Title: Nash Equilibria and Concentration in Reaction-Diffusion Equations: A Hamilton-Jacobi Approach
Speaker: Adrian Lam (OSU)
Abstract: We consider an integro-PDE model for a population structured by the spatial variables and a trait variable affecting the dispersal coefficients. Competition for resource is local in spatial variables, but nonlocal in the trait variable. We focus on the asymptotic profile of positive steady state solutions. Our result shows that in the limit of small mutation rate, the solution remains regular in the spatial variables and yet concentrates in the trait variable and forms Dirac concetrations (i) at one boundary point; (ii) the interior; or (iii) at both boundary points. The main techniques are the perturbed test function approach, a Liouville result on a cylinder, and elliptic DeGiorgi-Nash-Moser estimates for the obligue derivative problem. Finally, connections to notions and concepts in evolutionary game theory will be discussed. This is joint work with Wenrui Hao (MBI) and Yuan Lou(Ohio State).
Seminar URL: https://research.math.osu.edu/pde/
