COMPETING EFFECTS OF ATTRACTION VS. REPULSION IN CHEMOTAXIS

We consider the attraction-repulsion chemotaxis system {u(t) = Delta u - del . (chi u del v) + del . (xi u del w), x is an element of Omega, t > 0, tau v(t) = Delta v + alpha u - beta v, x is an element of Omega, t > 0, tau w(t) = Delta w + gamma u - delta w, x is an element of Omega, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Omega subset of R-n with smooth boundary, where chi >= 0, xi >= 0, alpha > 0, beta > 0, gamma > 0, delta > 0 and tau = 0, 1. We study the global solvability, boundedness, blow-up, existence of non-trivial stationary solutions and asymptotic behavior of the system for various ranges of parameter values. Particularly, we prove that the system with tau = 0 is globally well-posed in high dimensions if repulsion prevails over attraction in the sense that xi gamma - chi alpha > 0, and that the system with tau = 1 is globally well-posed in two dimensions if repulsion dominates over attraction in the sense that xi gamma - chi alpha > 0 and beta = delta. Hence our results confirm that the attraction-repulsion is a plausible mechanism to regularize the classical Keller-Segel model whose solution may blow up in higher dimensions.