Lévy flights in evolutionary ecology

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• 作者：Benjamin Jourdain (1) jourdain@cermics.enpc.fr
  Sylvie Méléard (2) sylvie.meleard@polytechnique.edu
  Wojbor A. Woyczynski (3) waw@case.edu
• 关键词：Darwinian evolution – Punctuated equilibria – Mutation law with heavy tail – Birth ; death ; mutation ; competition point process – Mutation–selection dynamics – Nonlinear fractional reaction–diffusion equations – Nonlinear superprocesses with fractional diffusion
• 刊名：Journal of Mathematical Biology
• 出版年：2012
• 出版时间：October 2012
• 年：2012
• 卷：65
• 期：4
• 页码：677-707
• 全文大小：320.6 KB
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• 作者单位：1. Université Paris Est, CERMICS, 6 et 8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France2. CMAP UMR 7641, Ecole Polytechnique, Route de Saclay, Palaiseau Cedex, France3. Department of Statistics, Center for Stochastic and Chaotic Processes in Science and Technology, Case Western Reserve University, Cleveland, OH 44122, USA
• ISSN：1432-1416

文摘

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 and the corresponding diffusion admits jumps. This could be seen as an alternative to Gould and Eldredge’s model of evolutionary punctuated equilibria. 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 non-local 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 non-trivial fractional objects; their uniqueness is also proved.