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

A simple, general result for the variance of substitution number in molecular evolution.

International audienceThe number of substitutions (of nucleotides, amino acids, ...) that take place during the evolution of a sequence is a stochastic variable of fundamental importance in the field of molecular evolution. Although the mean number of substitutions during molecular evolution of a sequence can be estimated for a given substitution model, no simple solution exists for the variance of this random variable. We show in this article that the computation of the variance is as simple as that of the mean number of substitutions for both short and long times. Apart from its fundamental importance, this result can be used to investigate the dispersion index R , i.e. the ratio of the variance to the mean substitution number, which is of prime importance in the neutral theory of molecular evolution. By investigating large classes of substitution models, we demonstrate that although R\ge1 , to obtain R significantly larger than unity necessitates in general additional hypotheses on the structure of the substitution model

Theory of neutral clustering for growing populations.

International audienceThe spatial distribution of most species in nature is nonuniform. We have shown recently [B. Houchmandzadeh, Phys. Rev. Lett. 101, 078103 (2008)] on an experimental ecological community of amoeba that the most basic facts of life--birth and death--are enough to cause considerable aggregation which cannot be smoothened by random movements of the organisms. This clustering, termed neutral and always present, is independent of external causes and social interaction. We develop here the theoretical groundwork of this phenomenon by explicitly computing the pair-correlation function and the variance to mean ratio of the above neutral model and its comparison to numerical simulations

Selection for altruism through random drift in variable size populations.

International audienceAltruistic behavior is defined as helping others at a cost to oneself and a lowered fitness. The lower fitness implies that altruists should be selected against, which is in contradiction with their widespread presence is nature. Present models of selection for altruism (kin or multilevel) show that altruistic behaviors can have 'hidden' advantages if the 'common good' produced by altruists is restricted to some related or unrelated groups. These models are mostly deterministic, or assume a frequency dependent fitness. Evolutionary dynamics is a competition between deterministic selection pressure and stochastic events due to random sampling from one generation to the next. We show here that an altruistic allele extending the carrying capacity of the habitat can win by increasing the random drift of {}''selfish'' alleles. In other terms, the \emph{fixation probability} of altruistic genes can be higher than those of a selfish ones, even though altruists have a smaller fitness. Moreover when populations are geographically structured, the altruists advantage can be highly amplified and the fixation probability of selfish genes can tend toward zero. The above results are obtained both by numerical and analytical calculations. Analytical results are obtained in the limit of large populations. The theory we present does not involve kin or multilevel selection, but is based on the existence of random drift in variable size populations. The model is a generalization of the original Fisher-Wright and Moran models where the carrying capacity depends on the number of altruists

Alternative to the diffusion equation in population genetics.

International audienceSince its inception by Kimura in 1955 [M. Kimura, Proc. Natl. Acad. Sci. U.S.A. 41, 144 (1955)], the diffusion equation has become a standard technique of population genetics. The diffusion equation is however only an approximation, valid in the limit of large populations and small selection. Moreover, useful quantities such as the fixation probabilities are not easily extracted from it and need the concomitant use of a forward and backward equation. We show here that the partial differential equation governing the probability generating function can be used as an alternative to the diffusion equation with none of its drawbacks: it does not involve any approximation, it has well-defined initial and boundary conditions, and its solutions are finite polynomials. We apply this technique to derive analytical results for the Moran process with selection, which encompasses the Kimura diffusion equation

The fixation probability of a beneficial mutation in a geographically structured population

International audienceOne of the most fundamental concepts of evolutionary dynamics is the 'fixation' probability, i.e. the probability that a gene spreads through the whole population. Most natural communities are geographically structured into habitats exchanging individuals among themselves. The topology of the migration patterns is believed to influence the spread of a new mutant, but no general analytical results were known for its fixation probability. We show that, for large populations, the fixation probability of a beneficial mutation can be evaluated for any migration pattern between local communities. Specifically, we demonstrate that for large populations, in the framework of the Voter model of the Moran model, the fixation probability is always smaller than or, at best, equal to the fixation probability of a non-structured population. In the 'invasion processes' version of the Moran model, the fixation probability can exceed that of a non-structured population; our method allows migration patterns to be classified according to their amplification effect. The theoretical tool we have developed in order to perform these computations uses the fixed points of the probability-generating function which are obtained by a system of second-order algebraic equations

Species abundance distribution and population dynamics in a two-community model of neutral ecology

International audienceExplicit formulas for the steady-state distribution of species in two interconnected communities of arbitrary sizes are derived in the framework of Hubbell's neutral model of biodiversity. Migrations of seeds from both communities as well as mutations in both of them are taken into account. These results generalize those previously obtained for the "island-continent" model and they allow an analysis of the influence of the ratio of the sizes of the two communities on the dominance/diversity equilibrium. Exact expressions for species abundance distributions are deduced from a master equation for the joint probability distribution of species in the two communities. Moreover, an approximate self-consistent solution is derived. It corresponds to a generalization of previous results and it proves to be accurate over a broad range of parameters. The dynamical correlations between the abundances of a species in both communities are also discussed

Stabilité du rythme circadien des cyanobactéries (investigation d un couplage entre oscillateurs)

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