We consider patch-based population dynamics, called metapopulations, and investigate the spread of two species and their competition for habitat areas. Let 0 ≤ p1,p2 denote the proportion (density) of the areas occupied by the species 1 and 2, respectively. Obviously, 0 ≤ p1 + p2 ≤ 1. Both species can locally extinct and colonize empty patches or patches occupied by the other with different rates. Hence the model is P1′ = κ1P1 (1 - P1 - P2) - e1p1p2-C1P1P2 - C2P1P2 P2′ = κ1P1 (1 - P1 - P2) - e1p1p2-C1P1P2 - C2P1P2 where ej are the extinction rates, κ1 are the rates of colonization of empty patches, and species "i" overcolonize "j" with the rate c1(i,j=1,2; i≠j) We investigate the asymptotic stability properties of the equilibria of such systems and we give a complete characterization of the parameter space. We prove that there can appear a globally asymptotically stable coexisting equilibrium. On the other hand, this equilibrium can become unstable and then the exclusive equilibria are only locally asymptotically stable. We have also developed dynamic applications in Wolfram Mathematica to illustrate our results.