Boundedness and asymptotically stability to chemotaxis system with competitive kinetics and nonlocal terms

Abstract

This paper deals with the solution of following chemotaxis system with competitive kinetics and nonlocal terms \begin{eqnarray*} \left\{ \begin{array}{llll} u_t=d_1\Delta u-\chi_1\nabla\cdot(u\nabla w)+u\left(a_0-a_1u-a_{2}v-a_3\int_\Omega u-a_4\int_\Omega v\right), &x\in \Omega, t>0,\\ v_t=d_2\Delta v-\chi_2\nabla\cdot(v\nabla w)+v\left(b_0-b_1u-b_{2}v-b_3\int_\Omega u-b_4\int_\Omega v\right), &x\in \Omega, t>0,\\ w_t=d_3\Delta w-\lambda w+k u+l v, &x\in\Omega, t>0, \end{array} \right. \end{eqnarray*} in a smoothly bounded domain $\Omega\subset \mathbb{R}^N, N\geq1$, where $a_0, a_1, a_2, b_0, b_1, b_2>0$ and $a_3, a_4, b_3, b_4\in\mathbb{R}$. The purpose of this paper is to investigate the impact on nonlocal terms of the system, and to find clear conditions on parameters such that the system possesses a unique global bounded solution. Our conclusion quantitatively suggests that the nonlocal competitions can contribute to global and uniformly bounded solutions, while the global cooperations are adverse to boundedness of system. That is: o If $a_3, a_4, b_3, b_4>0$, i.e., there are nonlocal intraspecific and interspecific competitions, when $N\leq2$, then for any positive parameters the solution of system is globally bounded; when $N\geq3$, suitable large $a_1, b_2$ (local intraspecific competitions) ensure there is no blowup. o If $a_3, a_4, b_3, b_4<0$, i.e., there are globally intraspecific and interspecific cooperations, then for any $N\geq1$, a clear and largeness condition on $a_1, b_2$ is obtained which makes the system admits a boundedness solution. Furthermore, we consider the globally asymptotically stability of spatially homogeneous equilibrium with weak and strongly asymmetric competition cases, respectively.