A Primer on Population Dynamics Modeling: Basic Ideas for Mathematical Formulation

Description

This textbook provides an introduction to the mathematical models of population dynamics in mathematical biology. The focus of this book is on the biological meaning/translation of mathematical structures in mathematical models, rather than simply explaining mathematical details and literacies to analyze a model. In some recent usages of the mathematical model simply with computer numerical calculations, the model includes some inappropriate mathematical structure concerning the reasonability of modeling for the biological problem under investigation. For students and researchers who study or use mathematical models, it is important and helpful to understand what mathematical setup could be regarded as reasonable for the model with respect to the relation between the biological factors involved in the assumptions and the mathematical structure of the model. Topics covered in this book are; modeling with geometric progression, density effect in population dynamics, deriving continuous time models from discrete time models, basic modeling for birth-death stochastic processes, continuous time models, modeling interspecific reaction for the continuous time population dynamics model, competition and prey-predator dynamics, modeling for population dynamics with a heterogeneous structure of population, qualitative analysis on the discrete time dynamical system, necessary knowledge about fundamental mathematical theories to understand the dynamical nature of continuous time models. The book includes popular topics in ecology and mathematical biology, as well as classic theoretical topics. By understanding the biological meaning of modeling for simple models, readers will be able to derive a specific mathematical model for a biological problem by reasonable modeling. The contents of this book is made accessible for readers without strong Mathematical background. Preface Contents Part I Modeling Biological Factors 1 Application of Geometric Progression 1.1 Geometric Growth Model 1.2 Immature Period 1.3 Fibonacci Sequence 1.4 Life Span 1.5 Survival Probability 1.6 Sexual Difference Answer to Exercise Exercise 1.1 (p. 5) Exercise 1.2 (p. 18) References 2 Influence From Surrounding 2.1 Negative Density Effect 2.1.1 Beverton-Holt Type Model 2.1.2 Ricker Model 2.1.3 Logistic Map Model 2.2 Positive Density Effect 2.2.1 Allee Effect 2.2.2 Scenarios of Extinction with Allee Effect 2.2.3 Weak Allee Effect 2.2.4 Reproduction Curve 2.3 Competition 2.3.1 Leslie-Gower Model 2.3.2 Ricker Type of Competition Model 2.4 Enemy 2.4.1 Dynamics of Enemy Population 2.4.2 Nicholson-Bailey Model 2.4.3 Synergy with the Density Effect for Prey 2.5 Harvesting/Culling 2.5.1 Beverton-Holt Type Model with Harvesting/Culling 2.5.2 Maximum Sustainable Yield 2.5.3 Cost for the Harvesting/Culling 2.5.4 Ricker Type Model with Harvesting/Culling 2.6 Semi-Spatial Modeling 2.6.1 Royama's Idea of Modeling 2.6.2 Skellam Model 2.6.3 Site-Based Model Answer to Exercise Exercise 2.1 (p.34) Exercise 2.2 (p.43) References 3 From Discrete Time Model to Continuous Time Model 3.1 Geometric Growth to Exponential Growth 3.2 Beverton-Holt Model to Logistic Growth 3.3 Time-Step-Zero Limit 3.3.1 Geometric Growth Model to Malthus Model 3.3.2 Beverton-Holt Model to Logistic Equation 3.3.3 Logistic Map Model to Logistic Equation 3.3.4 Skellam Model to Logistic Equation 3.4 Momental Velocity of Population Size Change Answer to Exercise Exercise 3.1 (p.105) References 4 Continuous Time Modeling for Birth and Death Processes 4.1 Yule-Furry Process 4.1.1 Probability Distribution for Population Size 4.1.2 Expected Population Size 4.2 Malthus Growth with Death 4.3 Death Process 4.3.1 Survival Probability 4.3.2 Expected Life Span 4.3.3 Probability Distribution for Cohort Size 4.3.4 Expected Population Size 4.3.5 Average Life Span for Extinct Population 4.3.6 Expected Extinction Time 4.4 Net Reproduction Rate 4.5 Logistic Equation Answer to Exercise Exercise 4.1 (p. 112) Exercise 4.2 (p.115) Exercise 4.3 (p.117) Exercise 4.4 (p.119) Exercise 4.5 (p. 121) Exercise 4.6 (p.124) Exercise 4.7 (p.125) References 5 Continuous Time Modeling for Single Species Population Dynamics 5.1 Malthus Model 5.2 Gompertz Curve 5.3 Logistic Equation 5.4 Verhulst Model 5.5 Logistic Equation to Logistic Map 5.6 Allee Effect 5.7 Metapopulation Model 5.7.1 Levins Model 5.7.2 3-State Metapopulation Model Answer to Exercise Exercise 5.1(p.142) Exercise 5.2 (p.148) Exercise 5.3 (p.149) Exercise 5.4 (p.155) Exercise 5.5 (p.158) References 6 Modeling of Interspecific Reaction 6.1 Mass Action Type of Interaction 6.1.1 Mass Action Assumption 6.1.2 Lokta-Volterra Type of Interaction 6.1.3 Logistic Equation by Lotka-Volterra Type of Interaction 6.1.4 Intraspecific Reaction and Density Effect 6.1.5 Consumer Population of Exhaustible Resource 6.2 Michaelis-Menten Type of Interaction 6.2.1 Michaelis-Menten Reaction Velocity Equation 6.2.2 Reaction Velocity Equation with Inhibitor 6.2.3 Application for Population Dynamics Answer to Exercise Exercise 6.1 (p.178) Exercise 6.2 (p.182) References 7 Modeling for Competitive Relation 7.1 Lotka-Volterra Competition Model 7.1.1 Influence of Competition 7.1.2 Competing Two Species System 7.2 Competition for Resource 7.2.1 MacArthur's Modeling 7.2.2 Tilman's Modeling Answer to Exercise Exercise 7.1 (p.195) Exercise 7.2 (p.195) References 8 Modeling for Prey-Predator Relation 8.1 Predator's Response 8.2 Prey-Predator Population Dynamics 8.3 Dynamics of Exhaustible Prey 8.4 Lotka-Volterra Prey-Predator Model 8.4.1 Trajectory in Phase Plane 8.4.2 Equilibrium and Averaged Population Size 8.4.3 Structural Stability 8.4.4 Predator vs Prey with Logistic Growth 8.5 Holling's Disc Equation 8.5.1 Disc Equation for a Single Prey Species Introduction of Handling Time 8.5.2 Disc Equation for Multiple Species 8.6 Rosenzweig-MacArthur Model 8.7 Regulation of Prey Use 8.7.1 Diet Selection: Which Should Be Used? Diet Selection for Two Prey Species 8.7.2 Switching Predation: How Much Should Be Used? Ideal Switching Response Biased Switching Response Answer to Exercise Exercise 8.1 (p.219) Exercise 8.2 (p.221) Exercise 8.3 (p.221) Exercise 8.4 (p.222) Exercise 8.5 (p.225) Exercise 8.6 (p.245) References 9 Modeling with Class Structure 9.1 Structured Population 9.2 Spread of Transmissible Disease 9.2.1 Generic Discrete Time Model 9.2.2 Invasion Success of Transmissible Disease 9.2.3 Reproduction Number of Infectives 9.2.4 SIR, SIS, and SIRS Models SIR Model Case of Uncertain Immunization SIS Model SIRS Model 9.3 Kermack-McKendrick Model 9.3.1 Infection Force 9.3.2 Invasion Success of Transmissible Disease 9.3.3 Final Epidemic Size 9.3.4 Reproduction Number of Infectives 9.3.5 Extension to SIS and SIRS Models SIS Model Case of Uncertain Immunization SIRS Model 9.3.6 Modeling with the Other Factors for Epidemic Dynamics Latent Period Vector-Borne Disease Long-Term Epidemic Dynamics Answer to Exercise Exercise 9.1 (p.294) Exercise 9.2 (p.316) References 10 Modeling for Age Structure 10.1 Discrete Time Model 10.1.1 Leslie Matrix Model 10.1.2 Lefkovitch Matrix Model 10.1.3 Stable Age Distribution 10.1.4 Reproductive Value 10.1.5 Sensitivity Analysis 10.2 Continuous Time Model 10.2.1 Age Distribution Function 10.2.2 von Foerster Equation 10.2.3 Population Renewal Process 10.2.4 Density Distribution Function on Characteristic Curve Mathematial Solution Along Characteristic Curve Two Kinds of Cohort 10.2.5 Renewal Equation 10.2.6 Stationary Age Distribution Case of Constant Death and Growth Rates Density Effect on Death and Growth Rates 10.3 Age Distribution from Death Process 10.4 Leslie Matrix and von Foerster Equation 10.4.1 From von Foerster Equation to Leslie Matrix Model 10.4.2 From Leslie Matrix Model to von Foerster Equation Answer to Exercise Exercise 10.1 (p.330) Exercise 10.2 (p.332) Exercise 10.3 (p.336) Exercise 10.4 (p.338) Exercise 10.5 (p.346) Exercise 10.6 (p.349) Exercise 10.7 (p.350) Exercise 10.8 (p.354) References Part II Mathematical Equipments 11 Homogeneous Linear Difference Equation 11.1 Second Order Linear Equation 11.2 Two Dimensional System of First Order Linear Equations 11.2.1 Simultaneous First Order Equations 11.2.2 Case of Distinct Real Eigenvalues 11.2.3 Case of Multiple Eigenvalues 11.2.4 Case of Imaginary Eigenvalues 11.2.5 Asymptotic Behavior of the Sequence References 12 Qualitative Analysis for Discrete Time Model 12.1 One Dimensional Discrete Time Model 12.1.1 Local Stability of Equilibrium 12.1.2 Cobwebbing Method 12.1.3 Logistic Map 12.1.4 Periodic Orbit 12.1.5 Period-Doubling Bifurcation 12.1.6 Tent Map Existence of Equilibrium Stability of Equilibrium Period-2 Solution Period-k Solution Chaotic Variation 12.2 Two Dimensional Discrete Time Model 12.2.1 Linearization Around Equilibrium 12.2.2 Classification of Equiliblium 12.2.3 Jury Stability Test Answer to Exercise Exercise 12.1 (p.389) Exercise 12.2 (p.391) References 13 First Order Linear Ordinary Differential Equation 13.1 One Dimensional First Order Linear Equation 13.1.1 First Order Ordinary Differential Equation 13.1.2 Separation of Variables 13.1.3 Linear Ordinary Differential Equation 13.1.4 Bernoulli Equation 13.2 Two Dimensional System of First Order Linear Equations 13.2.1 Simultaneous First Order Equations 13.2.2 Case of Distinct Real Eigenvalues 13.2.3 Case of Multiple Eigenvalues 13.2.4 Case of Imaginary Eigenvalues 13.2.5 Asymptotic Behavior of the Solution References 14 Qualitative Analysis for Continuous Time Model 14.1 Local Stability Analysis for One Dimensional Model 14.2 Linearization of Two Dimensional System around Equilibrium 14.3 Classification of Equiliblium 14.4 Lotka-Volterra Two Species Competition Model 14.5 Rosenzweig-Macarthur Model 14.6 Routh-Hurwitz Criterion 14.7 Isocline Method 14.8 Lyapunov Function 14.9 Poincaré-Bendixson Theorem Answer to Exercise Exercise 14.1 (p.424) Exercise 14.2 (p.439) Exercise 14.3 (p.442) References 15 Essentials of Poisson Process/Distribution 15.1 Poisson Process 15.2 Poisson Distribution 15.3 Interarrival Time References Index