An alternative to the breeder's and Lande's equations

The breeder's equation is a cornerstone of quantitative genetics and is widely used in evolutionary modeling. The equation which reads R=h^{2}S relates response to selection R (the mean phenotype of the progeny) to the selection differential S (mean phenotype of selected parents) through a simple proportionality relation. The validity of this relation however relies strongly on the normal (Gaussian) distribution of parent's genotype which is an unobservable quantity and cannot be ascertained. In contrast, we show here that if the fitness (or selection) function is Gaussian, an alternative, exact linear equation in the form of R'=j^{2}S' can be derived, regardless of the parental genotype distribution. Here R' and S' stand for the mean phenotypic lag behind the mean of the fitness function in the offspring and selected populations. To demonstrate this relation, we derive the exact functional relation between the mean phenotype in the selected and the offspring population and deduce all cases that lead to a linear relation between these quantities. These computations, which are confirmed by individual based numerical simulations, generalize naturally to the multivariate Lande's equation \Delta\mathbf{\bar{z}}=GP^{-1}\mathbf{S}

Neutral Aggregation in Finite Length Genotype space

The advent of modern genome sequencing techniques allows for a more stringent test of the neutrality hypothesis of Darwinian evolution, where all individuals have the same fitness. Using the individual based model of Wright and Fisher, we compute the amplitude of neutral aggregation in the genome space, i.e., the probability of finding two individuals at genetic (hamming) distance k as a function of genome size L, population size N and mutation probability per base \nu. In well mixed populations, we show that for N\nu\textless{}1/L, neutral aggregation is the dominant force and most individuals are found at short genetic distances from each other. For N\nu\textgreater{}1 on the contrary, individuals are randomly dispersed in genome space. The results are extended to geographically dispersed population, where the controlling parameter is shown to be a combination of mutation and migration probability. The theory we develop can be used to test the neutrality hypothesis in various ecological and evolutionary systems

General formulation of Luria-Delbr{\"u}ck distribution of the number of mutants

The Luria-Delbr{\"u}ck experiment is a cornerstone of evolutionary theory, demonstrating the randomness of mutations before selection. The distribution of the number of mutants in this experiment has been the subject of intense investigation during the last 70 years. Despite this considerable effort, most of the results have been obtained under the assumption of constant growth rate, which is far from the experimental condition. We derive here the properties of this distribution for arbitrary growth function, for both the deterministic and stochastic growth of the mutants. The derivation we propose uses the number of wild type bacteria as the independent variable instead of time. The derivation is surprisingly simple and versatile, allowing many generalizations to be taken easily into account

Exact results for fixation probability of bithermal evolutionary graphs

One of the most fundamental concepts of evolutionary dynamics is the "fixation" probability, i.e. the probability that a mutant spreads through the whole population. Most natural communities are geographically structured into habitats exchanging individuals among each other and can be modeled by an evolutionary graph (EG), where directed links weight the probability for the offspring of one individual to replace another individual in the community. Very few exact analytical results are known for EGs. We show here how by using the techniques of the fixed point of Probability Generating Function, we can uncover a large class of of graphs, which we term bithermal, for which the exact fixation probability can be simply computed

Fisher Waves: an individual based stochastic model

The propagation of a beneficial mutation in a spatially extended population is usually studied using the phenomenological stochastic Fisher-Kolmogorov (SFKPP) equation. We derive here an individual based, stochastic model founded on the spatial Moran process where fluctuations are treated exactly. At high selection pressure, the results of this model are different from the classical FKPP. At small selection pressure, the front behavior can be mapped into a Brownian motion with drift, the properties of which can be derived from microscopic parameters of the Moran model. Finally, we show that the diffusion coefficient and the noise amplitude of SFKPP are not independent parameters but are both determined by the dispersal kernel of individuals

Exact results for a noise-induced bistable system

A stochastic system where bistability is caused by noise has been recently investigated by Biancalani et al. (PRL 112:038101, 2014). They have computed the mean switching time for such a system using a continuous Fokker-Planck equation derived from the Taylor expansion of the Master equation to estimate the parameter of such a system from experiment. In this article, we provide the exact solution for the full discrete system without resorting to continuous approximation and obtain the expression for the mean switching time. We further extend this investigation by solving exactly the Master equation and obtaining the expression of other quantities of interests such as the dynamics of the moments and the equilibrium time

Fluctuation driven fixation of cooperative behavior

International audienceCooperative behaviors are defined as the production of common goods benefit-ting all members of the community at the producer's cost. They could seem to be in contradiction with natural selection, as non-cooperators have an increased fitness compared to cooperators. Understanding the emergence of cooperation has necessitated the development of concepts and models (inclusive fitness, mul-tilevel selection, ...) attributing deterministic advantages to this behavior. In contrast to these models, we show here that cooperative behaviors can emerge by taking into account only the stochastic nature of evolutionary dynamics: when cooperative behaviors increase the population size, they also increase the genetic drift against non-cooperators. Using the Wright-Fisher models of population ge-netics, we compute exactly this increased genetic drift and its consequences on the fixation probability of both types of individuals. This computation leads to a simple criterion: cooperative behavior dominates when the relative increase in population size caused by cooperators is higher than the selection pressure against them. This is a purely stochastic effect with no deterministic interpre-tation

Large deviation of long time average for a stochastic process : an alternative method.

We present here a simple method for computing the large deviation of long time average for stochastic jump processes. We show that the computation of the rate function can be reduced to that of a partial differential equation governing the evolution of the probability generating function. The long time limit of this equation, which in many cases can be easily obtained, leads naturally to the rate function