The present work investigates the Neumann initial boundary value problem for the following three-species predator–prey system \(\begin{aligned} {\left\{ \begin{array}{ll} u_t = \nabla \cdot \left( |\nabla u|^{p-2}\nabla u \right) - \chi \nabla \cdot (u \nabla v_1) - \eta \nabla \cdot (u \nabla v_2) \\ \quad + \beta u (F_1(v_1) + F_2(v_2)) - \theta u - \alpha u^r,& x\in \Omega ,t>0, \\ v_{1t} = \Delta v_1 - u F_1(v_1) + f_1(v_1, v_2),& x\in \Omega ,t>0, \\ v_{2t} = \Delta v_2 - u F_2(v_2) + f_2(v_1, v_2),& x\in \Omega ,t>0 \end{array}\right. } \end{aligned}\) under no-flux boundary conditions for \(u, v_1, v_2\) in a bounded domain \(\Omega \subset \mathbb {R}^{N}(N\ge 1)\) with smooth boundary, where \(\chi \) , \(\eta \) , \(\beta \) , \(\theta \) , \(\alpha \) , r, p are non-negative constants. For any choice of the initial datum, it is proved in this paper that the corresponding problem permits at least one global bounded weak solution provided that one of the following conditions holds: \(\begin{aligned} (i)\ r>2,\ p>1, \quad (ii)\ r>1,\ p>\frac{3N}{N+1} ,\quad (iii)\ r=2,\ p>2,\quad (iv)\ r=2, \ \alpha \ is \ suitably\ large. \end{aligned}\)
