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.

Automated summary

This paper examines a two-species chemotaxis-consumption system involving nonlinear diffusion and chemotaxis in a smooth bounded domain in 2D and 3D space. It is shown that the corresponding initial-boundary value problem has a unique global bounded classical solution for sufficiently large values of mu(1), mu(2). The paper also establishes asymptotic stabilization of solutions to the system, with convergence properties dependent on the parameters a(1), a(2).