Global Dynamical Properties of Lotka-Volterra Systems

The following sections are included:

• Contents

• Preface

The first part of this book consists of two chapters, one on single-species growth and one on multiple-species growth. In Chapter 1, we restrict our attention on an ecological system composed of only one species and consider its growth. This system may be regarded as just one treated in a laboratory. But this is our starting point for the analysis of ecological systems. Almost all real ecological systems, even if they have small sizes, are composed of multiple-species and each species interacts with some of them. In Chapter 2, we consider systems composed of multiple-species and introduce the most fundamental but important systems in mathematical ecology, Lotka-Volterra systems.

The following sections are included:

• Introduction

• Logistic Growth

• General Growth

The following sections are included:

• Introduction

• Lotka-Volterra Systems

• Lotka-Volterra Systems with Diffusion

• Lotka-Volterra Systems with Delay

In the next two chapters, Lotka-Volterra systems are discussed. In Chapter 3, the systems are composed of n populations and we analyze their global stability. The relationship between the global stability conditions and systems' structures is considered. Global stability for the L-V systems with general growth terms is also discussed. In Chapter 4, several specific predator-prey L-V systems are treated and their periodic and chaotic behaviors are investigated.

The following sections are included:

• Introduction

• Global Stability of L-V Systems

• Globed Stability and Systems' Structure

• Extension of Global Stability Conditions

• Generalized L-V Systems

The following sections are included:

• Introduction

• Two-Prey, One-Predator Systems

• Three-Prey, One-predator Systems

• Two-Prey, Two-Predator Systems

In the next two chapters, a Lotka-Volterra system with diffusion is discussed. The system is composed of several subsystems ( called patches ) described by Lotka-Volterra equations and the species are allowed to disperse among the patches. In Chapter 5, we consider global stability and persistence of the system. In Chapter 6, periodic and chaotic behaviors for predator-prey or competitive systems are investigated.

The following sections are included:

• Introduction

• L-V Systems with Identical Patch

• Generalized Systems with Identical Patch

• Single-Species Systems

• Cooperative Systems

• Competition Systems

• Two-Competitor Systems with One Refuge

• Three-Competitor Systems with One Refuge

• N — Species Competition Systems

• General Predator-Prey Systems

The following sections are included:

• Introduction

• Homogeneous Prey-Predator Systems

• Heterogeneous Prey-Predator Systems

• Three-Competitor Systems

In the final chapter, we shall discuss effect of time delays on global stability and permanence of L-V systems. The systems considered are composed of a single-species or competitive or cooperative or predator-prey species. Except for a competitive system, we shall consider L-V systems with continuous time delays which can be transformed into ODE by using a linear chain trick. For a competitive case, we analyze the system with discrete time delays.

The following sections are included:

• Introduction

• L-V Systems with Identical Patches

• Cooperative L-V Systems

• Single-Species Systems

• Predator-Prey Systems I

• Predator-Prey Systems II

• Competition Systems without Diffusion

The following sections are included:

• References

• Appendix
  • Appendix A Fundamental Properties of Solutions of ODE
    • Existence
    • Uniqueness
    • Nonnegativeness
  • Appendix B Stability of Equilibria
    • Local Stability
    • Global Stability
  • Appendix C Hopf Bifurcation Theorem
  • Appendix D Linear Complementarity Problem

• Index