Boundedness and stabilization in a three-species chemotaxis-competition system

Abstract

In this paper, we study the following chemotaxis-competition system of parabolic-parabolic-parabolic-elliptic type{ut=d1Δu−χ1∇⋅(u∇z)+μ1u(1−u−a1v),(x,t)∈Ω×(0,∞),vt=d2Δv−χ2∇⋅(v∇z)+μ2v(1−a2u−v),(x,t)∈Ω×(0,∞),wt=d3Δw−χ3∇⋅(w∇z)+μ3w(1−b1u−b2v−w),(x,t)∈Ω×(0,∞),0=d4Δz−αz+β1u+β2v+β3w,(x,t)∈Ω×(0,∞), under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω⊂Rn, n≥2, where χi>0,μi>0,βi>0,i=1,2,3 and α,a1,a2,b1,b2,d1,d2,d3,d4>0. Firstly, it is shown that this system possesses a global bounded nonnegative classical solution for any nonnegative initial data (u0,v0,w0)∈(C0(Ω‾))3 provided that χi are small enough or μi are large enough. Moreover, this paper proceeds to establish asymptotic stabilization of global bounded solutions to the above system as follows:• When a1,a2∈(0,1), the global bounded classical solution (u,v,w,z) converges to (u⁎,v⁎,w⁎,β1u⁎+β2v⁎+β3w⁎α) if b1(1−a1)+b2(1−a2)<1−a1a2 or (u⁎,v⁎,0,β1u⁎+β2v⁎α) if b1(1−a1)+b2(1−a2)≥1−a1a2 in L∞-norm as t→∞, where u⁎=1−a11−a1a2,v⁎=1−a21−a1a2,w⁎=1−a1a2−b1(1−a1)−b2(1−a2)1−a1a2.• When a1≥1>a2>0, the global bounded classical solution (u,v,w,z) converges to (0,1,1−b2,β2+β3(1−b2)α) if b2∈(0,1) or (0,1,0,β2α) if b2≥1 in L∞-norm as t→∞.