Competitive exclusion for a two-species chemotaxis system with two chemicals

In this paper we consider the following competitive two-species chemotaxis system with two chemicals ut=Δu−χ1∇⋅(u∇v)+μ1u(1−u−a1w),x∈Ω,t>0,0=Δv−v+w,x∈Ω,t>0,wt=Δw−χ2∇⋅(w∇z)+μ2w(1−a2u−w),x∈Ω,t>0,0=Δz−z+u,x∈Ω,t>0in a smooth bounded domain Ω⊂Rn with n≥1, where χi≥0, ai≥0 and μi>0(i=1,2). For the case a1>1>a2≥0, it will be proved that if χ1χ2<μ1μ2, χ1≤a1μ1 and χ2<μ2, then the initial-boundary value problem with homogeneous Neumann boundary condition admits a unique global bounded solution and (u,v,w,z)→(0,1,1,0) uniformly on Ω̄ as t→∞.