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


ISBN 978-93-91595-28-9 (Print)
ISBN 978-93-91595-29-6 (eBook)
DOI: 10.9734/bpi/mono/978-93-91595-28-9

Mathematical modeling of population dynamics has been attracted by many researchers over the last few decades. Specially, exponential difference equations have been used to model the interactions between different kind of population dynamics. Among these population models, Host-Parasitoid interactions play an important role in the ecosystem. A parasitoid is an organism that feeds another organism. The host is the organism which the parasitoid feeds. We have two different approaches to model these interactions, discrete and continuous. However, the discrete time models governed by difference equations are more realistic, rational and applicable rather than the continuous systems specially when the populations follow the non-overlapping generations. A well-known example of this non-overlapping is Insects which adults lay eggs in spring/summer and then die. The eggs hatch into larvae which eat and grow and then in the winter they fall in a pupal stage. The adults appear from the pupae in spring. Moreover, discrete models can present much more dynamical behaviors compared to the continuous-time model.

 

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

Tahmineh Azizi, Bacim Alali, Gabriel Kerr

Discrete Dynamical Systems: With Applications in Biology -2nd Edition, 3 August 2021, Page 1-103
https://doi.org/10.9734/bpi/mono/978-93-91595-28-9

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.