An Eco-Evolutionary approach of Adaptation and Recombination in a large population of varying size

We identify the genetic signature of a selective sweep in a population described by a birth-and-death process with density dependent competition. We study the limit behaviour for large K, where K scales the population size. We focus on two loci: one under selection and one neutral. We distinguish a soft sweep occurring after an environmental change, from a hard sweep occurring after a mutation, and express the neutral proportion variation as a function of the ecological parameters, recombination probability r\_K, and K. We show that for a hard sweep, two recombination regimes appear according to the order of r\_K log K.Comment: Accepted in SP

The effect of recurrent mutations on genetic diversity in a large population of varying size

Recurrent mutations are a common phenomenon in population genetics. They may be at the origin of the fixation of a new genotype, if they give a phenotypic advantage to the carriers of the new mutation. In this paper, we are interested in the genetic signature left by a selective sweep induced by recurrent mutations at a given locus from an allele A to an allele a, depending on the mutation frequency. We distinguish three possible scales for the mutation probability per reproductive event, which entail distinct genetic signatures. Besides, we study the hydrodynamic limit of the A- and a-population size dynamics when mutations are frequent, and find non trivial equilibria leading to several possible patterns of polymorphism

Genealogies of two linked neutral loci after a selective sweep in a large population of stochastically varying size

We study the impact of a hard selective sweep on the genealogy of partially linked neutral loci in the vicinity of the positively selected allele. We consider a sexual population of stochastically varying size and, focusing on two neighboring loci, derive an approximate formula for the neutral genealogy of a sample of individuals taken at the end of the sweep. Individuals are characterized by ecological parameters depending on their genetic type and governing their growth rate and interactions with other individuals (competition). As a consequence, the "fitness" of an individual depends on the population state and is not an intrinsic characteristic of individuals. We provide a deep insight into the dynamics of the mutant and wild type populations during the different stages of a selective sweep

Crossing a fitness valley as a metastable transition in a stochastic population model

We consider a stochastic model of population dynamics where each individual is characterised by a trait in {0,1,...,L} and has a natural reproduction rate, a logistic death rate due to age or competition and a probability of mutation towards neighbouring traits at each reproduction event. We choose parameters such that the induced fitness landscape exhibits a valley: mutant individuals with negative fitness have to be created in order for the population to reach a trait with positive fitness. We focus on the limit of large population and rare mutations at several speeds. In particular, when the mutation rate is low enough, metastability occurs: the exit time of the valley is random, exponentially distributed.Comment: Second round of revision. 40 pages, 4 figure

Asymptotic behaviour of exponential functionals of L\'evy processes with applications to random processes in random environment

Let $\xi=(\xi_t, t\ge 0)$ be a real-valued L\'evy process and define its associated exponential functional as follows $$I_t(\xi):=\int_0^t \exp\{-\xi_s\}{\rm d} s, \qquad t\ge 0.$$ Motivated by important applications to stochastic processes in random environment, we study the asymptotic behaviour of $$\mathbb{E}\Big[F\big(I_t(\xi)\big)\Big] \qquad \textrm{as}\qquad t\to \infty,$$ where $F$ is a non-increasing function with polynomial decay at infinity and under some exponential moment conditions on $\xi$. In particular, we find five different regimes that depend on the shape of the Laplace exponent of $\xi$. Our proof relies on a discretisation of the exponential functional $I_t(\xi)$ and is closely related to the behaviour of functionals of semi-direct products of random variables. We apply our main result to three {questions} associated to stochastic processes in random environment. We first consider the asymptotic behaviour of extinction and explosion for {stable} continuous state branching processes in a L\'evy random environment. Secondly, we {focus on} the asymptotic behaviour of the mean of a population model with competition in a L\'evy random environment and finally, we study the tail behaviour of the maximum of a diffusion process in a L\'evy random environment.Comment: arXiv admin note: text overlap with arXiv:1512.07691, arXiv:math/0511265 by other authors. Results are improve

On the extinction of Continuous State Branching Processes with catastrophes

We consider continuous state branching processes (CSBP) with additional multiplicative jumps modeling dramatic events in a random environment. These jumps are described by a L\'evy process with bounded variation paths. We construct a process of this class as the unique solution of a stochastic differential equation. The quenched branching property of the process allows us to derive quenched and annealed results and to observe new asymptotic behaviors. We characterize the Laplace exponent of the process as the solution of a backward ordinary differential equation and establish the probability of extinction. Restricting our attention to the critical and subcritical cases, we show that four regimes arise for the speed of extinction, as in the case of branching processes in random environment in discrete time and space. The proofs are based on the precise asymptotic behavior of exponential functionals of L\'evy processes. Finally, we apply these results to a cell infection model and determine the mean speed of propagation of the infection