The effective strength of selection in random environment

We analyse a family of two-types Wright-Fisher models with selection in a random environment and skewed offspring distribution. We provide a calculable criterion to quantify the impact of different shapes of selection on the fate of the weakest allele, and thus compare them. The main mathematical tool is duality, which we prove to hold, also in presence of random environment (quenched and in some cases annealed), between the population's allele frequencies and genealogy, both in the case of finite population size and in the scaling limit for large size. Duality also yields new insight on properties of branching-coalescing processes in random environment, such as their long term behaviour.Comment: 36 pages; v2 corrects an error in the proof of Thm 3.

A survey of random processes with reinforcement

The models surveyed include generalized P\'{o}lya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.Comment: Published at http://dx.doi.org/10.1214/07-PS094 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Simulation of ancestral selection graphs for Monte Carlo integration

An ancestral selection graph is a realization of an genealogy-process model which incorporates natural selection. The space of ancestral graphs is a countable union of spaces of unequal dimensions. We give a Markov Chain Monte Carlo algorithm simulating ancestral selection graphs. Output can be used to estimate expectations for functions defined on the space of ancestral graphs

Hierarchical Bayesian Models for Predicting The Spread of Ecological Processes

This is the pre-print version of the article found in Ecology (http://www.esajournals.org/loi/ecol).There is increasing interest in predicting ecological processes. Methods to accomplish such predictions must account for uncertainties in observation, sampling, models, and parameters. Statistical methods for spatio-temporal processes are powerful, yet difficult to implement in complicated, high-dimensional settings. However, recent advances in hierarchical formulations for such processes can be utilized for ecological prediction. These formulations are able to account for the various sources of uncertainty, and can incorporate scientific judgment in a probabilistically consistent manner. In particular, analytical diffusion models can serve as motivation for the hierarchical model for invasive species. We demonstrate by example that such a framework can be utilized to predict spatially and temporally, the house finch relative population abundance over the eastern United States.This research has been supported by a grant from the U.S. Environmental Protection Agency's Science to Achieve Results (STAR) program, Assistance Agreement No. R827257-01-0

Combinatorics of ancestral lines for a Wright-Fisher diffusion with selection in a L\'evy environment

Wright-Fisher diffusions describe the evolution of the type composition of an infinite haploid population with two types, type $0$ and type $1$, subject to neutral reproductions, selection, and sometimes mutations. In the present paper we study a Wright-Fisher diffusion in a L\'evy environment that gives a selective advantage to both types. Classical methods using the Ancestral Selection Graph (ASG) fail in the study of this model because of the complexity of the combinatorics of the ASG, resulting from the two-sided selection. We propose a new method that consists in encoding the relevant information of the ASG into a function. We show that the expectations of the coefficients of this function form a (non-stochastic) semi-group and deduce that they satisfy a linear system of differential equations. As a result we obtain a series representation for the fixation probability $h(x)$ (where $x$ is the initial proportion of individuals of type $0$ in the population) as a infinite sum of polynomials whose coefficients satisfy explicit linear relations. Our approach then allows to derive Taylor expansions at any order for $h(x)$ near $x=0$ and to obtain an explicit recursive formula for the coefficients