Methods for Mathematical Modeling I

Abstract

Mathematical modeling has become an essential component of research and studies in ecology. This book is intended for undergraduate and master’s level students who wish to acquire mathematical modeling techniques in ecology and epidemiology. It introduces fundamental concepts of mathematical modeling, focusing on deterministic dynamic systems, particularly ordinary differential equations. The book also presents a series of classical models in population dynamics and ecology. It aims to provide a rigorous yet accessible introduction to these methods, making them understandable not only for mathematicians but also for students from various scientific backgrounds, including life sciences, who may not have prior training in dynamic systems. Numerous examples and exercises illustrate the techniques presented, allowing students to practice and apply them to real ecological problems. We hope that students with a mathematical background will find clear explanations of qualitative analysis methods for dynamic systems—methods they may already be familiar with—along with numerous applications in ecology. Likewise, we hope that students with a biological background will find a comprehensive and accessible introduction to the main techniques used to study dynamic systems, as well as their implementation in classical ecological models such as the Lotka-Volterra model, Holling’s model, and many others. This book is a synthesis of the authors’ teaching experience in mathematical modeling applied to ecology. While primarily intended for students, doctoral candidates, postdoctoral researchers, and academics looking to acquire or deepen their knowledge in this field will also find it useful. Many researchers in both public and private institutions study complex natural and social systems, and mathematical modeling has become an indispensable tool in modern research to understand the mechanisms governing these systems’ dynamics. Although several books cover similar topics, most of them are written in English. This book aims to make mathematical modeling methods in ecology more accessible to a wider audience. It brings together a broad range of classical mathematical models in ecology, some of which are traditionally scattered across different sources, while also introducing some original models. Students will find a comprehensive collection of commonly used models in ecology, while researchers will have a fundamental reference for constructing and analyzing mathematical models relevant to their work. The book is organized into chapters that are either methodological or applied. The methodological chapters introduce techniques for analyzing mathematical models which includes the continuous-time models. The applied chapters use these techniques to study population and community dynamics. We provide an overview of population growth models and interaction models between two species (e.g., predator-prey, host-parasitoid, competition, mutualism). We also discuss models of multi-species interactions within trophic networks and structured population models incorporating age classes.