1,035 research outputs found
A new model for evolution in a spatial continuum
We investigate a new model for populations evolving in a spatial continuum.
This model can be thought of as a spatial version of the Lambda-Fleming-Viot
process. It explicitly incorporates both small scale reproduction events and
large scale extinction-recolonisation events. The lineages ancestral to a
sample from a population evolving according to this model can be described in
terms of a spatial version of the Lambda-coalescent. Using a technique of
Evans(1997), we prove existence and uniqueness in law for the model. We then
investigate the asymptotic behaviour of the genealogy of a finite number of
individuals sampled uniformly at random (or more generally `far enough apart')
from a two-dimensional torus of side L as L tends to infinity. Under
appropriate conditions (and on a suitable timescale), we can obtain as limiting
genealogical processes a Kingman coalescent, a more general Lambda-coalescent
or a system of coalescing Brownian motions (with a non-local coalescence
mechanism).Comment: 63 pages, version accepted to Electron. J. Proba
Coalescent simulation in continuous space:Algorithms for large neighbourhood size
Many species have an essentially continuous distribution in space, in which there are no natural divisions between randomly mating subpopulations. Yet, the standard approach to modelling these populations is to impose an arbitrary grid of demes, adjusting deme sizes and migration rates in an attempt to capture the important features of the population. Such indirect methods are required because of the failure of the classical models of isolation by distance, which have been shown to have major technical flaws. A recently introduced model of extinction and recolonisation in two dimensions solves these technical problems, and provides a rigorous technical foundation for the study of populations evolving in a spatial continuum. The coalescent process for this model is simply stated, but direct simulation is very inefficient for large neighbourhood sizes. We present efficient and exact algorithms to simulate this coalescent process for arbitrary sample sizes and numbers of loci, and analyse these algorithms in detail
The infinitesimal model with dominance
The classical infinitesimal model is a simple and robust model for the
inheritance of quantitative traits. In this model, a quantitative trait is
expressed as the sum of a genetic and a non-genetic (environmental) component
and the genetic component of offspring traits within a family follows a normal
distribution around the average of the parents' trait values, and has a
variance that is independent of the trait values of the parents. In previous
work, Barton et al.(2017), we showed that when trait values are determined by
the sum of a large number of Mendelian factors, each of small effect, one can
justify the infinitesimal model as limit of Mendelian inheritance.
In this paper, we show that the robustness of the infinitesimal model extends
to include dominance. We define the model in terms of classical quantities of
quantitative genetics, before justifying it as a limit of Mendelian inheritance
as the number, M, of underlying loci tends to infinity. As in the additive
case, the multivariate normal distribution of trait values across the pedigree
can be expressed in terms of variance components in an ancestral population and
identities determined by the pedigree. In this setting, it is natural to
decompose trait values, not just into the additive and dominance components,
but into a component that is shared by all individuals within the family and an
independent `residual' for each offspring, which captures the randomness of
Mendelian inheritance. We show that, even if we condition on parental trait
values, both the shared component and the residuals within each family will be
asymptotically normally distributed as the number of loci tends to infinity,
with an error of order 1/\sqrt{M}.
We illustrate our results with some numerical examples.Comment: 62 pages, 8 figure
Mutually catalytic branching in the plane: Finite measure states
We study a pair of populations in ℝ2 which undergo diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. For a diffusion rate sufficiently large compared with the branching rate, the model is constructed as the unique pair of finite measure-valued processes which satisfy a martingale problem involving the collision local time of the solutions. The processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of the two populations. In the long-term limit, global extinction of one type is shown. The process constructed is a rescaled limit of the corresponding ℤ2-lattice model studied by Dawson and Perkins (1998) and resolves the large scale mass-time-space behavior of that model
Clustering, advection and patterns in a model of population dynamics with neighborhood-dependent rates
We introduce a simple model of population dynamics which considers birth and
death rates for every individual that depend on the number of particles in its
neighborhood. The model shows an inhomogeneous quasistationary pattern with
many different clusters of particles.
We derive the equation for the macroscopic density of particles, perform a
linear stability analysis on it, and show that there is a finite-wavelength
instability leading to pattern formation. This is the responsible for the
approximate periodicity with which the clusters of particles arrange in the
microscopic model.
In addition, we consider the population when immersed in a fluid medium and
analyze the influence of advection on global properties of the model.Comment: Some typos and some problems with the figures correcte
- …