Candace M. Kent

The aim of this monograph is to initiate and facilitate a systematic study of the global asymptotic behavior of nonlinear nonautonomous difference equations. In addition to general classes of nonautonomous equations, a focus lies on some special classes including periodic and asymptotically autonomous equations. The authors' primary goal is to study global stability, in particular the extreme stability (path stability), as well as the global asymptotic stability in periodic equations. In addition, they explore the oscillatory character of solutions including semicycles analysis, boundednes and persistence, permanence, the existence and the character of solutions having special properties such as periodicity, unboundedness, or monotonicity. Special attention is placed on the notion of attenuance and resonance of periodic cycles and their generalizations. This monograph presents to the reader a broad spectrum of approaches to the mentioned problems and contains several important developments in this area with applications mainly to Mathematical Biology as well as to other disciplines. In the Appendix some issues about the dynamics of discontinuous population models and discontinuous difference equations are addressed. The reader is exposed to the frontiers of the discipline, including a variety of important open problems and conjectures that require immediate attention.

In the 1960s and 1970s, mathematical biologists Sir Robert M. May, E.C. Pielou, and others utilized di?erence equations as models of ecological and epidemiological phenomena. Since then, with or without applications, the mathematics of di?erence equations has evolved into a ?eld unto itself. Di?erence equations with the maximum (or the minimum or the "rank-type") function were rigorously investigated from the mid-1990s into the 2000s, without any applications in mind. These equations often involved arguments varying from reciprocal terms with parameters in the numerators to other special functions.Recently, the authors of Analysis of a Model for Epilepsy: Application of a Max-Type Difference Equation to Mesial Temporal Lobe Epilepsy and their colleagues investigated the ?rst known application of a "max-type" di?erence equation. Their equation is a phenomenological model of epileptic seizures. In this book, the authors expand on that research and present a more comprehensive development of mathematical, numerical, and biological results. Additionally, they describe the first documented instance of a novel dynamical behavior that they call rippled almost periodic behavior, which can be described as an unpredictable pseudo-periodic behavior.Features: Suitable for researchers in mathematical neuroscience and potentially as supplementary reading for postgraduate students Thoroughly researched and replete with references