Global Stability in a Two-species Attraction–Repulsion System with Competitive and Nonlocal Kinetics

This paper deals with a two-species attraction–repulsion chemotaxis system { u_t=Δ u-ξ _1∇· (u∇ v)+χ _1∇· (u∇ z)+f_1(u,w), (x,t)∈Ω× (0,∞ ), τ v_t=Δ v+w-v, (x,t)∈Ω× (0,∞ ), w_t=Δ w-ξ _2∇· (w∇ z)+χ _2∇· (w∇ v)+f_2(u,w), (x,t)∈Ω× (0,∞ ), τ z_t=Δ z+u-z, (x,t)∈Ω× (0,∞ ), . under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω⊆ℝ^n , where τ∈{0,1},ξ _i,χ _i>0 and f_i(u,w)(i=1,2) satisfy { f_1(u,w)=u (a_0-a_1u-a_2w+a_3∫ _Ωudx+a_4∫ _Ωwdx ), f_2(u,w)=w (b_0-b_1u-b_2w+b_3∫ _Ωudx+b_4∫ _Ωwdx ) . with a_i,b_i>0(i=0,1,2),a_j,b_j∈ℝ(j=3,4) . It is proved that in any space dimension n≥ 1 , the above system possesses a unique global and uniformly bounded classical solution regardless of τ =0 or τ =1 under some suitable assumptions. Moreover, by constructing Lyapunov functionals, we establish the globally asymptotic stabilization of coexistence and semi-coexistence steady states.