The Brownian Net and Selection in the Spatial Lambda-Fleming-Viot Process

Etheridge, Alison; Freeman, Nic; Straulino, Daniel

doi:10.48550/arxiv.1506.01158

Cited by 2 publications

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“…Convergence to the Brownian net so far has only been established by Sun and Swart [SS08] for branching-coalescing simple random walk paths with asymptotically vanishing branching probability (see Figure 4), and recently by Etheridge, Freeman and Straulino [EFS15] for the genealogies of a spatial Lambda-Fleming-Viot process. We identify below the key conditions and formulate them as convergence criteria for the Brownian net, which can be applied to random sets of paths with certain non-crossing properties.…”

Section: Convergence To the Brownian Netmentioning

confidence: 99%

“…Another natural extension of the paths topology is to consider the space of compact sets of càdlàg paths equipped with the Hausdorff topology, where the space of càdlàg paths is equipped with the Skorohod metric after compactification of space-time as done in Figure 2. Such an extension has been carried out by Etheridge, Freeman and Straulino [EFS15] in the study of scaling limits of spatial Lambda-Fleming-Viot processes.…”

Section: Alternative Topologiesmentioning

confidence: 99%

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The Brownian Web, the Brownian Net, and their Universality

Schertzer¹,

Sun

Swart

2016

Advances in Disordered Systems, Random Processes and Some Applications

107

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The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from everywhere in space and time, and the Brownian net is a generalization that also allows branching. They appear in the diffusive scaling limits of many one-dimensional interacting particle systems with branching and coalescence. This article gives an introduction to the Brownian web and net, and how they arise in the scaling limits of various one-dimensional models, focusing mainly on coalescing random walks and random walks in i.i.d. space-time random environments. We will also briefly survey models and results connected to the Brownian web and net, including alternative topologies, population genetic models, true self-repelling motion, planar aggregation, drainage networks, oriented percolation, black noise and critical percolation. Some open questions are discussed at the end.MSC 2000. Primary: 82C21 ; Secondary: 60K35, 60D05.

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Section: Convergence To the Brownian Netmentioning

confidence: 99%

Section: Alternative Topologiesmentioning

confidence: 99%

The Brownian Web, the Brownian Net, and their Universality

Schertzer¹,

Sun

Swart

2016

Advances in Disordered Systems, Random Processes and Some Applications

107

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“…without selection), but keeping the impact parameter macroscopic. The behaviour of the SLFVS in the corresponding regime is studied in [EFS15] and [EFPS15].…”

Section: Introductionmentioning

confidence: 99%

A central limit theorem for the spatial $\Lambda $-Fleming-Viot process with selection

Forien¹,

Penington²

2017

Electron. J. Probab.

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We study the evolution of gene frequencies in a population living in R d , modelled by the spatial Λ-Fleming-Viot process with natural selection ([BEV10], [EVY14]). We suppose that the population is divided into two genetic types, a and A, and consider the proportion of the population which is of type a at each spatial location. If we let both the selection intensity and the fraction of individuals replaced during reproduction events tend to zero, the process can be rescaled so as to converge to the solution to a reaction-diffusion equation (typically the Fisher-KPP equation, [EVY14]). We show that the rescaled fluctuations converge in distribution to the solution to a linear stochastic partial differential equation. Depending on whether offspring dispersal is only local or if large scale extinction-recolonization events are allowed to take place, the limiting equation is either the stochastic heat equation with a linear drift term driven by space-time white noise or the corresponding fractional heat equation driven by a coloured noise which is white in time. If individuals are diploid (i.e. either AA, Aa or aa) and if natural selection favours heterozygous (Aa) individuals, a stable intermediate gene frequency is maintained in the population. We give estimates for the asymptotic effect of random fluctuations around the equilibrium frequency on the local average fitness in the population. In particular, we find that the size of this effect -known as the drift load -depends crucially on the dimension d of the space in which the population evolves, and is reduced relative to the case without spatial structure.AMS 2010 subject classifications. Primary: 60G57 60F05 60J25 92D10. Secondary: 60G15.2. Update q as in (6).Note that if we let w = |B(x, r)| −1 B(x,r) q t − (z) dz, then at a neutral reproduction event, P (k = a) = w and at a selective event, P (k = a) = w − F (w).Remark. The existence of a unique Ξ-valued process following these dynamics under condition (4) is proved in [EVY14, Theorem 1.2] in the special case F (w) = w(1 − w) (in the neutral case s = 0, this was done in [BEV10]). In our general case, the condition on w − F (w) allows us to define a dual process and hence prove existence and uniqueness in the same way as in [EVY14].We shall consider two different distributions µ for the radii of events, i) the fixed radius case : µ(dr) = δ R (dr) for some R > 0, ii) the stable radius case : µ(dr) = 1 r≥1 r d+α+1 dr for a fixed α ∈ (0, 2 ∧ d).

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The Brownian Net and Selection in the Spatial Lambda-Fleming-Viot Process

Cited by 2 publications

References 0 publications

The Brownian Web, the Brownian Net, and their Universality

The Brownian Web, the Brownian Net, and their Universality

A central limit theorem for the spatial $\Lambda $-Fleming-Viot process with selection

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