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

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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.


Author(s): Hiromi Seno
Series: Theoretical Biology
Publisher: Springer
Year: 2022

Language: English
Pages: 469
City: Singapore

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