Spatiotemporal Dynamics of a General Two-Species System with Taxis Term

In this paper, we investigate the spatiotemporal dynamics in a diffusive two-species system with taxis term and general functional response, which means the directional movement of one species upward or downward the other one. The stability of positive equilibrium and the existences of Turing bifurcation, Turing-Hopf bifurcation and Turing-Turing bifurcation are investigated. An algorithm for calculating the normal form of the Turing-Hopf bifurcation induced by the taxis term and another parameter is derived. Furthermore, we apply our theoretical results to a cooperative Lotka-Volterra system and a predator-prey system with prey-taxis. For the cooperative system, stable equilibrium becomes unstable by taxis-driven Turing instability, which is impossible for the cooperative system without taxis. For a predator-prey system with prey-taxis, the dynamical classification near the Turing-Hopf bifurcation point is clearly described. Near the Turing-Hopf point, there are spatially inhomogeneous steady-state solution, spatially homogeneous/nonhomogeneous periodic solution and pattern transitions from one spatiotemporal state to another one.