Non-constant steady states and Hopf bifurcation of a species interaction model

Abstract

In this paper, we are concerned with the species interaction model with the ratio-dependent Holling III functional response and strong Allee effect. To explore nonhomogeneous solutions of the model, we consider the existence and non-existence of non-constant steady states and temporal bifurcation. Then we show the boundedness of the global positive solutions for the parabolic system and present the upper and lower bounds of positive solutions for the associated elliptic system. The non-existence and existence of the non-constant steady states of the elliptic system with the homogeneous Neumann boundary conditions are obtained by using the priori estimates, maximum principle and index theory. Furthermore, the existence and the direction of Hopf bifurcation are investigated via the stability analysis, center manifold theory and normal form reduction. Numerical simulations are carried out to verify our theoretical analysis and to illustrate that the ratio-dependent Holling III functional response and strong Allee effect have strong impact on dynamical behaviors of the species interaction systems.