Stochastic measure-valued models for populations expanding in a continuum

Abstract

We model spatially expanding populations by means of a spatial Λ-Fleming Viot process (SLFV) with selection : the k-parent SLFV. We fill empty areas with type 0 "ghost" individuals, which have a strong selective disadvantage against "real" type 1 individuals. This model is a special case of the SLFV with selection introduced in [19,22] : natural selection acts during all reproduction events, and the fraction of individuals replaced during a reproduction event is constant equal to 1. Letting the selective advantage k of type 1 individuals over type 0 individuals grow to +∞, and without rescaling time nor space, we obtain a new model for expanding populations, the ∞-parent SLFV.This model is reminiscent of the Eden growth model [13], but with an associated dual process of potential ancestors, making it possible to investigate the genetic diversity in a population sample. In order to obtain the limit k → +∞ of the k-parent SLFV, we introduce an alternative construction of the k-parent SLFV adapted from [38], which allows us to couple SLFVs with different selection strengths.

Growth properties of the infinite-parent spatial Lambda-Fleming Viot process

“…The ∞-parent spatial Λ-Fleming Viot process, or ∞-parent SLFV, was introduced in [32] as a model for expanding populations in R 2 . Its main feature is the use of "ghost" individuals (thereafter referred to as "type 0" individuals) to fill empty areas, adapting ideas from [13,23].…”

“…Its main feature is the use of "ghost" individuals (thereafter referred to as "type 0" individuals) to fill empty areas, adapting ideas from [13,23]. In [13,23,32], ghost individuals can reproduce as well, modelling stochastic fluctuations in population sizes, but with a selective disadvantage against real individuals (thereafter referred to as "type 1" individuals), ensuring that population expansions can indeed occur. In [32], it is shown that the ∞-parent SLFV corresponds to the limit of an SLFV with strong selection when the selective advantage of type 1 individuals over type 0 individuals becomes infinitely strong.…”

“…In [13,23,32], ghost individuals can reproduce as well, modelling stochastic fluctuations in population sizes, but with a selective disadvantage against real individuals (thereafter referred to as "type 1" individuals), ensuring that population expansions can indeed occur. In [32], it is shown that the ∞-parent SLFV corresponds to the limit of an SLFV with strong selection when the selective advantage of type 1 individuals over type 0 individuals becomes infinitely strong. Therefore, in the limiting regime, we no longer observe local extinctions due to the stochasticity in reproduction at the front, where densities of type 1 individuals are lower.…”

“…In particular, this means that the definition we just provided is only informal: when Vol(E 0 ) = +∞, an infinite number of reproduction events intersect the occupied area in any time interval, and the jump rate of the process is infinite. Rigorously, the ∞-parent SLFV is a measure-valued process, which can be defined as the limit of a sequence of SLFVs with strong selection, or (under stricter conditions which will be satisfied in this article) as the unique solution to a well-posed martingale problem [32]. If Vol(E 0 ) < +∞, it can also be defined as a process taking its values in the class of measurable subsets of R d .…”

Extinction threshold and large population limit of a plant metapopulation model with recurrent extinction events and a seed bank component

“…In order to incorporate extinction event-induced fluctuations in a Wright-Fisher model, it is possible to adopt the approach used in [13,22]: assign a maximal population size to each patch, and fill the remaining space with "ghost", or type 0, individuals. In this framework, each patch contains both type 1 "real" individuals and type 0 "ghost" individuals, the former having a very strong selective advantage over the latter (in the spirit of [30]).…”