Introduction to Reaction-Diffusion Equations: Theory and Applications to Spatial Ecology and Evolutionary Biology

✍ Scribed by King-Yeung Lam, Yuan Lou

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 316
• Series: Lecture Notes on Mathematical Modelling in the Life Sciences
• Category: Library

✦ Synopsis

This book introduces some basic mathematical tools in reaction-diffusion models, with applications to spatial ecology and evolutionary biology. It is divided into four parts.

The first part is an introduction to the maximum principle, the theory of principal eigenvalues for elliptic and periodic-parabolic equations and systems, and the theory of principal Floquet bundles.

The second part concerns the applications in spatial ecology. We discuss the dynamics of a single species and two competing species, as well as some recent progress on N competing species in bounded domains. Some related results on stream populations and phytoplankton populations are also included. We also discuss the spreading properties of a single species in an unbounded spatial domain, as modeled by the Fisher-KPP equation.

The third part concerns the applications in evolutionary biology. We describe the basic notions of adaptive dynamics, such as evolutionarily stable strategies and evolutionary branching points, in the context of a competition model of stream populations. We also discuss a class of selection-mutation models describing a population structured along a continuous phenotypical trait.

The fourth part consists of several appendices, which present a self-contained treatment of some basic abstract theories in functional analysis and dynamical systems. Topics include the Krein-Rutman theorem for linear and nonlinear operators, as well as some elements of monotone dynamical systems and abstract competition systems.

Most of the book is self-contained and it is aimed at graduate students and researchers who are interested in the theory and applications of reaction-diffusion equations.

 

 

 

 

 

✦ Table of Contents

Preface
Contents
Part I Linear Theory
Chapter 1 The Maximum Principle and the Principal Eigenvalues for Single Equations
1.1 The Maximum Principle for Single Parabolic Equations
1.2 The Comparison Principle for Semilinear Equations
1.3 The Principal Eigenvalue for Linear Elliptic Operators
1.4 Further Reading
Problems
References
Chapter 2 The Principal Eigenvalue for Periodic-Parabolic Problems
2.1 Existence of the Principal Eigenvalue for Periodic-Parabolic Problems
2.2 Qualitative Properties of Periodic Principal Eigenvalues
2.2.1 The Hutson–Shen–Vickers Lemma
2.2.2 Small diffusion limit
2.2.3 Large diffusion limit
2.2.4 Monotonicity in frequency
2.3 Applications to Two-Species Competition Models in a Spatially and Temporally Varying Environment
2.4 Further Reading
Problems
References
Chapter 3 The Maximum Principle and the Principal Eigenvalue for Systems
3.1 Comparison Principle of Cooperative Parabolic Systems
3.2 The Principal Eigenvalue of Cooperative Systems
3.2.1 Existence results
3.2.2 Asymptotic behavior of the principal eigenvalue
3.3 Comparison Principle and Principal Eigenvalue for Competitive Parabolic Systems
3.4 Further Reading
Problems
References
Chapter 4 The Principal Floquet Bundle for Parabolic Equations
4.1 Existence Results for Non-Divergence Form Parabolic Equations
4.2 Existence Results for Divergence Form Parabolic Equations
4.3 The Generalized Relative Entropy
4.4 Further Reading
Problems
References
Part II Ecological Dynamics
Chapter 5 The Logistic EquationWith Diffusion
5.1 A Reaction-Diffusion Model for a Single Species
5.2 The Logistic Equation
5.3 Critical Domain Size
5.4 Further Reading
Problems
References
Chapter 6 Spreading in Homogeneous and Shifting Environments
6.1 The Fisher–KPP Equation and the Definition of Spreading Speed
6.2 A Maximum Principle for Unbounded Domains
6.3 Homogeneous Environments
Traveling wave solutions
Periodically Varying Environments
6.4 Shifting Environments
Shifting environments with a moving source patch
Shifting boundary connecting an unbounded sink and an unbounded source patch
Shifting boundary connecting two unbounded source patches and nonlocally pulling
6.5 Further Reading
Problems
References
Chapter 7 The Lotka–Volterra Competition-Diffusion Systems for Two Species
7.1 Elements from the Theory of Monotone Dynamical Systems
7.2 Lotka–Volterra Systems with Constant Coefficients
7.3 Lotka–Volterra Systems with Heterogeneous Coefficients
7.3.1 Slow vs fast diffusing populations
7.3.2 Weak competition in a heterogeneous environment
7.4 Competition in an Advective Environment
7.5 Further Reading
Problems
References
Chapter 8 Dynamics of Phytoplankton Populations
8.1 Introduction
8.2 Single Species in a EutrophicWater Column
8.2.1 Monotonicity of the single species model
8.2.2 Long-time dynamics of the single species model
8.3 Dynamics for Two Competing Phytoplankton Species
Selection for more buoyant phytoplankton species
8.4 The N--Species Model – Application of the Principal Floquet Bundle
8.4.1 A priori estimates
8.4.2 A rough estimate
8.4.3 The normalized principal bundle
8.4.4 A general exclusion criterion
8.5 Further Reading
Problems
References
Part III Evolutionary Dynamics
Chapter 9 Elements of Adaptive Dynamics
9.1 Introduction
9.2 Evolution of Dispersal in Advective Environments
The invasion exponent
The selection gradient
Singular strategy
Convergence stable strategy
Evolutionarily stable strategy
Continuously stable strategy
Neighborhood invader strategy
Dimorphism (coexistence of phenotypes)
Evolutionary branching point
9.3 Further Reading
Problems
References
Chapter 10 Selection-Mutation Models
10.1 Populations Structured by a Phenotypic Trait
The Case Ω = RN
10.2 Populations Structured by Space and a Phenotypic Trait
10.3 Further Reading
Problems
References
Appendices
Appendix A The Fixed Point Index
A.1 Properties of the Leray–Schauder Degree
A.2 The Fixed Point Index
References
Appendix B The Krein–Rutman Theorem
B.1 Introduction
B.2 Cones and Orderings
B.3 The Classical Krein–Rutman theorem
B.4 The Generalized Krein–Rutman theorem for Homogeneous Maps
B.5 Further Reading
Problems
References
Appendix C Subhomogeneous Dynamics
C.1 Subhomogeneous Maps
C.2 Subhomogeneous Semiflows
C.3 Further Reading
Problems
References
Appendix D Existence of Connecting Orbits
D.1 Discrete-Time Monotone Dynamical Systems
D.2 Continuous-Time Monotone Dynamical Systems
References
Appendix E Abstract Competition Systems in Ordered Banach Spaces
E.1 Discrete-Time Competition Systems
E.2 Continuous-Time Competition Systems
E.3 Further Reading
Problems
References
Index