Spatial pattern formation in pine wilt disease model with prey-taxis and nonlocal competition

Prey-taxis is a biological phenomenon, which plays a key role in biological control and ecological balance. For the controlling of pine wilt disease, we establish a reaction-diffusion equation with prey-taxis and nonlocal intraspecific competition of prey in this paper. We investigate the spatial formation of the spread system of pine wilt disease. It is concluded that the spatial pattern formation induced by the prey-taxis and time delay is characterized by Turing, Hopf, and Turing-Hopf bifurcation. Moreover, we extend the multiple time scales method to derive the normal form of the co-dimension-2 Turing-Hopf bifurcation for the system with prey-taxis and nonlocal effect. Through analyzing the normal form near the critical point of Turing-Hopf bifurcation, we obtain that there exist a pair of spatially nonhomogeneous non-constant steady states. Especially, Turing instability does not occur when prey-taxis coefficient is greater than the critical value. The time delay influences spatiotemporal patterns arising from Hopf bifurcation and Turing-Hopf bifurcation. It is noticed that there exist spatially nonhomogeneous periodic solutions with large amplitude when parameters are selected to some specific values. Furthermore, we also reveal some biological explanations and provide some theoretical support for controlling of pine wilt disease.