Stability and Oscillations in Delay Differential Equations of Population Dynamics

✍ Scribed by K. Gopalsamy

• Publisher: Springer
• Year: 2010
• Tongue: English
• Leaves: 514
• Series: Mathematics and Its Applications
• Edition: 1st Edition.
• Category: Library

✦ Synopsis

This monograph provides a definitive overview of recent advances in the stability and oscillation of autonomous delay differential equations. Topics include linear and nonlinear delay and integrodifferential equations, which have potential applications to both biological and physical dynamic processes.
Chapter 1 deals with an analysis of the dynamical characteristics of the delay logistic equation, and a number of techniques and results relating to stability, oscillation and comparison of scalar delay and integrodifferential equations are presented. Chapter 2 provides a tutorial-style introduction to the study of delay-induced Hopf bifurcation to periodicity and the related computations for the analysis of the stability of bifurcating periodic solutions. Chapter 3 is devoted to local analyses of nonlinear model systems and discusses many methods applicable to linear equations and their perturbations. Chapter 4 considers global convergence to equilibrium states of nonlinear systems, and includes oscillations of nonlinear systems about their equilibria. Qualitative analyses of both competitive and cooperative systems with time delays feature in both Chapters 3 and 4. Finally, Chapter 5 deals with recent developments in models of neutral differential equations and their applications to population dynamics. Each chapter concludes with a number of exercises and the overall exposition recommends this volume as a good supplementary text for graduate courses.
For mathematicians whose work involves functional differential equations, and whose interest extends beyond the boundaries of linear stability analysis.

✦ Table of Contents

Cover......Page 1
Mathematics and Its Applications Volume 74......Page 2
Stability and Oscillations in Delay Differential Equations of Population Dynamics......Page 4
Copyright - ISBN: 0792315944......Page 5
Series Editor's Preface......Page 6
Contents......Page 8
Preface......Page 10
1.1. Introduction......Page 14
1.2. Linear stability criteria......Page 16
1.3. Linear oscillators and comparison......Page 50
1.4. Global stability......Page 68
1.5. Oscillation and nonoscillation......Page 79
1.6. Piecewise constant arguments and impulses......Page 91
1.7. Feedback control......Page 108
Exercises I......Page 115
2.1. Introduction......Page 137
2.2. Loss of linear stability......Page 141
2.3. Delay induced bifurcation to periodicity......Page 143
2.4. Stability of the bifurcating periodic solution......Page 149
2.5. An example......Page 156
2.6. Coupled oscillators......Page 161
Exercises II......Page 173
3.1. Preliminary remarks......Page 185
3.2. Delays in production......Page 188
3.3. Competition and cooperation......Page 195
3.4. Prey-predator systems......Page 209
3.5. Delays in production and destruction......Page 217
3.6. \dot{X}(t) = AX(t) + BX(t - τ)......Page 223
3.7. Stability switches......Page 252
3.8. Oscillations in linear systems......Page 266
3.9. Simple stability criteria......Page 276
Exercises III......Page 286
4.1. Some preliminaries......Page 305
4.2. Competition : exploitation and interference......Page 311
4.3. Delays in competition and cooperation......Page 319
4.4. Method of Lyapunov functional......Page 340
4.5. Oscillations in Lotka-Volterra systems......Page 353
4.6. Why positive steady states ?......Page 359
4.7. Dynamics in compartments......Page 368
Exercises IV......Page 383
5.1. Linear scalar equations......Page 406
5.2. Oscillation criteria......Page 412
5.3. Neutral logistic equation......Page 431
5.4. A neutral Lotka-Volterra system......Page 443
5.5. \dot{X}(t) = AX(t) + BX(t - τ) + C\dot{X}(t - τ)......Page 449
5.6. Large scale systems......Page 460
Exercises V......Page 475
References......Page 487
Index......Page 510