This paper deals with a two-species attraction–repulsion chemotaxis system ut=d1Δu-ξ1∇·(u∇v)+χ1∇·(u∇z)+g1(u,w),(x,t)∈Ω×(0,∞),τvt=d2Δv+w-v,(x,t)∈Ω×(0,∞),wt=d3Δw-ξ2∇·(w∇z)+χ2∇·(w∇v)+g2(u,w),(x,t)∈Ω×(0,∞),τzt=d4Δz+u-z,(x,t)∈Ω×(0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{aligned}{}&u_t=d_{1}\Delta u-\xi _{1}\nabla \cdot (u\nabla v)+\chi _{1}\nabla \cdot (u\nabla z)+g_{1}(u,w),&(x,t)\in \Omega \times (0,\infty ), \\&\tau v_{t}=d_{2}\Delta v+w-v,&(x,t)\in \Omega \times (0,\infty ),\\&w_t=d_{3}\Delta w-\xi _{2}\nabla \cdot (w\nabla z)+\chi _{2}\nabla \cdot (w\nabla v)+g_{2}(u,w),&(x,t)\in \Omega \times (0,\infty ), \\&\tau z_{t}=d_{4}\Delta z+u-z,&(x,t)\in \Omega \times (0,\infty ) \end{aligned} \right. \end{aligned}$$\end{document}under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω⊂Rn\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \subset {\mathbb {R}}^{n}$$\end{document} for n≥1\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 1$$\end{document}, where τ∈{0,1}\documentclass[12pt]{minimal}
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\begin{document}$$\tau \in \{0,1\}$$\end{document}, the parameters di(i=1,2,3,4),ξj,χj(j=1,2)\documentclass[12pt]{minimal}
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\begin{document}$$d_{i}(i=1,2,3,4),\xi _{j},\chi _{j}(j=1,2)$$\end{document} are positive and the kinetic terms g1(u,w),g2(u,w)\documentclass[12pt]{minimal}
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\begin{document}$$g_{1}(u,w),g_{2}(u,w)$$\end{document} satisfy g1(u,w)=u(a0-a1u-a2w-a3∫Ωudx-a4∫Ωwdx),g2(u,w)=w(b0-b1u-b2w-b3∫Ωudx-b4∫Ωwdx)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{aligned}{}&g_{1}(u,w)=u\bigg (a_{0}-a_{1}u-a_{2}w-a_{3}\int _{\Omega }u{\text {d}}x-a_{4}\int _{\Omega }w{\text {d}}x\bigg ),\\&g_{2}(u,w)=w\bigg (b_{0}-b_{1}u-b_{2}w-b_{3}\int _{\Omega }u{\text {d}}x-b_{4}\int _{\Omega }w{\text {d}}x\bigg )\\ \end{aligned} \right. \end{aligned}$$\end{document}with a0,a1,b0,b2>0,a2,a3,a4,b1,b3,b4∈R\documentclass[12pt]{minimal}
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\begin{document}$$a_{0},a_{1},b_{0},b_{2}>0,a_{2},a_{3},a_{4},b_{1},b_{3},b_{4}\in {\mathbb {R}}$$\end{document}. It is shown that under some suitable parameter conditions, the above system possesses a unique global and uniformly bounded solution in any spatial dimension. Moreover, we investigate the asymptotic stability of solutions under the locally intraspecific competition and globally interspecific cooperation. Finally, we present some numerical simulations, which not only support our analytically theoretical results, but also find some new and interesting phenomena.
