In this talk we will present some two phase free boundary problems arising from population dynamics. We will focus on systems with fully nonlinear diffusion and local interaction, and linear systems with a (non local) long range interaction. In the long range model, the growth of a population ui at x is inhibited by the populations uj in a full area surrounding x. This will force the populations to stay at distance 1 from each other in the limit configuration, so the free boundary will be a strip along the support of the population with size exactly one. This is a joint work with Luis Caffarelli and Stefania Patrizi. We will also motivate the need of a system of partial differential equations that models the propagation of activity in the brain, incorporating the role of the neurons as well as the volume propagation. This is a joint work with Aaron Yip, Zoltan Nadasdy and Silvia Barbeiro.
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