Dirac concentrations in Lotka-Volterra parabolic PDEs Guy BarlesBenoit Perthame 35B2535K5749L2592D15integral parabolic equationsadaptive dynamicsasymptotic behaviorDirac concentrationspopulation dynamics We consider parabolic partial differential equations of Lotka-Volterra type, with a non-local nonlinear term. This models, at the population level, the darwinian evolution of a population; the Laplace term represents mutations and the nonlinear birth/death term represents competition leading to selection. Once rescaled with a small diffusion, we prove that the solutions converge to a moving Dirac mass, this can be interpreted as well separated populations. The velocity and weights cannot be obtained by a simple expression, e.g., an ordinary differential equation. We show that they are given by a constrained Hamilton-Jacobi equation. This extends several earlier results to the parabolic case and to general nonlinearities. Technical new ingredients are a $BV$ estimate in time on the non-local nonlinearity, a characterization of the concentration point (in a monomorphic situation) and, surprisingly, some counter-examples showing that jumps on the Dirac locations are indeed possible. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3398 10.1512/iumj.2008.57.3398 en Indiana Univ. Math. J. 57 (2008) 3275 - 3302 state-of-the-art mathematics http://iumj.org/access/