Concentration phenomenon in some nonlocal equation
Olivier Bonnefon  BioSP, INRA Centre de Recherche PACA, 228 route de l'Aérodrome, Domaine Saint Paul  Site Agroparc, 84914 AVIGNON Cedex 9, France (email) Abstract: We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the integrodifferential equation \begin{align*} &\partial_t u(t,x)\\ &=\int_{\Omega}m(x,y)\left(u(t,y)u(t,x)\right)\,dy+\left(a(x)\int_{\Omega}k(x,y)u(t,y)\,dy\right)u(t,x), \end{align*} supplemented by the initial condition $u(0,\cdot)=u_0$ in $\Omega$ . 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 $\Omega$ is a bounded subset of $\mathbb{R}^N$, 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 $\Omega$. Moreover, in the case where this measure is regular and bounded, we prove its uniqueness and show that, for any nonnegative initial datum in $L^1(\Omega)\cap L^{\infty}(\Omega)$, the solution of the Cauchy problem converges to this limit measure in $L^2(\Omega)$. We also exhibit an example for which the measure is singular and nonunique, and investigate numerically the long time behaviour of the solution in such a situation. The numerical simulations seem to reveal a dependence of the limit measure with respect to the initial datum.
Keywords: Nonlocal equation, demogenetics, concentration phenomenon, asymptotic behaviour.
Received: September 2015; Revised: April 2016; Available Online: January 2017. 
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