Global boundedness in a two-competing-predator and one-prey system with indirect prey-taxis

This paper is concerned with a three-species predator-prey model with indirect prey-taxis$ \begin{equation*} \begin{aligned} \left\lbrace \begin{split} &u_t = \Delta u -\chi_{1}\nabla\cdot\left(u\nabla z\right)+a_{1}uw-u(\theta+b_{1}u)-\alpha_{1}uv, & x \in \varOmega, t>0, \\ &v_t = \Delta v -\chi_{2}\nabla\cdot\left(v\nabla z\right)+a_{2}vw-v(\theta+b_{2}v)-\alpha_{2}uv , & x \in \varOmega, t>0, \\ &{w_t} = \Delta w-uw-vw+\beta w(1-w), & x \in \varOmega, t>0 , \\ &\tau z_{t} = \Delta z-z+w, & x \in \varOmega, t>0 , \ \end{split} \right. \end{aligned} \end{equation*} $under homogeneous Neumann boundary conditions in a smoothly bounded domain $ \varOmega \subset \mathbb{R}^{n}(n\geq1) $, where $ \tau \in\{0, 1\} $ and the parameters $ \chi_{1}, \chi_{2}, a_{1}, a_{2}, \theta, b_{1}, \ b_{2}, $ $ \alpha_{1}, \alpha_{2}, \beta $ are positive. First, under the assumption on the arbitrary spatial dimension $ n $, we prove that the system admits a unique globally bounded classical solution when $ \tau = 0 $. Moreover, for the system with $ \tau = 1 $ and $ n = 2 $, the system possesses a classical solution, which is global in time and bounded. Finally, when $ b_{i}\geq11\chi_{i}^2+\frac{a_{i}}{2}+1(i = 1, 2) $, we study the global existence and boundedness of classical solutions to the above system in a three-dimensional bounded domain.