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Book review

Author(s) Gopalsamy, K.
Title Stability and oscillations in delay equations of population dynamics
Publisher Kluwer Academic Publishers
Year of publication 1992
Reviewed by Anton S. Muresan

Recent papers in the domain of differential equations with deviating argument are deal, not only with the effective resolution of some equations or systems, but also with the properties of solutions, as stability, oscillarity, maximum principles, and so on.

The purpose of this monograph is the study of the aspects mentioned and in particular, the study of stability and oscillarity of solutions of delay differential equations in background population dynamics.

In the first chapter, The delay logistic equation, are contained, in main, the paragraphs: linear stability criteria, linear oscillators and comparison, global stability, oscillation and nonoscillation and feedback control. One treat in consequently the analysis of the dynamical characteristics of the delay logistic equation and several of its variants, in particular, a series of techniques and results related to stability, oscillation and comparison of scalar delay and integrodifferential equations are presented.

The main paragraphs of the chapter 2, Delay induced bifurcation to periodicity, are loss of linear stability, delay induced bifurcation to periodicity, stability of the biturcating periodic solution, coupled oscillators. There are therefore studied the delay induced Hopf-bifurcation to periodicity and the related computations for the analysis of stability of bifurcating periodic solutions.

The chapter 3, Methods of linear analysis, has a special importance to all those interested in applications of delay and integrodifferential equations. Though in dynamical real world phenomena are rarely linear, the study of linear systems, with delays discrete, continuous piecewise constant and even unbounded, is make in the main paragraphs: delays in production, competition and cooperation, prey-predator systems, delays in production and destruction, stability switches and simple stability criteria for linear systems.

The chapter 4, Global attractivity, contains the main paragraphs: competition: exploitation and interference, delay in competition and cooperation, methods of Lyapunov functionals, oscillations in Lotka-Volterra systems, dynamics in compartments. The author consider here the global convergence to equilibrium states of nonlinear systems and a brief analysis of compartmental systems.

The last chapter, Models of neutral differential systems are paragraphs: linear scalar equations, oscillation criteria, neutral logistic equation, a neutral Lotka-Volterra system and large scale systems. Here, in the chapter 5, one present recent development in model of neutral differential equations and their applications to population dynamics.

At the end of each chapters are presented many exercices, some very simple, others more difficult and some included, after the author assertion, "not even he not know how to solve".

The references are extensive and been selected of towards author without being completely. The numbers of titles quote is 307 from among which we notice those 26 of titles of author.

We consider that this monograph is a source of several analytical techniques and is an extent of guide to the relevant literature or those interested in the applications of delay differential equations. We consider that this monograph will be accesible to advanced graduate students trained in differential equations, and to research worker engaged in the study of qualitative behavior of model systems involving delay differential equations.