Asymptotic behavior in a two-species chemotaxis system with Lotka-Volterra type competition and two signals

Abstract

This paper deals with the study of asymptotic behavior of the following two-species chemotaxis system with Lotka-Volterra type competition and two signals{∂tu1=Δu1−∇⋅(χ11(υ1)u1∇υ1)−∇⋅(χ12(υ2)u1∇υ2)+μ1u1(1−u1−a1u2),x∈Ω,t>0,∂tu2=Δu2−∇⋅(χ21(υ1)u2∇υ1)−∇⋅(χ22(υ2)u2∇υ2)+μ2u2(1−u2−a2u1),x∈Ω,t>0,∂tυ1=Δυ1−λ1υ1+α11u1+α12u2,x∈Ω,t>0,∂tυ2=Δυ2−λ2υ2+α21u1+α22u2,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn, where parameters μi≥0,ai≥0,λi>0, αij>0(i,j=1,2) and χij(i,j=1,2) are the chemotactic sensitivity functions. The boundedness and global existence of solutions to this system have been studied [22]. By constructing appropriate energy function, the purpose of this paper is to prove the asymptotic behavior for the cases•If a1,a2∈(0,1) and μ1,μ2 are suitably large, then the solution (u1,u2,υ1,υ2) is exponentially converges to (1−a11−a1a2,1−a21−a1a2,α11(1−a1)+α12(1−a2)λ1(1−a1a2),α21(1−a1)+α22(1−a2)λ2(1−a1a2)) as t→∞.•If a1>1>a2>0 and μ1,μ2 are suitably large, then the solution (u1,u2,υ1,υ2) is exponentially converges to (0,1,α12λ1,α22λ2) as t→∞.•If a1=1>a2>0 and μ1,μ2 are suitably large, then the solution (u1,u2,υ1,υ2) is algebraically converges to (0,1,α12λ1,α22λ2) as t→∞.