Summary: This article considers a two competitive biological species system involving signal-dependent motilities and sensitivities and nonlinear productions \[ \begin{cases} u_t = \nabla \cdot \big (D_1(v)\nabla u-uS_1(v)\nabla v\big )+\mu_1u(1-u^{\alpha_1}-a_1w),\quad & x\in \Omega,\ t>0,\\ v_t=\Delta v-v+b_1w^{\gamma_1}, & x\in \Omega,\ t>0,\\ w_t = \nabla \cdot \big (D_2(z)\nabla w-wS_2(z)\nabla z\big )+\mu_2w(1-w^{\alpha_2}-a_2u), & x\in \Omega,\ t>0,\\ z_t=\Delta z-z+b_2u^{\gamma_2}, & x\in \Omega,\ t>0 \end{cases} \] in a bounded and smooth domain \(\Omega \subset \mathbb{R}^2\), where the parameters \(\mu_i, \alpha_i, a_i, b_i, \gamma_i\) \((i=1,2)\) are positive constants, and the functions \(D_1(v),S_1(v),D_2(z),S_2(z)\) fulfill the following hypotheses: \( \Diamond D_i(\psi),S_i(\psi)\in C^2([0,\infty)), D_i(\psi),S_i(\psi)>0\) for all \(\psi \ge 0, D_i^{\prime}(\psi)<0\) and \(\underset{\psi \rightarrow \infty}{\lim} D_i(\psi)=0; \Diamond \underset{\psi \rightarrow \infty}{\lim} \frac{S_i(\psi)}{D_i(\psi)}\) and \(\underset{\psi \rightarrow \infty}{\lim} \frac{D^{\prime}_i(\psi)}{D_i(\psi)}\) exist. We first confirm the global boundedness of the classical solution provided that the additional conditions \(2\gamma_1\le 1+\alpha_2\) and \(2\gamma_2\le 1+\alpha_1\) hold. Moreover, by constructing several suitable Lyapunov functionals, it is demonstrated that the global solution exponentially or algebraically converges to the constant stationary solutions and the corresponding convergence rates are determined under some specific stress conditions.