This paper deals with a two-competing-species chemotaxis system with indirect signal production $$\left\{\begin{array}{cc}{u}_t=d_1\mathrm{\Delta }u-{\chi }_1\nabla \cdot (u\nabla w)+f_1(u,v),\hfill & (x,t)\in \mathrm{\Omega }\times (0,\infty ),\hfill \\ v_t=d_2\mathrm{\Delta }v-{\chi }_2\nabla \cdot (v\nabla w)+f_2(u,v),\hfill & (x,t)\in \mathrm{\Omega }\times (0,\infty ),\hfill \\ w_t=\mathrm{\Delta }w-w+z,\hfill & (x,t)\in \mathrm{\Omega }\times (0,\infty ),\hfill \\ z_t=\mathrm{\Delta }z-z+u+v,\hfill & (x,t)\in \mathrm{\Omega }\times (0,\infty ),\hfill \end{array}\right.$$ under homogeneous Neumann boundary conditions in a smoothly bounded domain $\mathrm{\Omega }\subset {ℝ}^n(n\ge 1)$, with the nonnegative initial data $(u_0,v_0,w_0,z_0)$, where $d_i>0$ and ${\chi }_i>0$, $i=1,2$.

• When $f_1(u,v)={\mu }_{1}u(1-u-a_{1}v),f_2(u,v)={\mu }_{2}v(1-v-a_{2}u)$, we study the global existence and boundedness of classical solutions to the above system in a three-dimensional bounded domain under some suitable conditions of parameters.

• When $f_1(u,v)=u(a_0-a_{1}u-a_{2}v-a_3{\int }_{\mathrm{\Omega }}u-a_4{\int }_{\mathrm{\Omega }}v)$, $f_2(u,v)=v(b_0-b_{1}u$$-b_{2}v-b_3{\int }_{\mathrm{\Omega }}u-b_4{\int }_{\mathrm{\Omega }}v)$, we discuss the existence of globally bounded solution for all $n\le 3$ provided that the parameters satisfy some suitable assumptions. Furthermore, we consider the asymptotic stabilization of globally bounded solutions to the above system under weak competition and strongly asymmetric competition cases, respectively.

本文处理具有间接信号产生的两个竞争物种趋化系统$$\left\{\begin{array}{cc}{你}_{吨}=d_1\mathrm{\Delta }你-{\chi }_1\nabla \cdot (你\nabla w)+F_1(你,v),\hfill & (X,吨)\in \mathrm{\Omega }\times (0,\infty ),\hfill \\ v_{吨}=d_2\mathrm{\Delta }v-{\chi }_2\nabla \cdot (v\nabla w)+F_2(你,v),\hfill & (X,吨)\in \mathrm{\Omega }\times (0,\infty ),\hfill \\ w_{吨}=\mathrm{\Delta }w-w+z,\hfill & (X,吨)\in \mathrm{\Omega }\times (0,\infty ),\hfill \\ z_{吨}=\mathrm{\Delta }z-z+你+v,\hfill & (X,吨)\in \mathrm{\Omega }\times (0,\infty ),\hfill \end{array}\right.$$在平滑有界域中的齐次 Neumann 边界条件下$\mathrm{\Omega }\subset {ℝ}^n(n\ge 1)$，具有非负初始数据$({你}_0,v_0,w_0,z_0)$， 在哪里$d_{\mathrm{一世}}>0$和${\chi }_{\mathrm{一世}}>0$,$\mathrm{一世}=1,2$.

• 什么时候$F_1(你,v)={\mu }_1你(1-你-{\mathrm{一个}}_{1}v),F_2(你,v)={\mu }_{2}v(1-v-{\mathrm{一个}}_2你)$，我们研究了在一些合适的参数条件下，在三维有界域中，上述系统的经典解的全局存在性和有界性。

• 什么时候$F_1(你,v)=你({\mathrm{一个}}_0-{\mathrm{一个}}_1你-{\mathrm{一个}}_{2}v-{\mathrm{一个}}_3{\int }_{\mathrm{\Omega }}你-{\mathrm{一个}}_4{\int }_{\mathrm{\Omega }}v)$,$F_2(你,v)=v(b_0-b_1你$$-b_{2}v-b_3{\int }_{\mathrm{\Omega }}你-b_4{\int }_{\mathrm{\Omega }}v)$，我们讨论所有全局有界解的存在$n\le 3$前提是参数满足一些合适的假设。此外，我们分别考虑了在弱竞争和强不对称竞争情况下上述系统的全局有界解的渐近稳定。