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Register a new userStochastic growth rates for populations in random environments with rare migration
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The growth of a population divided among spatial sites, with migration between the sites, is sometimes modelled by a product of random matrices, with each diagonal elements representing the growth rate in a given time period, and off-diagonal elements the migration rate. The randomness of the matrices then represents stochasticity of environmental conditions. We consider the case where the off-diagonal elements are small, representing a situation where migration has been introduced into an otherwise sessile meta-population. We examine the asymptotic behaviour of the long-term growth rate. When there is a single site with the highest growth rate, under the assumption of Gaussian log growth rates at the individual sites (or having Gaussian-like tails) we show that the behavior near zero is like a power of $epsilon$, and derive upper and lower bounds for the power in terms of the difference in the growth rates and the distance between the sites. In particular, when the difference in mean log growth rate between two sites is sufficiently small, or the variance of the difference between the sites sufficiently large, migration will always be favored by natural selection, in the sense that introducing a small amount of migration will increase the growth rate of the population relative to the zero-migration case.
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The growth of a population divided among spatial sites, with migration between the sites, is sometimes modelled by a product of random matrices, with each diagonal elements representing the growth rate in a given time period, and off-diagonal elements the migration rate. If the sites are reinterpreted as age classes, the same model may apply to a single population with age-dependent mortality and reproduction. We consider the case where the off-diagonal elements are small, representing a situation where there is little migration or, alternatively, where a deterministic life-history has been slightly disrupted, for example by introducing a rare delay in development. We examine the asymptotic behaviour of the long-term growth rate. We show that when the highest growth rate is attained at two different sites in the absence of migration (which is always the case when modelling a single age-structured population) the increase in stochastic growth rate due to a migration rate $epsilon$ is like $(log epsilon^{-1})^{-1}$ as $epsilondownarrow 0$, under fairly generic conditions. When there is a single site with the highest growth rate the behavior is more delicate, depending on the tails of the growth rates. For the case when the log growth rates have Gaussian-like tails we show that the behavior near zero is like a power of $epsilon$, and derive upper and lower bounds for the power in terms of the difference in the growth rates and the distance between the sites.
This chapter gives a synopsis of recent approaches to model and analyse the evolution of microbial populations under selection. The first part reviews two population genetic models of Lenskis long-term evolution experiment with Escherichia coli, where models aim at explaining the observed curve of the evolution of the mean fitness. The second part describes a model of a host-pathogen system where the population of pathogenes experiences balancing selection, migration, and mutation, as motivated by observations of the genetic diversity of HCMV (the human cytomegalovirus) across hosts.
In this paper, we wish to investigate the dynamics of information transfer in evolutionary dynamics. We use information theoretic tools to track how much information an evolving population has obtained and managed to retain about different environments that it is exposed to. By understanding the dynamics of information gain and loss in a static environment, we predict how that same evolutionary system would behave when the environment is fluctuating. Specifically, we anticipate a cross-over between the regime in which fluctuations improve the ability of the evolutionary system to capture environmental information and the regime in which the fluctuations inhibit it, governed by a cross-over in the timescales of information gain and decay.
In order to analyze data from cancer genome sequencing projects, we need to be able to distinguish causative, or driver, mutations from passenger mutations that have no selective effect. Toward this end, we prove results concerning the frequency of neutural mutations in exponentially growing multitype branching processes that have been widely used in cancer modeling. Our results yield a simple new population genetics result for the site frequency spectrum of a sample from an exponentially growing population.
Living species, ranging from bacteria to animals, exist in environmental conditions that exhibit spatial and temporal heterogeneity which requires them to adapt. Risk-spreading through spontaneous phenotypic variations is a known concept in ecology, which is used to explain how species may survive when faced with the evolutionary risks associated with temporally varying environments. In order to support a deeper understanding of the adaptive role of spontaneous phenotypic variations in fluctuating environments, we consider a system of non-local partial differential equations modelling the evolutionary dynamics of two competing phenotype-structured populations in the presence of periodically oscillating nutrient levels. The two populations undergo spontaneous phenotypic variations at different rates. The phenotypic state of each individual is represented by a continuous variable, and the phenotypic landscape of the populations evolves in time due to variations in the nutrient level. Exploiting the analytical tractability of our model, we study the long-time behaviour of the solutions to obtain a detailed mathematical depiction of evolutionary dynamics. The results suggest that when nutrient levels undergo small and slow oscillations, it is evolutionarily more convenient to rarely undergo spontaneous phenotypic variations. Conversely, under relatively large and fast periodic oscillations in the nutrient levels, which bring about alternating cycles of starvation and nutrient abundance, higher rates of spontaneous phenotypic variations confer a competitive advantage. We discuss the implications of our results in the context of cancer metabolism.
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