Spatiotemporal patterns induced by Turing and Turing-Hopf bifurcations in a predator-prey system

The Turing and Turing-Hopf bifurcations of a Leslie-Gower type predator-prey system with ratio-dependent Holling III functional response are investigated in this paper. Complex and interesting patterns induced by the bifurcations are identified theoretically and numerically. First the existence conditions of the Turing instability and the Turing-Hopf bifurcation are established from the theoretical analysis, respectively. Then by employing the technique of weakly nonlinear analysis, amplitude equations generated near the Turing instability critical value are derived. Various spatiotemporal patterns, such as homogeneous stationary state patterns, hexagonal patterns, coexisting patterns, stripe patterns, and their stability are determined via analyzing the obtained amplitude equations. Numerical simulations are presented to illustrate the theoretical analysis. Especially, the analogous-spiral and symmetrical wave patterns can be found near the codimension-two Turing-Hopf bifurcation point. A security center of the prey species can be found as well. These spatiotemporal patterns are explained from the perspective of the predators and prey species.