Stability Analysis Of Several Predator-prey Models

Biomathematics is an interdisciplinary subject between ecology and mathematics.Population ecology is an important branch of ecology,and the most popular model is the predator-prey model.The research on the survival and extinction conditions of biological populations is of great significance to ecology.Therefore,this paper studies the stability of several predator-prey models.Chapter 1: We mainly introduce the research background,research status and the main work of this dissertation.Chapter 2: We study a Leslie-Gower predator-prey model with linear harvesting effects.Firstly,the local bifurcation analysis and stability analysis of the model are carried out.Secondly,the direction of Hopf bifurcation at the positive equilibrium point of the system is discussed.Finally,the theoretical results are verified by numerical simulation.Chapter 3: We study a predator-prey model with Beddington-DeAngelis functional response of prey with refuge and nonlinear harvesting term.By analyzing the corresponding characteristic equation,the local stability condition of the equilibrium point is obtained.The existence of Hopf bifurcation is analyzed by using Routh-Hurwitz criterion and implicit function theorem.The ecological balance is analyzed by using the optimal control theory and Pontyragin’s maximum principle.Chapter 4: We study a Beddington-DeAngelis functional response predatorprey model with SIRS disease.By analyzing the corresponding characteristic equation,the existence and local stability conditions of the equilibrium point are obtained.By constructing a new Lyapunov function combined with LaSalle invariance principle,the global asymptotic stability conditions of partial equilibrium points are obtained.Chapter 5: We study a Beddington-DeAngelis functional response predatorprey model with age stage structure and SIS disease.The existence and local stability conditions of the equilibrium point are obtained by analyzing the corresponding characteristic equation.By constructing a new Lyapunov function and combining with the LaSalle invariance principle,the global asymptotic stability conditions of some equilibrium points are obtained.