Global Attractivity and Singular Perturbation for a Class of Nonlinear Cooperative Systems

In the first part of this paper it is proved a general principle for reaction-diffusion cooperative systems with concave nonlinearities. It is proved that there exists a unique coexistence state (u, v) if, and only if, the trivial solution is linearly unstable. In this case, (u, v) is globally asymptotically stable with respect to positive initial data. Moreover, if the trivial solution is linearly stable, it is globally asymptotically stable with respect to non-negative initial data. In the second part it is investigated the singular perturbation problem, it is shown that the positive solution of the diffusion model tends to the positive equilibrium of the purely kinetic model as the diffusion coefficients tend to zero, uniformly in compact subsets of the domain.