The following two-species chemotaxis system with signal-dependent motilities and indirect signal consumption \(\begin{aligned} \left\{ \begin{array}{llll} u_{t}=\Delta \left( \gamma _{1}\left( w\right) u\right) +\mu _{1}u\left( 1-u-a_{1}v\right) \!,\quad & x\in \Omega ,\quad t>0,\\ v_{t}=\Delta \left( \gamma _{2}\left( w\right) v\right) +\mu _{2}v\left( 1-a_{2}u-v\right) \!,\quad & x\in \Omega ,\quad t>0,\\ w_{t}=\Delta w-wz,\quad & x\in \Omega ,\quad t>0,\\ z_{t}=\Delta z-z+u+v,\quad & x\in \Omega ,\quad t>0,\\ \end{array} \right. \end{aligned}\) is considered in a smooth bounded domain \(\Omega \in \mathbb {R}^n(n\ge 1)\) with homogeneous Neumann boundary conditions, where the parameters \(a_i>0\) , \(\mu _i\ge 0 (i=1, 2)\) and the motility functions satisfy that \(\gamma _i(w)\in C^3([0, \infty ]), \gamma _i(w)>0 \) for all \(w\ge 0\) . It is proved that the associated initial-boundary value problem possesses a unique global and uniformly bounded classical solution in higher space dimensions for \(\mu _i>0 \) properly large and in any dimension for small \(||w_0||_{L^\infty (\Omega )}\) with \(\mu _i=0\) . Furthermore, it is shown that the global bounded solution of this system converges to different steady states under the different values of \(\mu _i\) and \(a_i\) . Meanwhile, the precise convergence rates of global solutions are derived in the paper.
