Summary: We study the existence of positive solutions of a population model with diffusion of the form \[ \begin{cases}-\Delta_pu = au^{p-1} - f(u) - \dfrac{c}{u^{\alpha}}, & x \in \Omega, \\ u=0, & x \in \partial\Omega,\end{cases} \] where \(\Delta_{p}\) denotes the \(p\)-Laplacian operator defined by \(\Delta_{p}z = \operatorname{div}(|\nabla z|^{p-2}\nabla z)\), \(p > 1\), \(\Omega\) is a bounded domain of \(\mathbb{R}^{N}\) with smooth boundary, \(\alpha \in (0, 1)\), \(a\) and \(c\) are positive constants. Here, \(f : [0, \infty) \to \mathbb{R}\) is a continuous function. This model arises in the studies of population biology of one species with \(u\) representing the concentration of the species. We discuss the existence of positive solution when \(f\) satisfies certain additional conditions. We use the method of sub- and super-solutions to establish our results.