Differential Equations and Population Dynamics I: Introductory Approaches

Dynamical Systems with Applications in Population Dynamics discusses existence, uniqueness, stability, global attractors, bifurcations, center manifold, and normal form theories with cutting-edge applications, such as a Holling's predator-prey model and COVID-19 epidemic projections. It bridges mathematics, biology, and medicine, making it a valuable resource for interdisciplinary research.

Format: Paperback / softback
Length: 458 pages
Publication date: 21 June 2022
Publisher: Springer Nature Switzerland AG

This comprehensive book delves into the fundamental theoretical concepts of dynamical systems, with a focus on their applications in population dynamics. It explores the concepts of existence, uniqueness, and stability of solutions, global attractors, bifurcations, center manifold, and normal form theories, while showcasing cutting-edge applications in fields such as predator-prey models, epidemics, and public health interventions for COVID-19. By seamlessly integrating relevant concepts from mathematics, biology, and medicine, this book serves as a valuable resource for bridging the gap between these disciplines. It is designed to be self-sufficient for readers, catering to both graduate and advanced undergraduate students interested in interdisciplinary research in mathematics and population dynamics.

The book begins by introducing the basic theoretical framework of dynamical systems, covering topics such as differential equations, linear algebra, and complex analysis. It then delves into the study of dynamical systems in biology, focusing on models of population growth, disease transmission, and ecological dynamics. The authors employ mathematical tools such as differential equations, vector analysis, and matrix theory to analyze these models and derive their analytical solutions.

One of the key themes of the book is the exploration of global attractors, which are regions in the state space of a dynamical system where solutions tend to persist over time. The authors discuss the properties of global attractors, including their stability, uniqueness, and the existence of limit cycles. They also illustrate how global attractors can be used to predict the long-term behavior of complex systems, such as the spread of diseases and the evolution of ecosystems.

Another important topic covered in the book is bifurcations, which occur when a system undergoes a sudden change in its behavior due to the interaction of multiple parameters. The authors discuss the types of bifurcations that can occur, including saddle-node, pitchfork, and fold bifurcations, and their implications for the stability and dynamics of systems. They also provide examples of bifurcations in real-world systems, such as the collapse of financial markets and the emergence of new species in biology.

The center manifold theory is another key concept discussed in the book. It is a mathematical tool used to study the structure of the state space of a dynamical system and to identify the stable and unstable manifolds. The authors demonstrate how the center manifold theory can be used to analyze the behavior of complex systems, including the Lorenz system, the Kuramoto-Sivashinsky equation, and the Hénon-Hilbert equation.

In addition to theoretical discussions, the book also includes practical applications of dynamical systems in population dynamics. The authors discuss the use of predator-prey models to study the dynamics of animal populations, including the effects of population density, food availability, and predation. They also explore the application of epidemic models to predict the spread of diseases, such as COVID-19, and to evaluate the effectiveness of public health interventions.

Throughout the book, the authors emphasize the interdisciplinary nature of dynamical systems and their relevance to various fields. They highlight the connections between mathematics, biology, and medicine, and demonstrate how these fields can be combined to address complex problems. The book is designed to be accessible to students with a background in mathematics, biology, or medicine, and it includes numerous examples and exercises to reinforce the concepts discussed.

In conclusion, this book presents a comprehensive and up-to-date introduction to the theoretical concepts of dynamical systems with applications in population dynamics. It serves as a valuable resource for students, researchers, and practitioners interested in interdisciplinary research in mathematics and population dynamics. By integrating relevant concepts from mathematics, biology, and medicine, the book bridges the gap between these disciplines and provides a solid foundation for understanding and analyzing complex systems.

Weight: 730g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030981358
Edition number: 1st ed. 2022