Global existence and stability of three species predator-prey system with prey-taxis

In this paper, we study the following initial-boundary value problem of a three species predator-prey system with prey-taxis which describes the indirect prey interactions through a shared predator, i.e., $ \begin{align*} \begin{cases} u_t = d\Delta u+u(1-u)- \frac{a_1uw}{1+a_2u+a_3v}, & \; \mbox{in}\ \ \Omega, t>0, \\ v_t = \eta d\Delta v+rv(1-v)- \frac{a_4vw}{1+a_2u+a_3v}, & \; \mbox{in}\ \ \Omega, t>0, \\ w_t = \nabla\cdot(\nabla w-\chi_1 w\nabla u-\chi_2 w\nabla v) -\mu w+ \frac{a_5uw}{1+a_2u+a_3v}+\frac{a_6vw}{1+a_2u+a_3v}, & \mbox{in}\ \ \Omega, t>0, \ \ \label{II} \end{cases} \end{align*} $ under homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset \mathbb{R}^n (n \geqslant 1) $ with smooth boundary, where the parameters $ d, \eta, r, \mu, \chi_1, \chi_2, a_i > 0, i = 1, \ldots, 6. $ We first establish the global existence and uniform-in-time boundedness of solutions in any dimensional bounded domain under certain conditions. Moreover, we prove the global stability of the prey-only state and coexistence steady state by using Lyapunov functionals and LaSalle's invariance principle.