Global dynamics in a two-species chemotaxis-competition system with two signals

In this paper, we consider a chemotaxis-competition system of parabolic-elliptic-parabolic-elliptic type $\begin{eqnarray*}\label{1}\left\{\begin{array}{llll}u_t = Δ u-χ_{1}\nabla·(u\nabla v)+μ_{1}u(1-u-a_{1}w), &x∈ Ω, ~~~t>0, \\0 = Δ v-v+w, &x∈Ω, ~~~t>0, \\w_t = Δ w-χ_{2}\nabla·(w\nabla z)+μ_{2}w(1-a_{2}u-w), &x∈ Ω, ~~~ t>0, \\0 = Δ z-z+u, &x∈Ω, ~~~t>0, \\\end{array}\right.\end{eqnarray*}$ with homogeneous Neumann boundary conditions in an arbitrary smooth bounded domain $Ω\subset R^n$, $n≥2$, where $χ_{i}$, $μ_{i}$ and $a_{i}$ $(i = 1, 2)$ are positive constants. It is shown that for any positive parameters $χ_{i}$, $μ_{i}$, $a_{i}$ $(i = 1, 2)$ and any suitably regular initial data $(u_{0}, w_{0})$, this system possesses a global bounded classical solution provided that $\frac{χ_{i}}{μ_{i}}$ are small. Moreover, when $a_{1}, a_{2}∈ (0, 1)$ and the parameters $μ_{1}$ and $μ_{2}$ are sufficiently large, it is proved that the global solution $(u, v, w, z)$ of this system exponentially approaches to the steady state $\left(\frac{1-a_{1}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{1}}{1-a_{1}a_{2}}\right)$ in the norm of $L^{∞}(Ω)$ as $t\to ∞$; If $a_{1}≥1>a_{2}>0$ and $μ_{2}$ is sufficiently large, the solution of the system converges to the constant stationary solution $\left(0, 1, 1, 0\right)$ as time tends to infinity, and the convergence rates can be calculated accurately.