Differential Equations and Population Dynamics I: Introductory Approaches

ISBN-13: 9783030981358 / Angielski / Miękka / 2022

Arnaud Ducrot;Quentin Griette; Zhihua Liu

Opis

This book provides an introduction to the theory of ordinary differential equations and its applications to population dynamics. Part I focuses on linear systems. Beginning with some modeling background, it considers existence, uniqueness, stability of solution, positivity, and the Perron–Frobenius theorem and its consequences.Part II is devoted to nonlinear systems, with material on the semiflow property, positivity, the existence of invariant sub-regions, the Linearized Stability Principle, the Hartman–Grobman Theorem, and monotone semiflow.Part III opens up new perspectives for the understanding of infectious diseases by applying the theoretical results to COVID-19, combining data and epidemic models. Throughout the book the material is illustrated by numerical examples and their MATLAB codes are provided.Bridging an interdisciplinary gap, the book will be valuable to graduate and advanced undergraduate students studying mathematics and population dynamics.

Spis treści

Part I Linear Differential and Difference Equations: 1 Introduction to Linear Population Dynamics.- 2 Existence and Uniqueness of Solutions.- 3 Stability and Instability of Linear.- 4 Positivity and Perron-Frobenius's Theorem.- Part II Non-Linear Differential and Difference Equations: 5 Nonlinear Differential Equation.- 6 Omega and Alpha Limit.- 7 Global Attractors and Uniformly.- 8 Linearized Stability Principle and Hartman-Grobman's Theorem.- 9 Positivity and Invariant Sub-region.- 10 Monotone semiflows.- 11 Logistic Equations with Diffusion.- 12 The Poincare-Bendixson and Monotone Cyclic Feedback Systems.- 13 Bifurcations.- 14 Center Manifold Theory and Center Unstable Manifold Theory.- 15 Normal Form Theory.- Part III Applications in Population Dynamics: 16 A Holling's Predator-prey Model with Handling and Searching Predators.- 17 Hopf Bifurcation for a Holling's Predator-prey Model with Handling and Searching Predators.- 18 Epidemic Models with COVID-19.

O autorze

Arnaud Ducrot is professor of mathematics at the University Le Havre Normandie, France. His research interests include analysis, dynamical systems and mathematical aspects of population dynamics and the natural sciences.

Quentin Griette is an associate professor in mathematics at the University of Bordeaux, France. His areas of expertise include ordinary differential equations, reaction-diffusion systems and the numerical computation of their solutions.

Zhihua Liu is a professor of mathematics at Beijing Normal University, China. Her research interests include differential equations, dynamical systems and applications in epidemics and population dynamics.

Pierre Magal is professor of mathematics at the University of Bordeaux, France. His research interests include differential equations, dynamical systems, numerical simulations and mathematical biology.