Inside dynamics of solutions of integro-differential equations

Pages: 3057 - 3085,
Volume 19,
Issue 10,
December
2014 doi:10.3934/dcdsb.2014.19.3057

Olivier Bonnefon - INRA, UR 546 Biostatistique et Processus Spatiaux (BioSP), F-84914 Avignon, France (email)

Jérôme Coville - INRA, UR 546 Biostatistique et Processus Spatiaux (BioSP), F-84914 Avignon, France (email)

Jimmy Garnier - INRA, UR 546 Biostatistique et Processus Spatiaux (BioSP), F-84914 Avignon, France (email)

Lionel Roques - INRA, UR 546 Biostatistique et Processus Spatiaux (BioSP), F-84914 Avignon, France (email)

Abstract:
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: Traveling waves, integro-differential equation, pulled and pushed solutions, monostable, long distance dispersal, thin-tailed/fat-tailed kernel.

Mathematics Subject Classification: Primary: 35R09, 45K05; Secondary: 35B06, 35K57.

Received: July 2013;
Revised:
January 2014;
Available Online: October 2014.

References