Hopf Bifurcation And Turing Instability Of A Predator-prey Model

In the field of biological mathematics,the Lotka-Volterra model is a classical mathematical model.It has a history of nearly 100 years,and it still has great vitality,attracting many scholars'attention and research.Classical Lotka-Volterra models have been studied more thoroughly.However,the Lotka-Volterra model with harvesting term is seldom studied in the existing literature.Generally,only the case with Constant Harvesting term is stucdied.Lotka-Volterra models with diffusion terms are few.In this paper,we mainly study the stability and Hopf bifurcation of a Lotka-Volterra predator-prey model with a Michaelis-Menten harvest term and a diffusion term by using the qualitative theory of differential equations and bifurcation theory.In this paper,the stability,Hopf bifurcation and Turing instability of a Lotka-Volterra predator-prey model with a Michaelis-Menten predator harvest term with diffusion term are studied by using the qualitative theory of differential equations,stability theory,normal form and central manifold theorem.The details are as follows.1.The stability of Lotka-Volterra predator-prey model with Michaelis-Menten type harvesting term without diffusion term and the condition of Hopf bifurcation are studied.The problem of determining the direction of Hopf bifurcation by using the normal form method is discussed.The problem of generating limit cycles from Hopf bifurcation is also discussed2.The influence of diffusion term on the stability of equilibrium point and bifurcation limit cycle is studied.3.The results are verified by numerical simulation.