Pattern selection in a predator-prey model with Michaelis-Menten type nonlinear predator harvesting

In this paper, the effect of diffusion on pattern selection in a predator-prey model with Michaelis-Menten type predator nonlinear harvesting is investigated. From the analysis of stability and bifurcation, the stability of the positive equilibrium and the occurrence conditions for the Hopf bifurcation, as well as the Turing instability, are obtained. Consequently, different patterns will occur in the system. It is necessary and imperative to derive the amplitude equations for understanding pattern selection. To this end, the multiple time scale analysis technique is employed to obtain the amplitude equations at the critical value of the Turing instability. Further, by analyzing the amplitude equations, we could identify different patterns, such as spot-like, stripe-like, hexagon-like and mixed state patterns. Finally, numerical simulations show that the rationality of the theoretical analysis and the obtained results will enrich the dynamical investigation of the predator-prey model.