Turing–Hopf Bifurcation and Spatio-Temporal Patterns of a Ratio-Dependent Holling–Tanner Model with Diffusion

Abstract

In this paper, the dynamics of a diffusive ratio-dependent Holling–Tanner model subject to Neumann boundary conditions is considered. We derive the conditions for the existence of Hopf, Turing, Turing–Hopf, Turing–Turing, Hopf-double-Turing and triple-Turing bifurcations at the unique positive equilibrium. Furthermore, we study the detailed dynamics in the neighborhood of the Turing–Hopf bifurcation by using the normal form method. Our results show that the Turing–Hopf bifurcation can give rise to the formation of the temporal and spatio-temporal patterns. In particular, we theoretically prove the existence of the spatially inhomogeneous periodic and quasi-periodic solutions, which can be used to explain the phenomenon of spatio-temporal resonance of the populations. Finally, the numerical simulations are given to illustrate the analytical results.