Dynamic Behavior Of A Diffusive Rosenzweig-MacArthur Predator-prey Model With Holling Type-?

In this paper,the dynamic behavior of a diffusive Rosenzweig-MacArthur predator-prey model with Holling type-II is studied.For the spatially homoge-neous model,the conditions for stability of equilibria are analysed and the existence of Hopf bifurcation is given by taking the ratio of the death rate of the hunting predators to the net growth rate of the preys as the bifurcation parameter.For the weakly-coupled reaction-diffusion model,we analyze the stability of the positive equilibrium,it is shown that Turing(diffusion-driven)instability does not occur.Moreover,the existence of nonconstant positive solutions are established by the fixed point index theory when the positive constant equilibrium is unstable.For the strongly-coupled cross-diffusion model,we first prove that cross-diffusion can give rise to Turing instability,which leads to the occurrence of spatially inhomo-geneous patterns.Meanwhile,it is demonstrated that the model exhibits Hopf bifurcation under suitable conditions.Then we deduce that the model undergoes Turing-Hopf bifurcation and presents spatiotemporal patterns at the intersections of Turing instability curve and Hopf bifurcation curve.Furthermore,the existence of nonconstant positive solutions are established by Leray-Schauder degree theory.Finally,theoretical results are explained by numerical simulations.