Convergence of an infinite dimensional stochastic process to a spatially structured trait substitution sequence

Leman, Hélène

doi:10.1007/s40072-016-0077-y

Cited by 7 publications

(6 citation statements)

References 33 publications

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“…Similar convergence results have been shown for many variations of the original individual-based model under the same scaling, including small mutational effects, fast phenotypic switches, spatial aspects, and also diploid organisms [2,1,13,30,23,15,26].…”

Section: Introductionsupporting

confidence: 77%

From adaptive dynamics to adaptive walks

Kraut

Bovier

2019

J. Math. Biol.

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We consider an asexually reproducing population on a finite trait space whose evolution is driven by exponential birth, death and competition rates, as well as the possibility of mutation at a birth event. On the individual-based level this population can be modeled as a measure-valued Markov process. Multiple variations of this system have been studied in the limit of large populations and rare mutations, where the regime is chosen such that mutations are separated. We consider the deterministic system, resulting from the large population limit, and let the mutation probability tend to zero. This corresponds to a much higher frequency of mutations, where multiple subcritical traits are present at the same time. The limiting process is a deterministic adaptive walk that jumps between different equilibria of coexisting traits. The graph structure on the trait space, determined by the possibilities to mutate, plays an important role in defining the adaptive walk. In a variation of the above model, where the radius in which mutants can be spread is limited, we study the possibility of crossing valleys in the fitness landscape and derive different kinds of limiting walks.2010 Mathematics Subject Classification. 37N25, 60J27, 92D15, 92D25.

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Section: Introductionsupporting

confidence: 77%

From adaptive dynamics to adaptive walks

Kraut

Bovier

2019

J. Math. Biol.

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“…and |j| is the number of terms in j (for instance |11221| = 5). These processes can be constructed in such a way that, for all j ∈ ∪ n∈N N n , i ∈ N, 36) where N (i) j (t) is the number of mutants of type ( α + |j| + 2) produced by the X (i) j population (which is of type ( α + |j| + 1)) until time t. Recall that among the offsprings produced by the population X (i) j , a fraction (1 − µ) is constituted by newborn individuals of type α + |j| + 1, and a fraction µ by new born individuals of type α + |j| + 2, and that at each birth event the probability to have a mutation is independent from the past.…”

Section: Poisson Representationmentioning

confidence: 99%

Crossing a fitness valley as a metastable transition in a stochastic population model

Bovier¹,

Coquille²,

Smadi³

2019

Ann. Appl. Probab.

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We consider a stochastic model of population dynamics where each individual is characterised by a trait in {0, 1, ..., L} and has a natural reproduction rate, a logistic death rate due to age or competition, and a probability of mutation towards neighbouring traits at each reproduction event. We choose parameters such that the induced fitness landscape exhibits a valley: mutant individuals with negative fitness have to be created in order for the population to reach a trait with positive fitness. We focus on the limit of large population and rare mutations at several speeds. In particular, when the mutation rate is low enough, metastability occurs: the exit time of the valley is an exponentially distributed random variable.2010 Mathematics Subject Classification. 92D25, 60J80, 60J27, 92D15, 60F15, 37N25.

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“…The class of stochastic individual-based models with competition and varying population size we are studying have been introduced in [6,20] and made rigorous in a probabilistic setting in the seminal paper of Fournier and Méléard [22]. Then they have been studied by many authors (see [10,11,18,41] and references therein for instance). Initially restricted to asexual populations, such models have evolved to incorporate the case of sexual reproduction, in both haploid [63] and diploid [15,16,48] populations.…”

Section: Introductionmentioning

confidence: 99%

A stochastic model for speciation by mating preferences

et al. 2017

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Abstract. Mechanisms leading to speciation are a major focus in evolutionary biology. In this paper, we present and study a stochastic model of population where individuals, with type a or A, are equivalent from ecological, demographical and spatial points of view, and differ only by their mating preference: two individuals with the same genotype have a higher probability to mate and produce a viable offspring. The population is subdivided in several patches and individuals may migrate between them. We show that mating preferences by themselves, even if they are very small, are enough to entail reproductive isolation between patches, and we provide the time needed for this isolation to occur as a function of the population size. Our results rely on a fine study of the stochastic process and of its deterministic limit in large population, which is given by a system of coupled nonlinear differential equations. Besides, we propose several generalisations of our model, and prove that our findings are robust for those generalisations.

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Convergence of an infinite dimensional stochastic process to a spatially structured trait substitution sequence

Cited by 7 publications

References 33 publications

From adaptive dynamics to adaptive walks

From adaptive dynamics to adaptive walks

Crossing a fitness valley as a metastable transition in a stochastic population model

A stochastic model for speciation by mating preferences

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