On a Two-Species Attraction-Repulsion Chemotaxis System with Nonlocal Terms.

This paper deals with a two-species attraction-repulsion chemotaxis system [GRAPHICS] under homogeneous Neumann boundary conditions in a smoothly bounded domain Omega subset of R-n for n >= 1, where tau is an element of{0,1}, the parameters d(i) (i = 1, 2, 3, 4), xi(j), chi(j) (j = 1, 2) are positive and the kinetic terms g(1) (u, w), g(2) (u, w) satisfy [GRAPHICS] with a(0), a(1), b(0), b(2) > 0, a(2), a(3), a(4), b(1), b(3), b(4) is an element of R. It is shown that under some suitable parameter conditions, the above system possesses a unique global and uniformly bounded solution in any spatial dimension. Moreover, we investigate the asymptotic stability of solutions under the locally intraspecific competition and globally interspecific cooperation. Finally, we present some numerical simulations, which not only support our analytically theoretical results, but also find some new and interesting phenomena.