Stability And Hopf Bifurcation Analysis Of Some Predator-prey Models

Many phenomena can be described by mathematical models in the natural sciences and social sciences,such as,the Logistic model for studying population growth,the predatory-prey model for describing the growth of predatory and prey fish,and the SIR model for studying the spread of infectious diseases.At the same time,these mathematical models can also study a variety of issues in physics,chemistry,biology and other disciplines,and the content and methods of their research are varied.By using the mathematical model of this tool,we can effectively describe the development of things in real life,and then can guide the practice of production through the research and application of mathematical models,the research results have important theoretical and practical significance.In this paper,we consider the stability and Hopf bifurcation of some reaction-diffusion predator-prey models.Firstly,a time-delay predator-prey model with Holling III is considered.By analyzing the corresponding characteristic equations,we judge the local stability of the positive equilibrium point.The properties of Hopf bifurcation are given by using Hopf bifurcation theorem,the center manifold theorem and normal form theory.Numerical simulations are carried out to illustrate our results.Next,we investigate a predator-prey model with herd behavior,the stability of the positive equilibrium point is given by choosing the appropriate parameters.The properties of Hopf bifurcation are obtained by choosing₀s and the delay as bifuacation parameter.Numerical simulations are carried out to illustrate our results.Finally,we discuss a predator-prey model with prey harvesting.The stability of the positive constant and the existence,stability and bifurcation direction of the periodic solution are obtained respectively by analyzing the corresponding characteristic equations,the global asymptotical stability of positive constant equilibrium of the diffusive model is obtained by iterative technique.Numerical simulations are carried out to illustrate our results.