Competing effects of attraction vs. repulsion in chemotaxis

We consider the attraction-repulsion chemotaxis system {equation presented} under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ Rnwith smooth boundary, where χ ≥ 0, ξ ≥ 0, α > 0, β > 0, γ > 0, δ > 0 and τ = 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 τ = 0 is globally well-posed in high dimensions if repulsion prevails over attraction in the sense that ξγ - χα > 0, and that the system with τ = 1 is globally well-posed in two dimensions if repulsion dominates over attraction in the sense that ξγ - χα > 0 and β = δ. 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.