Roff D.A. Modeling Evolution: An Introduction to Numerical Methods
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Modeling Evolution
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Modeling
Evolution
an introduction to
numerical methods
D. A. Roff
1
3
Great Clarendon Street, Oxford OX26DP
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# D. A. Roff 2010
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First published 2010
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Typeset by SPI Publisher Services, Pondicherry, India
Printed in Great Britain
on acid-free paper by
CPI Antony Rowe Chippenham, Wiltshire
ISBN 978–0–19–957114–7
13579108642
Contents
1 Overview 1
1.1 Introduction 1
1.1.1 The aim of this book 1
1.1.2 Why R and MATLAB? 2
1.2 Operational definitions of fitness 3
1.2.1 Constant environment, density‐independent, stable‐age distribution 5
1.2.2 Demographic stochasticity 5
1.2.3 Environments of fixed length (e.g., deterministic seasonal environments) 7
1.2.4 Constant environment, density‐dependence with a stable equilibrium 7
1.2.5 Constant environment, variable population dynamics 9
1.2.6 Temporally stochastic environments 10
1.2.7 Temporally variable, density‐dependent environments 12
1.2.8 Spatially variable environments 13
1.2.9 Social environment 14
1.2.10 Frequency‐dependence 15
1.3 Some general principles of model building 16
1.4 An introduction to modeling in R and MATLAB 17
1.4.1 General assumptions 17
1.4.2 Mathematical assumptions of model 1 18
1.4.3 Mathematical assumptions of model 2 25
1.4.4 Mathematical assumptions of model 3 40
1.4.5 Mathematical assumptions of model 4 43
1.4.6 Mathematical assumptions of model 5 45
1.4.7 Mathematical assumptions of model 6 51
1.5 Summary of modeling approaches described in this book 55
1.5.1 Fisherian optimality analysis (Chapter 2) 55
1.5.2 Invasibility analysis (Chapter 3) 56
1.5.3 Genetic models (Chapter 4) 56
1.5.4 Game theoretic models (Chapter 5) 57
1.5.5 Dynamic programming (Chapter 6) 57
2 Fisherian optimality models 59
2.1 Introduction 59
2.1.1 Fitness measures 59
2.1.2 Methods of analysis: introduction 61
2.1.3 Methods of analysis: W ¼ f ðy
1
; y
2
; …; y
k
; x
1
; x
2
; …; x
n
Þ and well‐behaved 62
2.1.4 Methods of analysis: W ¼ f ðy
1
; y
2
; …; y
k
; x
1
; x
2
; …; x
n
Þ and not well‐behaved 65
2.1.5 Methods of analysis: gðW Þ¼f ðy
1
; y
2
; …; y
k
; x
1
; x
2
; …; x
n
; W Þ 67
2.2 Summary of scenarios (Table 2.1) 69
2.3 Scenario 1: A simple trad e‐off model 71
2.3.1 General assumptions 71
2.3.2 Mathematical assumptions 72
2.3.3 Plotting the fitness function 72
2.3.4 Finding the maximum using the calculus 73
2.3.5 Finding the maximum using a numerical approach 75
2.4 Scenario 2: Adding age structure may not affect the optimum 75
2.4.1 General assumptions 75
2.4.2 Mathematical assumptions 75
2.5 Scenario 3: Adding age‐specific mortality that affects the optimum 76
2.5.1 General assumptions 76
2.5.2 Mathematical assumptions 76
2.5.3 Plotting the fitness function 77
2.5.4 Finding the maximum using the calculus 79
2.5.5 Finding the maximum using a numerical approach 81
2.6 Scenario 4: Adding age‐specific mortality that affects the optimum and using
integration rather than summation
81
2.6.1 General assumptions 81
2.6.2 Mathematical assumptions 82
2.6.3 Plotting the fitness function 82
2.6.4 Finding the maximum using the calculus 84
2.6.5 Finding the maximum using a numerical approach 85
2.7 Scenario 5: Maximizing the Malthusian parameter,
r
, rather than expected
lifetime reproductive success, R
0
86
2.7.1 General assumptions 87
2.7.2 Mathematical assumptions 87
2.7.3 Plotting the fitness function 88
2.7.4 Finding the maximum using the calculus 89
2.7.5 Finding the maximum using a numerical approach 92
2.8 Scenario 6: Stochastic variation in parameters 93
2.8.1 General assumptions 94
2.8.2 Mathematical assumptions 94
2.8.3 Plotting the fitness function 95
2.8.4 Finding the maximum using the calculus 97
2.8.5 Finding the maximum using a numerical approach 99
2.9 Scenario 7: Discrete temporal variation in parameters 100
2.9.1 General assumptions 100
2.9.2 Mathematical assumptions 100
2.9.3 Plotting the fitness function 101
2.9.4 Finding the maximum using the calculus 102
2.9.5 Finding the maximum using numerical methods 104
vi CONTENTS
2.10 Scenario 8: Continuous temporal variation in parameters 105
2.10.1 General assumptions 105
2.10.2 Mathematical assumptions 105
2.10.3 Plotting the fitness function 106
2.10.4 Finding the maximum using a numerical approach 107
2.11 Scenario 9: Maximizing two traits simultaneously 108
2.11.1 General assumptions 108
2.11.2 Mathematical assumptions 109
2.11.3 Plotting the fitness function 110
2.11.4 Finding the maximum using the calculus 112
2.11.5 Finding the maximum using a numerical approach 112
2.12 Scenario 10: Two traits may covar y but optima are independent 113
2.12.1 General assumptions 113
2.12.2 Mathematical assumptions 113
2.13 Scenario 11: Two traits may be resolved into a single trait 114
2.13.1 General assumptions 115
2.13.2 Mathematical assumptions 115
2.13.3 Plotting the fitness function 116
2.13.4 Finding the optimum using the calculus 117
2.13.5 Finding the optimum using a numerical approach 119
2.14 Scenario 12: The importance of plotting and the utility of brute force 119
2.14.1 General assumptions 119
2.14.2 Mathematical assumptions 120
2.14.3 Plotting the fitness function 120
2.14.4 Finding the maximum using the calculus 123
2.14.5 Finding the maximum using a numerical approach 128
2.15 Scenario 13: Dealing with recursion by brute force 130
2.15.1 General assumptions 130
2.15.2 Mathematical assumptions 131
2.15.3 Plotting the fitness function 132
2.15.4 Finding the maximum using the calculus 134
2.15.5 Finding the maximum using a numerical approach 134
2.16 Scenario 14: Adding a third variable and more 135
2.16.1 General assumptions 136
2.16.2 Mathematical assumptions 136
2.16.3 Plotting the fitness function 137
2.16.4 Finding the maximum using the calculus 137
2.16.5 Finding the maximum using a numerical approach 137
2.17 Some exemplary papers 139
2.18 MATLAB code 140
2.18.1 Scenario 1: Plotting the fitness function 140
2.18.2 Scenario 1: Finding the maximum using the calculus 140
2.18.3 Scenario 1: Finding the maximum using a numerical approach 141
2.18.4 Scenario 3: Plotting the fitness function 141
2.18.5 Scenario 3: Finding the maximum by the calculus 142
CONTENTS vii
2.18.6 Scenario 3: Finding the maximum using a numerical approach 142
2.18.7 Scenario 4: Plotting the fitness function 142
2.18.8 Scenario 4: Finding the maximum using the calculus 143
2.18.9 Scenario 4: Finding the maximum using a numerical approach 144
2.18.10 Scenario 5: Plotting the fitness function 144
2.18.11 Scenario 5: Finding the maximum using the calculus 145
2.18.12 Scenario 5: Finding the maximum using a numerical approach 145
2.18.13 Scenario 6: Plotting the fitness function 146
2.18.14 Scenario 6: Finding the maximum using the calculus 147
2.18.15 Scenario 6: Finding the maximum using a numerical approach 147
2.18.16 Scenario 7: Plotting the fitness function 148
2.18.17 Scenario 7: Finding the maximum using the calculus 149
2.18.18 Scenario 7: Finding the maximum using numerical methods 150
2.18.19 Scenario 8: Plotting the fitness function 150
2.18.20 Scenario 8: Finding the maximum using a numerical approach 151
2.18.21 Scenario 9: The derivative can also be determined using MATLAB 151
2.18.22 Scenario 9: Plotting the fitness function 151
2.18.23 Scenario 9: Finding the maximum using the calculus 152
2.18.24 Scenario 9: Finding the maximum using a numerical approach 152
2.18.25 Scenario 11: Plotting the fitness function 153
2.18.26 Scenario 11: Finding the optimum using the calculus 153
2.18.27 Scenario 11: Finding the optimum using a numerical approach 154
2.18.28 Scenario 12: Plotting the fitness function 154
2.18.29 Scenario 12: Finding the maximum using the calculus 155
2.18.30 Scenario 12: Finding the maximum using a numerical approach 158
2.18.31 Scenario 13: Plotting the fitness function 160
2.18.32 Scenario 13: Finding the maximum using a numerical approach 162
2.18.33 Scenario 14: Finding the maximum using a numerical approach 163
3 Invasibility analysis 165
3.1 Introduction 165
3.1.1 Age‐ or stage‐structured models 165
3.1.2 Modeling evolution using the Leslie matrix 169
3.1.3 Stage‐structured models 170
3.1.4 Adding density‐dependence 170
3.1.5 Estimating fitness 173
3.1.6 Pairwise invasibility analysis 174
3.1.7 Elasticity analysis 180
3.1.8 Multiple invasibility analysis 181
3.2 Summary of scenarios 184
3.3 Scenario 1: Comparing approaches 184
3.3.1 General assumptions 184
3.3.2 Mathematical assumptions 184
3.3.3 Solving using the methods of Chapter 2 185
3.3.4 Solving using the eigenvalue of the Leslie matrix 186
viii CONTENTS
3.4 Scenario 2: Adding density‐dependence 188
3.4.1 General assumptions 188
3.4.2 Mathematical assumptions 189
3.4.3 Solving using
R
0
as the fitness measure 189
3.4.4 Pairwise invasibility analysis 189
3.4.5 Elasticity analysis 193
3.5 Scenario 3: Functional dependence in the Ricker model 194
3.5.1 General assumptions 195
3.5.2 Mathematical assumptions 195
3.5.3 Pairwise invasibility analysis 195
3.5.4 Elasticity analysis 198
3.5.5 Multiple invasibility analysis 201
3.6 Scenario 4: The evolution of reproductive effort 203
3.6.1 General assumptions 203
3.6.2 Mathematical assumptions 203
3.6.3 Pairwise invasibility analysis 204
3.6.4 Elasticity analysis 206
3.7 Scenario 5: A two stage model 208
3.7.1 General assumptions 208
3.7.2 Mathematical assumptions 208
3.7.3 Elasticity analysis 210
3.7.4 Pairwise invasibility analysis 211
3.8 Scenario 6: A case in which the putative ESS is not stable 213
3.8.1 General assumptions 213
3.8.2 Mathematical assumptions 213
3.8.3 Pairwise invasibility analysis 213
3.8.4 Elasticity analysis 215
3.8.5 Multiple invasibility analysis 219
3.9 Some exemplary papers 221
4 Genetic models 223
4.1 Introduction 223
4.1.1 Population variance components (PVC) models 223
4.1.2 Individual variance components (IVC) models 228
4.1.3 Individual locus (IL) models 233
4.2 Summary of scenarios 243
4.3 Scenario 1: Stabilizing selection on two traits using a PVC model 243
4.3.1 General assumptions 244
4.3.2 Mathematical assumptions 244
4.3.3 Analysis 244
4.4 Scenario 2: Stabilizing selection using an IVC model 245
4.4.1 General assumptions 246
4.4.2 Mathematical assumptions 246
4.4.3 Analysis 246
CONTENTS ix