Steady state bifurcation and pattern formation of a diffusive population model

The steady state bifurcation and spatiotemporal patterns are induced by prey-taxis in a population model, in which prey, predators and scavengers are involved. Effects of prey-taxis are manifested from the obtained results. By using the prey-taxis coefficient as the bifurcation parameter, the occurrence conditions of the steady state bifurcation is established. It is found that there is no steady state bifurcation without prey-taxis or with the attractive type prey-taxis, meanwhile, the steady state bifurcation will occur when the repulsive type prey-taxis is present. In the sequel, by employing the Crandall-Rabinowitz local bifurcation theory, the existence of the bifurcating solution and its stability are further explored. It is stable if the second-order perturbation term is less than zero, and unstable if greater than zero. The resulting nonconstant steady states are displayed through numerical simulation, which are in agreement with theoretical analysis. No steady state bifurcation or patterns will appear in simulation when the prey-taxis is absent.