Global existence of classical solutions to a predator–prey model with nonlinear prey-taxis

This paper deals with a predator–prey model consisting of a 2×2 reaction–diffusion-taxis system recently proposed by Ainseba, Bendahmane and Noussair [B.E. Ainseba, M. Bendahmane, A. Noussair, A reaction–diffusion system modeling predator–prey with prey-taxis, Nonlinear Anal. RWA 9 (2008) 2086–2105]. The central point of this system is that the spatial–temporal variations of the predators’ velocity are directed by the prey gradient. The global existence and uniqueness of classical   solutions to this system are proved by the contraction mapping principle, together with LpLp estimates and Schauder estimates of parabolic equations. The crucial point of the proof is to deal with the prey-tactic term with a nonlinear tactic function.