Discrete Dynamical Systems: With Applications in Biology -2nd Edition

Abstract

Discrete-time dynamical systems or difference equations have been increasingly used to model the biological and ecological systems for which there is time interval between each measurement. This modeling approach is done through using the iterative maps. Iterative maps are an essential part of nonlinear systems dynamics as they allow us to take the output of the previous state of the system and fit it back to the next iteration. In general, it is not easy to explicitly solve a system of difference equations. There are different methods of solving different types of difference equations. This book introduces concepts, theorems, and methods in discreet-time dynamical systems theory which are widely used in studying and analysis of local dynamics of biological systems and provides many traditional applications of the theory to different fields in biology. Our focus in this book is covering three important parts of discrete-time dynamical systems theory: Stability theory, Bifurcation theory and Chaos theory. Mathematically speaking, stability theory in the field of discrete-time dynamical systems deals with the stability of solutions of difference equations and of orbits of dynamical systems under small perturbations of initial conditions. In dynamical systems point of view, bifurcation theory addresses the changes in the qualitative behavior or topological structure of the solutions of a family of difference equations. Finally, chaos theory is a branch of dynamical systems which focuses on the study of chaotic states of a dynamical system which is often governed by deterministic laws and its solutions demonstrate irregular behavior and are highly sensitive to initial conditions. Therefore, this book is a blend of three important parts of discrete-time dynamical systems theory and their exciting applications to biology.