Boundedness and Stabilization in a Two-Species and Two-Stimuli Chemotaxis System with Signaling Loop

This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} u_{t}=\Delta u-\chi _{1}\nabla \cdot (u\nabla v)+\mu _{1} u(1-u-a_{1}w), \quad &x\in \Omega ,\quad t>0, \\ v_{t}=\Delta v-v+w,\quad &x\in \Omega ,\quad t>0, \\ w_{t}=\Delta w-\chi _{2}\nabla \cdot (w\nabla z)-\chi _{3}\nabla \cdot (w\nabla v)+\mu _{2} w(1-w-a_{2}u),\quad &x\in \Omega ,\quad t>0, \\ z_{t}=\Delta z-z+u,\quad &x\in \Omega ,\quad t>0 \end{array}\displaystyle \right . \end{aligned}$$ in a bounded domain $\Omega \subset \mathbb{R}^{n}$ with $n\geq 1$ , where $\chi _{1},\chi _{2},\chi _{3}>0$ , $a_{1},a_{2}>0$ and $\mu _{1},\mu _{2}>0$ . The system models the communication between macrophages and breast tumor cells. It will be proved that if $n\leq 2$ , then for all appropriately regular nonnegative initial data $u_{0}, v_{0}, w_{0}$ and $z_{0}$ , the solution to the corresponding Neumann initial-boundary value problem is global and bounded. Moreover, the asymptotic stabilization of arbitrary global bounded solutions for any $n\geq 1$ under some explicit conditions will be investigated.