DCDS-B
Inside dynamics of solutions of integro-differential equations
Olivier Bonnefon Jérôme Coville Jimmy Garnier Lionel Roques
In this paper, we investigate the inside dynamics of the positive solutions of integro-differential equations \begin{equation*} \partial_t u(t,x)= (J\star u)(t,x) -u(t,x) + f(u(t,x)), \ t>0 \hbox{ and } x\in\mathbb{R}, \end{equation*} with both thin-tailed and fat-tailed dispersal kernels $J$ and a monostable reaction term $f.$ The notion of inside dynamics has been introduced to characterize traveling waves of some reaction-diffusion equations [23]. Assuming that the solution is made of several fractions $\upsilon^i\ge 0$ ($i\in I \subset \mathbb{N}$), its inside dynamics is given by the spatio-temporal evolution of $\upsilon^i$. According to this dynamics, the traveling waves can be classified in two categories: pushed and pulled waves. For thin-tailed kernels, we observe no qualitative differences between the traveling waves of the above integro-differential equations and the traveling waves of the classical reaction-diffusion equations. In particular, in the KPP case ($f(u)\leq f'(0)u$ for all $u\in(0,1)$) we prove that all the traveling waves are pulled. On the other hand for fat-tailed kernels, the integro-differential equations do not admit any traveling waves. Therefore, to analyse the inside dynamics of a solution in this case, we introduce new notions of pulled and pushed solutions. Within this new framework, we provide analytical and numerical results showing that the solutions of integro-differential equations involving a fat-tailed dispersal kernel are pushed. Our results have applications in population genetics. They show that the existence of long distance dispersal events during a colonization tend to preserve the genetic diversity.
keywords: long distance dispersal integro-differential equation monostable pulled and pushed solutions thin-tailed/fat-tailed kernel. Traveling waves
DCDS-B
Concentration phenomenon in some non-local equation
Olivier Bonnefon Jérôme Coville Guillaume Legendre
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.
keywords: Non-local equation demo-genetics concentration phenomenon asymptotic behaviour

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