Stability and Oscillations in Delay Differential Equations of Population Dynamics

This monograph provides a definitive overview of recent advances in the stability and oscillation of autonomous delay differential equations. Topics covered include linear and nonlinear delay and integrodifferential equations, which have potential applications to both biological and physical dynamic processes. The first chapter 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. The next section 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 co-operative systems with time delays feature in both Chapters 3 and 4. Finally, the last chapter 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 supplementary text for graduate courses. The book is aimed at mathematicians whose work involves functional differential equations, and whose interest extends beyond the boundaries of linear stability analysis.