polycopiés pédagogique 2024-2025

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Browsing polycopiés pédagogique 2024-2025 by Author "DJILALI, Salih"

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    Applications of bifurcation theory in biomathematics
    (University Hassiba Benbouali of Chlef, 2025) DJILALI, Salih
    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
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    Methods for Mathematical Modeling I
    (University Hassiba Benbouali of Chlef, 2025) DJILALI, Salih
    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.