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

Baar, Martina; Bovier, Anton; Champagnat, Nicolas

doi:10.1214/16-aap1227

Cited by 16 publications

(24 citation statements)

References 22 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: 78%

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: 78%

From adaptive dynamics to adaptive walks

Kraut

Bovier

2019

J. Math. Biol.

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“…Let us now introduce a finite subset of N containing the equilibrium size of the 0-population, 3) and the stopping times T K ε and S K ε , which denote respectively the hitting time of εK by the total mutant population (X 1 + ... + X L ) and the exit time of I K ε by the resident 0-population,…”

Section: Poisson Representationmentioning

confidence: 99%

“…A rigorous derivation of the theory was achieved over the last decade in the context of stochastic individual-based models, where the evolution of a population of individuals characterised by their phenotypes under the influence of the evolutionary mechanisms of birth, death, mutation, and ecological competition in an inhomogeneous "fitness landscape" is described as a measure valued Markov process. Using various scaling limits involving large population size, small mutation rates, and small mutation steps, key features described in the biological theory of adaptive dynamics, in particular the canonical equation of adaptive dynamics (CEAD), the trait substitution sequence (TSS), and the polymorphic evolution sequence (PES) were recovered, see [15,14,28,16,17,3]. Extensions of those results for more structured populations were investigated, for example, in [47,36].…”

Section: Introductionmentioning

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 random variables A, A 1,K, and A 2,K, can easily be constructed by using the pathwise description of ν K (cf. [3] or [9]).…”

Section: )mentioning

confidence: 99%

The polymorphic evolution sequence for populations with phenotypic plasticity

Baar¹,

Bovier²

2018

Electron. J. Probab.

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In this paper we study a class of stochastic individual-based models that describe the evolution of haploid populations where each individual is characterised by a phenotype and a genotype. The phenotype of an individual determines its natural birth-and death rates as well as the competition kernel, c(x, y) which describes the induced death rate that an individual of type x experiences due to the presence of an individual or type y. When a new individual is born, with a small probability a mutation occurs, i.e. the offspring has different genotype as the parent. The novel aspect of the models we study is that an individual with a given genotype may express a certain set of different phenotypes, and during its lifetime it may switch between different phenotypes, with rates that are much larger then the mutation rates and that, moreover, may depend on the state of the entire population. The evolution of the population is described by a continuous-time, measure-valued Markov process. In [4], such a model was proposed to describe tumor evolution under immunotherapy. In the present paper we consider a large class of models which comprises the example studied in [4] and analyse their scaling limits as the population size tends to infinity and the mutation rate tends to zero. Under suitable assumptions, we prove convergence to a Markov jump process that is a generalisation of the polymorphic evolution sequence (PES) as analysed in [8,10].

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

Cited by 16 publications

References 22 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

The polymorphic evolution sequence for populations with phenotypic plasticity

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