Our interest is to study the existence of positive solutions to the following elliptic system involving competing interaction $$ (1) { -partial(x,u,upsilon)Delta u = uf(x,u,v) { - psi(x,u,upsilon)Delta upsilon = upsilon g(x,u,upsilon) { frac{partial n}{partial u} + ku = 0 on partialOmega { frac{partial n}{partialupsilon} + sigmaupsilon = 0 $$ in a bounded region $Omega$ in $R^n$ with a smooth boundary, where the diffusion terms $varphi, psi$ are strictly positive nondecreasing function, and k, $sigma$ are positive constants. Also we assume that the growth rates f, g are $C^1$ monotone functions. The variables u, $upsilon$ may represent the population densities of the interacting species in problems from ecology, microbiology, immunology, etc.