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We follow up on a companion work that considered growth rates of populations growing at different sites, with different randomly varying growth rates at each site, in the limit as migration between sites goes to 0. We extend this work here to the special case where the maximum average log growth rate is achieved at two different sites. The primary motivation is to cover the case where `sites are understood as age classes for the same individuals. The theory then calculates the effect on growth rate of introducing a rare delay in development, a diapause, into an otherwise fixed-length semelparous life history. Whereas the increase in stochastic growth rate due to rare migrations was found to grow as a power of the migration rate, we show that under quite general conditions that in the diapause model --- or in the migration model with two or more sites having equal individual stochastic growth rates --- the increase in stochastic growth rate due to diapause at rate $epsilon$ behaves like $(log epsilon^{-1})^{-1}$ as $epsilondownarrow 0$. In particular, this implies that a small random disruption to the deterministic life history will always be favored by natural selection, in the sense that it will increase the stochastic growth rate relative to the zero-delay deterministic life history.
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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.
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.
Empirical observations show that ecological communities can have a huge number of coexisting species, also with few or limited number of resources. These ecosystems are characterized by multiple type of interactions, in particular displaying cooperative behaviors. However, standard modeling of population dynamics based on Lotka-Volterra type of equations predicts that ecosystem stability should decrease as the number of species in the community increases and that cooperative systems are less stable than communities with only competitive and/or exploitative interactions. Here we propose a stochastic model of population dynamics, which includes exploitative interactions as well as cooperative interactions induced by cross-feeding. The model is exactly solved and we obtain results for relevant macro-ecological patterns, such as species abundance distributions and correlation functions. In the large system size limit, any number of species can coexist for a very general class of interaction networks and stability increases as the number of species grows. For pure mutualistic/commensalistic interactions we determine the topological properties of the network that guarantee species coexistence. We also show that the stationary state is globally stable and that inferring species interactions through species abundance correlation analysis may be misleading. Our theoretical approach thus show that appropriate models of cooperation naturally leads to a solution of the long-standing question about complexity-stability paradox and on how highly biodiverse communities can coexist.
We propose an alternative to the prevailing two origin of life narratives, one based on a replicator first hypothesis, and one based on a metabolism first hypothesis. Both hypotheses have known difficulties: All known evolvable molecular replicators such as RNA require complex chemical (enzymatic) machinery for the replication process. Likewise, contemporary cellular metabolisms require several enzymatically catalyzed steps, and it is difficult to identify a non-enzymatic path to their realization. We propose that there must have been precursors to both replication and metabolism that enable a form of selection to take place through action of simple chemical and physical processes. We model a concrete example of such a process, repeated sequestration of binary molecular combinations after exposure to an environment with a broad distribution of chemical components, as might be realized experimentally in in a repeated wet-dry cycle. We show that the repeated sequestration dynamics results in a selective amplification of a very small subset of molecular species present in the environment, thus providing a candidate primordial selection process.
Inference with population genetic data usually treats the population pedigree as a nuisance parameter, the unobserved product of a past history of random mating. However, the history of genetic relationships in a given population is a fixed, unobserved object, and so an alternative approach is to treat this network of relationships as a complex object we wish to learn about, by observing how genomes have been noisily passed down through it. This paper explores this point of view, showing how to translate questions about population genetic data into calculations with a Poisson process of mutations on all ancestral genomes. This method is applied to give a robust interpretation to the $f_4$ statistic used to identify admixture, and to design a new statistic that measures covariances in mean times to most recent common ancestor between two pairs of sequences. The method more generally interprets population genetic statistics in terms of sums of specific functions over ancestral genomes, thereby providing concrete, broadly interpretable interpretations for these statistics. This provides a method for describing demographic history without simplified demographic models. More generally, it brings into focus the population pedigree, which is averaged over in model-based demographic inference.
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