This book concerns the behaviour of a particular class of delay differential equations, namely, those related to the (undelayed) logistic equation: \(dN/dt=rN(1-N/K)\) where \(N\) is the population size and \(r\) and \(K\) are positive constants. Various generalisations of this equation, mostly involving the inclusion of delayed terms, are analysed in this book. The contents are summarised below.
Chapter 1 is a short introduction to logistic models, and also lists several theorems that are used many times in the rest of the book. Chapter 2 concerns proving that a given delayed logistic equation has, or does not have, periodic solutions. Models considered are of Hutchinson type, with delayed feedback, with nonlinear delays, with harvesting, with varying capacity, and \(\alpha\)-delay models (with one or more delays) and hyperlogistic models. Chapter 3 concerns the local and global stability of a nontrivial steady state of various delayed logistic equations. Models with impulses are also introduced. Chapter 4 discusses delayed logistic models (autonomous and nonautonomous) with piecewise arguments and gives proofs as to whether such models have oscillatory solutions, and whether non-trivial fixed points are locally or globally stable. Chapter 5 concerns “food-limited” population models, again with delays. Similar results regarding the existence of oscillatory solutions and the local and global stability of fixed points are given. Food-limited models with impulses and those with periodic coefficients (i.e., nonautonomous) are also considered. In Chapter 6, spatially-extended models are considered. These are formed by adding a spatial diffusive term to some of the simpler models already considered. Proofs that such models have temporally periodic solutions are given, as results on the local and global stability of fixed points.
The content is highly mathematical, contains more equations than text, and is organized in lemmas and theorems, each of which is proved in detail. The book should be of interest to those interested in proving results about the behaviour of delay differential equations.