In this paper, we consider the following forager-exploiter system \(\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u-\chi \nabla \cdot (u\nabla w), & x\in \Omega ,\,\,t>0, \\ v_t=\Delta v-\xi \nabla \cdot (v\nabla u)+av-bv^{\beta }, & x\in \Omega ,\,\,t>0,\\ w_t=\Delta w-\frac{u+v}{(1+u+v)^\gamma }w-\mu w+r(x,t),& x\in \Omega ,\,\,t>0, \end{array}\right. } \end{aligned}\) in a smooth bounded domain \(\Omega \subset \mathbb {R}^n\,\,(n\ge 2)\) with homogeneous Neumann boundary conditions, where \(\chi ,\,\xi ,\,\gamma ,\,\mu >0\) , and r is a given, non-negative function. It is shown that the initial-boundary value problem admits a unique global bounded classical solution if \(\beta >\frac{(n+2)(1-\gamma )}{2}\) under the condition that \(0<\gamma <\frac{n}{n+2}\) , or if \(\beta >1\) under the condition that \(\gamma \ge \frac{n}{n+2}\) . Furthermore, the asymptotic behaviour can be thoroughly investigated and analysed by imposing additional hypotheses on r(x, t).
