This paper deals with a two-species competition system with density-dependent motility and indirect signal production \(\begin{aligned} \left\{ \begin{array}{ll} u_t=\Delta \left( \gamma _1\left( w\right) u\right) +\mu _1u\left( 1-u-a_1v\right) ,\quad & x\in \Omega ,\quad t>0,\\ v_t=\Delta \left( \gamma _2\left( w\right) v\right) +\mu _2v\left( 1-v-a_2u\right) ,\quad & x\in \Omega ,\quad t>0,\\ w_t=\Delta w-w+z,\quad & x\in \Omega ,\quad t>0,\\ z_t=\Delta z-z+u+v,\quad & x\in \Omega ,\quad t>0, \end{array}\right. \end{aligned}\) under a smooth bounded domain \(\Omega \subset \mathbb {R}^2\) with homogeneous Neumann boundary conditions. For \(i=1,2\) , the parameters \(\mu _{i}\) and \(a_{i}\) are assumed to be positive. By some appropriate conditions on the density-dependent motility functions \(\gamma _{i}(w)\) , we establish the global boundedness of the classical solutions of the system for any \(\mu _{i}\) . Moreover, through the utilization of specific Lyapunov functionals, we establish the asymptotic stability of the solution under suitably chosen parameter conditions.
