Voter models with heterozygosity selection

Sturm, Anja; Swart, Jan M.

doi:10.1214/07-aap444

Cited by 21 publications

(71 citation statements)

References 27 publications

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“…Coexistence in suitable notions should hold except in very few, if not pathological, cases. Some resolutions of Conjecture 3.8 are achieved in [5,7,6,28], since [22, Theorem 1] takes the first few steps to do so. Yet to the knowledge of the authors, the full resolution of Conjecture 3.8 remains open.…”

Section: Invariance Of Equiparity Coexistencementioning

confidence: 99%

“…The notion of extinction versus unbounded growth is introduced by Sturm and Swart in [28]. We will discuss this notion in more detail later on.…”

Section: Invariance Of Equiparity Coexistencementioning

confidence: 99%

“…More generally, the proof of Theorem 3.9 can be modified in an obvious way to show a similar formula if we use Bernoulli product measures with site-dependent densities, where the densities = 0, 1 are bounded away from 0 and 1. Theorem 3.9 and Lemma 3.11 should provide the basic idea behind the related methods in [3,22,28]. There the method to resolve the main technical issue for not starting with Bernoulli measures is to identify 'Bernoulli subsystems' which have sizes growing to infinity as t → ∞.…”

Section: Invariance Of Equiparity Coexistencementioning

confidence: 99%

“…Here, (η t ) and ( η t ) can be coupled on the same probability space. See [9,14,28]. For the case of interacting diffusions in [1], we discuss large-time behavior of the model using duality in the form (1.1).…”

Section: Introductionmentioning

confidence: 99%

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Coexistence and Duality in Competing Species Models

Chen

Hammer

2020

Genealogies of Interacting Particle Systems

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We discuss some stochastic spatial generalizations of the Lotka-Volterra model for competing species. The generalizations take the forms of spin systems on general discrete sets and interacting diffusions on integer lattices. Methods for proving coexistence in these generalizations and some related open questions are discussed. We use duality as the central point of view. It relates coexistence of the models to survival of their dual processes.

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Section: Invariance Of Equiparity Coexistencementioning

confidence: 99%

“…The notion of extinction versus unbounded growth is introduced by Sturm and Swart in [28]. We will discuss this notion in more detail later on.…”

Section: Invariance Of Equiparity Coexistencementioning

confidence: 99%

Section: Invariance Of Equiparity Coexistencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Coexistence and Duality in Competing Species Models

Chen

Hammer

2020

Genealogies of Interacting Particle Systems

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“…With some care, it can be shown that this implies that the process spends a zero fraction of time in states y with |y| = N + 2, completing the induction step. This argument is written down more carefully in [SS08a]. Formula (3.65) of that paper has to be slightly modified in our situation since we only know that the boundaries of type 01 are unstable.…”

Section: Proof Of Lemma 26mentioning

confidence: 99%

Equilibrium interfaces of biased voter models

Sun¹,

Swart²,

Yu³

2019

Ann. Appl. Probab.

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A one-dimensional interacting particle system is said to exhibit interface tightness if starting in an initial condition describing the interface between two constant configurations of different types, the process modulo translations is positive recurrent. In a biological setting, this describes two populations that do not mix, and it is believed to be a common phenomenon in one-dimensional particle systems. Interface tightness has been proved for voter models satisfying a finite second moment condition on the rates. We extend this to biased voter models. Furthermore, we show that the distribution of the equilibrium interface for the biased voter model converges to that of the voter model when the bias parameter tends to zero. A key ingredient is an identity for the expected number of boundaries in the equilibrium voter model interface, which is of independent interest. MSC 2010. Primary: 82C22; Secondary: 82C24, 82C41, 60K35.

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Numerical Analysis of the Rebellious Voter Model

Swart

Vrbenský

2010

J Stat Phys

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The rebellious voter model, introduced by , is a variation of the standard, one-dimensional voter model, in which types that are locally in the minority have an advantage. It is related, both through duality and through the evolution of its interfaces, to a system of branching annihilating random walks that is believed to belong to the 'parity-conservation' universality class. This paper presents numerical data for the rebellious voter model and for a closely related one-sided version of the model. Both models appear to exhibit a phase transition between noncoexistence and coexistence as the advantage for minority types is increased. For the one-sided model (but not for the original, two-sided rebellious voter model), it appears that the critical point is exactly a half and two important functions of the process are given by simple, explicit formulas, a fact for which we have no explanation.MSC 2000. Primary: 82C22; Secondary: 65C05, 82C26, 60K35

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Voter models with heterozygosity selection

Cited by 21 publications

References 27 publications

Coexistence and Duality in Competing Species Models

Coexistence and Duality in Competing Species Models

Equilibrium interfaces of biased voter models

Numerical Analysis of the Rebellious Voter Model

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