Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis

We propose a new reaction-diffusion predator-prey model system with predator-taxis in which the preys could move in the opposite direction of predator gradient. A similar situation also occurs when susceptible population avoids the infected ones in epidemic spreading. The global existence and boundedness of solutions of the system in bounded domains of arbitrary spatial dimension and any predator-taxis sensitivity coefficient are proved. It is also shown that such predator-taxis does not qualitatively affect the existence and stability of coexistence steady state solutions in many cases. For diffusive predator-prey system with diffusion-induced instability, it is shown that the presence of predator-taxis may annihilate the spatial patterns.