Global dynamics for a diffusive predator-prey model with prey-taxis and classical Lotka-Volterra kinetics

In this paper, we use energy method to study the global dynamical properties for nonnegative solutions of the following reaction-advection-diffusion system of predator-prey model with prey-taxis and the classical Lotka-Volterra kinetics: { u(t) = d(1)Delta u - chi del center dot (u del v) - a(1)u + b(1)uv, x is an element of Omega, t > 0, v(t) = d(2)Delta v + a(2)v - b(2)uv, x is an element of Omega, t > 0 in a bounded smooth but not necessarily convex domain Omega subset of R-2 with nonnegative initial data u(0), v(0) and homogeneous Neumann boundary data. Here, d(1), d(1), b(2) are positive, chi, a(1), b(1) are nonnegative and a(2) is allowed to be real. It is shown that, for any regular initial data, the system has a unique global smooth solution for arbitrary size of chi, and it is uniformly bounded in time in the case of a(2) <= 0. In the latter case, we further study its long time dynamics, which in particular imply that the prey-tactic cross-diffusion and even the linear instability of the semi-trivial constant steady states (0, v*) with v* > a(1)/b(1,) b(1) > 0 and a(2) = 0 still cannot induce pattern formation. More specifically, it is shown that (u, v) converges exponentially to (0, 0) in the case that the net growth rate of prey is negative, i.e., a(2) < 0. In the case of a(2) = 0, we obtain the following classification for its long time behavior. (P1) Case I: a(1) > 0, b(1) = 0, then u converges exponentially to 0 and v -> k in C-2((Omega) over bar), where k is a positive and finite number and it satisfies (ln k)vertical bar Omega vertical bar = d(2) integral(infinity)(0) integral(Omega) vertical bar del(v)vertical bar(2)/v(2) - b(2)/a(1) integral(Omega) u(0) + integral(Omega) ln v(0). (P2) Case II: a(1) > 0, b(1) > 0, then u -> 0 and v -> m in C-2((Omega) over bar), where m is a positive and finite number and it satisfies m vertical bar Omega vertical bar = integral(Omega) v(0) + b(2)/b(1) integral(Omega) u(0) - a(1)b(2)/b(1) integral(infinity)(0) integral(Omega) u. (P3) Case III: a(1) = 0, then u -> ((u) over bar (0) + b(1)/b(2)(v) over bar (0)) in C-2((Omega) over bar) and v -> 0 exponentially, where u(0) = 1/vertical bar Omega vertical bar integral(Omega) u(0) and (v) over bar (0) = 1/vertical bar Omega vertical bar integral(Omega) v(0). The convergence properties (P1) and (P2) imply that, spatial diffusion, especially, the random movement of prey plays a role in the long time behavior and that the chemotaxis mechanism may have certain influence on its long time behavior. In particular, the long time behavior may not always be determined by its corresponding ODE system, which seems to be a rarely occurring phenomenon. (C) 2017 Elsevier Ltd. All rights reserved.