Asymptotic estimates for the spatial segregation of competitive systems

For a class of population models of competitive type, we study the asymptotic behavior of the positive solutions as the competition rate tends to infinity. We show that the limiting problem is a remarkable system of differential inequalities, which defines the functional class S in (2). By exploiting the regularity theory recently developed in Conti et al. (Indiana Univ. Math. J., to appear) for the elements of functional classes of the form S, we provide some qualitative and regularity property of the limiting configurations. Besides, for the case of two competing species, we obtain a full description of the limiting states and we prove some quantitative estimates for the rate of convergence. Finally, we prove some new Liouville-type results which allow to have uniform regularity estimates of the solutions.