Global Behavior in a Two-Species Chemotaxis-Competition System with Signal-Dependent Sensitivities and Nonlinear Productions

This article considers a two competitive biological species system involving signal-dependent motilities and sensitivities and nonlinear productions \(\begin{aligned} \left\{ \begin{array}{l} \begin{aligned} &{}u_t = \nabla \cdot \big (D_1(v)\nabla u-uS_1(v)\nabla v\big )+\mu _1u(1-u^{\alpha _1}-a_1w),&{} x\in \Omega ,\ t>0&{},\\ &{} v_t=\Delta v-v+b_1w^{\gamma _1}, &{} x\in \Omega ,\ t>0&{},\\ &{}w_t = \nabla \cdot \big (D_2(z)\nabla w-wS_2(z)\nabla z\big )+\mu _2w(1-w^{\alpha _2}-a_2u),&{} x\in \Omega ,\ t>0&{},\\ &{} z_t=\Delta z-z+b_2u^{\gamma _2}, &{} x\in \Omega ,\ t>0&{}\\ \end{aligned} \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}\begin{array}{cccc}& u_t=\mathrm{\nabla }·(D_1(v)\mathrm{\nabla }u-u{S}_1(v)\mathrm{\nabla }v)+{\mu }_{1}u(1-u^{{\alpha }_1}-a_{1}w),\hfill & \hfill x\in \mathrm{\Omega },t>0& ,\hfill \\ \hfill & v_t=\mathrm{\Delta }v-v+b_1{w}^{{\gamma }_1},\hfill & \hfill x\in \mathrm{\Omega },t>0& ,\hfill \\ \hfill & w_t=\mathrm{\nabla }·(D_2(z)\mathrm{\nabla }w-w{S}_2(z)\mathrm{\nabla }z)+{\mu }_{2}w(1-w^{{\alpha }_2}-a_{2}u),\hfill & \hfill x\in \mathrm{\Omega },t>0& ,\hfill \\ \hfill & z_t=\mathrm{\Delta }z-z+b_2{u}^{{\gamma }_2},\hfill & \hfill x\in \mathrm{\Omega },t>0& \hfill \\ \hfill \end{array}\hfill \end{array}\right)\end{array}$$ in a bounded and smooth domain \(\Omega \subset \mathbb R^2\) $\mathrm{\Omega }\subset {\mathbb{R}}^2$ , where the parameters \(\mu _i, \alpha _i, a_i, b_i, \gamma _i\) ${\mu }_i,{\alpha }_i,a_i,b_i,{\gamma }_i$ \((i=1,2)\) $(i=1,2)$ are positive constants, and the functions \(D_1(v),S_1(v),D_2(z),S_2(z)\) $D_1(v),S_1(v),D_2(z),S_2(z)$ fulfill the following hypotheses: \(\Diamond \) $◊$ \(D_i(\psi ),S_i(\psi )\in C^2([0,\infty ))\) $D_i(\psi ),S_i(\psi )\in C^2([0,\infty ))$ , \(D_i(\psi ),S_i(\psi )>0\) $D_i(\psi ),S_i(\psi )>0$ for all \(\psi \ge 0\) $\psi \ge 0$ , \(D_i^{\prime }(\psi )<0\) $D_i^{\prime }(\psi )<0$ and \(\underset{\psi \rightarrow \infty }{\lim } D_i(\psi )=0\) $\underset{\psi \to \infty }{lim}{D}_i(\psi )=0$ ; \(\Diamond \) $◊$ \(\underset{\psi \rightarrow \infty }{\lim } \frac{S_i(\psi )}{D_i(\psi )}\) $\underset{\psi \to \infty }{lim}\frac{S_i(\psi )}{D_i(\psi )}$ and \(\underset{\psi \rightarrow \infty }{\lim } \frac{D^{\prime }_i(\psi )}{D_i(\psi )}\) $\underset{\psi \to \infty }{lim}\frac{D_i^{\prime }(\psi )}{D_i(\psi )}$ exist. We first confirm the global boundedness of the classical solution provided that the additional conditions \(2\gamma _1\le 1+\alpha _2\) $2{\gamma }_1\le 1+{\alpha }_2$ and \(2\gamma _2\le 1+\alpha _1\) $2{\gamma }_2\le 1+{\alpha }_1$ 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.