugi

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Behavioral Laterality in the Scale-Eating Cichlid Fish: Detailed Movement, Development, and Neuronal Mechanisms . . . . . . . 115 Yuichi Takeuchi

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Models of Lateral Asymmetry Dynamics: Realistic Oscillations by Time Delay and Frequency Dependence .. . . . . . . . . . . . . . 143 Satoshi Takahashi

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Mathematical Models of Predators and Prey with Laterality . . . . . . . . . 177 Mifuyu Nakajima

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Statistical Analysis of Lateral Asymmetry: Detect Antisymmetry and Oscillation from Unequal-Interval Binomial Data .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 193 Satoshi Takahashi

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Models of Genetic System of Lateral Asymmetry: Population Dynamics Drive Evolution of Genetic System . . . . . . .. . . . . . . . . . . . . . . . . . . . 227 Satoshi Takahashi

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Contents

Mechanisms Underlying Variations in the Dentition Asymmetry of Asian Snail-Eating Snakes . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 261 Masaki Hoso

10 Single-Gene Speciation, Balanced Polymorphism, and Antagonistic Coevolution in Left-Right Asymmetry of Land Snails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 275 Masato Yamamichi

Chapter 1

Laterality of Fish: Antisymmetry in Fish Populations Maintained by the Interspecific Interaction Michio Hori

Abstract This chapter tries to give a full picture of the laterality (antisymmetry or bilateral asymmetry) in fish populations, based mainly on a series of our field works. It covers a story of discovery of the phenomenon together with its background, dynamics of the ratios of laterality (frequency of righty morphs) in fish populations, and the causation of the dynamics. Our field works in Lake Tanganyika, east Africa, and Lake Biwa, Japan, revealed that the laterality ratios changed periodically within 0.3 and 0.7 for many years in every population studied in both lakes. The driving force of the periodical change in the laterality ratios seems to be predator-prey interactions, specifically the predominance of corss-predation in every predation incident, which operates a negative frequency-dependent selection between predator and prey populations. The effects of the selection may involve related groups of fishes with similar life-form along the structure of food web in the fish community. Keywords Lateral dimorphism · Exploitative mutualism · Frequency-dependent selection · Scale eater · Cross-predation

1.1 Introduction: Background of Discovery of the Lateral Dimorphism The main theme of this book is the lateral dimorphism in animals and its meaning in the biological community. One of the typical examples of the dimorphism was discovered in scale-eating cichlid fishes during a series of ecological studies on fish community of Lake Tanganyika in the Great Rift Valley of East Africa. The story of discovery will be shown in the next section. Here, we examine briefly the background of the discovery, that is to say, why and how such a dimorphism of fish

M. Hori () Kyoto University, Yoshida-Honmachi, Kyoto, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Hori, S. Takahashi (eds.), Lateral Asymmetry in Animals, Ecological Research Monographs, https://doi.org/10.1007/978-981-19-1342-6_1

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was focused in the studies on fish community, which may enrich the consideration of the meaning of the lateral dimorphism in animals.

1.1.1 Coexistence of Predatory Cichlid Fishes in Lake Tanganyika Lake Tanganyika is one of the oldest lakes in the world (Fig. 1.1). More than 200 fish species are known from the lake, of which 65% are cichlids, the vast majority of them being endemic (Lowe-McConnell 1987; Poll 1956, 1986). The cichlid fishes are most abundant in the littoral areas, especially on rocky shores (Coulter 1991) (Fig. 1.2). For example, our census study on a rocky shore of the east coast indicated that about 5700 adults and subadult fishes belonging to 48 species inhabited in a 20 × 20 m quadrat (14.3 indi./m2 in average), of which 44 species were cichlids (Hori 1987). In order to know the reason why such high level of biodiversity was maintained in the fish community in the rocky shore of the lake, we studied the feeding relationships in the fish community, by direct observation using scuba and by stomach contents analysis of fish samples. Figure 1.3 shows a food web for the fishes inhabiting the quadrat at the east central coast of the lake. A similar food web at the northernmost coast is shown by Hori et al. (1993), and a comparison of food webs among three sites including the southernmost coast was made by Hori (1997). There were several groups of feeding habit, such as three kinds of algal feeders (unicellular algal feeders, filamentous algal feeders, microfilamentous algal

Fig. 1.1 A view of Lake Tanganyika from Chipwa, Zambia

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Fig. 1.2 A view of a littoral fish community in water of Lake Tanganyika. Photo by Dr. Kazutaka Ota

feeders), shrimp eaters, and other benthos eaters, piscivores, and scale eaters. Scale eaters are predatory fishes which attack other live fishes and rob some scales from their flank. It is clear that such diverse types of food resources support the many fish species of various food habits. However, a more noteworthy point is that a number of the related fishes, usually congeneric species, live together depending on the same type of resources. In our study area, two species of scale eater, Perissodus microlepis and P. straeleni (Fig. 1.4), live together, sharing common prey species, such as large fishes including piscivores (Hori 1987). It has been widely accepted that related species with similar food requirements cannot easily coexist (e.g., Pianka 1988; Roughgarden 1976; Begon et al. 2006). Many studies on biological communities have supported the idea that interspecific competition is a major factor of community organization (e.g., Hairston et al. 1960; Connell 1975). In our studies, we have also found that the competitive relationships among species greatly affect the structure of the fish community (Hori et al. 1983; Takamura 1984). Nevertheless, many allied fishes, not only of the scale eaters but also of fishes of other feeding habits, coexist sharing similar food items along the rocky shores, and thus, some factors counteracting or balancing interspecific competition may also be operating. Our studies on this topic indicated that fishes obtained an advantage in feeding efficiency

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Fig. 1.3 Food web of a littoral fish community in Lake Tanganyika (Hori 1987). Only adult and subadult fishes are considered. Subadult fish of four piscivorous species, which feed mainly on benthic animals, are considered separately from their adults. Arrows show the food items for each species, and arrow thickness indicates the degree of utilization of each item. Data for piscivores and scale eaters are based on underwater observations, and those for fishes of other food habits are derived from stomach contents analysis of samples caught by the aid of village fishermen

from the activities of other species with similar food requirements (Hori 1987, 1991).

1.1.2 Exploitative Mutualism Mediated by the Differentiation in Attacking Behaviors Detailed observation on the hunting behavior of the two scale eaters in water indicated that the hunting success of P. microlepis was low; the ratio of the number of successes to the number of attacks was about 0.18, as potential prey fish were usually alert to approaching scale eaters (Hori 1987). The success ratio of P. straeleni was about 0.25, slightly higher than that of P. microlepis. P. microlepis, however, can take more scales in one bite than P. straeleni, and since the number of scales taken per hour was not different between the two species, the feeding efficiency was at almost the same level on average. Although many potential prey fish were present, scale feedings were concentrated on several species. The three most vulnerable species represent about 60% and 55% of the total scale feedings

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Fig. 1.4 Two scale eaters, Perissodus microlepis and P. straeleni, cohabiting in the littoral area of Lake Tanganyika

by P. microlepis and P. straeleni, respectively. About 70% of prey approached was common between the two scale eaters. The two scale eaters mainly attacked their prey when the victims were feeding or swimming in mid-water. As the success of scale-eating was dependent upon whether the scale eater could take advantage of the prey’s unguarded moments (Nshombo et al. 1985), the success ratio was high when the preys were engaged in territorial fights or mating behavior. Such opportunities, however, were rare. The prey’s behavior when being attacked by either of the two scale eaters was similar.

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Fig. 1.5 Sequence of hunting behavior (upper) and hunting success ratio (lower) of two scale eaters (Hori 1987). The success ratios of “tearing-off” and “dash” are shown as proportions of “approach,” which is set at 1.0. Marks * and - indicate that the ratios are significantly different at the 5% level from that in a solitary feeding situation. The significance level was tested with approximation to normal distribution. Both species obtained a significantly higher hunting success when another species was present within 0.5 m

Figure 1.5 shows the sequence of hunting behavior of the two scale eaters. For both species, the hunting behavior is composed of a search in mid-water, a stealthy approach, and a dash to the prey, although there is a difference between the two in the position relative to and distance from the prey. In general, P. microlepis, using the spindle shape of its body, approached and dashed from outside the range in

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which the prey was able to detect it. On the other hand, P. straeleni generally, using the shape and color pattern of its body being resemble to algal-feeding fishes, began to approach casually and then dashed from a position within the range in which the prey was able to detect it. P. microlepis was more successful in approaching than P. straeleni, but less successful in dashing. The difference in the hunting technique of the two scale caters corresponded to the difference in their body shape. Thus, these two scale eaters hunt common prey in similar situations but overcome the prey’s guard in different ways. Figure 1.5 also shows how the success ratio of scale-eating changed if another scale eater was present nearby. When a scale eater of another species was present within a radius of 50 cm, the success ratio was significantly higher than that in solitary hunting for both species. However, when the scale eater nearby was a conspecific one, the difference was not significant. Aggressive behavior was often observed between two species as well as between conspecifics, but it seemed to have no effect on the feeding success ratio of scale-eating. Thus, the result suggests that anticipation of and concentration upon attack by one scale-eating species resulted in a prey fish being less wary of attack by a different scale-eating species. A similar phenomenon can be seen among four piscivorous fishes in this area (Fig. 1.6). Though this advantage does not seem large, it is probable that the advantage is not restricted to situations when other predators are nearby. The coexistence of several expert hunters should itself prevent the development by prey fish of antipredator techniques against one specific predator, although it would be difficult to test this view. However, if this is true, such a mutualistic relationship between potentially competitive species may also be realized among other groups of carnivores. Actually, a similar phenomenon can be seen among benthos eaters, such as five species of shrimp-eating congeneric species in the rocky shore, which depend on the same species of shrimp with different hunting techniques from each other (Fig. 1.7). When benthos-eating species hunt in many different ways, their prey cannot develop antipredator defenses against one specific predator. Therefore, the mutualism seems to contribute to the coexistence of allied benthos feeders. Furthermore, mutualistic relationships were also found among prey fishes in regard to defense against predators (Hori 1987). Scale eaters often failed when hunting their prey due to lateral attack by other fish during their approach and while aiming at prey. As a result, the targeted prey was saved from scale-eating by the attacker. In the two scale eater species, no target fish was saved by a conspecific fish nearby, but filamentous and unicellular algal feeders were often saved by fish of the same and other groups of algal feeders. This phenomenon is reasonable, because the presence of conspecific fish nearby, especially when engaging in territorial fighting or mating behavior, provides an opportunity for scale-eating. However, other algalfeeding fishes may easily locate and repel the scale eater approaching its prey, because their ability to locate the enemy and their watching positions are somewhat different from those of the targeted prey species. This should also be true for prey fish of piscivores, because they often became aware of an approaching enemy due to the sudden avoidance movement of other fish. Thus, for both predators and preys, fish get some benefit from the presence of other species sharing common food. It is

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Fig. 1.6 Foraging behavior of three piscivore species of Lepidiolamprologus and the frequency of their aiming prey at the Mahale study site, Tanzania, in Lake Tanganyika (Hori 1997)

noteworthy that, nevertheless, the driving force that diversifies the behaviors should operate stronger among carnivores than among prey fishes, because each carnivore fish can become specialized to its hunting technique, while prey fishes have to guard themselves omnidirectionally against enemies anyway. In short, the mutualism among species of the same food habit found in this lake is a factor that positively facilitates their coexistence. The subject must be viewed from a new angle whereby mutualism raises the species richness of each food habit

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Fig. 1.7 Foraging behavior of six shrimp-eating cichlid fishes and the frequency of food items at the Mahale study site, Tanzania, in Lake Tanganyika (Hori 1997). Yuma et al. (1998) proved that the shrimps were composed of only one species, Limnocaridina latipes (Atyidae)

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group without any partitioning of ecological resources, if their feeding methods can diverge. Matsuda et al. (1993) termed the situation in which each of multiple carnivorous species with similar feeding habit but with differentiated attacking behaviors get some advantages from the presence of other species as “exploitative mutualism”, and provided a theoretical model in which this mutualism is a factor promoting the coexistence of species of similar food habit. An important point for community organization is that exploitative mutualism should operate within a food habit group as a stabilizing mechanism for relevant populations, unlike interspecific competition. The reason is that the numerically inferior populations will obtain a relatively higher feeding efficiency than a numerically superior one by circumventing the prey’s guard toward the more abundant predator. Using mathematical models, Matsuda et al. (1994, 1996) have further analyzed the conditions necessary for exploitative mutualism to be realized in a predatory food habit group. Their studies have demonstrated that the most important condition is that the antipredator defense be specific to each type of predator, i.e., an increase in a prey’s attention to one type of predator decreases its attention to another type of predator. These mathematical models have shown that predator-specific defense increases the community complexity, the resource overlap between predators, the total abundance of predators, and the predator/prey abundance ratio.

1.1.3 Exploitative Mutualism in Polymorphism The exploitative mutualism, in which one species can get benefit from the presence of other species of the same food habit, can promote the coexistence of such potential competitors. This mechanism, however, seems to operate not only in the interspecific interaction but also in the interaction within each species; that is to say, polymorphism may be promoted and maintained in a population of predator species. Kohda et al. (1997) reviewed interindividual variation in foraging behaviors and related polymorphisms in predatory cichlids in Lake Tanganyika. In the piscivorous species, Lepidiolamprologus profundicola, individuals specialize in one or several of nine distinctive foraging techniques over extended periods (Fig. 1.8). Although prey fishes are plentiful, availabilities of prey are restricted by their effective alert and escape. For successful hunting, elaborated foraging behaviors are inevitable, and individuals specialize in certain techniques mainly through learning. Such individual variations are also shown in other piscivorous fishes (Hori 1983; Nakai 1993), scale eaters (Nshombo 1994), and shrimp eaters (Kohda and Hori 1993). Those hunters are dependent upon evasive prey, and have a pale-dark dichromatism which functions as a hunting camouflage. At a given time, the less frequent morph holds an advantage over the more frequent morph, due to the prey being more vigilant against attacks from the more frequent morph. Similarly, being minority in specialized foraging repertoires would be advantageous. Thus, this mechanism is the exploitative mutualism in intraspecific polymorphism. Its frequency-dependent

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Fig. 1.8 Diagrams of the nine hunting techniques of L. profundicola (Kohda 1994). The large shaded object represents a rock

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advantage whereby the minority is favored may be an ecological key concept to explain the development and maintenance of the interindividual variation in foraging behaviors of predatory cichlids. Notwithstanding, the driving force which functions to diverse the hunting methods in one carnivorous species may be restricted or cancelled by the genetic constraint which governs the species-specific framework of morphology. Therefore, it is reasonable that the widespread and conspicuous intraspecific diversity in carnivorous species is the color polymorphism such as pale-dark dimorphism, and it is expected that the diversity in hunting methods associated with differential body forms is fulfilled in the interspecific diversity of carnivorous species. An exception of this, however, is the laterality of scale eaters, which possess lateral dimorphism and associated behavior. Now, let’s turn our attention to the laterality of scale eaters.

1.2 Laterality of Scale Eaters 1.2.1 Discovery of the Dimorphic Asymmetry of Scale Eaters The first and most notable example of laterally asymmetric bodies in fish was found in scale eaters in Lake Tanganyika (Liem and Stewart 1976; Hori 1991, 1993). The scale eaters are one of the most specialized cichlid fishes in the African Great Lakes in the Rift Valley (Lowe-McConnell 1987), which attack other fishes and tear off scales from the flunk of them. In Lake Tanganyika, seven species are known, all of which belong to the genus Perissodus (Liem and Stewart 1976). One of them, P. eccentricus, was originally thought to be unique, because an individual’s mouth opens either rightward or leftward as a result of an asymmetrical joint of the jaw to the suspensorium (Liem and Stewart 1976). This lateral asymmetry of the mouth opening was considered an adaptation for efficiently tearing off prey’s scales and the result of specialization among the lineage. As this species is very rare and inhabits in deep water (more than 100 m depth), further study on its ecology and observation of behavior seemed to be difficult.

1.2.2 All the Species of Scale Eater Share the Laterality After that, however, further examination revealed that all the seven species of the genus display asymmetrical mouth opening to some degree regardless of their developmental stage or sex (Hori 1991). The laterality of each fish can be defined by the direction of the mouth opening (Hori 1991, 1993; Hori et al. 2007) (Fig. 1.9). Morphologically, the asymmetric mouth opening is due to either side of the joint, say right joint, between mandible and suspensorium taking a position frontward, ventrally, and outside compared to the opposite side of the joint (Liem and Stewart

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Fig. 1.9 Antisymmetry of the mouth opening of scale-eating cichlid, Perissodus microlepis, of Lake Tanganyika (Hori 1993). A lefty morph (upper) and a righty one (lower) are shown from both sides

1976) (Fig. 1.10). The bending rightward should mean that the left side of its head and flank has developed more compared to the right side (Hori et al. 2007). This asymmetry is defined as antisymmetry, in which one side of the body is structurally and/or functionally more developed than the other side, and distinguished from fluctuating or directional asymmetry (Palmer and Strobeck 1986). In antisymmetry, each population is composed of “righty” and “lefty” individuals, and measurements of the character in question show a bimodal distribution. The functional morphology and the quantitative measurement of the asymmetric mouth opening have been developed with separate works (Hori et al. 2017; Yasugi and Hori 2012; Hata et al. 2013), which is reviewed in Chap. 2. Note that the definition of laterality used here and in recent studies differs from that used in earlier papers (Hori 1991, 1993; Liem and Stewart 1976; Seki et al. 2000), which defined individuals with the mouth opening to the right as “right-handed” or “dextral.” The terminology used in the present study, “lefty,” reflects the fact that the left mandible of such “right-handed” fish is larger than the right mandible (Hori et al. 2007, 2017) and the left eye is dominant (Matsui et al. 2013; Takeuchi and Hori 2008; Takeuchi et al. 2010a). A detailed study was made of the dimorphism in P. microlepis and P. straeleni, the most and the second abundant species of scale eater in the littoral area of the lake (Hori et al. 1983; Nshombo et al. 1985). Habitat conditions in this area are very stable, and most resident fish populations are highly persistent with respect to their density (Hori 1991; Hori et al. 1993; Takeuchi et al. 2010b). Therefore, in order to

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Fig. 1.10 CT scan image of mouthpart bones of Perissodus eccentricus from the ventral view, indicating the asymmetric positions of right and left articular between lower jaw and suspensory apparatus (red points). Blue, lower jaw (mandible); pink, ascending process of articular. Photo by Dr. Rieko Takahashi

consider any change in the frequency of scale eaters of each morph, it is sufficient to observe the ratio between the two morphs. The two species of scale eater approach its targeted prey stealthily from behind in order to snatch several scales from the flank of the prey (Fig. 1.5). A field experiment with the use of common prey species as live lures revealed that righty individuals always attack the victim’s right flank and lefty ones the left flank (Fig. 1.11). This correspondence was supported by examining the asymmetry of prey’s scales found in the stomachs of each individual of the scale eaters. The asymmetry of scales was particularly easy to determine for the pored scales of the lateral lines (Fig. 1.12). Table 1.1 shows that righty fish took scales from the right flank of prey and lefty fish from the left flank. The distorted mouth apparently enlarges the area of teeth in contact with the prey’s flank, but this is the case only when the scale eater attacks a corresponding side of the prey. Thus, the correspondence between the laterality and the attack side should be a functional requisite for the success in feeding of these scale eaters and may have evolved from the beginning of the lineage. Laterality is detectable even in fry stage under parental care (Hori 1991), a time when they feed exclusively on plankter (Nshombo et al. 1985), which indicates that the laterality is a genetic character. Perissodus is a substrate-brooder in which parents guard their eggs and fry. The genetics of laterality was investigated by

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Fig. 1.11 Correspondence between the laterality of scale eaters and the flank of prey attacked (modified from Hori (1993)). This experiment was done under natural condition using scuba utilizing with adults of Cyathopharynx furcifer, Tropheus moorii, and Neolamprologus modestus as decoys, all of which are abundant and common prey species of the two species of scale eater. Each live fish of these species was connected by a hook to a fishing line and allowed to swim. Each P. microlepis and P. straeleni that attacked the lure was caught by a short gill net in water

Fig. 1.12 Asymmetry of fish scale. The asymmetry of scales was determined under a binocular microscope on the basis of the shape of the exposed granulated portion and the number of basal ridges in the upper and lower part separated by the mucous tube

examination of sets of parents of P. microlepis and their fry collected from natural habitats. P. microlepis parents have the unusual habit of farming out their fry to other breeding pairs (Yanagisawa 1985), which makes it difficult to define the exact phenotype frequency in each brood. The results, however, strongly suggest that laterality is determined by a simple Mendelian one locus-two alleles system, in which righty is dominant over lefty (Table 1.2).

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M. Hori

Table 1.1 Occurrence of right and left pored scales in the stomach of P. microlepis (Hori 1993; the names of phenotypes are altered). Unknown scales are those misshapen as a result of partial digestion and those of abnormal shape Laterality of P. microlepis Righty Lefty

Fish (n) 24 32

Pored scales (n) Right Left 76 0 0 139

Unknown 23 31

Table 1.2 Phenotype frequency in broods of P. microlepis (Hori 1993, names of phenotypes are changed). The observed ratio in F1 is shown with the actual number (n) of righty (R) and lefty (L) fry guarded by an adult pair (R × R, R × L, L × L); expected ratios are theoretical ones that should appear from the parents if the genetic system follows the Mendelian one locus-two alleles system with lefty being dominant over righty. To exclude fry from other parents as much as possible, fry whose size conspicuously differed from the majority were omitted from the observed ratio, although they were included in the total Righty × Righty (5) Righty:Lefty (n) Observed 79: 0 (80) 29: 0 (30) 19: 1 (21) 79: 15 (99) 28: 12 (49) Expected 1: 0

Righty × Lefty (4) Righty:Lefty (n)

Lefty × Lefty (3) Righty:Lefty (n)

0: 74 (80) 2: 18 (20) 53: 55 (109) 29: 27 (59)

2: 23 (29) 11: 35 (46) 15: 39 (54)

0: 1 1: 1

0: 1 1: 3

1.2.3 Dynamics of Laterality of P. microlepis Because the hunting success of scale-eating is low and laterality is inherited, the ratio between the two phenotypes in a population theoretically should be balanced at the same frequency, as each phenotype will be at an advantage when rare. When one of the two phenotypes, say righty, is more abundant in the population, prey fishes will tend to guard more against attacks to their right side, which results in lefty individuals gaining greater hunting success and commensurate greater fitness. Should this simple frequency-dependent selection mechanism operate, then the polymorphism can be regarded as an evolutionary stable state (Cresswell and Sayre 1991; Maynard Smith 1991). To verify this prediction, the ratio of laterality was examined for samples of P. microlepis collected from two adjacent sites (about 7 km apart) along the northernmost shoreline of the lake, where P. microlepis was virtually the only scaleeating species. The survey was carried out in every year at 1- to 2-year intervals over an 11-year period (Hori 1993). As predicted, the ratio remained at around 0.5 and never deviated from it by more than 0.1 during the period, which provides

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Fig. 1.13 Change in ratio of laterality of two adjacent populations (open symbols, Luhanga coast; closed symbols, Bemba coast) of P. microlepis at the northwestern coast of Lake Tanganyika over an 11-year period (Hori 1993). Star symbols indicate breeding adults, selectively collected with the use of scuba

strong support for the presence of a balancing mechanism (Fig. 1.13). Moreover, closer examination of the temporal changes in the ratio of laterality demonstrates that it significantly oscillated with an amplitude of 0.15 and a period of 5 years. Both populations at the two sites showed this oscillation. The oscillation is also observable upon examination of the ratio of laterality in every size class in each year; the ratio in the larger individuals is opposite that in the smaller ones. Provided that the survival rate does not markedly differ between the two morphs and assuming that the size classes can be regarded as age classes, evidence for the oscillation can be detected in the difference of ratios among size classes in each year. No previous studies on balancing polymorphism have detected or predicted such an oscillation. Two hypotheses may be proposed to explain this oscillation: either (1) it is the result of a time lag between the differential hunting success and the resultant increase in progeny or (2) the prey does not respond in an exact frequency-dependent manner to the ratio of laterality in the scale eater population. The differential hunting success should result in a differential reproductive success between the two morphs. The ratio of laterality in samples of breeding pair collected on three occasions was all opposite the ambient ratio in the total population (Fig. 1.13). This fact strongly suggests that rare phenotype at a time enjoy some reproductive advantage. Almost all previous field studies of polymorphism maintained through predatorprey interaction have shown a combination of frequency-dependent and frequencyindependent selection as being necessary to maintain the polymorphism. For

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example, it has been shown that the polymorphism of snails of Cepaea is chiefly maintained by the temperature condition of habitat and predation from birds (Cain and Sheppard 1954; Clarke 1962, 1969). By contrast, the laterality of P. microlepis may be maintained solely by frequency-dependent selection. It seems implausible that either righty or lefty itself has any superiority over the other and pleiotropic gene effects seem most unlikely. Moreover, habitat heterogeneity should not favor either phenotypes. Therefore, the laterality of this scale eater is a documented example of a polymorphism maintained solely by frequency-dependent selection through predator-prey interaction. It is also the rare example to demonstrate that the minority advantage results in differential reproductive success in a natural population. The ratio of laterality, however, does not stay at an equilibrium level but oscillates around it, which contradicts an intuitive prediction. Here, this is attributed to a trait of the prey’s reaction and the inevitable time lag between the occurrence of the advantage and the resultant change in phenotypic frequency. At first glance, such an oscillation may appear specific to this system. However, as we cannot expect that both the predators and the prey attain the highest efficiency in their interaction nor that any action of an agent affects the reproduction without a time lag, we can suppose that the oscillation is not unique but rather a common process in polymorphic populations under natural conditions. We will see many cases of such oscillation of the laterality in fish communities later.

1.3 Dynamics of Laterality Along Predator-Prey Interactions As shown in the preceding section, the laterality of fish was firstly found in the scale eaters in Lake Tanganyika, and then also found in various other fish species. Eventually, it was found in all the extant fishes, including agnathan and cartilaginous fish (Hori et al. 2017). The issue how to evaluate the laterality of all the fishes will be explored in the next chapter, together with the laterality of invertebrates. Anyway, under the condition that all the fish have the antisymmetric laterality, we were forced to re-examine the relationship between the laterality and predator-prey interaction in fish communities. As like the scale eater, the laterality of fish has a genetic basis (Seki et al., 2000; Hori et al. 2007; Stewart and Albertson 2010; Hata and Hori 2012; Hata et al. 2012). Then, the examination may involve the analysis on the change in a time series of the ratio of laterality in populations concerned.

1.3.1 Laterality of Fish in Predator-Prey Interaction Nakajima and coworkers got a start on this topic (see Chap. 6 in this book). They made an experiment of lure fishing for the largemouth bass (Micropterus salmoides) in a pool, and found that righty morphs were hooked more frequently on the right side of their mouth and lefty ones on the left side, indicating that each morph uses

1 Laterality of Fish: Antisymmetry in Fish Populations Maintained. . .

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its specific side of the mouth when attacking on prey (Nakajima et al. 2007). It is the same situation to the scale eater which has its specific side for attacking depending on each morph. In situations where all fishes have laterality, two types of predation incidents can be distinguished: (1) a predator catches a prey of the same morph of laterality (righty predator catches righty prey, and lefty morph catches lefty prey), and (2) a predator catches a prey of the opposite morph (righty predator catches lefty prey, and lefty predator catches righty prey). Nakajima et al. (2004) called the former as “parallel-predation” and the latter as “cross-predation”. Provided that the ratios of laterality in both populations of predator and prey are maintained in any pattern, the predation incidents in an overall system (at a single point in time) are predicted to be biased toward an excess of one type of predation over the other; that is to say, either cross-predations are predominant over parallel-predationss (predominance of cross-predation) or the reverse (predominance of parallel-predation), but both types of predation are not predicted to occur at a similar frequency (random-predation). Nakajima et al. (2004) constructed mathematical model of population dynamics of one-predator-one-prey system with laterality each, and performed computer simulations. The results showed that, under the condition of predominance of crosspredation, the laterality of both predator and prey is maintained and the ratios of laterality (frequencies of righty morphs) of the both populations exhibit oscillation for a long period. They expanded the model to one-predator-two-prey system and a three-trophic-level system with omnivory (Nakajima et al. 2004, 2005), and got virtually the same results. These mathematical models imagined the scale eaters and their prey in Lake Tanganyika as a real system. At that time, however, few field studies had investigated the maintenance and dynamics of laterality in predator and prey over a long period in this lake. Another reason of the few field studies is that, because littoral fish communities in the lake harbor a large number of fish species and their interspecific interactions are very complex (Hori 1987, 1991, 1997), it is quite difficult to investigate the relationship of laterality between any predator and its specific prey species.

1.3.2 Dynamics of Laterality of Fish in a Temporal Lake, Lake Biwa This difficulty may be overcome by studying fish communities in a temporal lake, such as Lake Biwa, where the composition of the fish community is rather simple that the relationship of laterality between a predator and specific prey species can be investigated relatively easily. Yasugi and Hori (2011) studied the predator-prey relationship between the largemouth bass and its prey, freshwater goby, Rhinogobius spp., in Lake Biwa, Japan (Fig. 1.14a), with respect to their laterality (see Chap. 3 in this book). They revealed, with stomach contents analysis, that cross-predation

20

M. Hori

Fig. 1.14 Study site and two methods of sampling (Hori et al. 2021). (a) Map of Lake Biwa showing Wani Point, the study site of Yasugi and Hori (2011) and Hori et al. (2021); (b) view of “eri-ami” settle net at Wani Point; and (c) “okisukui-ami” fishing boat in operation off Wani Point

significantly predominated over parallel-predation. The largemouth bass is a sunfish (Centrarchidae) which originally comes from the USA to Japan before 1970 and recently increased in number in Lake Biwa (Maehata 2020). As the predominance of cross-predation was found in the relationship between the exotic largemouth bass and an endemic goby, the predominance may be caused by a kinematical interplay at each predation incident. Under an experimental condition, Yasugi and Hori (2012) tested the hunting success of largemouth bass on freshwater goby, and found that predation was more successful when a lefty (righty) predator met righty (lefty) prey and less successful when a lefty (righty) predator met a lefty (righty) prey, due to the kinematical concord or discord between strike and escape abilities of each morph. Yasugi and Hori (2011) also demonstrated, with annual sampling for 8 years in Lake Biwa, that in both largemouth bass and freshwater goby, the ratio of laterality fluctuated temporally around 0.5, though the relation between the ratios of predator and prey was not clear. One of the reasons of the unclear relationship may be that the largemouth bass is the predator which takes not only the freshwater goby but also any available prey of fish in the littoral area of the lake.

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1.3.3 Dynamics of Laterality Between a Piscivore, Hasu, and Its Prey, Ayu, in Lake Biwa In the pelagic area of Lake Biwa, the food web in the fish community (especially in open water) is much simpler. Specifically, the predominant predator is the piscivorous chub (hasu), Opsariichthys uncirostris (Cyprinidae), which is originally endemic in Lake Biwa and River Yodo Water System. The main prey of hasu is ayu, Plecoglossus altivelis (Plecoglossidae) (Tanaka 1964; Maehata 2020; Okuda et al. 2013). Ayu is distributed in Japan widely, but the population of Lake Biwa is confined in the lake and its tributaries, which depends mainly on plankters in the pelagic zone and partly on epilithic algae on rocks in the littoral zone (Miyadi et al. 1963). Ayu are an annual fish that breed in late summer (from August to October) in Lake Biwa (Miyadi et al. 1963). Hasu in the lake breed in summer (from May to August) and mature in the third or fourth season at lengths greater than 16 cm standard length (SL) for males and in the second or third season at lengths greater than 13 cm SL for females (Tanaka 1970). It has been proven that the two species exhibit laterality and that each population is composed of righty morphs and lefty morphs (Hori et al. 2017). Hori et al. (2021) performed a long period survey of temporal changes in the ratio of laterality of hasu and ayu. Adult fish samples of the two species were taken at a fixed pelagic site of the lake by purchasing from particular fishermen using the traditional settle net, “eri-ami,” once in a year of the same season (from middle June to early July) for a 20-year period (Fig. 1.14a, b). The result showed that, the dimorphism of each species was maintained dynamically throughout the period, and notably that the ratios of laterality of the two species changed in a cyclical manner with a similar amplitude throughout the study period (Fig. 1.15). Fourier transform analysis indicated that the ratio of hasu oscillated significantly (P < 0.001) with a period of 4.2 years throughout the study period, but that ayu showed no significant periodicity (P > 0.1). The ratio of ayu, however, exhibited some cyclic change similar to that of hasu; more specifically, it appeared to change in accordance with the oscillation of the ratio of hasu, suggesting the predator-prey interaction was in effect between the ratios of the two species. Fourier transform analysis attempts to detect any fixed periodicity. Thus, if the period of any involved oscillation changes during the research period, the analysis may fail to detect the periodicity. In the dynamics of laterality of interacting species, there is no logical reason that ratios of laterality should oscillate with a fixed periodicity. Using a mathematical model, Takahashi and Hori (1994) suggested that the periodicity is a function of the life history parameter of the interacting species. Thus, if the development rates change because of some factors, such as mean annual temperature, the periodicity of cyclic change may be altered. Furthermore, the impact of annual temperature on the development rate may be large for annual fish, i.e., ayu. The relationship of the cyclic change of laterality between the two fish species was examined by directly plotting the ratio of ayu against that of hasu (Fig. 1.16).

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M. Hori

Fig. 1.15 Temporal changes in the ratios of laterality of hasu and ayu from 1995 to 2014 in a pelagic area offshore from Wani Point, Lake Biwa (Hori et al. 2021)

Fig. 1.16 Relationship of ratios of laterality between hasu and ayu from 1995 to 2014 at Wani Point (Hori et al. 2021). Each dot is numbered in order from the first year (1995) to the last year (2014) of the survey

1 Laterality of Fish: Antisymmetry in Fish Populations Maintained. . .

23

The plots were scattered in a counterclockwise pattern of rotation, indicating that the ratio of ayu followed nearly 1 year behind the oscillation of the ratio of hasu. Such cyclic change suggests that some negative frequency-dependent selection was in effect. As pointed above, some empirical study (Yasugi and Hori 2011) and also theoretical ones (Nakajima et al. 2004, 2005) have suggested that the predominance of cross-predation is a critical factor in the resulting oscillations. The combination of laterality of each predator fish and fish preyed on can be investigated by a stomach contents analysis of the predator fish. However, fish caught by the settle net are unavailable for this examination, because the fish are left alive in the net for up to 2 days and the stomach contents are nearly fully digested. In contrast, hasu foraging on ayu can be obtained as a bycatch of “okisukui-ami” fishing, in which a special fishing boat equipped with a large scooping net at the bow rushes toward and scoops a shoal of ayu at the surface of the pelagic area of the lake (Fig. 1.14c). The fish caught are immediately kept on ice in the boat. Then, many adult fish of hasu were purchased from the okisukui-ami fishermen at a fishery port near the settle-net in June and July for the stomach contents analysis. The samples were dissected later in the laboratory, and fish preyed on, if any, were taken from the stomach (Fig. 1.17). Nearly all prey fish found in the hasu were ayu; 60 ayu (in total) with little damage from digestion were taken. The laterality of both the hasu and the ayu were identified based on the direction of the mouth

Fig. 1.17 Example of stomach contents analysis (Hori et al. 2021). Dissection revealed that the adult female hasu (Opsariichthys uncirostris, TL: 155 mm) had preyed on one subadult ayu (Plecoglossus altivelis, TL: 51 mm). The predation incident had occurred immediately before collection of the hasu, as the ayu remained at the anterior division of the intestine and had not undergone digestion

24 Table 1.3 Correspondence of morph types between each hasu and each prey ayu (Hori et al. 2021)

M. Hori

Hasu fish

Righty Lefty

Ayu fish Righty Lefty 7 20 24 9 Total: 60

opening, and the combinations of their laterality were examined. The stomach contents analysis of each hasu (Table 1.3) revealed that cross-predation occurred more frequently than the reverse combination (parallel-predation). A statistical test based on the common odds ratio revealed a significant probability, suggesting that cross-predation was 2.8-fold more frequent than parallel-predation. Under such situation, the majority morph of hasu at a given time, say righty, will exploit the lefty of ayu, and then the righty of ayu will attain higher fitness, i.e., minority advantage, and increase their frequency after a particular time span had elapsed, which will then favor the lefty morph of hasu. This differential predation is presumed to cause frequency-dependent selection on the two morphs of the predator and prey. The negative frequency-dependent selection for antisymmetric dimorphism involves a time lag, because the minority advantage exerted by the predation and the resultant increase in progeny presumably requires at least one generation. This time lag effect should cause the semi-synchronized cyclic changes in the ratios of laterality in predator and prey populations. It is noteworthy that the rotation in Fig. 1.16 did not occur around the equilibrium point, i.e., the coordinate [0.5, 0.5], but around the coordinate [0.57, 0.52], calculated as the average of all of the data points for both species. This suggests that the periodic changes of laterality of both populations were maintained such that righties tended to outnumber lefties slightly during the entire period, especially for hasu. It was an unexpected result. In fact, our mathematical models (Takahashi and Hori 1994, 1998, 2005; Nakajima et al. 2004) assumed a priori no differences in physical and/or sensory abilities between righty morphs and lefty morphs; the resultant time series of the ratios of laterality in these studies showed that all ratios of predator and prey oscillated around 0.5. Moreover, neither field nor experimental studies of fish have indicated superiority of righty morphs over lefty morphs in terms of abilities to hunt or escape. At present, it is difficult to interpret our findings in the relation between hasu and ayu. Abiotic ecological and environmental factors are likely irrelevant, because such factors, outside of the animal individuals, cannot affect the antisymmetric abilities of individuals; only interactions among individuals can affect these abilities. Quite interestingly, the same trend has been seen in the periodic change in the ratios of laterality of piscivore, and only piscivores, in a littoral fish community in Lake Tanganyika. There is a possibility that the lateralization of the brain and/or sensory organs is concerned. In mammals and birds, the selective involvement of the right side of the brain in spatial tasks has been known (Vallortigara and Bisazza 2002). But there is little evidence that it holds in teleost fishes, and, first of all, nothing is known about the lateralization of the brain

1 Laterality of Fish: Antisymmetry in Fish Populations Maintained. . .

25

system in piscivorous fishes. This consideration, therefore, is beyond the scope of the present review.

1.3.4 Laterality of Fish Is an Issue Always Concerned with Populations Notwithstanding the deviation from the even ratio, it remains noteworthy that the ratios of laterality of the predator and prey changed cyclically for a long period of time. This study proves that disproportional attacking success linked to laterality is the driving factor for cyclic changes. However, many studies of lateralized behavior in fish, such as turning mainly rightward in a maze or using the right eye for scouting behavior, overlooked the possibility of such cyclic change in the ratio of individuals that perform some lateralized behavior. Those studies concerning lateralized behavior in fish (e.g., Bisazza et al. 1998; Vallortigara et al. 1999; Vallortigara and Bisazza 2002) have considered that the behavioral laterality in the population is a fixed property, especially for fishes forming school, and referred to as the laterality at the population level, and few studies have traced the annual change in ratio of laterality of fish populations in the field. However, there may be no evidence for the ratio of laterality of a fish population being fixed under natural conditions. Thus, the present study may prompt reconsideration of the proposition that the behavioral laterality in some fish is fixed at the population level. The temporal change in the ratios of morphological laterality of many species in a cichlid fish community will be examined in the last section of this chapter. In the community, the ratio of laterality of all the species including fishes forming school changes periodically. It is also noteworthy that almost all the studies on lateralized behavior of fish have paid a few attentions to the morphological asymmetry of the body, except for the examination of fluctuating asymmetry. Matsui et al. (2013) tried to bridge the gap between such conventional studies of lateralized behavior and the recent morphological asymmetry of fish. They studied the relation between lateralized behaviors, such as detouring direction in maze and escape behavior from enemy (a dummy of predator), and morphological asymmetry using a poeciliid fish, Girardinus metallicus. A congeneric fish, G. falcatus, is familiar as a model animal to the conventional studies on the lateralized behavior in antisymmetric fashion, and G. metallicus is expected to shire the same functional lateralization. Matsui et al. (2013) showed that the lateralized behavior is associated with morphological antisymmetry, i.e., two types of morph were discriminated in regard to the head inclination, and the righty morphs significantly tended to detour leftward and escape rightward, whereas lefty morph tended to detour rightward and escape leftward. The authors investigated only the relation between the morphological and behavioral asymmetries of the fish, and did not mention the lateralization of the brain system.

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Then, it is obvious that more works are necessary to investigate the relation between the lateralization of the brain and the morphological asymmetry of fish.

1.4 Dynamics of Laterality Among Species of Competing Predators In the preceding section, we examined the dynamics of laterality in the predator-prey interaction under a new light that all the fishes have laterality. Here, let’s examine the dynamics of laterality among species in the same trophic group, especially species of competing predators. For this topic, we can now use a long-term data on the ratios of laterality of the two scale eaters cohabiting in a rocky shore of Lake Tanganyika. Hori (1993) demonstrated that the ratio of laterality (frequency of righty morph in a population) of P. microlepis oscillated with a period of about 5 years. But the survey was carried out in the area where P. microlepis was virtually the only species of scale eater in the fish community (a northernmost coast of the lake) and the period of survey was only 10 years.

1.4.1 Temporal Changes in the Laterality of Two Scale Eaters On a rocky shore of a southernmost coast of the lake, i.e., Kasenga Point, Zambia, two scale-eating species, P. microlepis and P. straeleni (Fig. 1.18), cohabit. A longterm census for 21 years carried out at the same cite (Takeuchi et al. 2010b) indicated that the densities of the two species were maintained rather constant and that of P. microlepis was 8.4 times higher than that of P. straeleni (the mean density of P. microlepis was 11.73/10 m2 (s.d., 28.76) and that of P. straeleni 1.39 (s.d., 3.98)). Hori et al. (2019) investigated temporal changes in the ratios

Fig. 1.18 The laterality of two scale-eating cichlids, Perissodus straeleni (upper) and P. microlepis (lower) in Lake Tanganyika (Hori et al. 2019). A lefty morph of the former and righty morph of the latter species are shown from both sides

1 Laterality of Fish: Antisymmetry in Fish Populations Maintained. . .

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Fig. 1.19 Temporal changes in the ratios of laterality in two scale eaters, P. microlepis and P. straeleni on the rocky shore of Kasenga Point, Zambia, in Lake Tanganyika over a 31-year period from 1988 to 2018 (Hori et al. 2019)

of laterality of the two scale eaters over a 31-year period on the site. The fish samples were taken with the aid of village fishermen in the same season of every year (November and December) over the study period. The result showed that the ratios of laterality of both species changed periodically around a value of 0.5 but were almost always maintained within a range between 0.4 and 0.6 (Fig. 1.19). The Fourier transformation detected a significant period of cycle in the ratio of each species, i.e., 3.9-year for P. microlepis (P-value < 0.001) and 4.1-year for P. straeleni (P-value < 0.01), indicating that the ratios of the two species oscillated with almost the same period (4 years). The result is just like as it was in the area where only one species of scale eater (P. microlepis) inhabited (Hori 1993), though the period of oscillation is 1 year shorter than the period observed in that study (about 5 years). The reason why the period of oscillation was different in the two studies is not clear. The ratios of laterality of the two species look to change in a semi-synchronized pattern, i.e., the ratio of P. microlepis changed seemingly following after the periodic change in P. straeleni, suggesting that the laterality of the two species interacted with each other; more specifically, the ratio of majority population (P. microlepis) changed following after the ratio of minority population (P. straeleni). Then, the relation of laterality between the two species was examined by directly plotting the ratio of P. microlepis against that of P. straeleni (Fig. 1.20). The plots scattered in a counterclockwise rotation around equilibrium point in both species, i.e., coordinate [0.5, 0.5]. It suggests that the ratio of P. microlepis followed nearly 1 year behind the periodic change of P. straeleni. The semi-synchronized pattern may be interpreted by the frequency-dependent selection (minority advantage) in scale-eating and also by the sharing of advantage between the minor morphs in frequency of the two species. If the minority advantage operates in this system, the minor morph at a time of minor species (P. straeleni), say righty morph, will get the higher fitness, and then the advantage should be shared by righty morph of P. microlepis. This sharing of minority advantage may be the most responsible for the ratio of laterality of majority scale eater in number, P. microlepis, following after the oscillation of the minority one, P. straeleni.

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Fig. 1.20 Relationship of laterality ratios between P. microlepis and P. straeleni (Hori et al. 2019). Red lines indicate that the change proceeds anticlockwise; thus, the change in P. microlepis follows that in P. straeleni, and blue lines proceed clockwise, indicating the opposite pattern

1.4.2 Relationship Between Lateralities of Hunting Scale Eater and Its Victim In the preceding section, we saw that the antisymmetric dimorphism of hasu and ayu in Lake Biwa is maintained by frequency-dependent selection between the predator and its prey species. Then, it is reasonable to suppose that the same mechanism is in effect between each scale-eating species and its prey fish. The combinations of laterality in each individual of scale eater and its victim were investigated (Hori et al. 2019). In this case, the stomach contents analysis is unavailable, because the scales from the stomach of scale eater tell us which body side was attached (see Sect. 1.2), but do not which type of morph (righty or lefty) was attacked. So, in this study, each hunting scale eater and its victim were collected in water just after the scale-eating. Using scuba, each foraging scale eater of either species was traced, and if the fish succeeded in scale-eating from any prey fish, both the scale eater and the victim were captured by a small gill-net. In most cases, the victim fishes were composed of Interochromis loocki, Petrochromis spp., Tropheus moorii, and Lamprologus callipterus, and the composition of the victim species looked to be little different between the two species of scale eater. The result showed that the

1 Laterality of Fish: Antisymmetry in Fish Populations Maintained. . .

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Table 1.4 The combination of morph types between each scale eater and its victim fish (Hori et al. 2019). R and L represent the righty and lefty, respectively

P. microlepis

R L

Victim fish R L 3 19 25 6 Total:

P. straeleni 53

R L

Victim fish R L 4 21 21 4 Total:

50

combination between the laterality of scale eater and its prey exhibited a significant bias toward cross-predation in both species (Table 1.4). The odds ratio was 26.4 for P. microlepis (95% confidential limit; 24.9–27.9) and 27.6 for P. straeleni (27.6– 29.1), indicating that cross-predation significantly occurred more frequently than parallel-predation (Mantel-Haenszel test; P 0 were designated as righties, and those with an index 0.05)

with higher motor performance should be advantageous for juveniles to succeed in foraging scales, as shown in adult fish. Interestingly, the lateral differences in kinetics were already significant during Session 1. That is, the lateral difference in kinetics is not explained by learning; instead, the finding strongly suggests that the scale eater intrinsically has a dominant side in terms of motor performance for predation. Therefore, these results indicate that scale-eating fish have a naturally stronger side for attacking prey fish and that they learn to use the dominant side through experience, with some adjustments in dynamics.

134

Y. Takeuchi

Body flexion amplitude (deg)

a

70 60

N = 16 *

** **

50

**

*

40 30 20 Dominant-side attack

10

Nondominant-side attack

b

7000

Maximum angular velocity (deg/s)

0

6000

* ** **

5000

**

*

4000 3000 2000 1000 0 Session 1 Session 2 Session 3 Session 4 Session 5

Fig. 4.10 Temporal change in the kinematic difference between dominant- and nondominant-side attacks (Takeuchi and Oda 2017). (a) Change in the amplitude of body flexion and (b) maximum angular velocity of predation from Sessions 1 to 5 (mean ± SE, N = 16 fish). P-values are derived from Wilcoxon rank-sum test between dominant and nondominant sides. *P < 0.05; **P < 0.01

4.4 The Neuronal Mechanisms of Behavioral Laterality 4.4.1 Comparison of Bending Movements During Scale-Eating and Escape Behavior It is not easy to identify neural circuits involved in a specific behavior, but by comparing the behavior with movements for which neural mechanisms are already understood, inferences can be made. Importantly, we noticed that the bending

4 Behavioral Laterality in the Scale-Eating Cichlid Fish: Detailed Movement,. . .

135

Prey fish

Scale eating

Scale eater

Escape

Fig. 4.11 Comparison of bending motions during scale-eating and escape (Takeuchi et al. 2012). Image shows the silhouette of the fish every 4 ms

movement during scale-eating resembles the bending movement observed during an escape response. Thus, to compare body flexion during scale-eating and escape in detail, we examined the escape behavior of the same scale eaters in response to a sound stimulus applied from below (Takeuchi et al. 2012). Escape behaviors were monitored with a high-speed video camera (1000 fps) set above the aquarium. Acoustic stimulation (500 Hz, about 120 dB) was delivered from an underwater speaker at the bottom of the aquarium. Acoustically elicited startle responses in scale eaters were initiated with a typical C-shaped bend of the body during the initial phase of escape behavior, followed by a counter-bend of the whole body and forward swimming, as previously reported in goldfish and zebra fish (Zottoli 1977; Eaton et al. 1981; Kohashi and Oda 2008). The onset latency (from sound presentation to the onset of the C-shaped bend) was as short as that observed in goldfish (9.8 ± 1.6 ms, mean ± SD, N = 63). Despite strict comparison, the quick bending movement during scale-eating was similar to the C-bend at escape and consisted of many of the same components but also included slightly different components (J-bend; Fig. 4.11). In scale-eating behavior, a combination of the S-shaped posture and the subsequently elicited Cbend may produce the J-bend, with bilateral contraction of the posterior trunk muscles. During a J-bend, the posterior body may function as a pivot for strong body bending. The similarity of body flexion exhibited during the two behaviors suggests that they may be elicited by shared neuronal networks, at least in part, as they both require nearly top speed. The kinematic parameters during both behaviors showed little fluctuation among individuals and trials, indicating that they are stereotypic behaviors.

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Y. Takeuchi

4.4.2 Estimation of a Neuronal Basis for Asymmetric Flexion Movement In teleost fish, the hindbrain reticulospinal (RS) neurons play an important role in the bending movement exhibited during an escape response (Ewert et al. 2001). In particular, a C-bend with a short response latency is driven by the paired giant Mauthner cells (M-cells), which receive visual and acoustic input (Zottoli 1977; Eaton et al. 1981; Canfield and Rose 1993). For example, when the left M-cell fires due to stimulation from the left side, the action potential crosses the midline and propagates along the contralateral spinal cord, allowing the trunk and tail muscles (right side) to contract simultaneously, and the body bends in a C-shape toward the right (Sillar 2009). M-cells may be involved in initiating the fast body bend during scale-eating as well as during an escape response. The maximum angular velocity (4010 ◦ /s) in the bending motion exhibited during scale-eating is fast enough to match the C-bend at escape. It would be more efficient for both mechanisms to share the same circuit, rather than requiring separate neural circuits depending on the situation. The innate superiority of the kinetics of dominant-side attacks may be explained by the lateralized strength of the trunk muscles or functionally lateralized control of the central nervous system (brain and spinal cord). Our previous study (Takeuchi et al. 2012) demonstrated that body flexion during a scale-eating attack is quite similar in kinetics (velocity and amplitude) to the C-bend exhibited at the beginning of fast escape behavior in adult P. microlepis and that lefty/righty individuals exhibit equivalent C-bends to left and right sides. In addition, no left-right difference was found in the maximum angular velocity and the change in the bending angle. That is, it is thought that there is no left/right difference in the bending motion when escaping. Therefore, muscle activity and basic neural mechanisms in the spinal cord to control the C-bend are bilaterally symmetrical in P. microlepis, and the behavioral laterality in scale-eating may be produced upstream of the common motor pathways. It is suggested that M-cells are involved in controlling the C-bend during scaleeating. Thus, if M-cells play a key role by triggering the body bending movement during an attack, one of the bilateral M-cell circuits might be more effective at propagating signals intrinsically and might have already been established before the start of scale-eating, because the dominant-side kinetics already exceeded those of the nondominant side in first-time scale-eating. Based on the results obtained so far, we propose the neural circuit model for the lateralized body flexion of scale-eating behavior shown in Fig. 4.12. Scale eaters obviously use visual cues at every stage of predation, from recognizing a prey fish to pursuing it, to moving to its flank, to targeting its scales. The visual information is received by the retina and processed within the optic tectum, the visual center in fish, via the optic nerve (Northmore 2011). Then, the processed signal propagates to the telencephalon and hindbrain involving M-cells and the contralateral spinal motor neuron, and the one side of trunk muscles contract rapidly. As the most important task, we focused on the dendrites of the M-cell receiving

4 Behavioral Laterality in the Scale-Eating Cichlid Fish: Detailed Movement,. . . Fig. 4.12 Neural circuit model of lateralized scale-eating behavior assumed for righty fish. The flow of information transmission from input to output is shown (1 → 10). There is a possibility that there are left-right differences in the size of neurons, the thickness of axons, and the structure of nerve dendrites

137

10. Attack 5.Telencephalon 1.Visual information of prey 2.Retina 3.Optic nerves 4.Optic tectum 6.Hindbrain 7.Mauthner cells 8.Spinal cord

9.Trunk muscles

input from the visual information and are conducting experiments incorporating new methods to compare with that of M-cell of the other side (Fig. 4.13). What is the level of the left-right differences? Another possibility is offered by the lateral differences in the optic tectum. A previous anatomical study (Sato et al. 2007) has revealed the existence of projections from the tectum to the RS neurons involving the M-cells in zebra fish. It has also been shown that two pairs of RS neurons (MeLc and MeLr) coordinate visually elicited prey capture movements (Gahtan et al. 2005) and that the ventromedial RS neurons (RoV3, MiV1, and MiV2) in the hindbrain play an important role in visually eliciting turning movements (Orger et al. 2008). In addition, the telencephalon possesses histologically primitive structures corresponding to the mammalian cerebral cortex (Wullimann and Mueller 2004), which is well known to have specific functions in learning and memory (Salas et al. 2006; Pollen et al. 2007). Learning plays an important role in acquiring attackside preference of P. microlepis (Takeuchi and Oda 2017; Takeuchi et al. 2022). These observations suggest that the visuomotor pathway controlling scale-eating behavior and circuits for modifying information processing are possible loci where the lateralized movements of scale-eating are initiated.

138 Fig. 4.13 Mauthner cells (M-cells) of P. microlepis (righty fish). (a) Dorsal view of the brain. Red dots represent the rostrocaudal position of M-cells. (b) Cross-sections of M-cells in the hindbrain shown after applying a transparency technique (thickness of section: 500 μm). M-cells were retrogradely labeled using Dextran-conjugated Texas Red injected into the spinal cord. Arrow heads, white arrows, yellow arrows, and blue arrows indicate the soma, ventral dendrites, lateral dendrites, and axons of M-cells, respectively. (c) Frontal, (d) sagittal (left and right sides), and (e) top views of 3D reconstruction image of M-cells using Neurolucida software (Microbrightfield, Williston, Vermont, USA)

Y. Takeuchi

b

a

Lef

Rostral

ight

Telencephalon

Optic tectum 100μm

Hindbrain Dorsal

Caudal

c

Ventral

Dorsal

Lef

ight

100μm

Ventral

d

Rostral Left side

Caudal

Caudal

100μm

e

100μm

Rostral

Lef

Rostral Right side

100μm

Caudal

ight

4 Behavioral Laterality in the Scale-Eating Cichlid Fish: Detailed Movement,. . .

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4.5 Conclusion It is important that the laterality of P. microlepis scale eaters closely resembles several features of handedness in humans. First, lefties and righties coexist within a single population of P. microlepis, although they represent opposite behavioral preferences. Like human handedness, then, behavioral laterality in scale eaters is subject to selection pressure for lateral dimorphism rather than uniformity. Second, complicated tasks such as scale-eating are generally performed more effectively when performed on the dominant side (Annett 1970). Third, behavioral laterality reflects morphological asymmetry (Chhibber and Singh 1970, 1972). Fourth, behavioral laterality is reinforced during development (Roy et al. 2003; Michel et al. 2006). However, unlike human handedness, behavioral laterality in scale eaters has been observed to increase both qualitatively and quantitatively throughout development in a short period, and the genetic system and molecular mechanisms underlying this remarkable laterality can be identified. The model system of scale eaters, which shows a clear functional laterality and consists of analyzable circuits, offers the possibility to obtain important knowledge about the relationship between lateralized behavior and functional differentiation of the brain.

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Chapter 5

Models of Lateral Asymmetry Dynamics: Realistic Oscillations by Time Delay and Frequency Dependence Satoshi Takahashi

Abstract In this chapter, we first review some population dynamics models of lateral asymmetry in fish, one-species model with discrete time and two species competition model. Then we focus on 1-prey–1-predator models with cross predation dominance, with or without time delay and frequency dependence. The equilibrium with equal morph frequencies is stable, though almost neutral, without time delay and frequency dependence. Time delays due to growth periods induce oscillation of laterality morph frequencies, which is too large and slow comparing to those observed in field. Introduction of frequency dependence due to prey’s vigilance to predator breaks left-right symmetry, which leads to the extinction of a prey morph. Only the model with cross predation dominance, time delay, and two frequency dependencies, that of predation success due to prey’s vigilance and that of prey choice, reveals oscillation with realistic amplitude and period. Keywords Lateral asymmetry · Oscillation · Frequency dependence · Time delay · Prey-predator

5.1 Population Dynamics Models of Lateral Asymmetry in Fish Hori (1993) observed that lefty and righty morphs of a scale eating cichlid Perissodus microlepis have almost same frequency, 0.5, and they oscillate around it in 5-year period with amplitude 0.15. Since prey’s scales are teared off more by minor morph, frequency dependent selection causes balanced polymorphism of laterality in P. microlepis. Takahashi and Hori (1994) constructed a one-species model of frequency dependent selection of lateral asymmetry in P. microlepis. The model showed

S. Takahashi () Nara Women’s University, Nara, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Hori, S. Takahashi (eds.), Lateral Asymmetry in Animals, Ecological Research Monographs, https://doi.org/10.1007/978-981-19-1342-6_5

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that strong frequency dependent selection by prey’s alert to major morph causes oscillation around the equilibrium 0.5, and it explains the amplitude and period of the oscillation. The detail of the model is as follows. Diploid model with lefty dominance is assumed. Note that the current definition of laterality is reversed from that in Hori (1993) and Takahashi and Hori (1994). Let Lt , Ht , Rt are fraction of lefty homozygote (genotype LL), heterozygote (Lr), and righty homozygote (rr) among adults, and L1t , H1t , R1t are those in 1-year old juveniles. Adults survive to the next year with probability q: Lt +1 = qLt + (1 − q)L1t

(5.1a)

Ht +1 = qHt + (1 − q)H1t

(5.1b)

Rt +1 = qRt + (1 − q)R1t .

(5.1c)

We assume that lefty and righty individuals reproduce with ratio f (Lt + Ht ) : 1 − f (Lt + Ht ), where function f (x) representing frequency dependence of reproduction is monotone decreasing and is symmetric around 0.5, f (1 − x) = 1 − f (x). An example of frequency dependence function f is f (x) = β −

2β − 1 , 1 + e−α(x−0.5)

(5.2)

(see Fig. 5.1). From the fraction of lefty gene st =

f (Lt + Ht )(Lt + 0.5Ht ) , f (Lt + Ht )(Lt + Ht ) + (1 − f (Lt + Ht ))Rt

Fig. 5.1 Function representing the ratio of frequency dependent reproduction for lefty to righty morph. Formula of function is given by Eq. (5.2) with α = 20, β = 0.9

(5.3)

5 Models of Lateral Asymmetry Dynamics

145

fractions of genotypes in 1-year old juveniles at the next year are calculated as L1(t +1) = st2

(5.4a)

H1(t +1) = 2st (1 − st )

(5.4b)

R1(t +1) = (1 − st )2 ,

(5.4c)

under the assumption of random mating.√ √ Equilibrium (L∗ , H ∗ , R ∗ ) = (1.5 − 2, 2 − 1, 0.5) ∼ (0.086, 0.414, 0.5) is unstable if √   2+1 1  −f (0.5) > +1 , 2 1−q oscillation period p is restricted to 4≤p≤

2π cos−1 (q/2)

< 6,

if the amplitude is sufficiently small (see Takahashi and Hori 1994). Comparison between field data and simulation result is shown in Fig. 5.2. Two scale-eaters, P. microlepis and P. straeleni, coexist at southern shore of Lake Tanganyika with similar densities. Both of them have lateral asymmetry dimorphism. Takahashi and Hori (1998, 2005) constructed models of competing two species with lateral asymmetry and showed that oscillation of morph fraction enhances the maintenance of lateral asymmetry dimorphism among species, as well as coexistence of two species. Ratio of two competing species is fixed to p1 : p2 in Takahashi and Hori (1998). Haploidy is assumed in the model. Let x and y are Fig. 5.2 Lefty fraction P. microlepis (Closed circle) and simulation result (solid line). Parameters in simulation are q = 0.6, b = 0.71, a = 60

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frequencies of lefty morph in the first and second species, respectively. The equation is    f (p1 xd + p2 yd )xd dx =r − x + ζ1 (t) xd (1 − xd ) dt f (p1 xd + p2 yd )xd + (1 − f (p1 xd + p2 yd ))(1 − xd )

(5.5a) 

f (p1 xd + p2 yd )yd dy =r dt f (p1 xd + p2 yd )yd + (1 − f (p1 xd + p2 yd ))(1 − yd )



 − y + ζ2 (t) yd (1 − yd ),

(5.5b) where xd = x(t − T ), yd = y(t − T ) are lefty fractions before growth period and f (x + y) : 1 − f (x + y) is the ratio of reproductive success of lefty morph to righty, ζ1 (t) and ζ2 (t) are white noises. Fraction term and minus term in the square bracket represent the reproduction and death, respectively. Term with white noise represents random genetic drift. If fraction of two morphs of lateral asymmetry oscillates by strong frequency dependence, sufficient condition for the stable coexistence of two laterality morphs in both of species is |x(0) − y(0)|
c, equilibrium of Eq. (5.9), corresponding to the positive one of Eq. (5.8), is given by  (X, Y, u, v) = where D = 1 −

 c D , , 0, 0 , A A

(5.10)

c . Jacobean matrix at this equilibrium point A ⎛ c ⎞ − −c 0 0 ⎜ A ⎟ ⎜ D 0 0 0 ⎟ ⎜ ⎟ Bc ⎟ J =⎜ ⎜ 0 0 ⎟ 0 ⎜ A⎟ ⎝ ⎠ BD 0 0 0 − A

has eigen values λ1 λ2 λ3 λ4

√ c2 − 4A2 cD = 2A √ −c − c2 − 4A2 cD = 2A √ B cD i = A √ B cD i. =− A −c +

Eigen values λ1 and λ2 have negative real part; λ3 and λ4 are pure imaginary, i.e., their real part equals zero. Therefore equilibrium (5.10) is neutrally stable at the first order of deviation. To investigate stability in higher order of deviation, we use asymptotic expansion. D c are deviation of X and Y from the equilibrium. Let p = X − , q = Y − A A Dynamics of deviations are given by Takahashi (1998) dp dt dq dt du dt dv dt

c = − p − cq − p2 − Apq + Buv A

(5.11a)

= Dp + Apq − Buv

(5.11b)

Bc v − pu + Bpv − Aqu A BD =− u + Apv − Bqu. A =

(5.11c) (5.11d)

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We expand p, q, u, v in orders of R, degree of deviation: p=

∞ 

R i pi ,

i=1

q=

∞ 

R i qi ,

i=1

u=

∞ 

R i ui ,

v=

i=1

i=1

First order of Eq. (5.11), dp1 dt dq1 dt du1 dt dv1 dt

c = − p1 − cq1 A = Dp1 Bc v1 A BD =− u1 , A =

has periodic solution  (p1 , q1 , u1 , v1 ) = 0, 0, sin ωt,



∞ 

 D cos ωt , c

√ B cD where ω = . A Second order of Eq. (5.11) is given by dp2 c = − p2 − cq2 − p12 − Ap1 q1 + Bu1 v1 dt A  c B D sin 2ωt = − p2 − cq2 + A 2 c dq2 = Dp2 + Ap1 q1 − Bu1 v1 dt  B D sin 2ωt = Dp2 − 2 c du2 Bc = v2 − p1 u1 + Bp1 v1 − Aq1u1 dt A Bc = v2 A BD dv2 =− u2 + Ap1 v1 − Bq1 u1 dt A BD =− u2 . A

R i vi .

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153

This has a solution in the form (p2 , q2 , u2 , v2 ) = (E1 cos 2ωt + E2 sin 2ωt, E3 cos 2ωt + E4 sin 2ωt, 0, 0), with coefficients of trigonometric functions E1 =

AB 2 (A2 − 2Ac − 4B 2 D) c(4B 2 c + D(A2 − 4B 2 )2 )

E2 =

A2 ω(4B 2 − 4AB 2 + A3 ) 2c(4B 2c + D(A2 − 4B 2 )2 )

E3 =

AB 2 (−A + 2c − D(A2 − 4B 2 )) c(4B 2 c + D(A2 − 4B 2 )2 )

E4 =

A3 ω(A2 − 2Ac − 4B 2 D) . 2c2(4B 2 c + D(A2 − 4B 2 )2 )

Since solution form of u2 , v2 , the second order terms of u, v is same to that of u1 , v1 , we include them to the first order term and set them to zero. We define a measure of the amplitude of oscillation by  s=

u2 +

c 2 v = R + O(R 3 ). D

Its dynamics is deduced by calculating

1 ds 2 . 2 dt

1 d  2 1 ds 2 c  = u + v2 2 dt 2 dt D c dv du + v =u dt D dt     c BD Bc v − pu + Bpv − Aqu + v − u + Apv − Bqu =u A D A Ac 2 Bc = −pu2 + Bpuv − Aqu2 + pv − quv D D   Ac Bc = −p2 u21 + Bp2 u1 v1 − Aq2 u21 + p2 v12 − q2 u1 v1 R 4 + O(R 5 ). D D (5.12) We define time average of function f (t) by ω [f (t)]t = 2π



2π ω

0

f (t) dt.

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Using [p2 ]t = 0, E1 , 2 E3 , [q2 cos 2ωt]t = 2

[p2 cos 2ωt]t =

[q2 ]t = 0, E2 2 E4 , [q2 sin 2ωt]t = 2

[p2 sin 2ωt]t =

we obtain time average of Eq. (5.12) as 



1 ds 2 2 dt

= − t

AB 2 (A2 c + 4B 2 D) R 4 + O(R 5 ). 8c(4B 2c + D(A2 − 4B 2 )2 )

Since ds 1 ds 2 ds =s = (R + O(R 3 )) , 2 dt dt dt oscillation amplitude changes in time average by 

ds dt

 = − t

AB 2 (A2 c + 4B 2 D) R 3 + O(R 4 ). 8c(4B 2c + D(A2 − 4B 2 )2 )

Therefore the equilibrium of equal laterality morphs is stable, though it is almost neutral and its damping is quite slow (Fig. 5.6). Nakajima (2003) and Nakajima et al. (2004) showed the global stability of the equilibrium of (5.8) (xL , xR , yL , yR ) = (x ∗ , x ∗ , y ∗ , y ∗ )      c 1 c 2c 1 2c , , = 1− , 1− a+b a+b a+b a+b a+b a+b Fig. 5.6 Dynamics of Eq. (5.7). Parameters are r = 2, K = 10, a = 0.3, b = 0.7, k = 1, c = 2

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by using the Lyapunov function L (xL , xR , yL , yR ) = x ∗ G

x  L x∗

+ x ∗G

x  R x∗

+ y ∗G



yL y∗



+ y ∗G



yL y∗

 ,

where G(z) = z − 1 − log z.

5.3 Prey–Predator Model with Time Delay In the previous section, we found that prey–predator laterality model without time delay has stable equilibrium and oscillation is not persistent. In this subsection, we include time delay due to growth period into the model and investigate the stability of the model. In Takahashi (1998), time delay is included into the term of prey’s density dependence as well as predator’s growth:   xL (t − τ ) + xR (t − τ ) dxL (t) = rxL (t) 1 − − akxL(t)yL (t) − bkxL(t)yR (t) dt K (5.13a)   xL (t − τ ) + xR (t − τ ) dxR (t) = rxR (t) 1 − − bkxR (t)yL (t) − akxR (t)yR (t) dt K (5.13b) dyL(t) = axL (t − τ )yL (t − τ ) + bxR(t − τ )yL (t − τ ) − cyL (t) dt dyR (t) = bxL(t − τ )yR (t − τ ) + axR (t − τ )yR (t − τ ) − cyR (t). dt

(5.13c) (5.13d)

The internal equilibrium of this system is always unstable. However, time delay terms of (5.13a), (5.13b) do not reflect growth period. Here, we investigate a model with time delays representing growth periods of prey and predator: dxL (t) rxL (t)(xL (t) + xR (t)) = rxL (t − τx ) − − akxL (t)yL (t) − bkxL(t)yR (t) dt K (5.14a) dxR (t) rxR (t)(xL (t) + xR (t)) = rxR (t − τx ) − − bkxR (t)yL (t) − akxR (t)yR (t) dt K (5.14b)

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dyL (t) = axL (t − τy )yL (t − τy ) + bxR (t − τy )yL (t − τy ) − cyL (t) dt dyR (t) = bxL(t − τy )yR (t − τy ) + axR (t − τy )yR (t − τy ) − cyR (t). dt

(5.14c) (5.14d)

Symbols τx and τy are growth periods of prey and predator, respectively. In (5.14), time delays are included only in population growth terms. Changing variables and parameters as in the previous subsection, xL + xR → X, K rt → t,

k(yL + yR ) → Y, K

rτx → τx ,

rτy → τy ,

xL − xR k(yL − yR ) → u, → v, K K bK c aK → a, → b, →c r r r

Eq. (5.14) is converted to dX dt dY dt du dt dv dt

= X(t − τx ) − X(t)X(t) − AX(t)Y (t) + Bu(t)v(t) = AX(t − τx )Y (t − τy ) + Bu(t − τx )v(t − τy ) − cY (t) = u(t − τx ) − u(t)X(t) − Au(t)Y (t) + BX(t)v(t) = AX(t − τy )v(t − τy ) − Bu(t − τy )Y (t − τy ) − cv(t).

 c D , , 0, 0 is given by The dynamics of deviation from the internal equilibrium A A 

dp dt dq dt du dt dv dt

 c p(t) − p(t)2 − cq(t) − Ap(t)q(t) + Bu(t)v(t) = p(t − τx ) − 1 + A = Dp(t − τy ) + cq(t − τy ) + Ap(t − τy )q(t − τy ) + Bu(t − τy )v(t − τy ) − cq(t) = u(t − τx ) − u(t) − u(t)p(t) − Au(t)q(t) + = cv(t − τy ) + Ap(t − τy )v(t − τy ) −

Bc v(t) + Bp(t)v(t) A

BD u(t − τy ) − Bu(t − τy )q(t − τy ) − cv(t), A

 c D , . This equation where p, q are deviations of X and Y from the equilibrium A A is linearized to 

 dp c = p(t − τx ) − 1 + p(t) − cq(t) dt A

(5.15a)

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dq = Dp(t − τy ) + cq(t − τy ) − cq(t) dt du Bc = u(t − τx ) − u(t) + v(t) dt A dv BD = cv(t − τy ) − u(t − τy ) − cv(t). dt A

(5.15b) (5.15c) (5.15d)

This deviation system is decomposed to that of prey and predator (5.15a), (5.15b) and of laterality (5.15c), (5.15d). We focus on the stability of laterality system (5.15c), (5.15d). We seek the solution of (5.15c), (5.15d) with eigenvalue λ = μ + iθ : u(t) = αeλt

(5.16a)

v(t) = eλt .

(5.16b)

If (5.15c), (5.15d) has a solution with eigenvalue of positive real part, then the internal equilibrium is unstable. Substituting (5.16) into (5.15c), (5.15d), we obtain Bc A BD −λτy αe − 1) − . A

λα = α(e−λτx − 1) + λ = c(e−λτy

(5.17a) (5.17b)

From (5.17), we have Bc =0 A BD α = c − (c + λ)eλτy . A

α(e−λτx − 1 − λ) +

Multiplying (5.18) by

(5.18) (5.19)

BD and substituting (5.19) to it, we obtain A

BD B 2 cD α(e−λτx − 1 − λ) + A A2 = (c − (c + λ)eλτy )(e−λτx − 1 − λ) + = 0.

B 2 cD A2

(5.20)

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Real and imaginary parts of (5.20) are given by (((c + μ)(1 + μ) − θ 2 ) cos θ τy − θ (1 + c + 2μ) sin θ τy )eμτy − ((c + μ) cos θ (τy − τx ) − θ sin θ (τy − τx ))eμ(τy −τx ) − c(1 + μ) +

B 2 cD A2

(5.21a)

+ (c cos θ τx )e−μτx = 0,

(((c + μ)(1 + μ) − θ 2 ) sin θ τy + θ (1 + c + 2μ) cos θ τy )eμτy − ((c + μ) sin θ (τy − τx ) + θ cos θ (τy − τx ))eμ(τy −τx )

(5.21b)

− cθ − (c sin θ τx )e−μτx = 0. By (5.21a) × (− sin θ τy ) + (5.21b) × cos θ τy and (5.21a) × (cos θ τy ) + (5.21b) × sin θ τy , we have θ (1 + c + 2μ)eμτy + ((c + μ) sin θ τx − θ (cos θ τx ))eμ(τy −τx ) + c(1 + μ) sin θ τy −

B 2 cD sin θ τy − cθ cos θ τy − c sin θ (τy + τx )e−μτx = 0 A2

((c + μ)(1 + μ) − θ 2 )eμτy − ((c + μ) cos θ τx + θ sin θ τx )eμ(τy −τx )

(5.22a)

(5.22b)

− c(1 + μ) cos θ τy +

B 2 cD cos θ τy − cθ sin θ τy + (c cos θ (τy + τx ))e−μτx = 0. A2

We define f (μ, θ ) by the division of (5.22a) by θ , and g(μ, θ ) by (5.22b), respectively. f (μ, θ ) = (1 + c + 2μ)e

μτy

  sin θ τx − (cos θ τx ) eμ(τy −τx ) + (c + μ) θ

sin θ τy B 2 cD sin θ τy − − c cos θ τy θ A2 θ sin θ (τy + τx ) −μτx e −c θ

+ c(1 + μ)

(5.23a)

g(μ, θ ) = ((c + μ)(1 + μ) − θ 2 )eμτy − ((c + μ) cos θ τx + θ sin θ τx )eμ(τy −τx ) − c(1 + μ) cos θ τy +

B 2 cD cos θ τy − cθ sin θ τy A2

+ (c cos θ (τy + τx ))e−μτx

(5.23b)

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Fig. 5.7 Zero curves of functions f (μ, θ) and g(μ, θ). Functions are given by (5.23). Parameter corresponds those of Fig. 5.6 and is A = 2.5, B = 1, c = 1, D = 0.6, τx = 2, τy = 4. (a) Wider range of μ and θ. (b) Magnification around the origin

f (μ, 0) is defined by the limit lim f (μ, θ ). θ→0

f (μ, 0) = (1 + c + 2μ)eμτy + ((c + μ)τx − 1)eμ(τy −τx ) + c(1 + μ)τy −

B 2 cD τy − c − c(τy + τx )e−μτx . A2

(5.23c)

f (μ, θ ) = 0

(5.24a)

g(μ, θ ) = 0.

(5.24b)

Eigenvalue λ = μ + iθ satisfies

Curves f (μ, θ ) = 0 and g(μ, θ ) = 0 cross each other at infinitely many points (Fig. 5.7a). Most of them have negative real part. They cross at positive real part in Fig. 5.7b. We show that (5.24) has solution of positive μ, so that equilibrium (u, v) = (0, 0) is unstable, if τx < τy . Since f (μ, 0) changes its sign at positive region and monotone increasing, i.e., f (0, 0) = −

B 2 cD τy < 0 A2

lim f (μ, 0) = lim 2μe μτy = ∞

μ→∞

μ→∞

(5.25) (5.26)

∂f (μ, 0) = ((c + 2μ + (1 − e −μτx ))τy + 2)e μτy + (2 + (c + μ)(τy − τx ))τx e μ(τy −τx ) ∂μ + cτy + c(τy + τx )τx e −μτx > 0,

equation f (μ, 0) = 0 has a unique positive solution μ1 > 0.

(5.27)

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We investigate which direction the curve f (μ, θ ) = 0 extends, which pass through the point (μ1 , 0). If μ is at least 

 log 2τx log 2cτy log 2(τy + τx ) B2D 1 , , ,c + + μ0 = max , τx τy τy + τx 2A2 2 then we have f (μ, θ ) ≥ e

μτy

 (1 + c + 2μ) − ((c + μ)τx + 1)e−μτx

   B 2 cD −μτy −μτx −μτy − c(1 + μ)τy + τy + c e − c(τy + τx )e A2   1 + μ B2D c c+μ μτy ≥e −1− − −c− (1 + c + 2μ) − 2 2 2A2 2   2 1 B D = eμτy μ − − −c 2 2A2 > 0. (5.28) On the other hand, we can determine the sign of f (μ, θ ) at θ = 

π f μ, τy

 = ((c + μ)e

μτy

π : τy

  sin θτx −μτx e + e μτy (μ + 1 − (cos θτx )e −μτx ) + c) 1 + θ

> 0.

(5.29) From (5.28) and (5.29), the curve f (μ, θ ) = 0 that passes through (μ1 , 0) should π cross θ -axis at (0, θ2 ) with 0 < θ2 < (Fig. 5.8). If the curve crosses θ -axis τy multiple times, we denote by (0, θ2 ) the first crossing point. The sign of g(μ, θ ) is positive at (μ1 , 0) as g(μ, 0) = (c(eμτy − 1) + μeμτy )(1 − e−μτx ) + cμ(eμτy − 1) + μ2 eμτy + > 0.

B 2 cD A2

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Fig. 5.8 Sign of functions f (μ, θ) and g(μ, θ), and curve f (μ, θ) = 0

Subtracting θ2 (cos θ2 τy )f (0, θ2 ) = 0 from (sin θ2 τy )g(0, θ2 ), we have sin θ2 τy g(0, θ2 ) = −θ22 sin θ2 τy − c sin θ2 (τy − τx )(1 − cos θ2 τx ) − θ2 (cos θ2 (τy − τx ) − cos θ2 τy ) − cθ2 (1 − cos θ2 τy ) − c sin θ2 τx (1 − cos θ2 (τy − τx ))

8, αg ∼ 0), negative feedback loop around equal frequencies of lefty and righty breaks. If there is more lefty prey than righty one, righty predator (that feeds on lefty prey) increases. Then due to frequency dependence of predation success, lefty predator, minority in predator, feeds more. Therefore righty prey, which is main target of lefty predator, decreases, and deviation from equal morphs in prey increases. Finally righty prey becomes extinct. This result contradicts to the fact that all fish investigated ever has lefty–righty dimorphism. Oscillation period and amplitudes in the system of larger frequency dependence (αf > 22, αg > 0.6) are realistic about 6 in period and 0 ∼ 0.25 in amplitude, though oscillation amplitude in prey is quite smaller than in predator. The reason of short oscillation period is that the feedback by frequency dependence is instantaneous. When lefty predator is more than righty one, it becomes disadvantageous by frequency dependent prey’s alert at that moment. Frequency dependence also restricts oscillation amplitudes. Disadvantage of major morph diminishes deviation of lefty fraction from 0.5. Realistic oscillation period and amplitude are also observed in age-structured PDE model, where period doubling or quasi-periodic oscillation appears in some parameter region (Ito 2016). While laterality oscillation in predator is induced by frequency dependence to predator, frequency dependence to prey do not induce laterality oscillation in prey. This is because frequency dependence operates to breeding in predator and to survival in prey. Frequency dependence to survival does not overshoot in continuous system. Laterality oscillation in prey is not induced by frequency dependence to prey, but transmitted from oscillation in predator by cross predation dominance (Figs. 5.18 and 5.19). Without cross predation dominance, lefty fraction oscillates

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Fig. 5.18 Dependence of lefty fraction dynamics to the cross predation dominance. (a) a = 1.5, b = 1.5, αf = 50, αg = 10. (b) a = 0.3, b = 2.7, αf = 50, αg = 10. (c) a = 0.15, b = 2.85, αf = 30, αg = 10. (d) a = 0.3, b = 2.7, αf = 50, αg = 50. Other parameters are K = 1, c = 0.5, k = 2, r = 1, τx = 1, τy = 2, βf = 0.8, βg = 0.2

only in predator (Fig. 5.18a). Amplitude of lefty fraction oscillation in prey becomes larger at large cross predation dominance (Figs. 5.18b and 5.19a). Frequency dependence to prey rather reduces oscillation amplitude (Fig. 5.18d). High cross predation dominance with intermediate frequency dependence to predator displace the lateral frequencies from equality and inhibit its oscillation (Figs. 5.18c and 5.19). Oscillation of morph frequencies in lateral asymmetry dimorphism is a coordinated work of cross predation dominance, time delay due to growth period and frequency dependence to both predator and prey. Strong frequency dependency to predator together with time delay generate oscillation of laterality morphs in predator. Laterality oscillation in predator is transmitted to prey by cross predation dominance. Finally, frequency dependence to prey maintains coexistence of laterality dimorphism in prey.

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Fig. 5.19 Dependence of oscillation amplitude and average of lefty fraction to cross predation fraction. (a) Oscillation amplitude of lefty fraction in prey. (b) Oscillation amplitude of lefty fraction in predator. (c) Average of lefty fraction in prey. (d) Average of lefty fraction in predator. Other parameters are K = 1, a + b = 3, c = 0.5, k = 2, r = 1, τx = 1, τy = 2, αg = 10, βf = 0.8, βg = 0.2

Acknowledgments Part of this chapter is a result of collaborations with Prof. Michio Hori, and my students Ms. Yumiko Takahashi, Ms. Mayuko Kodama, Ms. Lisa Kaichi, Ms. Masami Yata, and Ms. Tomoka Ito.

References Hori M (1993) Frequency-dependent natural selection in the handedness of scale-eating cichlid fish. Science 260:216–219 Ito T (2016) Age-structured model of laterality in prey and predator fish with fry eating and frequency dependence (in Japanese). Bachelor’s Thesis of Nara Women’s University

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Kaichi L (2005) Dynamics of laterality polymorphism model for cuttlefish (in Japanese). Bachelor’s Thesis of Nara Women’s University Kodama M (2004) Dynamics of laterality polymorphism model of prey predator system with time delay (in Japanese). Bachelor’s Thesis of Nara Women’s University Nakajima M (2003) Persistence and fluctuation of lateral dimorphism in fishes. Master Thesis of Tokyo University Nakajima M, Matsuda H, Hori M (2004) Persistence and fluctuation of lateral dimorphism in fishes. Am Nat 163:692–698 Nakajima M, Matsuda H, Hori M (2005) A population genetic model for lateral dimorphism frequency in fishes. Popul Ecol 47:83–90 Takahashi S, Hori M (1994) Unstable evolutionarily stable strategy and oscillation: a model of lateral asymmetry in scale-eating cichlids. Am Nat 144:1001–1020 Takahashi S, Hori M (1998) Oscillation maintains polymorphisms — a model of lateral asymmetry in two competing scale-eating cichlids. J Theor Biol 195:1–12 Takahashi Y (1998) Dynamics of prey predator model with laterality (in Japanese). Master Thesis of Osaka University Takahashi S, Hori M (2005) Coexistence of competing species by the oscillation of polymorphisms. J Theor Biol 235:591–596 Takeuchi Y, Hori M (2008) Behavioural laterality in the shrimp-eating cichlid fish, Neolamprologus fasciatus, in Lake Tanganyika. Anim Behav 75:1359–1366 Yasugi M, Hori M (2011) Predominance of cross-predation between lateral morphs in a largemouth bass and a freshwater goby. Zool Sci 28:869–874 Yata M (2011) Age-structured model of laterality in shrimp and shrimp eating fish (in Japanese). Master Thesis of Nara Women’s University

Chapter 6

Mathematical Models of Predators and Prey with Laterality Mifuyu Nakajima

Abstract Laterality in fish is considered one of the best examples of frequencydependent selection. In scale-eating fish, the most abundant morphological type (i.e., right or left type) of a species consumes less food because prey fish tend to defend themselves more against the dominant type; therefore, the rarer type of scale eater can consume more, becoming the dominant type. This idea is well supported by one mathematical model that illustrates changes in the fitness of each type of scale eater due to prey defense. Another model explains frequency-dependent selection between the two laterality types in both piscivorous fish and their prey. This model describes the population dynamics of each laterality type in predators and prey that exhibit biased predation, i.e., the right-type predator mainly consumes the left-type prey, and vice versa. The laterality ratio of each species obtained from this model oscillates periodically, showing continuous alternations of fitness between the two laterality types. In addition, the model suggests that monomorphism is not sustained in food webs with omnivory. Therefore, these models indicate that biased predation between different laterality types may maintain dimorphism. Keywords Dimorphism · Predatory behaviors · Frequency-dependent selection · Prey vigilance · Mathematical model · Biased predation

6.1 Introduction Laterality in scale-eating fish may be the simplest example of frequency-dependent selection (Bulmer 1994; Futuyma 2013; Lively 1993). Its general concept is that when one morphological type, for example, the right type, of a species of scaleeating fish is more abundant than the other morphological type, the left type, the prey species is attacked more frequently from the side that the dominant predator

M. Nakajima () Independent Researcher, Tokyo, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Hori, S. Takahashi (eds.), Lateral Asymmetry in Animals, Ecological Research Monographs, https://doi.org/10.1007/978-981-19-1342-6_6

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tends to use, the right flank in this example (see Chaps. 1–3 for details of predatory behaviors). The prey therefore defends the right flank more often and carefully, making it difficult for the dominant predator to harvest scales from the prey; this offers the rare type, the left type in this case, more opportunities to obtain scales. As a result, the rare type increases in numbers, eventually becoming the dominant type of the scale-eating species. This alternation of the fittest type between the left and right types continues over time. This theory explains how dimorphism is maintained in scale eaters and why the proportion of the two types oscillates periodically (see Chap. 1 for details). This chapter introduces a simple mathematical model that has been used to describe this mechanism of frequency-dependent selection and another model that expands this concept to piscivorous fish. The latter model focuses on cross predation, a type of biased predation between the two laterality types. These mathematical models do not explain laterality exclusively, but are also applicable to other types of dimorphism. In particular, the first model was developed to explain two antipredator defenses and can explain various defense behaviors, including vigilance, hiding, and fleeing; however, this chapter focuses on defense against scale-eating fish with laterality. In addition, neither model directly describes the morphological or phenotypic differences between the right and left types; rather, they describe differences in predatory and escape behaviors and the resulting effects on fitness. Furthermore, populations with different predation/escape strategies do not necessarily have to be different morphological types within the same species, but can be different species (Abrams and Nakajima 2007; Matsuda et al. 1993).

6.2 Model of Prey Vigilance Matsuda et al. (1993) created a mathematical model to investigate how population densities of two predators change in response to the time allocated for defense by prey against each predator. In the case of vigilance against scale-eating fish, the two predators can be interpreted as the right and left types of one predator species. As mentioned above and detailed in Chaps. 1–3, we expect that prey vigilance is concentrated toward the direction from which they are more frequently attacked. In other words, it is more adaptive for the prey to allocate more time to defending themselves against the more frequently attacked side. Matsuda et al. (1993) denoted the time to defend one side (i.e., vigilance against predator type i; i = 1, 2) as Vi . The mathematical model investigates how the optimal Vi , which maximizes prey fitness (denoted by Vi *), changes in response to predator population densities using partial differential equations. Prey fitness, W, is defined as follows: W = r (V1 , V2 , N) − F1 P1 − F2 P2 .

(6.1a)

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Table 6.1 Definitions of symbols used in Matsuda et al. (1993) Symbol aij Fi N Pi r Vi

Definition Availability to predator i of prey defending against predator j Risk of predation from predator i Prey density Density of predator i Prey fitness other than predation Antipredator effort toward predator i

All symbols are defined in Table 6.1. The Fi’ s are calculated as: F1 = a10 (1 − V1 − V2 ) + a11 V1 + a12 V2 ,

(6.1b)

F2 = a20 (1 − V1 − V2 ) + a21V1 + a22V2 .

(6.1c)

When the prey defends either side of its flank specifically, the availability to predator i of prey defending against the other predator j (i.e., aij ) should be larger than or equal to the availability of prey not defending at all (i.e., ai0 ). Therefore, we have aij ≥ ai0 . Given that aij > ai0 seldom occurs in our case of vigilance (Matsuda et al. 1993), we can assume that aij = ai0. As predator i is more successful in attacking prey when the prey is defending against the other predator, j, aij > aii . We assume that prey fitness, W, have a maximum value at positive V1 and V2 . Then, the following three conditions must be satisfied: (1) ∂W/∂Vi = 0, (2) ∂ 2 W/∂Vi 2 < 0, (3) ∂/∂V1 (∂W/∂V1 ) × ∂/∂V2 (∂W/∂V2 ) – ∂/∂V2 (∂W/∂V1 ) × ∂/∂V1 (∂W/∂V2 ) > 0. Condition (1) can be satisfied using Eq. 6.1a–6.1c: ∂W ∂r = + (a10 − a11 ) P1 + (a20 − a21 ) P2 = 0, ∂V1 ∂V1

(6.2a)

∂W ∂r = + (a10 − a12 ) P1 + (a20 − a22 ) P2 = 0. ∂V2 ∂V2

(6.2b)

Since ∂r/∂V1 should be negative, Eqs. 6.2a and 6.2b are satisfied with positive P1 andP2 under the following condition: (a10 − a11 ) (a20 − a22) > (a20 − a21) (a10 − a12) .

(6.2c)

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Substituting Eq. 6.1a–6.1c into condition (2) yields: ∂ 2r < 0. ∂Vi 2

(6.2d)

Finally, from condition (3), we expect: 

∂ 2r ∂V1 2



∂ 2r ∂V2 2



 >

∂ 2r ∂V1 ∂V2

2 .

(6.2e)

Next, we investigate the effect of predator density, P, on optimal antipredator effort, V*. Matsuda et al. (1993) obtained ∂Vi */∂Pj by implicitly differentiating Eqs. 6.2a and 6.2b with respect to Pj (see Appendix in Matsuda et al. (1993) for details):       1 ∂V1 ∗ ∂ 2r ∂ 2r = − (a10 − a11) + (a10 − a12 ) , ∂P1 Z ∂V1 ∂V2 ∂V22

(6.3a)

      1 ∂V2 ∗ ∂ 2r ∂ 2r = − (a10 − a12 ) + (a10 − a11 ) , ∂P1 Z ∂V1 ∂V2 ∂V12

(6.3b)

      1 ∂ 2r ∂V1 ∗ ∂ 2r = − (a20 − a21) + (a20 − a22 ) , ∂P2 Z ∂V1 ∂V2 ∂V22

(6.3c)

      ∂ 2r ∂V2 ∗ 1 ∂ 2r − = (a20 − a22 ) + (a20 − a21 ) , ∂P2 Z ∂V1 ∂V2 ∂V12

(6.3d)

where Z = (∂ 2 r/∂V1 2 )(∂ 2r/∂V2 2 ) − (∂ 2 r/∂V1 ∂V2 ) 2 , which is positive, as indicated by condition 2e. The first terms in [•] of Eqs. 6.3a and 6.3d are positive, and the second terms are 0 because of the assumption that aij = ai0 . Therefore, Eqs. 6.3a and 6.3d are always positive. Thus, the amount of time defending a predator increases with increasing density of the targeted predator type. The signs of Eqs. 6.3b and 6.3c depend on ∂ 2 r/∂V1 ∂V2 , i.e., the interaction of costs. In our case of a prey defending its flank against scale-eating fish, it is unlikely that defending the left or right makes a difference to r, which describes the prey’s fitness independent of predation. Therefore, ∂ 2 r/∂V1 ∂V2 is assumed to have the same sign as ∂ 2 r/∂Vi 2 , which is negative (Eq. 6.2d), and Eqs. 6.3b and 6.3c are always negative. Therefore, the effort of defending against one predator decreases with increasing density of the other type of predator, which is likely the case for scale-eating fish.

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Furthermore, Matsuda et al. (1993) examined the effect of predator density on the other predator’s density. Because the effect of predator density is reflected in the prey’s functional response, they derived ∂NF1 /∂P2 and ∂NF2 /∂P1 as: ∂NF1∗ ∂P2

=

∂NF2∗ ∂P1

=

N  Z

   ∂2 r (a10 − a11 ) (a20 − a21 ) ∂V 2 2

+ (a10 − a12 ) (a20 − a22 )



∂2r ∂V12



− {(a10 − a11 ) (a20 − a22 ) + (a10 − a12 ) (a20 − a21 )}



∂2 r ∂V1 ∂V2

 .

(6.4) Because the first two terms in [•] are 0, as aij = ai0 , and {•} in the last term is positive, ∂NF1 */∂P2 and ∂NF2 */∂P1 are positive. Therefore, the two predators are mutualistic: increase in the density of one predator promotes the consumption of prey by the other predator. Notably, both ∂NF1 /∂P1 and ∂NF2 /∂P2 are always negative because all of the terms in the following equations are less than or equal to 0: ∂NF1∗ ∂P1

=

N  Z

     2 ∂2r 2 ∂2r (a10 − a11 ) ∂V 2 + (a12 − a11 ) ∂V 2 2

+2 (a12 − a10 ) (a10 − a11 )

∂NF2∗ ∂P2

=

N  Z



1

∂2r ∂V1 ∂V2

 ,

     2 ∂2r 2 ∂2r (a20 − a22 ) ∂V 2 + (a21 − a22 ) ∂V 2 1

+2 (a21 − a20 ) (a20 − a22 )



2

∂2r ∂V1 ∂V2

 .

Based on Eqs. 6.3a–6.3c and 6.4, we can predict that when the right type is the dominant predator, the optimal amount of effort defending against the left-type predator decreases, and the left-type predator can consume more prey. Therefore, the left type becomes the majority among the predator, which then reduces the optimal amount of defense against the right-type predator, leading to an increase in prey consumption by the right-type predator. This continuous alternation of dominance between the right- and left-type predators explains the oscillation in the laterality ratio in scale-eating cichlids (Hori 1993). Moreover, the mutualistic relationship between the two predators in this model is suggestive of a mechanism of coexistence of right- and left-type predators based on prey switching their defenses.

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6.3 Model of Cross and Parallel Predation In the second model (Nakajima et al. 2004), both the predator and prey species exhibit dimorphism. After the first model and its subsequent papers (Matsuda et al. 1993; Matsuda et al. 1994, 1996) were published, laterality was reported or discovered in fishes other than scale-eating cichlids, for example, algae-eating cichlid (Mboko et al. 1998), goby (Seki et al. 2000), and some piscivorous fishes. As described by Yasugi and Hori (2011), some piscivorous fish consume more prey of the other morphological type than prey of the same type (see Chap. 2 for details). Nakajima et al. (2004) named this type of biased predation “cross predation”, while between fishes of the same type was named “parallel predation”. The model explicitly describes biased predation between two morphological types and mathematically shows the occurrence of laterality ratio oscillations. Nakajima et al. (2004) first constructed a model of a simple food web consisting of two trophic levels, piscivorous species y and its prey, species z. In this model, both species exhibit right and left types, denoted as yR , yL , zR , and zL . As cross predation dominates this food web, yR consumes zL more than it consumes zR . This system is described with a simple, classic equation describing predation, a Lotka-Volterra equation:     dyR = m Cyz zL + Pyz zR − dy yR , dt

(6.5a)

  dyL   = m Cyz zR + Pyz zL − dy yL , dt

(6.5b)

    dzR zL + zR = r 1− − Cyz yL − Pyz yR zR , dt K

(6.5c)

    dzL zL + zR = r 1− − Cyz yR − Pyz yL zL . dt K

(6.5d)

All symbols in the equations are explained in Table 6.2. It is assumed that prey species z increases logistically and that the two morphological types share the same carrying capacity, K. This system has an equilibrium where yR , yL , zR , and zL are all > 0 and satisfies dyR /dt = dyL /dt = dzR /dt = dzL /dt = 0. At this equilibrium, the right and left types of each species have the same population size, y† and z†, respectively, where:     Km Cyz + Pyz − 2dy r y = ,  2 Km Cyz + Pyz †

(6.6a)

6 Mathematical Models of Predators and Prey with Laterality

183

Table 6.2 Symbols used in Eqs. 6.5a–6.5d and 6.7a–6.7f Symbol m Cij Pij di rj Kj α jk

Definition Proportion of energy intake to total food consumed (≤1) Predation efficiency between different types of predator i and prey j Predation efficiency between the same types of predator i and prey j (Cij > Pij ) Mortality of predator i Intrinsic growth rate of prey j Carrying capacity of prey j Competition among prey species j and k

z† =

dy .  m Cyz + Pyz

(6.6b)

Using a local stability analysis after linearization (see Roughgarden 1979 and Nakajima et al. 2004 for details of this procedure), Nakajima et al. (2004) showed that the equilibrium in the linearized system is neutrally stable, which indicates the population sizes of yR , yL , zR , and zL change circularly around this equilibrium (Fig. 6.1a).1 In our computer simulation, these population sizes oscillated over time, as do the laterality ratios of both species (Fig. 6.1b). The proportion of the righttype predator (yR /(yR + yL )), shown as the black solid line in Fig. 6.1b, follows its prey (zL /(zR + zL )), shown as the gray solid line in Fig. 6.1b, much like the predator in the classic Lotka-Volterra predation model of two trophic levels. The equilibrium in the original system before linearization is globally stable, confirmed by a Lyapunov function (Nakajima et al. 2004). Therefore, the trajectory slowly dumps to the equilibrium. In addition, Nakajima et al. (2004) investigated laterality ratio behavior in a more complex food web comprising three trophic levels, in which both top and intermediate predators consume the bottom prey (Fig. 6.2), referred to as omnivory (Pimm and Lawton 1978). Without omnivory, this three-trophic-level system behaves as we simply expect from the two-trophic-level system shown above: increase of righty in the bottom species cause increase of lefty in the intermediate predator and hence increase of righty in the top predator. Similarly, we would predict in the system with omnivory that when the left type of the bottom prey is dominant, the right types of both the top and intermediate predator species have advantages and eventually increase in numbers. However, because the righttype intermediate predator increases, the left-type top predator can also increase.

1

Nakajima et al. (2004) linearized Eq. 6.2a–6.2e and obtained the Jacobian matrix with equilibrium (y†, y†, z†, z†). The eigenvalues of the Jacobian matrix consisted of a pair of purely imaginal numbers and a pair with negative real parts. Therefore, the equilibrium was neutrally stable

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(a) 0.90 0.85

yR

0.80 0.75 0.70 0.65

0.75

0.80

0.85

0.90

0.95

1.00

zL

(b) 0.65

Laterality ratio

0.60 0.55 0.50 0.45 0.40 0

10

20

30

40

50

Time Fig. 6.1 Numerical solution for a two-species system described by a Lotka-Volterra-type model (Eq. 6.5a–6.5d). (a) The solution trajectory (black line) shown in the zL –yR phase plane. The population sizes circulate around the equilibrium (gray dot). (b) Change in the laterality ratios yR /(yR + yL ) (black line), zR /(zR + zL ) (dashed line), and zL /(zR + zL ) (gray line) over time. The parameter values used were as follows: m = 1, Cyz = 1, Pyz = 0.15, dy = 0.9, K = 1000, r = 1

6 Mathematical Models of Predators and Prey with Laterality

xR

xL

yR

yL

zR

zL

185

Proportion of right type

0.65 0.60 0.55 0.50 0.45 0.40 0

10

20

30

40

50

Time Fig. 6.2 Simulated food web (upper) and numerical solution (lower) consisting of three trophic levels with omnivory described by a Lotka-Volterra-type model. The thick and thin arrows in the food web represent cross and parallel predation, respectively, where the directions of the arrows represent energy flow. The lines in the numerical solution describe the proportions of the right type in species x (black), y (gray), and z (dashed). The parameter values used were as follows: Cyz = 0.8, Pyz = 0.1, Cxz = 0.3, Pxz = 0.0525, Cxy = 0.3, Pxy = 0.1, m = 0.8, dx = 0.5, dy = 0.5, K = 10, r = 1

Nakajima et al. (2004) investigated this system numerically2 and found that the laterality ratios of all species oscillated periodically (Fig. 6.2), as can be observed in nature (Hori 1991, 1993). Another similar but different equations that described the fitness of right- and left-type genes in a stable population suggested that the 2

The function “DSolve” in the software Mathematica (ver. 9.0, Wolfram Research, Champaign, IL, USA) was used to solve the differential equations numerically

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laterality ratios in all three species continue to oscillate periodically3 (Nakajima et al. 2005). In the numerical simulations, the oscillations are not synchronized, likely reflecting slight differences in the timing between the advantages of right and left types of the two predators. In addition, monomorphism in any species in this food web is unstable. All possible equilibria with only one type in any species are locally unstable and can be invaded by the absent type of the monomorphic species, resulting in dimorphism. Interestingly, this is not true when parallel predation dominates over cross predation (e.g., Pyz > Cyz ). In numerical simulations using various values of parameters, food webs consisting of monomorphic species persist (Nakajima et al. 2004). This is probably because the dominance of parallel predation almost divides the food web into two food webs, consisting of only the right or left type, although minor interactions between the two food webs remain via cross predation (assume that the thin arrows dominate the food web in Fig. 6.2). For example, if xR were to disappear, the remaining food webs in the parallel predation-dominated system would consist of yR -zR and the left types, which could persist stably (the stability of the left-type food web is discussed by Holt and Polis (1997)). Therefore, species x could exist stably as a monomorphic species composed of only the left type. In addition, the results of the local stability analysis on the two trophic systems described above (Eq. 6.5a–6.5d) hold, even when Pyz > Cyz . Therefore, the model can also explain examples of lateral dimorphism in nature that show a dominance of parallel predation (e.g., angler fish (Yasugi and Hori 2016)). As shown above, the analysis of the laterality ratios of three trophic levels reveals an interference with laterality dynamics caused by additional predation, omnivory. Next, the effects of another primary form of species interaction, competition, are investigated in a system consisting of two trophic levels with more species using similar Lotka-Volterra predation equations that include competition in the bottom level (Nakajima, unpublished). The first system (Fig. 6.3) consists of one predator species (x) and two prey species (y and z):     dxR = m Cxy yL + Pxy yR + Cxz zL + Pxz zR − dx xR , dt

(6.7a)

  dxL   = m Cxy yR + Pxy yL + Cxz zR + Pxz zL − dx xL , dt

(6.7b)

    αyy (yL + yR ) + αyz (zR + zL ) dyR = ry 1 − − Cxy xL − Pxy xR yR , dt K (6.7c)

3

Local stability analysis on linearized equations showed that the equilibrium with the ratios of right type and left type equal to 1:1 in all three species is an unstable focus. The change of ratios shows a limit cycle

6 Mathematical Models of Predators and Prey with Laterality

x y

z

0.70 Proportion of right type

Fig. 6.3 Simulated food web (upper) and numerical solution (lower) of a three-species system described by a Lotka-Volterra-type model (Eq. 6.7a–6.7f). The letters in the food web indicate the species names. Arrows show the uptake of energy. The dashed line indicates competition between the prey species. The colors of the lines in the lower panel correspond to the colors of the species used in the food web in the upper panel. The parameter values used were as follows: m = 0.8, Cxy = Cxz = 0.15, Pxy = 0.01, Pxz = 0.1, dx = 0.5, Ky = 11, Kz = 5, ry = 1.5, rz = 2, α yy = α yz = α zz = 0.1

187

0.65

y

0.60 0.55

x

0.50 0.45

z

0.40 0.35 0

10

20

30

40

50

Time

    αyy (yL + yR ) + αyz (zR + zL ) dyL = ry 1 − − Cxy xR − Pxy xL yL , dt K (6.7d)     αyz (yL + yR ) + αzz (zR + zL ) dzR = rz 1 − − Cyz xL − Pyz xR zR , dt K (6.7e)     αyz (yL + yR ) + αzz (zR + zL ) dzL = rz 1 − − Cyz xR − Pyz xL zL . dt K (6.7f) All symbols are explained in Table 6.2. Nakajima et al. (2005) analyzed laterality ratio dynamics in the same food web using the equations of gene frequencies described above. This dimorphic equilibrium is neutrally stable, suggesting that the laterality ratios oscillate around the equilibrium. In accordance with this analysis, the numerical solution2 of the above model (Eq. 6.7a–6.7f) shows periodic oscillations in all species (Fig. 6.3)2 . In addition, the oscillations of the two prey species are synchronized. The oscillations of both prey species follow that of the predator, similar to the previous model with two species (Eq. 6.5a–6.5d, Fig. 6.1b).

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A model of more complex systems was developed by adding more predation in addition to competition (Nakajima, unpublished). This model consists of a food web containing two trophic levels with three species in each level. In the first scenario (Fig. 6.4a), carnivorous species 1, 2, and 3 consume herbivorous species 4, 5, and 6, respectively, while species 4, 5, and 6 compete for resources. Similar to Eq. 6.7a– 6.7f, the dynamics of predator species i and prey species j are described as:

dNjR dt

    dNiR = mi Cij Nj L + Pij Nj R − di NiR , dt

(6.8a)

  dNiL   = mi Cij Nj R + Pij Nj L − di NiL , dt

(6.8b)

  = rj 1 −

αj4 (N4L +N4R )+αj5 (N5R +N5L )+αj6 (N6R +N6L ) K

 (6.8c)



−Cij NiL − Pij NiR Nj R ,

dNjL dt

  = rj 1 −

αj4 (N4L +N4R )+αj5 (N5R +N5L )+αj6 (N6R +N6L ) K

 −Cij NiR − Pij NiL Nj L .

 (6.8d)

where NiR and NiL denote the population size of the right and left types of species i, respectively. In computer simulations of this model (Fig. 6.4a), the laterality ratios of both trophic levels oscillate, although the periods and amplitudes of these oscillations are not as regular as those observed in the previous models (Figs. 6.1b and 6.2). The oscillations of species 1 and 4 have longer periods and larger average amplitudes than those of other species. Predation may not be the cause because in the computer simulation shown in Fig. 6.4, the predation efficiencies between these species (C14 and P14 ) and those between species 2 and 5 (C25 and P25 ) are exactly the same. Irregular oscillations are observed in predator species 1 (blue line in Fig. 6.4a) whose death rate was relatively low and prey species 4 (red line in Fig. 6.4a) whose growth rate was relatively low. The population sizes of the right types in these species likely change slowly, and therefore the oscillation periods become longer. Other computer simulations with (a) species 1 and 2 having the same death rates (d1 = d2 ) and (b) species 4 and 6 having the same growth rates (r4 = r6 ) both supported this idea, showing disturbed oscillations in the species two to five pairs, as observed in one to four pairs in the simulation explained above (Fig. 6.4a). Each prey oscillation follows that of its predator by about a quarter period, similar to the one-predator-one-prey model (Eq. 6.5a–6.5d and Fig. 6.1b). After adding a predatory connection between species one and five (Fig. 6.4b), the oscillations become more irregular. Species 3 and 6, which are not directly affected by this change, exhibit disordered oscillations. This is likely due to indirect

6 Mathematical Models of Predators and Prey with Laterality (a)

( b)

1.0 Proportion of right type

189

1

2

3

1

2

3

4

5

6

4

5

6

4

4 5

0.8

2 3

0.6

3 5

6

6 1

0.4 0.2 0

1 20

2 40

60

80

100 0

20

40

60

80

100

Time

Fig. 6.4 Simulated food webs (upper) and the results of simulations (lower) with and without a prey shared between predators (a and b, respectively). The numbers in the food webs indicate the species number. The arrows show the uptake of energy. The dashed lines indicate competition among the prey species. The colors of lines shown in lower panels correspond to the colors of the species used in the food webs in the upper panels. The parameter values used were as follows: m = 1, C14 = C25 = 0.1, C36 = 0.2, P14 = P25 = 0.015, P36 = 0.03, C15 = 0.05, P15 = 0.0075, d1 = 0.2, d2 = 0.4, d3 = 1.1, r4 = 0.75, r5 = 0.8, r6 = 0.85, K4 = 150, K5 = 100, K6 = 50, α 44 = α 45 = α 46 = α 55 = α 56 = α 66 = 0.1

competition among the predators, as well as discordant alternations of fitness among the right and left types in all species. Nakajima (2006) analyzed the same food webs as shown in Fig. 6.4 using equations to describe the dynamics of the gene that determines laterality. Her results were consistent with the behavior of the population dynamics shown here.

6.4 Conclusion The mathematical models introduced in this chapter show that dimorphism can be maintained and ratios of two morphological types can oscillate because of biased predation. The models strongly support frequency-dependent selection between right and left types. Interestingly, the model with cross and parallel predations implies that biased predation can maintain dimorphism in complex food webs with omnivory. This may answer a basic question of laterality: Why do both right and left types exist in a population, rather than only one type?

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The last computer simulation suggests that a single addition of predation to a food web can disturb oscillations in species not directly connected to the new predator. This indicates that even if laterality ratios in the real world do not oscillate periodically, biased predation may still influence laterality ratio dynamics. In addition, other factors may cause laterality ratio fluctuations, such as timedelayed effects of biased predation, which is discussed in detail in Chap. 5. The time scale of oscillations of laterality frequencies in real world is not very short which makes us hard to observe. It took several years to complete one cycle in Lake Tanganyika cichlids (Hori 1993, also see Chaps. 1 and 5). Oscillations may especially be overlooked when (1) the oscillation period is long, (2) the food web has many species interactions, and (3) (a) species involved in the food web has a low death rate or a low growth rate, as described above (also see Fig. 6.4). Mathematical models can help understand and predict the mechanism and behaviors of dynamics.

References Abrams PA, Nakajima M (2007) Does competition between resources change the competition between their consumers to mutualism? Variations on two themes by Vandermeer. Am Nat 170(5):744–757 Bulmer M (1994) Theoretical evolutionary ecology. Sinauer Associates, Sunderlad, MA Futuyma DJ (2013) Evolution, 3rd edn. Sinauer Associates, Sunderland, MA Holt RD, Polis GA (1997) A theoretical framework for intraguild predation. Am Nat 149(4):745– 764 Hori M (1991) Feeding relationships among cichlid fishes in Lake Tanganyika: effects of intraand interspecific variations of feeding behavior on their coexistence. Ecol Int Bull 19:89–101 Hori M (1993) Frequency-dependent natural-selection in the handedness of scale-eating cichlid fish. Science 260(5105):216–219 Lively CM (1993) Rapid evolution by biological enemies. Trends Ecol Evol 8(10):345–346 Matsuda H, Abrams PA, Hori H (1993) The effect of adaptive antipredator behavior on exploitative competition and mutualism between predators. Oikos 68(3):549–559 Matsuda H, Hori M, Abrams PA (1994) Effects of predator-specific defense on community complexity. Evol Ecol 8(6):628–638 Matsuda H, Hori M, Abrams PA (1996) Effects of predator-specific defence on biodiversity and community complexity in two-trophic-level communities. Evol Ecol 10(1):13–28 Mboko SK, Kohda M, Hori M (1998) Asymmetry of mouth-opening of a small herbivorous cichlid fish Telmatochromis temporalis in Lake Tanganyika. Zool Sci 15(3):405–408 Nakajima M (2006) A theoretical study of population dynamics and frequency dependent selection due to asymmetric predation in laterally dimorphic fishes. PhD thesis, University of Tokyo, Tokyo, Japan Nakajima M, Matsuda H, Hori M (2004) Persistence and fluctuation of lateral dimorphism in fishes. Am Nat 163(5):692–698 Nakajima M, Matsuda H, Hori M (2005) A population genetic model for lateral dimorphism frequency in fishes. Popul Ecol 47(2):83–90 Pimm SL, Lawton JH (1978) Feeding on more than one trophic level. Nature 275(5680):542–544 Roughgarden J (1979) Theory of population genetics and evolutionary ecology: an introduction. Macmillan Publishing, New York, NY Seki S, Kohda M, Hori M (2000) Asymmetry of mouth morph of a freshwater goby, Rhinogobius flumineus. Zool Sci 17(9):1321–1325

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Yasugi M, Hori M (2011) Predominance of cross-predation between lateral morphs in a largemouth bass and a freshwater goby. Zool Sci 28:869–874 Yasugi M, Hori M (2016) Predominance of parallel- and cross-predation in anglerfish. Mar Ecol 37(3):576–587

Chapter 7

Statistical Analysis of Lateral Asymmetry: Detect Antisymmetry and Oscillation from Unequal-Interval Binomial Data Satoshi Takahashi

Abstract In this chapter, we apply statistical methods for two problems of laterality asymmetry. In the first part, we determine either the distribution of the index of the asymmetry is unimodal or bimodal, by using model selection. If laterality is not functional and caused by randomness or error, distribution of index of asymmetry should be symmetric and unimodal (fluctuating asymmetry); if it is functional and under the negative frequency dependent selection, distribution of index of asymmetry should be likely bimodal (antisymmetry). In the second part, we detect the oscillation of the dynamics of laterality fraction. We calculate the probability distribution of the coefficients of discrete Fourier transform under the assumption that laterality frequency is stationary. Then decide if the oscillation is significant. Keywords Antisymmetry · Model selection · Oscillation detection · Fourier transform · Lateral asymmetry · Unequal-intervals

7.1 Statistics of Lateral Asymmetry Distribution Lateral asymmetry in scale eaters was very clear and its level is not quantified at first (Hori 1993). As the study of lateral asymmetry extends beyond scale eaters, intensity of lateral asymmetry for each individual is quantified (Hata et al. 2013; Mboko et al. 1998; Seki et al. 200). While lateral asymmetry in scale eaters are clear dimorphism, those in other fish are more subtle and its distribution is statistically analyzed (Hata et al. 2011, 2013; Yasugi and Hori 2011). Typical distributions of lateral asymmetries are, fluctuating asymmetry (FA), directional asymmetry (DA), and antisymmetry (AS) (Palmer and Strobeck 1992). In FA, distribution is normal with mean 0. Deviations from symmetry are interpreted

S. Takahashi () Nara Women’s University, Nara, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Hori, S. Takahashi (eds.), Lateral Asymmetry in Animals, Ecological Research Monographs, https://doi.org/10.1007/978-981-19-1342-6_7

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as errors or the result of developmental perturbation when the distribution is FA. DA is normal distribution with non-zero mean. In DA, all the individuals are biased to the same direction. AS is represented as superposition of two normal distributions with means of same absolute value and different signs. Laterality in this book plays important roles in prey–predator interactions. They often show AS in fish or shrimps as in the first 4 chapters of this book, or DS in snails or snakes as in the last 4 chapters. In this section we consider the following four models with distribution density function f (x) (Fig. 7.1): 1. FA model fFA (x) = √

1 2πσ 2

e

−

x2 2σ 2

.

(7.1)

2. DA model 2 1 − (x−μ) fDA (x) = √ e 2σ 2 . 2πσ 2

(7.2)

Fig. 7.1 Density function for distribution models. (a) FA model, (b) DA model, (c) AS model, (d) Skewed AS model

7 Statistical Analysis of Lateral Asymmetry

195

3. AS model 2 2 0.5 0.5 − (x−μ) − (x+μ) fAS (x) = √ e 2σ 2 + √ e 2σ 2 . 2πσ 2 2πσ 2

(7.3)

4. Skewed AS model 2 r 1 − r − (x+μ)2 2 − (x−μ) fSAS (x) = √ e 2σ 2 + √ e 2σ . 2πσ 2 2πσ 2

(7.4)

σ 2 , μ are variance and mean of normal distribution. In addition to FA, DS, and AS models, we consider skewed AS model where the fraction of normal distribution with mean μ does not equal to that with mean μ, since lefty and righty morphs in scale eaters have different fractions and they oscillate (Hori 1993). Let the data be x1 , . . . , xn . Parameters of models, σ 2 , μ, r are determined so as to maximize the likelihood  L= f (xi ). (7.5) In FA model (7.1), maximum likelihood estimator of σ 2 is given by the second moment σ2 =

1 2 xi . n

In DA model (7.2), maximum likelihood estimators of μ and σ 2 are sample mean and sample variance. μ=

1 1 xi , σ 2 = (xi − μ)2 . n n

Maximum likelihood estimators of parameter μ, σ 2 , r in AS model and Skewed AS model are calculated numerically. Let Lˆ be the maximum of the likelihood, i.e., maximum value of (7.5) as changing parameters, σ 2 , μ, r. The model that best explains the data is selected by using AIC (Akaike information criterion) (Akaike 1974). AIC is calculated from maximum logarithmic likelihood log Lˆ the number of parameters k by the formula AIC = −2 log Lˆ + 2k.

(7.6)

The first term is small when the model fits the data. The second term represents the error from parameter estimation. It is known that the model with smaller AIC value k2 (Sakamoto et al. predicts better, if the number of data n is larger than 2k and 4 1986).

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In the following, we calculate AIC of 4 models for the index of Asymmetry (IAS) of Japanese amberjack (Seriola quinqueradiata). IAS is calculated from the height of the mandible posterior ends (HPME) in lower jawbone by x =2×

R−L × 100, R+L

where x is IAS, R and L are the heights of the right and left mandibles, respectively (Hori et al. 2007, Hata et al. 2013). Righty individuals have positive IAS, while lefty ones have negative. Assume the data file of IAS named “data.csv” is prepared in CSV format with contents: "Code","IAS" "06666-1",2.322 "06666-2",-4.54 "06666-3",-3.641 "470-13",0.648 :

The first column, Code, is individual code. The second column is IAS value. The following R script calculates AIC of four models and plot their density functions. library(stats4) data