Dynamics of a Harvested Predator-Prey Model with Predator-Taxis

Effects of predator-taxis on the dynamics of a predator-prey model with Michaelis-Menten type nonlinear harvesting are considered in this paper. Through theoretical analysis of quasilinear parabolic equations, the local existence, global existence and boundedness of solutions to the system are first established. Then the formation mechanisms of spatiotemporal patterns in such model are explored. It is found that only the attractive predator-taxis will lead to the Turing bifurcation and the corresponding spatiotemporal solutions; meanwhile, no such phenomenon occurs with the repulsive taxis or in the absence of predator-taxis. Moreover, the diffusion ratio can affect the Turing bifurcation thresholds, and nonlinear harvesting can affect the spatial distribution of prey and predator species. Further, the stability of the nonconstant steady state bifurcated from the Turing bifurcation is analyzed through the amplitude equation, so that the direction of the Turing bifurcation is determined. Effectiveness of the analysis is illustrated in the numerical simulations.