# From stochastic, individual-based models to the canonical equation of adaptive dynamics - In one step

2 Probabilités et statistiques
IECL - Institut Élie Cartan de Lorraine
3 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
Abstract : We consider a model for Darwinian evolution in an asexual population with a large but non-constant populations size characterized by a natural birth rate, a logistic death rate modelling competition and a probability of mutation at each birth event. In the present paper, we study the long-term behavior of the system in the limit of large population ($K\rightarrow\infty$) size, rare mutations ($u\rightarrow 0$), and small mutational effects ($\sigma\rightarrow 0$), proving convergence to the canonical equation of adaptive dynamics (CEAD). In contrast to earlier works, e.g. by Champagnat and Méléard, we take the three limits simultaneously, i.e. $u=u_K$ and $\sigma=\sigma_K$, tend to zero with $K$, subject to conditions that ensure that the time-scale of birth and death events remains separated from that of successful mutational events. This slows down the dynamics of the microscopic system and leads to serious technical difficulties that requires the use of completely different methods. In particular, we cannot use the law of large numbers on the diverging time needed for fixation to approximate the stochastic system with the corresponding deterministic one. To solve this problem we develop a "stochastic Euler scheme" based on coupling arguments that allows to control the time evolution of the stochastic system over time-scales that diverge with $K$.
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Article dans une revue
The Annals of Applied Probability : an official journal of the institute of mathematical statistics, The Institute of Mathematical Statistics, 2017, 27 (2), pp.1093-1170. 〈10.1214/16-AAP1227〉
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https://hal.inria.fr/hal-01188189
Contributeur : Nicolas Champagnat <>
Soumis le : mercredi 29 novembre 2017 - 12:41:57
Dernière modification le : samedi 27 janvier 2018 - 01:31:51

### Citation

Martina Baar, Anton Bovier, Nicolas Champagnat. From stochastic, individual-based models to the canonical equation of adaptive dynamics - In one step. The Annals of Applied Probability : an official journal of the institute of mathematical statistics, The Institute of Mathematical Statistics, 2017, 27 (2), pp.1093-1170. 〈10.1214/16-AAP1227〉. 〈hal-01188189〉

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