Abstract:
The bifurcation of a diffusive predator-prey models with Michaelis-Menten type prey harvesting and refuge is considered under homogeneous Neumann boundary condition.Firstly,the local stability of positive constant steady-state solution is obtained by using the theory of stability;secondly,by using the maximum principle,Harnack inequalities and the integral property,the priori estimates of positive steady-state solutions and the non-existence of the non-constant positive steady-state solutions are proved;the local bifurcation from the positive constant steady-state solutions is given by further applying simple eigenvalue bifurcation theory;finally,the positive constant steady-state solution where Hopf bifurcation occurs is investigated by using Hopf bifurcation theory.