Book contents
- Frontmatter
- Contents
- Preface
- Chapter 1 Introduction
- Chapter 2 Exponential Growth
- Chapter 3 Introduction to Differential Equations
- Chapter 4 Stability in a One-Component System
- Chapter 5 Systems of First-Order Differential Equations
- Chapter 6 Phase Plane Analysis
- Chapter 7 Introduction to Vectors
- Chapter 8 Equilibrium in Two-Component, Linear Systems
- Chapter 9 Stability in Nonlinear Systems
- Chapter 10 Nonlinear Stability Revisited
- Chapter 11 Matrix Notation
- Chapter 12 Remarks About Australian Predators
- Chapter 13 Introduction to Advection
- Chapter 14 Diffusion Equations
- Chapter 15 Two Key Properties of the Advection and Diffusion Equations
- Chapter 16 The No-Trawling Zone
- Chapter 17 Separation of Variables
- Chapter 18 The Diffusion Equation and Pattern Formation
- Chapter 19 Stability Criterion
- Chapter 20 Summary of Advection/Diffusion
- Chapter 21 Traveling Waves
- Chapter 22 Traveling Wave Velocities
- Chapter 23 Periodic Solutions
- Chapter 24 Fast and Slow
- Chapter 25 Estimating Elapsed Time
- Chapter 26 Switches
- Chapter 27 Testing for Periodicity
- Chapter 28 Causes of Chaos
- Extra Exercises and Solutions
- Index
Chapter 23 - Periodic Solutions
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Chapter 1 Introduction
- Chapter 2 Exponential Growth
- Chapter 3 Introduction to Differential Equations
- Chapter 4 Stability in a One-Component System
- Chapter 5 Systems of First-Order Differential Equations
- Chapter 6 Phase Plane Analysis
- Chapter 7 Introduction to Vectors
- Chapter 8 Equilibrium in Two-Component, Linear Systems
- Chapter 9 Stability in Nonlinear Systems
- Chapter 10 Nonlinear Stability Revisited
- Chapter 11 Matrix Notation
- Chapter 12 Remarks About Australian Predators
- Chapter 13 Introduction to Advection
- Chapter 14 Diffusion Equations
- Chapter 15 Two Key Properties of the Advection and Diffusion Equations
- Chapter 16 The No-Trawling Zone
- Chapter 17 Separation of Variables
- Chapter 18 The Diffusion Equation and Pattern Formation
- Chapter 19 Stability Criterion
- Chapter 20 Summary of Advection/Diffusion
- Chapter 21 Traveling Waves
- Chapter 22 Traveling Wave Velocities
- Chapter 23 Periodic Solutions
- Chapter 24 Fast and Slow
- Chapter 25 Estimating Elapsed Time
- Chapter 26 Switches
- Chapter 27 Testing for Periodicity
- Chapter 28 Causes of Chaos
- Extra Exercises and Solutions
- Index
Summary
In this and the subsequent chapters, I return to the milieu of the first 12 chapters, in which only dependence on time was at issue. However, whereas the first 12 chapters considered equilibrium issues almost exclusively, this chapter and the remaining chapters consider nonequilibrium phenomena. Here, the story is amazingly complicated and there is no sense in which it can be said that the interesting questions are all solved. Indeed, the complicated nonequilibrium dynamics that arise even from very simple models are still a wealthy source of very interesting mathematics. Meanwhile, similar appearing dynamics appears in real biological systems and the underlying causes are a subject of intense investigation.
I shall start by describing a predator-prey model that has a stable, time-dependent, periodic solution. The model presented provides one mathematical explanation for cyclic behavior. However, the model itself is not the main point of this chapter. Rather, you should focus on those aspects of the model that guarantee the existence of cyclic solutions, for those aspects are found in many other models. That is, there are certain generic properties of a differential equation that a priori imply that there are periodic solutions. In particular, the properties to notice are the existence of a basin of attraction in the phase plane that has inside a certain type of unstable equilibrium point.
- Type
- Chapter
- Information
- Modeling Differential Equations in Biology , pp. 375 - 392Publisher: Cambridge University PressPrint publication year: 2008