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 1 u - a 2 w + a 3 ∫ Ω u d x + a 4 ∫ Ω w d x) , f 2 (u , w) = w (b 0 - b 1 u - b 2 w + b 3 ∫ Ω u d x + b 4 ∫ Ω w d x) 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. [ABSTRACT FROM AUTHOR]