Global existence and boundedness of solutions in a reaction-diffusion system of Michaelis-Menten-type predator-prey model with nonlinear prey-taxis and random diffusion

Abstract

This article deals with a 2 × 2 reaction-diffffusion-taxis model consisting of Michaelis-Menten functional response predator-prey system proposed by Michaelis and Menten [L. Michaelis,M. L. Menten, Die Kinetik der Invertinwerkung, Biochemische Zeitschrift 49 (1913) 333–369]. The critical section of this model is that temporal-spatial evolution of the predators’ velocitydepends largely on the gradient of prey. But beyond that, this system also inscribes a prey-taxis mechanism that is an immediate movement of the predator u in response to a change of the prey v (which leads to the collection of u). By using contraction mapping principle, Lp estimates and Schauder estimates of parabolic equations, we prove the global existence and uniquenessof classical solutions to this model. In addition to this, we prove the global boundedness of solutions by overcome the diffiffifficulties brought by nonlinear prey-taxis.