Summary: The problem of interest is a discrete reaction-diffusion equation motivated by models in population biology. We consider \[ Au+\phi (u)+\lambda f(u)=0 \quad \text{for } u\in \mathbb{R}^{n-1}, \] where \(n\geq 3, A\) is an \((n-1) \times (n-1)\) matrix such that \(-A\) is monotone, \(\phi :\mathbb{R}^{n-1} \rightarrow \mathbb{R}^{n-1}\) and \(f:\mathbb{R}^{n-1}\rightarrow \mathbb{R}^{n-1}\) are smooth functions, and \(\lambda\) is a positive real constant. Of particular interest is the case where \(A\) is the discrete Laplacian and \(f\) is the vector-valued logistic function. The function \(\phi (u)\) will encode boundary conditions. Our primary goal is to establish the existence of nonnegative solutions for several interesting choices of \(\phi\). For each choice we use monotonicity methods to find nonnegative solutions for appropriate ranges of \(\lambda\).