In this work we study the existence and qualitative properties of non-negative solutions of the Lotka—Volterra models with nonlinear diffusion under homogeneous Dirichlet boundary conditions. We consider the three typical interactions: prey—predator, competition and symbiosis. Unlike the linear diffusion models, non-trivial non-negative solutions can exist which are not strictly positive. Sufficient conditions in terms of the coefficients involved in the setting of the models are given, assuring that one species (or both) does not survive on a set of its habitat (called ‘dead core’) of positive measure.