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

Yuan, Linglong

doi:10.1007/s10955-020-02609-w

Cited by 6 publications

(6 citation statements)

References 19 publications

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“…More realistic versions of the Kingman model include randomness in the mutation rate [50,51]. Alternatively, we could generalize the present model by introducing independent identically-distributed sources of noise to the local resetting rate.…”

Section: Discussionmentioning

confidence: 99%

Local resetting in non-conserving zero-range processes with extensive rates

Grange

2024

J. Phys. Commun.

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A non-conserving zero-range process with extensive creation, annihilation and hopping rates is subjected to local resetting. The model is formulated on a large, fully-connected network of states. The states are equipped with a (bounded) fitness level: particles are added to each state at a rate proportional to the fitness level of the state. Moreover, particles are annihilated at a constant rate, and hop at a fixed rate to a uniformly-drawn state in the network. This model has been interpreted in terms of population dynamics: the fitness is the reproductive fitness in a haploid population, and the hopping process models mutation. It has also been interpreted as a model of network growth with a fixed set of nodes (in which particles occupying a state are interpreted as links pointing to this state). In the absence of resetting, the model is known to reach a steady state, which in a certain limit may  exhibit a condensate at maximum fitness. If the model is subjected to global resetting by annihilating all particles at Poisson-distributed times, there is no condensation in the steady state. If the system is subjected to local resetting, the occupation numbers of each state are reset to zero at independent random times. These times are distributed according to a Poisson process whose rate (the resetting rate) depends on the fitness. We derive the evolution equation satisfied by the probability law of the occupation numbers. We calculate the average occupation numbers in the steady state. The existence of a condensate is found to depend on the local behavior of the resetting rate at maximum fitness: if the resetting rate vanishes at least linearly at high fitness, a condensate appears at maximum fitness in the limit where the sum of the annihilation and hopping rates is equal to the maximum fitness.

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Section: Discussionmentioning

confidence: 99%

Local resetting in non-conserving zero-range processes with extensive rates

Grange

2024

J. Phys. Commun.

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“…We refer to [16] for a reference on random measures. The definition of weak convergence for random measures stated in the follow-up paper [23] is incorrect. But this does not affect anything there as the weak convergence results are all proved in this paper.…”

Section: Kingman's Model With Random Mutation Probabilitiesmentioning

confidence: 99%

“…In the follow-up paper [23], we provide a matrix representation for I Q , so the condensation criterion can be written neatly (see [23,Corollary 2,p. 877]).…”

Section: If H > S Q Then There Is No Condensation At H If and Only Ifmentioning

confidence: 99%

“…877]). Moreover, using matrix analysis, we can compare the fitness of equilibria from different models (see [23,, pp. 878-879]).…”

Section: If H > S Q Then There Is No Condensation At H If and Only Ifmentioning

confidence: 99%

“…Remark 10. The equality (23) holds in the pointwise sense. In other words, given any realisation of (β j ) (or equivalently, conditioning on (β j )), the equality holds for any j as in the general model.…”

Section: Introducing the Finite Backward Sequencesmentioning

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

Cited by 6 publications

References 19 publications

Local resetting in non-conserving zero-range processes with extensive rates

Local resetting in non-conserving zero-range processes with extensive rates

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

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

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