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Concentration phenomenon in some non-local equation

  • * Corresponding author: Jérôme Coville

    * Corresponding author: Jérôme Coville 

Dedicated to Professor Stephen Cantrell, with all our admiration

The research leading to these results has received funding from the french ANR program under the "ANR JCJC" project MODEVOL: ANR-13-JS01-0009 held by Gael Raoul and the "ANR DEFI" project NONLOCAL: ANR-14-CE25-0013 held by Fran¸cois Hamel. J. Coville wants to thank G. Raoul for interesting discussions on this topic.
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  • We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the integro-differential equation

    $\partial_t u(t,x)\\=\int_{\Omega }m(x,y)\left(u(t,y)-u(t,x)\right)\,dy+\left(a(x)-\int_{\Omega }k(x,y)u(t,y)\,dy\right)u(t,x),$

    supplemented by the initial condition $u(0,\cdot)=u_0$ in $\Omega $, where the domain $\Omega $ is a, the functions $k$ and $m$ are non-negative kernels satisfying integrability conditions and the function $a$ is continuous. 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, that is, the kernel $k$ is independent of $x$, ($k(x,y)=k(y)$), we construct a positive stationary solution which belongs to $d\mu$ inthe space of Radon measures on $\Omega $. $\mathbb{M}(\Omega )$.Moreover, in the case where this measure $d\mu$ is regular and bounded, we prove its uniqueness and show that, for any non-negative 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 non-unique, 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.

    Mathematics Subject Classification: Primary:35R09, 45K05;Secondary:35B40, 35B44, 92D15.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  Numerical approximation of the solution to (10) at different times for two configurations, in which only the mutation rate differs. The competition rate is constant and set to $1$ and the growth rate function achieves its maximum only at the origin while the initial datum $u_0$ is uniform with value 1. We have set $\rho=1$ for the first simulation (subfigures (A) to (D)), and $\rho=0.1$ for the second one (subfigures (E) to (H)). In both situations, we observe the convergence to a stationary solution, either to a regular measure (see subfigure (D)) or to a singular measure with one Dirac mass at the origin (see subfigure (H)), the latter being characteristic of a concentration phenomenon. In the regular case (subfigures (A) to (D)), the stationarity being attained numerically around $t=590$

    Figure 2.  Numerical approximation of the solution of problem (10)-(11) at different times for two configurations, which differ only in their initial datum. The mutation and competition rates are constant and set respectively to $2$ and $1$, the growth rate function achieves its maximum at four points and the initial datum $u_0$ is such that it vanishes on three (subfigures (A) to (D)) or two (subfigures (E) to (H)) of these points. In both cases, rapid convergence of the approximate solution towards an identical regular stationary state is observed, the numerical stationarity being attained around $t=85$

    Figure 3.  Numerical approximation of the solution of problem (10)-(11) at different times. The mutation and competition rates are constant and set respectively to $0.01$ and $1$, the growth rate function achieves its maximum at four points and the initial datum $u_0$ is such that it vanishes on three of these four points. We observe a slow convergence of the numerical solution towards the approximation of a singular stationary measure containing a single Dirac mass. The approximate solution continues to take large increasing values in a single element at $t=1000$)

    Figure 4.  Numerical approximation of the solution of problem (10)-(11) at different times. The mutation and competition rates are constant and set respectively to $0.01$ and $1$, the growth rate function achieves its maximum at four points and the initial datum $u_0$ is such that it vanishes on two of these four points. We observe a slow convergence of the numerical solution towards the approximation of a singular stationary measure containing two Dirac masses

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