Concentration phenomenon in some non-local equation (original) (raw)

Discrete and Continuous Dynamical Systems - Series B

We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the following integro-differential equation ∂tu(t, x) = a(x) −ˆΩ k(x, y)u(t, y) dy u(t, x) +ˆΩ m(x, y)[u(t, y) − u(t, x)] dy (t, x) ∈ R+ × Ω, together with the initial condition u(0, •) = u0 in Ω. Such a problem is used in population dynamics models to capture the evolution of a clonal population structured with respect to a phenotypic trait. In this context, the function u represents the density of individuals characterized by the trait, the domain of trait values Ω is a bounded subset of R N , the kernels k and m respectively account for the competition between individuals and the mutations occurring in every generation, and the function a represents a growth rate. When the competition is independent of the trait, we construct a positive stationary solution which belongs to the space of Radon measures on Ω. Morever, when this "stationary" measure is regular and bounded, we prove its uniqueness and show that, for any non negative initial datum in L ∞ (Ω) ∩ L 1 (Ω), the solution of the Cauchy problem converges to this limit measure in L 2 (Ω). We also construct an example for which the measure is singular and non-unique, and investigate numerically the long time behaviour of the solution in such a situation. These numerical simulations seem to reveal some dependence of the limit measure with respect to the initial datum.