# American Institute of Mathematical Sciences

## Journals

DCDS-B
Discrete & Continuous Dynamical Systems - B 2017, 22(3): 763-781 doi: 10.3934/dcdsb.2017037
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.
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