Asymptotic analysis of a quantitative genetics model with nonlinear integral operator

Cálvez, Vincent; Garnier, Jimmy; Patout, Florian

doi:10.5802/jep.100

Cited by 20 publications

(36 citation statements)

References 40 publications

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“…The above sexual reproduction operator B has recently pulled the attention of both the applied and more theoretical communities, cf. [4,9,15,16,17,18]. In this paper we shall restrict to Gaussian mixing kernel and quadratic selection function, i.e.,…”

mentioning

confidence: 99%

Ergodicity of the Fisher infinitesimal model with quadratic selection

Cálvez¹,

Lepoutre²,

Poyato³

2021

Preprint

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We study the convergence towards a unique equilibrium distribution of the solutions to a time-discrete model with non-overlapping generations arising in quantitative genetics. The model describes the dynamics of a phenotypic distribution with respect to a multi-dimensional trait, which is shaped by selection and Fisher's infinitesimal model of sexual reproduction. We extend some previous works devoted to the time-continuous analogues, that followed a perturbative approach in the regime of weak selection, by exploiting the contractivity of the infinitesimal model operator in the Wasserstein metric. Here, we tackle the case of quadratic selection by a global approach. We establish uniqueness of the equilibrium distribution and exponential convergence of the renormalized profile. Our technique relies on an accurate control of the propagation of information across the large binary trees of ancestors (the pedigree chart), and reveals an ergodicity property, meaning that the shape of the initial datum is quickly forgotten across generations. We combine this information with appropriate estimates for the emergence of Gaussian tails and propagation of quadratic and exponential moments to derive quantitative convergence rates. Our result can be interpreted as a generalization of the Krein-Rutman theorem in a genuinely non-linear, and non-monotone setting.

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confidence: 99%

Ergodicity of the Fisher infinitesimal model with quadratic selection

Cálvez¹,

Lepoutre²,

Poyato³

2021

Preprint

Self Cite

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“…It describes a trait deviation of the offspring around the mean of the phenotype of the parents, drawn from a Gaussian distribution. Mathematically, few studies have tackled the operator, with the notable only exceptions of the derivation from a microscopic point of view of Barton et al (2017), the small variance and stability analysis of Calvez et al (2019); Patout (2020), and finally in Mirrahimi and Raoul (2013); Raoul (2017), with an additional spatial structure, the convergence of the model towards the Kirkpatrick-Barton model when the reproduction rate is large. In all those cases, the reproduction term is assumed to be constant.…”

Section: Discussionmentioning

confidence: 99%

The emergence of a birth-dependent mutation rate in asexuals: causes and consequences

Patout

Forien

Alfaro

et al. 2021

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In unicellular organisms such as bacteria and in most viruses, mutations mainly occur during reproduction. Thus, genotypes with a high birth rate should have a higher mutation rate. However, standard models of asexual adaptation such as the 'replicator-mutator equation' often neglect this generation-time effect. In this study, we investigate the emergence of a positive dependence between the birth rate and the mutation rate in models of asexual adaptation and the consequences of this dependence. We show that it emerges naturally at the population scale, based on a large population limit of a stochastic time-continuous individual-based model with elementary assumptions. We derive a reaction-diffusion framework that describes the evolutionary trajectories and steady states in the presence of this dependence. When this model is coupled with a phenotype to fitness landscape with two optima, one for birth, the other one for survival, a new trade-off arises in the population. Compared to the standard approach with a constant mutation rate, the symmetry between birth and survival is broken. Our analytical results and numerical simulations show that the trajectories of mean phenotype, mean fitness and the stationary phenotype distribution are in sharp contrast with those displayed for the standard model. The reason for this is that the usual weak selection limit does not hold in a complex landscape with several optima associated with different values of the birth rate. Here, we obtain trajectories of adaptation where the mean phenotype of the population is initially attracted by the birth optimum, but eventually converges to the survival optimum, following a hook-shaped curve which illustrates the antagonistic effects of mutation on adaptation.

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“…Here a female of type x ′ mates with a male of type x ′ * , chosen with the probability u t (•), the newborn is distributed with type x according to the law K(x; x ′ , x ′ * ). As often in this theory, we assume the distribution of males and females are identical, and rely on the formalism which can be found in [7,25,9] for instance. The (homogeneous) model reads…”

Section: Sexually Structured Populationsmentioning

confidence: 99%

A non-expanding transport distance for some structured equations

Fournier¹,

Perthame²

2021

Preprint

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Structured equations are a standard modeling tool in mathematical biology. They are integrodifferential equations where the unknown depends on one or several variables, representing the state or phenotype of individuals. A large literature has been devoted to many aspects of these equations and in particular to the study of measure solutions. Here we introduce a transport distance closely related to the Monge-Kantorovich distance, which appears to be non-expanding for several (mainly linear) examples of structured equations.

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Asymptotic analysis of a quantitative genetics model with nonlinear integral operator

Cited by 20 publications

References 40 publications

Ergodicity of the Fisher infinitesimal model with quadratic selection

Ergodicity of the Fisher infinitesimal model with quadratic selection

The emergence of a birth-dependent mutation rate in asexuals: causes and consequences

A non-expanding transport distance for some structured equations

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