submit
english version rss feed
HAL: hal-00560633, version 1

Detailed view  Export this paper
Journal of Mathematical Biology (2011) 31 p.
Lévy flights in evolutionary ecology
Benjamin Jourdain 1, 2, Sylvie Méléard 3, Wojbor Woyczynski 4
(2011)

We are interested in modeling Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individual's trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a random mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. In the case we are interested in, the probability distribution of mutations has a heavy tail and belongs to the domain of attraction of a stable law. We investigate the large-population limit with allometric demographies: larger populations made up of smaller individuals which reproduce and die faster, as is typical for micro-organisms. We show that depending on the allometry coefficient the limit behavior of the population process can be approximated by nonlinear Lévy flights of different nature: either deterministic, in the form of nonlocal fractional reaction-diffusion equations, or stochastic, as nonlinear super-processes with the underlying reaction and a fractional diffusion operator. These approximation results demonstrate the existence of such nontrivial fractional objects; their uniqueness is also proved.
1:  Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique (CERMICS)
INRIA – École des Ponts ParisTech (ENPC)
2:  MATHRISK (INRIA Paris-Rocquencourt)
INRIA – École des Ponts ParisTech (ENPC) – Université Paris-Est Marne-la-Vallée (UPEMLV)
3:  Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP)
Polytechnique - X – CNRS : UMR7641
4:  Department of Statistics and Center for Stochastic and Chaotic Processes in Science and Technology
Case – Western Reserve University, Cleveland
Mathematics/Probability
Attached file list to this document: 
PDF
ecolevy28012011-fin.pdf(261.5 KB)
PS
ecolevy28012011-fin.ps(615.6 KB)

all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...