Modeling Population Dynamics - Home - Universiteit van Amsterdam

222 Pages·2014·5.21 MB·English

Modeling Population Dynamics
Andr´e M. de Roos

Modeling Population Dynamics
Andr´e M. de Roos
Institute for Biodiversity and Ecosystem Dynamics
University of Amsterdam
Science Park 904, 1098 XH Amsterdam, The Netherlands
E-mail: [email protected]
December 4, 2019

Contents
I Preface and Introduction 1
1 Introduction 3
1.1 Some modeling philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
II Unstructured Population Models in Continuous Time 5
2 Modelling population dynamics 7
2.1 Describing a population and its environment . . . . . . . . . . . . . . . . . . 7
2.1.1The population or p-state . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2The individual or i-state . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.3The environmental or E-condition . . . . . . . . . . . . . . . . . . . . . 9
2.2 Population balance equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Characterizing the population . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Population-level and per capita rates . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Model building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5.1Exponential population growth . . . . . . . . . . . . . . . . . . . . . . . 15
2.5.2Logistic population growth. . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.3Two-sexes population growth . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Parameters and state variables . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.7 Deterministic and stochastic models . . . . . . . . . . . . . . . . . . . . . . . 19
3 Single ordinary differential equations 21
3.1 Explicit solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Analyzing flow patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Steady states and their stability . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5 Units and non-dimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.6 Existence and uniqueness of solutions . . . . . . . . . . . . . . . . . . . . . . 35
3.7 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Competing for resources 37
4.1 Intraspecific competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
i
ii CONTENTS
4.1.1Growth of yeast in a closed container . . . . . . . . . . . . . . . . . . . . 38
4.1.2Bacterial growth in a chemostat . . . . . . . . . . . . . . . . . . . . . . . 40
4.1.3Asymptotic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.4Phase-plane methods and graphical analysis . . . . . . . . . . . . . . . . 44
4.2 Interspecific competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.1Lotka-Volterra competition model . . . . . . . . . . . . . . . . . . . . . . 52
4.2.2Competition for resources . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5 Systems of ordinary differential equations 77
5.1 Computation of steady states . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Linearization of dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Characteristic equation, eigenvalues and eigenvectors . . . . . . . . . . . . . . 81
5.4 Phase portraits of dynamics in planar systems . . . . . . . . . . . . . . . . . . 84
5.4.1Two real eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4.2Two complex eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.5 Stability of steady states in planar ODE systems . . . . . . . . . . . . . . . . 94
5.6 Models with 3 or more variables. . . . . . . . . . . . . . . . . . . . . . . . . . 96
6 Predator-prey interactions 99
6.1 The Lotka-Volterra predator-prey model . . . . . . . . . . . . . . . . . . . . . 100
6.1.1Incorporating prey logistic growth . . . . . . . . . . . . . . . . . . . . . 104
6.1.2Incorporating a predator type II functional response . . . . . . . . . . . 108
6.2 Confronting models and experiments . . . . . . . . . . . . . . . . . . . . . . . 117
6.2.1Predator-controlled, steady state abundance of prey . . . . . . . . . . . 119
6.2.2Oscillatory dynamics at high prey carrying capacities . . . . . . . . . . . 123
6.2.3Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
III Bifurcation theory 129
7 Continuous time models 131
7.1 General setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.2 Transcritical bifurcation and branching point . . . . . . . . . . . . . . . . . . 140
7.3 Saddle-node bifurcation and limit point . . . . . . . . . . . . . . . . . . . . . 141
7.4 Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
IV Exercises 145
8 Exercises 147
8.1 ODEs (differential equations) in 1 dimension . . . . . . . . . . . . . . . . . . 147
8.1.1Population size and population growth rate . . . . . . . . . . . . . . . . 147
CONTENTS iii
8.1.2Graphical analysis of differential equations . . . . . . . . . . . . . . . . . 147
8.1.3Formulation and graphical analysis of a model . . . . . . . . . . . . . . . 149
8.2 Equilibria in 1D ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.2.1Mathematical analysis of a differential equation . . . . . . . . . . . . . . 151
8.3 Studying ODEs in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . 153
8.3.1Graphical analysis of a system of differential equations . . . . . . . . . . 153
8.3.2Mathemetical analysis of a system of differential equations . . . . . . . . 154
8.4 Sample Exam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.5 Answers to exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
V Computer Labs 173
9 Computer Labs 175
9.1 Harvesting Cod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
9.2 The spruce-budworm(Choristoneura fumiferana) . . . . . . . . . . . . . . . . 179
9.3 Interspecific competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
9.4 Vegetation catastrophes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
9.5 Lotka-Volterra Predation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
9.6 The Paradox of Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
9.7 The Return of the Paradox of Enrichment . . . . . . . . . . . . . . . . . . . . 186
9.8 Cannibalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
9.9 A predator-prey model with density-dependent prey development . . . . . . . 189
9.10 Chaotic dynamics of Hare and Lynx populations . . . . . . . . . . . . . . . . 190
Bibliography 193
iv CONTENTS
List of Figures
2.1 Growth of the world population over the last century . . . . . . . . . . . . . . 8
2.2 Population age distribution in the Netherlands in 1950, 1975 and 2000 . . . . 9
2.3 Total number of births and death per 5 year age class in the Netherlands in 1999 10
4.1 Growth of the yeast Schizosaccharomyces kephir over a period of 160 h . . . . 38
4.2 Schematic layout of a chemostat for continuous culture of micro-organisms . . 41
4.3 Geometric representation of a system of ODEs in the phase plane . . . . . . . 46
4.4 The phase plane of the bacterial growth model in a chemostat . . . . . . . . . 50
4.5 Operating diagram of a chemostat . . . . . . . . . . . . . . . . . . . . . . . . 52
4.6 Four isocline cases for the Lotka-Volterra competition model. . . . . . . . . . 54
4.7 Four isocline cases with steady states and flow patterns for the Lotka-Volterra
competition model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.8 Michaelis-Menten relationship as a function of resource concentration . . . . . 59
4.9 Zero net growth isoclines for a one consumer-two resource model . . . . . . . 62
4.10 Steady state location in a one consumer-two resource model . . . . . . . . . . 63
4.11 Steady state location in a two consumer-two resource model . . . . . . . . . . 66
4.12 Predicted and observed outcomes of competition for phosphate and silicate by
Asterionella formosa and Cyclotella meneghiniana . . . . . . . . . . . . . . . 70
4.13 Single-species Monod growth experiments for four diatoms under conditions of
limited silicate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.14 Single-species Monod growth experiments for four diatoms under conditions of
limited phosphate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.15 Predicted and observed outcomes of competition between Asterionella formosa
and Synedra filiformis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.16 Predicted and observed outcomes of competition between Asterionella formosa
and Tabellaria flocculosa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.17 Predicted and observed outcomes of competition between Asterionella formosa
and Fragilaria crotonensis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.18 Predicted and observed outcomes of competition between Synedra andFragilaria 74
4.19 PredictedandobservedoutcomesofcompetitionbetweenSynedra andTabellaria 75
4.20 Predicted and observed outcomes of competition between Fragilaria and Tabel-
laria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.21 The four cases of resource competition in a two consumer-two resource model 76
v
vi LIST OF FIGURES
5.1 Geometric representation of two real-valued eigenvectors in a planar system . 87
5.2 Characteristic flow patterns in the neighborhood of the steady state with real-
valued eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3 Geometricrepresentationofthetwobasisvectorsinaplanarsystemwithcomplex
eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4 Characteristicflowpatternsintheneighborhoodofthesteadystatewithcomplex
eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.5 Summary of stability properties for planar ODE systems . . . . . . . . . . . . 95
6.1 Solution curves in the phase plane of the Lotka-Volterra predator-prey model 102
6.2 Prey dynamics predicted by the Lotka-Volterra predator-prey model . . . . . 103
6.3 SolutioncurvesinthephaseplaneoftheLotka-Volterrapredator-preymodelwith
logistic prey growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.4 Functional response of Daphnia pulex on three algal species . . . . . . . . . . 109
6.5 Nullclines of the Rosenzweig-MacArthur, predator-prey model . . . . . . . . . 112
6.6 Solution curves in the phase plane of the Rosenzweig-MacArthur predator-prey
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.7 Solution curve in the phase plane of the Rosenzweig-MacArthur predator-prey
model for high carrying capacity . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.8 Oscillatory dynamics of the Rosenzweig-MacArthur predator-prey model for high
carrying capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.9 Daphnia pulex carrying asexual eggs in the brood pouch . . . . . . . . . . . . 119
6.10 Schematic setup of the experiments by Arditi et al. (1991) . . . . . . . . . . . 120
6.11 Population dynamics of Daphnia, Ceriodaphnia and Scapholeberis in a chain of
semi-chemostats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.12 PopulationequilibriaofDaphnia andCeriodaphnia inachainofsemi-chemostats 123
6.13 Predicted and observed equilibrium values of algae in the presence of Daphnia 124
6.14 Population dynamics of the snowshoe hare and the lynx in northern Canada . 125
6.15 Population dynamics of two species of voles in northern Finland . . . . . . . 125
6.16 Cycle amplitudes (log) observed for Daphnia and algal populations from lakes
and ponds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.17 Examples of the dynamics of Daphnia and algae in nutrient-rich and nutrient-
poor, experimental tanks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.18 Large- and small-amplitude cycles of Daphnia and edible algae in the same global
environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.19 Energy channelling towards sexual reproduction prevents the occurrence of large-
amplitude predator-prey cycles in Daphnia . . . . . . . . . . . . . . . . . . . 128
7.1 Eigenvalue positions in the complex plane corresponding to a stability change in
a 2 ODE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.2 Bifurcation structure of the cannibalism model for α < ρ . . . . . . . . . . . . 138
7.3 Bifurcation structure of the cannibalism model for α > ρ . . . . . . . . . . . . 139
7.4 Schematic,3-dimensionalrepresentationoftheHopf-bifurcationintheRosenzweig-
MacArthur model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142