Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals

We consider the two-species-two-chemical chemotaxis system $\left\{ \begin{array}{l}{u_t}\; = \Delta u - {\chi _1}\nabla \cdot (u\nabla v) + {\mu _1}u(1 - u - {a_1}w),\;\;\;\;\;\;\;x \in \Omega ,t > 0,\\{u_t}\; = \Delta v - v + w,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \Omega ,t > 0,\\{w_t}\; = \Delta w - {\chi _2}\nabla \cdot (w\nabla z) + {\mu _2}w(1 - w - {a_2}u),\;\;\;x \in \Omega ,t > 0,\\{z_t}\; = \Delta z - z + u,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \Omega ,t > 0,\end{array} \right.$ where $Ω\subset\mathbb{R}^n$ is a bounded domain with smooth boundary. The system models Lotka-Volterra competition of two species coupled with an additional chemotactic influence. In this model each species is attracted by the signal produced by the other.We firstly show that if $n=2$ and the parameters in the system above are positive, the solution to the corresponding Neumann initial-boundary value problem, emanating from appropriately regular and nonnegative initial data, is global and bounded.Furthermore, we prove asymptotic stabilization of arbitrary global bounded solutions for any $n≥q2$ , in the sense that:• If $a_1<1$ , $a_2<1$ and both $\frac{μ_1}{χ_1^2}$ and $\frac{μ_2}{χ_2^2}$ are sufficiently large, then any global solution satisfying $u\not\equiv0\not\equiv w$ converges towards the unique positive spatially homogeneous equilibrium of the system given above.and• If $a_1≥q 1$ , $a_2<1$ and $\frac{μ_2}{χ_2^2}$ is sufficiently large any global solution satisfying $w\not\equiv0$ tends to $(0,1,1,0)$ as $t\to∞$ .