A stochastic analysis of resource sharing with logarithmic weights

The paper investigates the properties of a class of resource allocation algorithms for communication networks: if a node of this network has $x$ requests to transmit, then it receives a fraction of the capacity proportional to $\log(1+x)$, the logarithm of its current load. A detailed fluid scaling analysis of such a network with two nodes is presented. It is shown that the interaction of several time scales plays an important role in the evolution of such a system, in particular its coordinates may live on very different time and space scales. As a consequence, the associated stochastic processes turn out to have unusual scaling behaviors. A heavy traffic limit theorem for the invariant distribution is also proved. Finally, we present a generalization to the resource sharing algorithm for which the $\log$ function is replaced by an increasing function. Possible generalizations of these results with $J>2$ nodes or with the function $\log$ replaced by another slowly increasing function are discussed.Comment: Published at http://dx.doi.org/10.1214/14-AAP1057 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The role of mode switching in a population of actin polymers with constraints

In this paper, we introduce a stochastic model for the dynamics of actin polymers and their interactions with other proteins in the cellular envelop. Each polymer elongates and shortens, and can switch between several modes depending on whether it is bound to accessory proteins that modulate its behaviour as, for example, elongation-promoting factors. Our main aim is to understand the dynamics of a large population of polymers, assuming that the only limiting quantity is the total amount of monomers, set to be constant to some large N. We first focus on the evolution of a very long polymer, of size O(N), with a rapid switch between modes (compared to the timescale over which the macroscopic fluctuations in the polymer size appear). Letting N tend to infnity, we obtain a fluid limit in which the effect of the switching appears only through the fraction of time spent in each mode at equilibrium. We show in particular that, in our situation where the number of monomers is limiting, a rapid binding-unbinding dynamics may lead to an increased elongation rate compared to the case where the polymer is trapped in any of the modes. Next, we consider a large population of polymers and complexes, represented by a random measure on some appropriate type space. We show that as N tends to infinity, the stochastic system converges to a deterministic limit in which the switching appears as a flow between two categories of polymers. We exhibit some numerical examples in which the limiting behaviour of a single polymer differs from that of a population of competing (shorter) polymers for equivalent model parameters. Taken together, our results demonstrate that under conditions where the total number of monomers is limiting, the study of a single polymer is not sufficient to understand the behaviour of an ensemble of competing polymers

Modelling evolution in a spatial continuum

We survey a class of models for spatially structured populations which we have called spatial Λ-Fleming–Viot processes. They arise from a flexible framework for modelling in which the key innovation is that random genetic drift is driven by a Poisson point process of spatial ‘events’. We demonstrate how this overcomes some of the obstructions to modelling populations which evolve in two- (and higher-) dimensional spatial continua, how its predictions match phenomena observed in data and how it fits with classical models. Finally we outline some directions for future research

The fate of recessive deleterious or overdominant mutations near mating-type loci under partial selfing

Large regions of suppressed recombination having extended over time occur in many organisms around genes involved in mating compatibility (sex-determining or mating-type genes). The sheltering of deleterious alleles has been proposed to be involved in such expansions. However, the dynamics of deleterious mutations partially linked to genes involved in mating compatibility are not well understood, especially in finite populations. In particular, under what conditions deleterious mutations are likely to be maintained for long enough near mating-compatibility genes remains to be evaluated, especially under selfing, which generally increases the purging rate of deleterious mutations. Using a branching process approximation, we studied the fate of a new deleterious or overdominant mutation in a diploid population, considering a locus carrying two permanently heterozygous mating-type alleles, and a partially linked locus at which the mutation appears. We obtained analytical and numerical results on the probability and purging time of the new mutation. We investigated the impact of recombination between the two loci and of the mating system (outcrossing, intra and inter-tetrad selfing) on the maintenance of the mutation. We found that the presence of a fungal-like mating-type locus (i.e. not preventing diploid selfing) always sheltered the mutation under selfing, i.e. it decreased the purging probability and increased the purging time of the mutations. The sheltering effect was higher in case of automixis (intra-tetrad selfing). This may contribute to explain why evolutionary strata of recombination suppression near the mating-type locus are found mostly in automictic (pseudo-homothallic) fungi. We also showed that rare events of deleterious mutation maintenance during strikingly long evolutionary times could occur, suggesting that deleterious mutations can indeed accumulate near the mating-type locus over evolutionary time scales. In conclusion, our results show that, although selfing purges deleterious mutations, these mutations can be maintained for very long times near a mating-type locus, which may contribute to promote the evolution of recombination suppression in sex-related chromosomes

Quenched convergence of a sequence of superprocesses in R^d among Poissonian obstacles

We prove a convergence theorem for a sequence of super-Brownian motions moving among hard Poissonian obstacles, when the intensity of the obstacles grows to infinity but their diameters shrink to zero in an appropriate manner. The superprocesses are shown to converge in probability for the law $\mathbf{P}$ of the obstacles, and $\mathbf{P}$-almost surely for a subsequence, towards a superprocess with underlying spatial motion given by Brownian motion and (inhomogeneous) branching mechanism $\psi(u,x)$ of the form $\psi(u,x)= u^2+ \kappa(x)u$, where $\kappa(x)$ depends on the density of the obstacles. This work draws on similar questions for a single Brownian motion. In the course of the proof, we establish precise estimates for integrals of functions over the Wiener sausage, which are of independent interest.Comment: 22 page

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Quenched convergence of a sequence of superprocesses in among Poissonian obstacles

We prove a convergence theorem for a sequence of super-Brownian motions moving among hard Poissonian obstacles, when the intensity of the obstacles grows to infinity but their diameters shrink to zero in an appropriate manner. The superprocesses are shown to converge in probability for the law of the obstacles, and -almost surely for a subsequence, towards a superprocess with underlying spatial motion given by Brownian motion and (inhomogeneous) branching mechanism [psi](u,x) of the form [psi](u,x)=u2+[kappa](x)u, where [kappa](x) depends on the density of the obstacles. This work draws on similar questions for a single Brownian motion. In the course of the proof, we establish precise estimates for integrals of functions over the Wiener sausage, which are of independent interest.Super-Brownian motion Random obstacles Quenched convergence Brownian motion Wiener sausage

Growth properties of the infinite-parent spatial Lambda-Fleming-Viot process

The infinite-parent spatial Lambda-Fleming Viot process, or infinite-parent SLFV, is a model for spatially expanding populations in which empty areas are filled with "ghost" individuals. The interest of this process lies in the fact that it is akin to a continuous-space version of the classical Eden growth model, while being associated to a dual process encoding ancestry and allowing one to study the evolution of the genetic diversity in such a population. In this article, we focus on the growth properties of the infinite-parent SLFV in two dimensions. To do so, we first define the quantity that we shall use to quantify the speed of growth of the area covered with the subpopulation of real individuals. Using the associated dual process and a comparison with a first-passage percolation problem, we show that the growth of the "occupied" region in the infinite-parent SLFV is linear in time. We use numerical simulations to approximate the growth speed, and conjecture that the actual speed is higher than the speed expected from simple first-moment calculations due to the characteristic front dynamics. We then study a toy model of two interacting growing piles of cubes in order to understand how the growth dynamics at the front edge can increase the global growth speed of the "occupied" region. We obtain an explicit formula for this speed of growth in our toy model, using the invariant distribution of a discretised version of the model. This study is of interest on its own right, and its implications are not restricted to the case of the infinite-parent SLFV