Boundedness and stabilization in a two-species chemotaxis-competition system with indirect signal production

Abstract

This paper deals with a two-competing-species chemotaxis system with indirect signal production{ut=d1Δu−χ1∇⋅(u∇w)+μ1u(1−u−a1v),(x,t)∈Ω×(0,∞),vt=d2Δv−χ2∇⋅(v∇w)+μ2v(1−v−a2u),(x,t)∈Ω×(0,∞),τwt=Δw−w+z,(x,t)∈Ω×(0,∞),τzt=Δz−z+u+v,(x,t)∈Ω×(0,∞), under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn, where χi,μi,ai,di>0(i=1,2), τ∈{0,1}. When τ=1 and n≤2, we study global boundedness of classical solutions in this system. When τ=0, based on some a priori estimates, it is shown that regardless of the size of u0,v0, this system possesses a global bounded classical solution if n=2. This paper also proceeds to establish asymptotic stabilization of global bounded solutions to the above system as follows:(i) when a1,a2∈(0,1), if μ1 and μ2 are sufficiently large, the global bounded classical solution (u,v,w,z) exponentially converges to (1−a11−a1a2,1−a21−a1a2,2−a1−a21−a1a2,2−a1−a21−a1a2) in L∞-norm as t→∞;(ii) when a1>1>a2>0, if μ2 is sufficiently large, the global bounded classical solution (u,v,w,z) exponentially converges to (0,1,1,1) in L∞-norm as t→∞;(iii) when a1=1>a2>0, if μ2 is sufficiently large, the global bounded classical solution (u,v,w,z) polynomially converges to (0,1,1,1) in L∞-norm as t→∞.