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