L\'evy flights in ecology

TL;DR

This paper models evolutionary dynamics using stochastic processes with heavy-tailed mutation distributions, revealing that large-population limits can be described by nonlinear fractional equations or super-processes, highlighting the role of Le9vy flights.

Contribution

It introduces a novel framework linking heavy-tailed mutation distributions to fractional PDEs and super-processes in evolutionary modeling.

Findings

Large populations can be approximated by nonlinear Le9vy flights.

The limit behaviors include deterministic fractional equations and stochastic super-processes.

Existence and uniqueness of these fractional objects are established.

Abstract

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 L\'evy 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.