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Modeling Population Dynamics

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Modeling Population Dynamics Modeling Population Dynamics Andr´eM. de Roos Modeling Population Dynamics Andr´eM. 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 2.1.1 The population or p-state . .7 2.1.2 The individual or i-state . .8 2.1.3 The 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 2.5.1 Exponential population growth . 15 2.5.2 Logistic population growth . 16 2.5.3 Two-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 4.1.1 Growth of yeast in a closed container . 38 4.1.2 Bacterial growth in a chemostat . 40 4.1.3 Asymptotic dynamics . 43 4.1.4 Phase-plane methods and graphical analysis . 44 4.2 Interspecific competition . 52 4.2.1 Lotka-Volterra competition model . 52 4.2.2 Competition 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 5.4.1 Two real eigenvalues . 85 5.4.2 Two 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 6.1.1 Incorporating prey logistic growth . 104 6.1.2 Incorporating a predator type II functional response . 108 6.2 Confronting models and experiments . 117 6.2.1 Predator-controlled, steady state abundance of prey . 119 6.2.2 Oscillatory dynamics at high prey carrying capacities . 123 6.2.3 Concluding 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 8.1.1 Population size and population growth rate . 147 CONTENTS iii 8.1.2 Graphical analysis of differential equations . 147 8.1.3 Formulation and graphical analysis of a model . 149 8.2 Equilibria in 1D ODEs . 151 8.2.1 Mathematical analysis of a differential equation . 151 8.3 Studying ODEs in two dimensions . 153 8.3.1 Graphical analysis of a system of differential equations . 153 8.3.2 Mathemetical 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 and Fragilaria 74 4.19 Predicted and observed outcomes of competition between Synedra and Tabellaria 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 Geometric representation of the two basis vectors in a planar system with complex eigenvectors . 91 5.4 Characteristic flow patterns in the neighborhood of the steady state with complex 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 Solution curves in the phase plane of the Lotka-Volterra predator-prey model with 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 Population equilibria of Daphnia and Ceriodaphnia in a chain of semi-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-dimensional representation of the Hopf-bifurcation in the Rosenzweig- MacArthur model . 142 List of Tables 4.1 State variables and parameters of a bacterial growth model in a chemostat . 42 4.2 State variables and parameters of Tilman's resource competition model ..
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