Global dynamics in a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop

This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop

$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \Delta u-\chi_1\nabla\cdot(u\nabla v)+\mu_1 u(1-u-a_1w),\, x\in \Omega,\, t>0,\\ 0 = \Delta v-v+w,\,x\in\Omega,\, t>0,\\ w_t = \Delta w-\chi_2\nabla\cdot(w\nabla z)-\chi_3\nabla\cdot(w\nabla v)+\mu_2 w(1-w-a_2u), \,x\in \Omega,\,t>0,\\ 0 = \Delta z-z+u, \,x\in\Omega,\, t>0, \end{array} \right. \end{eqnarray*} $

under homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset \mathbb{R}^n $ with $ n\geq1 $, where the parameters $ a_1,a_2 $, $ \chi_1, \chi_2, \chi_3 $, $ \mu_1, \mu_2 $ are positive constants. We first showed some conditions between $ \frac{\chi_1}{\mu_1} $, $ \frac{\chi_2}{\mu_2} $, $ \frac{\chi_3}{\mu_2} $ and other ingredients to guarantee boundedness. Moreover, the large time behavior and rates of convergence have also been investigated under some explicit conditions.