Global boundedness of a three species predator-prey model with power-like interspecific interaction

In this paper, we study the global existence and boundedness of a Neumann initial-boundary value problem for the following three-species predator-prey model with power-like inter-specific interaction mechanism and prey-taxis:

$ \begin{align*} \begin{cases} u_t = d_1\Delta u+u(1-u)-b_1u^{1+\alpha}v^{1-\alpha}, \; &x\in{\Omega}, t>0, \\ v_t = d_2\Delta v-\nabla\cdot(\xi v\nabla u)+u^{1+\alpha}v^{1-\alpha}-b_2v^{1+\beta}w^{1-\beta}-\theta_1v, \; &x\in{\Omega}, t>0, \\ w_t = \Delta w-\nabla\cdot(\chi w\nabla v)+v^{1+\beta}w^{1-\beta}-\theta_2w, \; &x\in{\Omega}, t>0, \ \end{cases} \end{align*} $

in a bounded smooth domain $ \Omega\subset \mathbb{R}^2 $. The inter-specific interaction of the forms $ b_1u^{1+\alpha}v^{1-\alpha} $ and $ b_2v^{1+\beta}w^{1-\beta} $ with $ \alpha\in (0, 1) $, $ \beta\in(0, 1) $ is a natural extension of the classical predator-prey types $ b_1uv $ and $ b_2vw $. By delicate coupling energy estimates, we first establish the global existence of classical solutions in two dimensional spaces for appropriate initial data.