Global dynamics for a two-species chemotaxis-competition system with loop and nonlocal kinetics

In this paper, we consider the two-species chemotaxis-competition system with loop and nonlocal kinetics$\{\begin{array}{cc}{u}_t=\Delta u-{\chi }_{11}\nabla \cdot \left(u\nabla v\right)-{\chi }_{12}\nabla \cdot \left(u\nabla z\right)+f_1(u,w),& x\in \Omega ,t>0,\\ 0=\Delta v-v+u+w,& x\in \Omega ,t>0,\\ w_t=\Delta w-{\chi }_{21}\nabla \cdot \left(w\nabla v\right)-{\chi }_{22}\nabla \cdot \left(w\nabla z\right)+f_2(u,w),& x\in \Omega ,t>0,\\ 0=\Delta z-z+u+w,& x\in \Omega ,t>0,\end{array}$ subject to homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset R^n(n\ge 1)$, where ${\chi }_{ij}>0(i,j=1,2)$, $f_1(u,w)=u(a_0-a_{1}u-a_{2}w-a_3{\int }_{\Omega }udx-a_4{\int }_{\Omega }wdx)$, $f_2(u,w)=w(b_0-b_{1}u-b_{2}w-b_3{\int }_{\Omega }udx-b_4{\int }_{\Omega }wdx)$ with $a_i,b_i>0(i=0,1,2),a_j,b_j\in R(j=3,4)$. It is shown that if the parameters satisfy certain conditions, then the corresponding initial boundary value problem admits a unique global-in-time classical solution in any spatial dimension, which is uniformly bounded. Moreover, based on the construction of suitable energy functionals, the globally asymptotic stabilization of coexistence and semi-coexistence steady states is considered. Our results generalize and improve some previous results in the literature.