Abstract
It is well known that the dynamics of a subpopulation of individuals of a rare type in a Wright-Fisher diffusion can be approximated by a Feller branching process. Here we establish an analogue of that result for a spatially distributed population whose dynamics are described by a spatial Lambda-Fleming-Viot process (SLFV). The subpopulation of rare individuals is then approximated by a superBrownian motion. This result mirrors [10], where it is shown that when suitably rescaled, sparse voter models converge to superBrownian motion. We also prove the somewhat more surprising result, that by choosing the dynamics of the SLFV appropriately we can recover superBrownian motion with stable branching in an analogous way. This is a spatial analogue of (a special case of) results of [6], who show that the generalised Fleming-Viot process that is dual to the beta-coalescent, when suitably rescaled, converges to a continuous state branching process with stable branching mechanism.
Highlights
Our aim in this paper is to establish a relationship between two, at first sight, very different classes of measure-valued processes
The first, the spatial Lambda-Fleming-Viot processes, is a collection of models for the evolution of frequencies of different genetic types in a population that is dispersed across a spatial continuum
E-mail: etheridg@stats.ox.ac.uk on the other, we address a question of some interest in population genetics: how does the frequency of a rare neutral mutation evolve in a spatially distributed population?
Summary
Our aim in this paper is to establish a relationship between two, at first sight, very different classes of measure-valued processes. If type 1 is rare, the absolute number of type 1 individuals evolves approximately according to a branching process which, under the same scaling, converges to a Feller diffusion This branching process approximation for the rare type has been used extensively in the population genetics literature and so it is natural to try to establish analogous results for spatially distributed populations. The spatial Lambda-Fleming-Viot process (SLFV) introduced in [16], overcomes the pain in the torus to provide a class of models for allele frequencies in populations distributed across spatial continua of any dimension. We can recover superBrownian motion from the SLFV in arbitrary spatial dimensions, but only if we take a sufficiently ‘sparse’ initial condition This should be compared to the results of [10], who recover superBrownian motion from sparse voter models and our analysis in the finite variance case owes a great deal to that paper.
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