The Shape of the Emerging Condensate in Effective Models of Condensation

Betz, Volker; Dereich, Steffen; Mörters, Peter

doi:10.1007/s00023-018-0673-7

Cited by 7 publications

(8 citation statements)

References 17 publications

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“…Later on, recombination was incorporated into the model [10]. The model of Betz et al [2] generalised a continuous-time version of Kingman's model and other models arising from physics.…”

Section: Motivation and Backgroundmentioning

confidence: 99%

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

Yuan

2022

Adv. Appl. Probab.

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For a one-locus haploid infinite population with discrete generations, the celebrated model of Kingman describes the evolution of fitness distributions under the competition of selection and mutation, with a constant mutation probability. This paper generalises Kingman’s model by using independent and identically distributed random mutation probabilities, to reflect the influence of a random environment. The weak convergence of fitness distributions to the globally stable equilibrium is proved. Condensation occurs when almost surely a positive proportion of the population travels to and condenses at the largest fitness value. Condensation may occur when selection is favoured over mutation. A criterion for the occurrence of condensation is given.

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“…Later on, recombination was incorporated into the model [10]. The model of Betz et al [2] generalised a continuous-time version of Kingman's model and other models arising from physics.…”

Section: Motivation and Backgroundmentioning

confidence: 99%

“…where the inequality is due to the fact that y 2 ν(dy) ≥ ( yν(dy)) 2 . Then we obtain yQ(dy) ≤ E yν(dy) ≤ 1.…”

Section: Some Properties Of Invariant Measuresmentioning

confidence: 99%

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

Yuan

2022

Adv. Appl. Probab.

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“…A series of papers [1,8,10,11,15] were written later on to investigate the shape of traveling wave in other models where condensation appears and have proved that gamma distribution is universal. Park and Krug [16] adapted Kingman's model to a finite population with unbounded fitness distribution and observed in a particular case emergence of Gaussian distribution as the wave travels to infinity.…”

Section: Perspectivesmentioning

confidence: 99%

Kingman’s Model with Random Mutation Probabilities: Convergence and Condensation II

Yuan

2020

J Stat Phys

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A generalisation of Kingman’s model of selection and mutation 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 was proved. The 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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“…Let M 1 be the space of probability measures on [0, 1] endowed with the topology of weak convergence. Let (b n ) = (b n ) n≥1 be a sequence of numbers in [0, 1), and P 0 , Q ∈ M 1 . Kingman's model with time-varying mutation probabilities or simply the general model has parameters (b n ), Q, P 0 .…”

Section: Two Deterministic Modelsmentioning

confidence: 99%

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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The Shape of the Emerging Condensate in Effective Models of Condensation

Cited by 7 publications

References 17 publications

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

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

Kingman’s Model with Random Mutation Probabilities: Convergence and Condensation II

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

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