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

Abstract

This paper deals with a two-species attraction–repulsion chemotaxis system

$$\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta u-\xi _{1}\nabla \cdot (u\nabla v)+\chi _{1}\nabla \cdot (u\nabla z)+f_{1}(u,w),&(x,t)\in \Omega \times (0,\infty ), \\&\tau v_{t}=\Delta v+w-v,&(x,t)\in \Omega \times (0,\infty ),\\&w_t=\Delta w-\xi _{2}\nabla \cdot (w\nabla z)+\chi _{2}\nabla \cdot (w\nabla v)+f_{2}(u,w),&(x,t)\in \Omega \times (0,\infty ), \\&\tau z_{t}=\Delta z+u-z,&(x,t)\in \Omega \times (0,\infty ), \end{aligned} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions in a smoothly bounded domain $\Omega \subseteq {\mathbb {R}}^{n}$, where $\tau \in \{0,1\},\xi _{i},\chi _{i}>0$ and $f_{i}(u,w)(i=1,2)$ satisfy

$$\begin{aligned} \left\{ \begin{aligned}&f_{1}(u,w)=u\bigg (a_{0}-a_{1}u-a_{2}w+a_{3}\int _{\Omega }udx+a_{4}\int _{\Omega }wdx\bigg ),\\&f_{2}(u,w)=w\bigg (b_{0}-b_{1}u-b_{2}w+b_{3}\int _{\Omega }udx+b_{4}\int _{\Omega }wdx\bigg )\\ \end{aligned} \right. \end{aligned}$$

with $a_{i},b_{i}>0(i=0,1,2),a_{j},b_{j}\in {\mathbb {R}}(j=3,4)$. It is proved that in any space dimension $n\ge 1$, the above system possesses a unique global and uniformly bounded classical solution regardless of $\tau =0$ or $\tau =1$ under some suitable assumptions. Moreover, by constructing Lyapunov functionals, we establish the globally asymptotic stabilization of coexistence and semi-coexistence steady states.