Global dynamics of a two-species chemotaxis-consumption system with signal-dependent motilities

This paper deals with a two-species chemotaxis-consumption system involving nonlinear diffusion and chemotaxis { u(t) = Delta(gamma(1)(w)u) + mu(1)u(1- u - a(1)v), x is an element of Omega, t > 0, u(t) = Delta(gamma(2)(w)v) + mu(2)v(1- a(2)u - v), x is an element of Omega, t > 0, w(t) = Delta w - (alpha u + beta v)w, x is an element of Omega, t > 0, in an arbitrary smooth bounded domain Omega subset of R-n (n = 2, 3) under homogeneous Neumann boundary conditions, where mu(i), a(i), alpha, beta are positive constants and the motility functions gamma(i)(w) is an element of C-3 ([0, infinity)), gamma(i)(w) > 0, y(i)'(w) < 0 for all w >= 0, lim(w ->infinity )gamma(i)(w)=0 and lim(w ->infinity) gamma i'(w)/gamma i(w) exist for i = 1, 2. It is proved that the corresponding initial-boundary value problem possesses a unique global bounded classical solution in 2-D and in 3-D for mu(1), mu(2) being sufficiently large. Furthermore, in a spatially three-dimensional setting, the paper also proceeds to establish asymptotic stabilization of solutions to the above system, and the following properties hold: when a(1), a(2) is an element of (0, 1), the global bounded classical solution (u, v, w) exponentially converges to (1-a(1)/1-a(1) a(2), 0) as t -> infinity; when a(1) > 1 > a(2) , the global bounded classical solution (u, v, w) exponentially converges to (0, 1, 0) as t -> infinity; when a(1) = 1 > a(2), the global bounded classical solution (u, v, w) polynomially converges to (0,1, 0) as t ->infinity. (C) 2020 Elsevier Ltd. All rights reserved.