Turing-Hopf bifurcation and bi-stable spatiotemporal periodic orbits in a delayed predator-prey model with predator-taxis

Abstract

When Turing-Hopf bifurcation occurs, it is naturally worth discussing the types of spatiotemporal patterns and their stabilities. In this paper, for partial functional differential equations with nonlinear diffusion, we establish the explicit formulae for the coefficients of normal form associated with Turing-Hopf bifurcation. This provides an explicit connection between the original system and the third-order ordinary differential equations (i.e. normal form) restricted on center manifold. Subsequently, with the aid of extended formulae, we discuss a delayed predator-prey model with predator-taxis and find that Turing-Hopf bifurcation leads to the emergence of a pair of stable single-peak spatially inhomogeneous steady states, a pair of stable multi-peak spatially inhomogeneous periodic orbits and so on. Also, the parameter regions in which these phenomena arise are given quantitatively. It is further shown that the pair of bistable spatially inhomogeneous periodic orbits arise due to the interaction of predator-taxis and time delay, which won't emerge in the same system without predator-taxis.