Introduction to Quantitative Ecology: Mathematical and Statistical Modelling for Beginners 2021937956, 9780192843470, 9780192843487, 0192843478

Table of contents :
Cover
Introduction to Quantitative Ecology: Mathematical and Statistical Modelling for Beginners
Copyright
Acknowledgements
About This Book
“Can I handle this?”
How to use this book
Contents
Part I: Fundamentals of Dynamic Models
1: Why Do We Model?
1.1 Myths of modeling
1.2 Types of models
1.2.1 Organizational scales
1.2.2 Purpose
1.2.3 Model endpoint
1.2.4 Statistical or mathematical?
1.3 Developing your model
1.3.1 Steps in model development
1.3.1.1 Identifying your question
1.3.1.2 Defining the model world
1.3.1.2.1 Trade-offs Levins (1966) proposed three main dimensions of trade-offs
1.3.1.3 Building a conceptual model
1.3.2 Advanced applications and topics
1.3.2.1 Interacting species
1.3.2.2 What happens when two state variables share a rate?
1.3.2.3 Translating conceptual models into math
1.3.2.4 Drawing function shapes
1.3.3 The rest of the process
1.3.3.1 Model coding and diagnostics
1.3.3.2 Interpreting and analyzing the model’s output
1.3.4 Modeling is not a linear process
Summary
Exercises
2: Introduction to Population Models
2.1 Introduction to population dynamics
2.2 Fundamental structure of population models
2.3 Density-independent models
2.3.1 Continuous-time density-independent models
2.4 Developing a density-dependent model
2.4.1 The logistic model
2.4.2 Advanced: Continuous logistic models
2.4.3 Other density-dependent models
2.4.3.1 The Ricker model
2.4.3.2 The Beverton-Holt model
2.4.3.3 The Gompertz model
2.4.3.4 The theta-logistic model
2.4.4 Allee effects
2.5 Dynamic behavior of density-dependent models
2.6 Cobwebbing
2.6.1 Interpreting cobweb diagrams
2.6.2 Advanced:What governs dynamic behavior?
Summary
Exercises
3: Structured Population Models
3.1 Types of population structure: Age versus stage structure
3.1.1 An example of age structure
3.1.2 An example of stage structure
3.1.3 Transient and stable behaviors of structured models
3.1.4 Advanced: Deriving recursive equations
3.1.5 Advanced: Life-table analysis, reproductive outputs, and Euler’s equation
3.2 Modeling using matrix notation
3.2.1 Why do we bother with matrix representation?
3.3 Characteristics of structured models
3.4 Elasticity analysis
3.4.1 Advanced: How to calculate elasticities
3.5 Advanced: Structured density-dependent models
Summary
Exercises
4: Competition and Predation Models
4.1 Introduction
4.1.1 Consider the following two ecological scenarios
4.1.2 What does “stability”mean?
4.2 Solving for equilibria
4.2.1 Continuous-time model
4.3 Isocline analysis
4.3.1 Continuous-time model
4.3.2 Key points
4.3.2.1 Identify equilibrium
4.3.2.2 Predicting stability
4.4 Analytic stability analysis
4.4.1 Motivation
4.4.2 A brief aside on predator-prey models
4.4.3 Background and framework
4.4.3.1 Key points
4.4.4 Calculating stability
4.4.4.1 Calculate the Jacobian matrix
4.4.4.2 How do I know which derivative goes where?
4.4.4.3 Calculate the eigenvalues of the Jacobian matrix
4.4.4.4 Interpreting eigenvalues
4.4.4.5 What about continuous-time models?
4.4.4.6 Key points
4.4.5 Advanced:Why does the eigenvalue predict stability?
4.4.5.1 Using the Taylor approximation
4.4.5.2 Multiple state variables
4.5 Larger models
Summary
Exercises
5: Stochastic Population Models
5.1 Introduction
5.1.1 What causes stochasticity?
5.2 Why consider stochasticity?
5.2.1 Reason 1
5.2.2 Reason 2
5.2.3 So, why aren’t all models stochastic?
5.2.4 Advanced:Why does stochasticity lower population abundance?
5.2.5 Why was the arithmetic mean incorrect?
5.3 Density-independent predictions: An analytic result
5.3.1 Projecting forward with unknown future stochasticity
5.4 Eastern Pacific Southern Resident killer whales
5.5 Estimating extinction risk
5.5.1 Advanced: Autocorrelation
5.6 Uncertainty in model parameters
5.7 Density-dependent stochastic models
5.7.1 Allee effects
5.8 Structured stochastic models
Summary
Exercises
Part II: Fitting Models to Data
6: Why Fit Models to Data?
7: Random Variables and Probability
7.1 Introduction
7.1.1 What is a random variable?
7.2 Binomial
7.2.1 Key things about this distribution
7.2.2 The probability mass function
7.2.3 When would I use this?
7.2.4 Properties of the function
7.2.5 Example
7.3 Poisson
7.3.1 Key things about this distribution
7.3.2 The probability function
7.3.3 When would I use this?
7.3.4 Properties of the function
7.3.5 Example
7.4 Negative binomial
7.4.1 Key things about this function
7.4.2 When would I use this?
7.4.3 The probability function
7.4.4 Properties of the function
7.4.5 Example
7.5 Normal
7.5.1 Key things about this distribution
7.5.2 When would I use this?
7.5.3 The probability density function
7.5.4 Properties of the function
7.5.5 Example
7.6 Log-normal
7.6.1 Key things about this distribution
7.6.2 When would I use this?
7.6.3 The probability density function
7.6.4 Properties of the function
7.6.5 Example
7.7 Advanced: Other distributions
7.7.1 The gamma distribution
7.7.2 The beta distribution
7.7.3 Student’s t-distribution
7.7.4 The beta-binomial distribution
7.7.5 Zero-inflated models
Summary
Exercises
8: Likelihood and Its Applications
8.1 Introduction
8.1.1 Was this a fair coin?
8.1.2 Likelihood to the rescue
8.1.3 Maximum likelihood estimation
8.1.4 What likelihood is not
8.2 Parameter estimation using likelihood
8.3 Uncertainty in maximum likelihood parameter estimates
8.3.1 Calculating confidence intervals using likelihoods
8.3.2 To summarize
8.3.3 Practice example 1
8.4 Likelihood with multiple observations
8.5 Advanced: Nuisance parameters
8.5.1 What is a likelihood profile?
8.5.2 Example
8.5.3 The likelihood profile
8.6 Estimating parameters that do not appear in probability functions
8.6.1 Entanglements of Hector’s dolphins
8.6.2 Practice example 2
8.7 Estimating parameters of dynamic models
8.8 Final comments on maximum likelihood estimation
8.9 Overdispersion and what to do about it
Summary
Exercises
9: Model Selection
9.1 Framework
9.2 An intuitive method: Cross validation
9.3 The Akaike information criterion as a measure of model performance
9.3.1 Take-home points
9.3.2 Advanced: Theoretical underpinnings of information theory
9.3.2.1 What is the distance from truth?
9.3.3 Alternatives to the AIC
9.4 Interpreting AIC values
9.4.1 Nested versus nonnested
9.4.2 Fit to data
9.5 Final thoughts
9.5.1 Model selection practices to avoid
9.5.2 True story
9.5.2.1 What is wrong with this scenario?
Summary
Exercises
10: Bayesian Statistics
10.1 Introduction
10.1.1 Are you a frequentist or a Bayesian?
10.1.2 Wait, what are you talking about?
10.1.3 So, how is this different from likelihood?
10.2 What is Bayes’ theorem, and how is it used in statistics and model selection?
10.2.1 Doesn’t the prior probability influence the posterior probability?
10.3 Practice example: The prosecutor’s fallacy
10.4 The prior
10.4.1 Example: Do people have extrasensory perception?
10.4.2 Criticisms
10.5 Bayesian parameter estimation
10.5.1 How do we do that?
10.5.2 Monte Carlo methods
10.5.2.1 Software to do MCMC
10.5.2.2 Key points
10.5.3 Laplace approximation
10.5.4 Mechanics of using the prior
10.5.4.1 Example:The beta distribution
10.6 Final thoughts on Bayesian approaches
Part III: Skills
11: Mathematics Refresher
11.1 Logarithms
11.1.1 Common operations with logarithms
11.2 Derivatives and integrals
11.3 Matrix operations
11.3.1 Dimensions of matrices
11.3.2 Adding two matrices
11.3.3 Multiplying two matrices
11.3.3.1 The identity matrix
12: Modeling in Spreadsheets
12.1 Practicum: A logistic population model in Excel
12.1.1 Naming spreadsheet cells
12.2 Useful spreadsheet functions
12.2.1 Exercise
12.3 Array formulas
12.3.1 Exercise
12.4 The data table
12.4.1 Exercise
12.5 Programming in Visual Basic
12.5.1 Creating your own functions in Visual Basic
13: Modeling in R
13.1 The basics
13.1.1 First, some orientation
13.1.2 Writing and running R code
13.1.3 Statistical functions
13.1.4 Basic plotting
13.1.5 Data input and output
13.1.6 Looping
13.1.7 Loops within loops
13.2 Practicum: A logistic population model in R
13.3 Creating your own functions
14: Skills for Dynamic Models
14.1 Skills for population models
14.1.1 Implementing structured population models
14.1.1.1 Age structure in spreadsheets
14.1.1.2 Structured population modeling using matrices
14.1.1.3 Age structure in R
14.1.1.4 Structured population modeling using matrices
14.1.1.5 Advanced: Calculating elasticities in R
14.1.2 Cobwebbing
14.2 Skills for multivariable models
14.2.1 Calculating isoclines
14.2.1.1 The X isocline
14.2.1.2 The case of the disappearing state variable
14.2.2 Calculating Jacobian matrices
14.2.2.1 What about discrete-time models?
14.2.2.2 Calculating eigenvalues
14.3 Monte Carlo methods
14.3.1 Monte Carlo example: What is p?
14.3.2 Monte Carlo simulation of population models
14.3.3 Spreadsheet guidance
14.3.4 R guidance
14.4 Skills for stochastic models
14.4.1 Stochastic models in spreadsheets
14.4.1.1 Calculating extinction risk
14.4.2 Stochastic models in R
14.4.2.1 Calculating extinction risk
14.4.3 Advanced: Adding autocorrelation
14.4.3.1 Spreadsheet guidance
14.4.3.2 R guidance
14.4.4 Propagating uncertainty
14.4.4.1 Spreadsheet guidance
14.4.4.2 R guidance
14.5 Numerical solutions to differential equations
14.5.1 The Euler method
14.5.2 The Adams-Bashford method
14.5.3 Runge-Kutta methods
14.5.3.1 The fourth-order Runge-Kutta method
15: Sensitivity Analysis
15.1 Introduction
15.1.1 Types of sensitivity analysis
15.1.2 Steps in sensitivity analysis
15.1.3 Example: Tree snakes
15.2 Individual parameter perturbation
15.2.1 Quantifying sensitivity
15.2.2 Global sensitivity analysis
15.3 The Monte Carlo method
15.4 Structural uncertainty
16: Skills for Fitting Models to Data
16.1 Maximum likelihood estimation
16.1.1 Maximum likelihood estimation: Direct method
16.1.1.1 Excel guidance
16.1.1.2 R guidance
16.1.1.3 More precise confidence interval bounds
16.1.2 Maximum likelihood estimation: Numerical methods
16.1.2.1 Spreadsheet guidance
16.1.2.2 R guidance
16.2 Estimating parameters that do not appear in probability functions
16.3 Likelihood profiles
16.3.1 Profiles in spreadsheets
16.3.2 Profiles in R
Part IV: Putting It All Together and Next Steps
17: Putting It Together: Fitting a Dynamic Model
17.1 Fitting the observation error model
17.1.1 Observation error model in spreadsheets
17.1.2 Observation error model in R
17.1.3 Evaluating fits of the observation error models
17.2 Fitting the process error model
17.2.1 Process error model in spreadsheets
17.2.2 Process error model in R
17.2.3 Evaluating fits of the process error models
17.3 Parameter estimates and model selection
17.4 Can this population exhibit complex population dynamics?
18: Next Steps
18.1 Reality check
18.2 Learn by doing
18.2.1 True story
Bibliography
Index