We consider the discrete systems of prescribed mean curvature equations with Lane-Emden type nonlinearities in Minkowski spaces \(\begin{aligned} \left\{ \begin{array}{ll} \nabla \big (\frac{\Delta u(x)}{\sqrt{1-|\Delta u(x)|^2}}\big )+\lambda _1\mu _1(x)u^{p_1}v^{q_1}=0,\ \ \ \ x\in [1, N-1]_{\mathbb Z},\\ \nabla \big (\frac{\Delta v(x)}{\sqrt{1-|\Delta v(x)|^2}}\big )+\lambda _2\mu _2(x)u^{p_2}v^{q_2}=0,\ \ \ \ x\in [1, N-1]_{\mathbb Z},\\ u(0)=u(N)=0,\ v(0)=v(N)=0, \end{array}\right. \end{aligned}\) where \(\lambda _1, \lambda _2>0\) are real parameters, and \(\mu _1, \mu _2: [1, N-1]_{\mathbb Z}\rightarrow [0,\infty )\) are continuous functions. Based on the lower and upper solutions method and fixed point index, we prove that there exists a continuous curve \(\Gamma \) , such that the first quadrant is divided into two disjoint unbounded open sets \(\mathcal {O}_1\) and \(\mathcal {O}_2\) by this curve, and the discrete system has no positive solutions if \((\lambda _1, \lambda _2)\in \mathcal {O}_1\) , at least one positive solution if \((\lambda _1, \lambda _2)\in \Gamma \) or at least two positive solutions if \((\lambda _1, \lambda _2)\in \mathcal {O}_2\) .
