Kingman’s model with random mutation probabilities: convergence and condensation I

Yuan, Linglong

doi:10.1017/apr.2021.33

Cited by 5 publications

(8 citation statements)

References 20 publications

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“…In this paper, we continue to study the equilibrium and the condensation phenomenon in the first random model. By Corollary 4 in [21], if Q(h) = 0, then I(h) > 0 a.s. or I(h) = 0 a.s.. We say there is a condensation on {h} in the first random model if Q(h) = 0 but I(h) > 0 a.s.. We call I(h) the condensate size on {h} if Q(h) = 0. A condensation criterion, which relies on a function of β and I, was established in [21].…”

Section: Two Random Modelsmentioning

confidence: 92%

“…Then by Theorem 4 in [21], H j d = I j,SQ . By ( 5) and ( 20), for both sequences, the multi-dimensional distributions are determined in the same way by one dimensional distribution.…”

Section: Proofs Of Theorem 3 and Corollarymentioning

confidence: 95%

“…This section is mainly a summarisation of Section 2 in [21], in addition to the introduction of a new random model where all mutation probabilities are equal but random.…”

Section: Modelsmentioning

confidence: 99%

“…If b n = β n for any n we call it Kingman's model with random mutation probabilities or simply the first random model. It has been proved in [21] that (P n ) converges weakly to a globally stable equilibrium, that we denote by I whose distribution depends on β, Q, h but not on P 0 .…”

Section: Two Random Modelsmentioning

confidence: 99%

“…Recall that E 1−β yIQ exists and does not depend on the joint law of β, I Q . Using (21) in the first random model, together with Corollary 1, we can rewrite Theorem 2 into Corollary 2.…”

Section: Proof Of Corollarymentioning

confidence: 99%

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Kingman's model with random mutation probabilities: convergence and condensation II

Yuan

2020

Preprint

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Kingman's model describes the evolution of a one-locus haploid population of infinite size and discrete generations under the competition of selection and mutation. A random generalisation has been made in a previous paper which assumes all mutation probabilities to be i.i.d.. The weak convergence of fitness distributions to a globally stable equilibrium for any initial distribution was proved. A condensation occurs if almost surely a positive proportion of the population travels to and condensates on the largest fitness value due to the dominance of selection over mutation. A criterion of condensation was given which relies on the equilibrium whose explicit expression is however unknown. This paper tackles these problems based on the discovery of a matrix representation of the random model. An explicit expression of the equilibrium is obtained and the key quantity in the condensation criterion can be estimated. Moreover we examine how the design of randomness in Kingman's model affects the fitness level of the equilibrium by comparisons between different models. The discovered facts are conjectured to hold in other more sophisticated models.

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Section: Two Random Modelsmentioning

confidence: 92%

“…Then by Theorem 4 in [21], H j d = I j,SQ . By ( 5) and ( 20), for both sequences, the multi-dimensional distributions are determined in the same way by one dimensional distribution.…”

Section: Proofs Of Theorem 3 and Corollarymentioning

confidence: 95%

“…This section is mainly a summarisation of Section 2 in [21], in addition to the introduction of a new random model where all mutation probabilities are equal but random.…”

Section: Modelsmentioning

confidence: 99%

Section: Two Random Modelsmentioning

confidence: 99%

Section: Proof Of Corollarymentioning

confidence: 99%

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Kingman's model with random mutation probabilities: convergence and condensation II

Yuan

2020

Preprint

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Branching with Selection and Mutation I: Mutant Fitness of Fréchet Type

et al. 2023

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We investigate two stochastic models of a growing population with discrete and non-overlapping generations, subject to selection and mutation. In our models each individual carries a fitness which determines its mean offspring number. Many of these offspring inherit their parent’s fitness, but some are mutants and obtain a fitness randomly sampled, as in Kingman’s house-of-cards model, from a distribution in the domain of attraction of the Fréchet distribution. We give a rigorous proof for the precise rate of superexponential growth of these stochastic processes and support the argument by a heuristic and numerical study of the mechanism underlying this growth. This study yields in particular that the empirical fitness distribution of one model in the long time limit displays periodic behaviour.

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Non-conserving zero-range processes with extensive rates under resetting

Grange

2020

J. Phys. Commun.

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We consider a non-conserving zero-range process with hopping rate proportional to the number of particles at each site. Particles are added to the system with a site-dependent creation rate, and vanish with a uniform annihilation rate. On a fully-connected lattice with a large number of sites, the meanfield geometry leads to a negative binomial law for the number of particles at each site, with parameters depending on the hopping, creation and annihilation rates. This model can be mapped to population dynamics (if the creation rates are reproductive fitnesses in a haploid population, and the hopping rate is the mutation rate). It can also be mapped to a Bianconi-Barabási model of a growing network with random rewiring of links (if creation rates are the rates of acquisition of links by nodes, and the hopping rate is the rewiring rate). The steady state has recently been worked out and gives rise to occupation numbers that reproduce Kingman's house-of-cards model of selection and mutation. In this paper we solve the master equation using a functional method, which yields integral equations satisfied by the occupation numbers. The occupation numbers are shown to forget initial conditions at an exponential rate that decreases linearly with the fitness level. Moreover, they can be computed exactly in the Laplace domain, which allows to obtain the steady state of the system under resetting. The result modifies the house-of-cards result by simply adding a skewed version of the initial conditions, and by adding the resetting rate to the hopping rate.

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Kingman’s model with random mutation probabilities: convergence and condensation I

Cited by 5 publications

References 20 publications

Kingman's model with random mutation probabilities: convergence and condensation II

Kingman's model with random mutation probabilities: convergence and condensation II

Branching with Selection and Mutation I: Mutant Fitness of Fréchet Type

Non-conserving zero-range processes with extensive rates under resetting

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