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### Open Access Journals

DCDS-B

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.
DCDS-B

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

in

, where the domain

is a, the functions

and

are non-negative kernels satisfying integrability conditions and the function

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

represents the density of individuals characterized by the trait, the domain of trait values

is a bounded subset of

, the kernels

and

respectively account for the competition between individuals and the mutations occurring in every generation, and the function

represents a growth rate. When the competition is independent of the trait, that is, the kernel

is independent of

, (

), we construct a positive stationary solution which belongs to

inthe space of Radon measures on

.

.Moreover, in the case where this measure

is regular and bounded, we prove its uniqueness and show that, for any non-negative initial datum in

, the solution of the Cauchy problem converges to this limit measure in

. 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.

$u(0,\cdot)=u_0$ |

$\Omega $ |

$\Omega $ |

$k$ |

$m$ |

$a$ |

$u$ |

$\Omega $ |

$\mathbb{R}^N$ |

$k$ |

$m$ |

$a$ |

$k$ |

$x$ |

$k(x,y)=k(y)$ |

$d\mu$ |

$\Omega $ |

$\mathbb{M}(\Omega )$ |

$d\mu$ |

$L^1(\Omega )\cap L^{\infty}(\Omega )$ |

$L^2(\Omega )$ |

## Year of publication

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