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Register a new userGenealogies and inference for populations with highly skewed offspring distributions
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We review recent progress in the understanding of the role of multiple- and simultaneous multiple merger coalescents as models for the genealogy in idealised and real populations with exceptional reproductive behaviour. In particular, we discuss models with `skewed offspring distribution (or under other non-classical evolutionary forces) which lead in the single locus haploid case to multiple merger coalescents, and in the multi-locus diploid case to simultaneous multiple merger coalescents. Further, we discuss inference methods under the infinitely-many sites model which allow both model selection and estimation of model parameters under these coalescents.
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Fernando Cordero
, Adrian Gonzalez Casanova
, Jason Schweinsberg andn Maite Wilke-Berenguer
2020
Consider a population evolving from year to year through three seasons: spring, summer and winter. Every spring starts with $N$ dormant individuals waking up independently of each other according to a given distribution. Once an individual is awake, it starts reproducing at a constant rate. By the end of spring, all individuals are awake and continue reproducing independently as Yule processes during the whole summer. In the winter, $N$ individuals chosen uniformly at random go to sleep until the next spring, and the other individuals die. We show that because an individual that wakes up unusually early can have a large number of surviving descendants, for some choices of model parameters the genealogy of the population will be described by a $Lambda$-coalescent. In particular, the beta coalescent can describe the genealogy when the rate at which individuals wake up increases exponentially over time. We also characterize the set of all $Lambda$-coalescents that can arise in this framework.
Full likelihood inference under Kingmans coalescent is a computationally challenging problem to which importance sampling (IS) and the product of approximate conditionals (PAC) method have been applied successfully. Both methods can be expressed in terms of families of intractable conditional sampling distributions (CSDs), and rely on principled approximations for accurate inference. Recently, more general $Lambda$- and $Xi$-coalescents have been observed to provide better modelling fits to some genetic data sets. We derive families of approximate CSDs for finite sites $Lambda$- and $Xi$-coalescents, and use them to obtain approximately optimal IS and PAC algorithms for $Lambda$-coalescents, yielding substantial gains in efficiency over existing methods.
A population is considered stationary if the growth rate is zero and the age structure is constant. It thus follows that a population is considered non-stationary if either its growth rate is non-zero and/or its age structure is non-constant. We propose three properties that are related to the stationary population identity (SPI) of population biology by connecting it with stationary populations and non-stationary populations which are approaching stationarity. One of these important properties is that SPI can be applied to partition a population into stationary and non-stationary components. These properties provide deeper insights into cohort formation in real-world populations and the length of the duration for which stationary and non-stationary conditions hold. The new concepts are based on the time gap between the occurrence of stationary and non-stationary populations within the SPI framework that we refer to as Oscillatory SPI and the Amplitude of SPI. This article will appear in Bulletin of Mathematical Biology (Springer)
Moran or Wright-Fisher processes are probably the most well known model to study the evolution of a population under various effects. Our object of study will be the Simpson index which measures the level of diversity of the population, one of the key parameter for ecologists who study for example forest dynamics. Following ecological motivations, we will consider here the case where there are various species with fitness and immigration parameters being random processes (and thus time evolving). To measure biodiversity, ecologists generally use the Simpson index, who has no closed formula, except in the neutral (no selection) case via a backward approach, and which is difficult to evaluate even numerically when the population size is large. Our approach relies on the large population limit in the weak selection case, and thus to give a procedure which enable us to approximate, with controlled rate, the expectation of the Simpson index at fixed time. Our approach will be forward and valid for all time, which is the main difference with the historical approach of Kingman, or Krone-Neuhauser. We will also study the long time behaviour of the Wright-Fisher process in a simplified setting, allowing us to get a full picture for the approximation of the expectation of the Simpson index.
We study simple stochastic scenarios, based on birth-and-death Markovian processes, that describe populations with Allee effect, to account for the role of demographic stochasticity. In the mean-field deterministic limit we recover well-known deterministic evolution equations widely employed in population ecology. The mean-time to extinction is in general obtained by the Wentzel-Kramers-Brillouin (WKB) approximation for populations with strong and weak Allee effects. An exact solution for the mean time to extinction can be found via a recursive equation for special cases of the stochastic dynamics. We study the conditions for the validity of the WKB solution and analyze the boundary between the weak and strong Allee effect by comparing exact solutions with numerical simulations.
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