In the present research, we consider a boundary layer originated system of reaction diffusion problems whose boundary conditions are of mixed type. This problem is singularly perturbed nature and the diffusion parameters are considered to be of different magnitude. We develop two adaptive methods based on r-refinement strategy, which move the mesh points toward the boundary layers. Here, we use a curvature based monitor function for adaptive moving mesh generation. Based on a combination of forward and backward difference schemes on this adaptively generated equidistributed mesh, we obtain a first order uniformly accurate solution. The discrete solution can be enhanced to a second order uniform accuracy by our proposed cubic spline based scheme. Numerically, we provide a comparison of our present method on the proposed adaptive moving meshes with the commonly used meshes. Experiments show the highly effective behavior of the present method.