Random Models in Biology, Ecology and Evolution.

Format:

Book

Author/Creator:

Méléard, Sylvie.

Series:

Texts in Applied Mathematics Series

Texts in Applied Mathematics Series ; v.87

Language:

English

Physical Description:

1 online resource (292 pages)

Edition:

1st ed.

Place of Publication:

Berlin, Heidelberg : Springer, 2026.

Summary:

This book defines and develops probabilistic tools for modeling populations, describing the dynamics of biological quantities such as population size, allele proportions, and individual locations.

Contents:

Intro

Preface

Contents

Chapter 1 Introduction

1.1 Introduction to the Course

1.2 The Importance of Modeling

1.3 Mathematical Modeling in Ecology and Evolution

1.3.1 Ecology and Evolution

1.3.2 Dispersion and Colonization

1.3.3 Population Dynamics

1.3.4 Population Genetics

Chapter 2 Spatial Populations and Discrete Time

2.1 Random Walks and Markov Chains

2.2 The Study of Passage Times

2.2.1 Stopping Time and the Strong Markov Property

2.2.2 Law of the First Return Time to 0 of a Simple RandomWalk

2.3 Recurrence and Transience - Ergodic Theorems

2.4 Absorbed or Reflected Random Walks

2.4.1 Absorbing Barriers

2.4.2 Reflecting Barriers

2.5 Discrete Time Martingales

2.6 Exercises

Chapter 3 Population Dynamics in Discrete Time

3.1 Birth and Death Markov Processes

3.2 The Galton-Watson or Bienaymé-Galton-Watson Process

3.2.1 Definition

3.2.2 Elementary Results

3.2.3 Behavior at Infinity

3.2.4 Subcritical Case: Detailed Analysis of Extinction

3.3 Relationship Between the Bienaymé-Galton-Watson Process and the Genealogy Model

3.4 Quasi-Stationary Behavior

3.4.1 Quasi-Stationary Distributions and the Yaglom Limit

3.4.2 Quasi-Stationary Distributions for a BGWChain

3.5 Extension 1: Density-Dependent Chains

3.6 Extension 2: BGW Process With Immigration

3.7 Extension 3: The Multi-Type GW process

3.8 Exercises

Chapter 4 Brownian Motion and Diffusion Processes

4.1 Finite-Dimensional Convergence of Renormalized Random Walks

4.2 Random Processes and Brownian Motion

4.3 Some Properties of Brownian Motion

4.4 The Markov Property and Brownian Motion

4.5 Continuous Time Martingales and Stopping Times

4.5.1 Continuous Time Martingales

4.5.2 Fundamental Inequalities and Behavior at Infinity

4.5.3 Stopping Theorems.

4.5.4 Applications to Brownian Motion

4.6 Stochastic Integrals and Stochastic Differential Equations

4.6.1 Stochastic Integrals

4.6.2 Stochastic Differential Equations (SDEs)

4.6.3 Stochastic Differential Systems

4.6.4 The Markov Property for a Solution to an SDE

4.6.5 Itô's Formula

4.6.6 Generators - A Link With Partial Differential Equations

4.6.7 Applications to Hitting Times of Barriers

4.7 Stochastic Differential Equations for Population Studies

4.7.1 Feller's Diffusion Equation

4.7.2 The Logistic Feller Equation

4.7.3 Ornstein-Uhlenbeck Processes

4.7.4 Other Examples of Spatial Movements

4.7.5 TheWright-Fisher Diffusion Process

4.8 Exercises

Chapter 5 Continuous Time Population Processes

5.1 Markov Pure Jump Processes

5.2 A Prototype: The Poisson Process

5.2.1 Definition of a Poisson Process

5.2.2 The Strong Markov Property

5.2.3 Asymptotic Behavior of a Poisson Process

5.2.4 Compound Poisson Processes

5.3 The Generator of a Pure Jump Markov Process

5.3.1 The Infinitesimal Generator

5.3.2 Embedded Markov Chains

5.4 Continuous Time Branching Processes

5.4.1 Definition and Branching Property

5.4.2 Equation for the Generating Function

5.4.3 Non-Explosion Criterion

5.4.4 The Moment Equation - Probability and Extinction Time

5.4.5 The Binary Case

5.4.6 Extensions

5.5 Birth and Death Processes

5.5.1 Definition and Non-Explosion Criterion

5.5.2 Kolmogorov Equations and Invariant Measure

5.5.3 Extinction Criterion - Extinction Time

5.5.4 Trajectorial Representation of Birth and Death Processes

5.6 Continuous Approximations: Deterministic and Stochastic Models

5.6.1 Deterministic Approximations - Malthusian and Logistic Equations

5.6.2 Stochastic Approximation - Demographic Stochasticity, Feller's Equation.

5.6.3 The Lotka-Volterra and Predator-Prey Models

5.7 Exercises

Chapter 6 Genetic Evolution Processes

6.1 An Idealized Model of Infinite Population: The Hardy-Weinberg Model

6.2 Finite Population: The Wright-Fisher Model

6.2.1 Wright-Fisher Process

6.2.2 Quasi-Stationary Distribution for aWright-Fisher Process

6.2.3 TheWright-Fisher Process With Mutation

6.2.4 TheWright-Fisher ProcessWith Selection

6.3 Demographic Diffusion Models

6.3.1 TheWright-Fisher Diffusion Process

6.3.2 TheWright-Fisher Diffusion ProcessWith Mutation or Selection

6.3.3 Another Time Scale Change

6.4 Coalescence: Description of Genealogies

6.4.1 Asymptotic as Tends to Infinity: The Kingman Coalescent

6.4.2 The CoalescentWith Mutation

6.4.3 Law of the Number of Distinct Alleles, Ewens' Formula

6.4.4 A Branching Process With Immigration Point of View

6.5 Exercises

Chapter 7 Some Modern Developments in Ecology-Evolution

7.1 Survival and Growth of Meta-Populations Distributed on a Graph

7.1.1 First Approach: Multitype Galton-Watson Process

7.1.2 Second Approach - Markov Chain on a Graph

7.2 Abundance in a Random Environment

7.3 Study of the Invasion of a Mutant in a Large Resident Population at Equilibrium

7.4 A Stochastic Model for the Self-Incompatibility of Flowering Plants

7.5 Modeling a Diploid Population

7.5.1 A Multi-Type Birth and Death Process

7.5.2 Large Population Approximations

7.6 Genealogical Trees of Sexual Populations

7.6.1 DiploidWright-Fisher ModelWith Recombination

7.6.2 Number of Generations to Reach the Common Ancestor

Appendix A Measures, Integration and Probability Measures

A.1 -Fields and Measures

A.1.1 -Fields and Measurable Space

A.1.2 Positive Measures on -Fields

A.1.3 Definition of the Lebesgue Measure

A.1.4 Measurable Functions.

A.2 Probability Measures and Expectation

A.2.1 Probability Measures

A.2.2 Random Variables

A.2.3 Expectation of Random Variables

A.2.4 Convergence Theorems

Appendix B Poisson Point Measures

References

Index.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

3-662-73483-4