Asymptotic stability of the two-dimensional two-species attraction-repulsion chemotaxis system with two chemicals

We were concerned with the following two-species attraction-repul-sion chemotaxis system with two chemicals$ \begin{equation} \begin{cases} u_{t} = \triangle u+\chi\nabla\cdot(u\nabla z)-\xi\nabla\cdot(u\nabla v), &(x, t)\in\Omega\times(0, \infty), \\ v_{t} = \triangle v-\beta v+\alpha\omega, &(x, t)\in\Omega\times(0, \infty), \\ \omega_{t} = \triangle\omega+\tilde{\chi}\nabla\cdot(\omega\nabla v)-\tilde{\xi}\nabla\cdot(\omega\nabla z), &(x, t)\in\Omega\times(0, \infty), \\ z_{t} = \triangle z-\delta z+\gamma u, &(x, t)\in\Omega\times(0, \infty), \end{cases} \end{equation} \;\;\;\;\;\;\;\;\;\;\;\;(1) $in a smoothly bounded domain $ \Omega\subset \mathbb{R}^{2} $ subject to homogeneous Neumann boundary conditions. By constructing Lyapunov function and its differential governing inequality, for the first time we showed that the solutions with initial data $ (u_{0}, v_{0}, \omega_{0}, z_{0})\in C^{0}(\bar{\Omega})\times C^{1}(\bar{\Omega})\times C^{0}(\bar{\Omega})\times C^{1}(\bar{\Omega}) $ converged exponentially to the constant steady state $ (\bar{u}_{0}, \frac{\alpha}{\beta}\bar{\omega}_{0}, \bar{\omega}_{0}, \frac{\gamma}{\delta}\bar{u}_{0}) $ as $ t\rightarrow \infty $ under the case of $ \frac{\chi\tilde{\chi}}{\xi\tilde{\xi}}>\max\{\frac{\beta}{\delta}, \frac{\delta}{\beta}\} $ for all positive parameters $ \chi, \xi, \alpha, \beta, \gamma, \delta $. Here, $ \bar{u}_{0} = \frac{1}{|\Omega|}\int_{\Omega}u_{0} $ and $ \bar{\omega}_{0} = \frac{1}{|\Omega|}\int_{\Omega}\omega_{0} $.