Applications of bifurcation theory in biomathematics

Abstract

Bifurcation theory is a powerful mathematical tool used to study the qualitative changes in the dynamics of a system as a parameter is varied. In the context of biomathematics, bifurcation analysis provides critical insights into the behavior of biological and ecological models, such as predator-prey interactions, epidemiological spread, and pattern formation in biological tissues. The techniques of bifurcation theory allow us to understand how small changes in parameters can lead to significant changes in the system’s dynamics—such as the transition from steady states to oscillatory behavior (Hopf bifurcation), the sudden appearance or disappearance of equilibria (saddle-node bifurcation), or the onset of spatial patterns (Turing instability). This course is designed for first-year Master’s students in Biomathematics and is structured to cover a broad spectrum of topics in bifurcation theory applied to systems of ordinary and partial differential equations, as well as delay differential equations. The applications include ecological and epidemiological models, where spatial diffusion and time delays are critical in reproducing realistic dynamics observed in nature. The course is organized as follows: – 1. Applications of Bifurcation Theory on Systems of Equations: This section introduces the fundamental notions of bifurcation theory, starting with onedimensional ordinary differential equations. We study classic bifurcations such as saddle-node, pitchfork, and transcritical bifurcations, and extend these concepts to two-dimensional systems. Later, we discuss bifurcations in delay differential equations. – 2. Bifurcation Analysis for Ecological Models: In this section, the focus shifts to ecological models such as predator-prey systems. We examine both delay differential equations (DDEs) and ordinary differential equations (ODEs) in the context of population dynamics. Detailed analysis of the delayed Volterra predatorprey model is presented, including exercises to reinforce the concepts. – 3. Bifurcation Analysis for Epidemiological Models: Here, we apply bifurcation theory to models arising in epidemiology. The course covers topics such as Hopf bifurcation in delayed epidemiological SIS models, and includes several exercises to enhance understanding. – 4. Bifurcation Theory for Partial Differential Equations (PDEs): This section extends the bifurcation analysis to spatially extended systems governed by partial differential equations. We cover the existence and uniqueness of solutions for parabolic problems, the spectral analysis of the Laplacian operator, and methods of separation of variables. Finally, the course discusses Hopf and Turing bifurcations in spatial models. Exercises: Each section is supplemented with exercises that challenge the students to apply the theory to various models, thereby deepening their understanding of both the mathematical techniques and their biological applications. Throughout the course, our aim is to not only present the theoretical framework but also to provide practical examples and computational techniques that reveal how bifurcation theory can be used to predict and analyze complex spatiotemporal dynamics in biological systems. The interplay between delay, diffusion, and nonlinear interactions is central to these analyses, and this course equips students with the necessary tools to explore these phenomena in dept