Splitting trees with neutral Poissonian mutations I: Small families

We consider a neutral dynamical model of biological diversity, where individuals live and reproduce independently. They have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate b. Such a genealogical tree is usually called a splitting tree, and the population counting process (N_t;t\ge 0) is a homogeneous, binary Crump--Mode--Jagers process. We assume that individuals independently experience mutations at constant rate \theta during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called allele, to its carrier. We are interested in the allele frequency spectrum at time t, i.e., the number A(t) of distinct alleles represented in the population at time t, and more specifically, the numbers A(k,t) of alleles represented by k individuals at time t, k=1,2,...,N_t. We mainly use two classes of tools: coalescent point processes and branching processes counted by random characteristics. We provide explicit formulae for the expectation of A(k,t) in a coalescent point process conditional on population size, which apply to the special case of splitting trees. We separately derive the a.s. limits of A(k,t)/N_t and of A(t)/N_t thanks to random characteristics. Last, we separately compute the expected homozygosity by applying a method characterizing the dynamics of the tree distribution as the origination time of the tree moves back in time, in the spirit of backward Kolmogorov equations.Comment: 32 pages, 2 figures. Companion paper in preparation "Splitting trees with neutral Poissonian mutations II: Large or old families

Evolution of discrete populations and the canonical diffusion of adaptive dynamics

The biological theory of adaptive dynamics proposes a description of the long-term evolution of a structured asexual population. It is based on the assumptions of large population, rare mutations and small mutation steps, that lead to a deterministic ODE describing the evolution of the dominant type, called the ``canonical equation of adaptive dynamics.'' Here, in order to include the effect of stochasticity (genetic drift), we consider self-regulated randomly fluctuating populations subject to mutation, so that the number of coexisting types may fluctuate. We apply a limit of rare mutations to these populations, while keeping the population size finite. This leads to a jump process, the so-called ``trait substitution sequence,'' where evolution proceeds by successive invasions and fixations of mutant types. Then we apply a limit of small mutation steps (weak selection) to this jump process, that leads to a diffusion process that we call the ``canonical diffusion of adaptive dynamics,'' in which genetic drift is combined with directional selection driven by the gradient of the fixation probability, also interpreted as an invasion fitness. Finally, we study in detail the particular case of multitype logistic branching populations and seek explicit formulae for the invasion fitness of a mutant deviating slightly from the resident type. In particular, second-order terms of the fixation probability are products of functions of the initial mutant frequency, times functions of the initial total population size, called the invasibility coefficients of the resident by increased fertility, defence, aggressiveness, isolation or survival.Comment: Published at http://dx.doi.org/10.1214/105051606000000628 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Adaptive dynamics in logistic branching populations

We consider a trait-structured population subject to mutation, birth and competition of logistic type, where the number of coexisting types may fluctuate. Applying a limit of rare mutations to this population while keeping the population size finite leads to a jump process, the so-called `trait substitution sequence', where evolution proceeds by successive invasions and fixations of mutant types. The probability of fixation of a mutant is interpreted as a fitness landscape that depends on the current state of the population. It was in adaptive dynamics that this kind of model was first invented and studied, under the additional assumption of large population. Assuming also small mutation steps, adaptive dynamics' theory provides a deterministic ODE approximating the evolutionary dynamics of the dominant trait of the population, called `canonical equation of adaptive dynamics'. In this work, we want to include genetic drift in this models by keeping the population finite. Rescaling mutation steps (weak selection) yields in this case a diffusion on the trait space that we call `canonical diffusion of adaptive dynamics', in which genetic drift (diffusive term) is combined with directional selection (deterministic term) driven by the fitness gradient. Finally, in order to compute the coefficients of this diffusion, we seek explicit first-order formulae for the probability of fixation of a nearly neutral mutant appearing in a resident population. These formulae are expressed in terms of `invasibility coefficients' associated with fertility, defense, aggressiveness and isolation, which measure the robustness (stability w.r.t. selective strengths) of the resident type. Some numerical results on the canonical diffusion are also given

Birth and death processes with neutral mutations

In this paper, we review recent results of ours concerning branching processes with general lifetimes and neutral mutations, under the infinitely many alleles model, where mutations can occur either at birth of individuals or at a constant rate during their lives. In both models, we study the allelic partition of the population at time t. We give closed formulae for the expected frequency spectrum at t and prove pathwise convergence to an explicit limit, as t goes to infinity, of the relative numbers of types younger than some given age and carried by a given number of individuals (small families). We also provide convergences in distribution of the sizes or ages of the largest families and of the oldest families. In the case of exponential lifetimes, population dynamics are given by linear birth and death processes, and we can most of the time provide general formulations of our results unifying both models.Comment: 20 pages, 2 figure

Splitting trees with neutral Poissonian mutations II: Largest and Oldest families

We consider a supercritical branching population, where individuals have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate. We assume that individuals independently experience neutral mutations, at constant rate $\theta$ during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called allele or haplotype, to its carrier. The type carried by a mother at the time when she gives birth is transmitted to the newborn. We are interested in the sizes and ages at time $t$ of the clonal families carrying the most abundant alleles or the oldest ones, as $t\to\infty$, on the survival event. Intuitively, the results must depend on how the mutation rate $\theta$ and the Malthusian parameter $\alpha>0$ compare. Hereafter, $N\equiv N_t$ is the population size at time $t$, constants $a,c$ are scaling constants, whereas $k,k'$ are explicit positive constants which depend on the parameters of the model. When $\alpha>\theta$, the most abundant families are also the oldest ones, they have size $cN^{1-\theta/\alpha}$ and age $t-a$. When $\alpha<\theta$, the oldest families have age $(\alpha /\theta)t+a$ and tight sizes; the most abundant families have sizes $k\log(N)-k'\log\log(N)+c$ and all have age $(\theta-\alpha)^{-1}\log(t)$. When $\alpha=\theta$, the oldest families have age $kt-k'\log(t)+a$ and tight sizes; the most abundant families have sizes $(k\log(N)-k'\log\log(N)+c)^2$ and all have age $t/2$. Those informal results can be stated rigorously in expectation. Relying heavily on the theory of coalescent point processes, we are also able, when $\alpha\leq\theta$, to show convergence in distribution of the joint, properly scaled ages and sizes of the most abundant/oldest families and to specify the limits as some explicit Cox processes

Motivational Ratings

Rating systems not only provide information to users but also motivate the rated agent. This paper solves for the optimal (effort-maximizing) rating system within the standard career concerns framework. It is a mixture two-state rating system. That is, it is the sum of two Markov processes, with one that re-effects the belief of the rater and the other the preferences of the rated agent. The rating, however, is not a Markov process. Our analysis shows how the rating combines information of different types and vintages. In particular, an increase in effort may affect some (but not all) future ratings adversely

Automatic difference measure between movies using dissimilarity measure fusion and rank correlation coefficients

International audienceWhen considering multimedia database growth, one current challenging issue is to design accurate navigation tools. End user basic needs, such as exploration, similarity search and favorite suggestions, lead to investigate how to find semantically resembling media. One way is to build numerous continuous dissimilarity measures from low-level image features. In parallel, an other way is to build discrete dissimilarities from textual information which may be available with video sequences. However, how such different measures should be selected as relevant and be fused ? To this aim, the purpose of this paper is to compare all those various issimilarities and to propose a suitable ranking fusion method for several dissimilarities. Subjective tests with human observers on the CITIA animation movie database have been carried out to validate the model

Using simple pseudo-3D hydrogeological modelling and a simplified agronomical representation to build a pertinent decision-making tool for local stakeholders: the Vivier karstic spring (France) case study

International audienceThe Syndicat des Eaux du Vivier (Vivier Water Agency-SEV) is a public agency monitoring the production, treatment, distribution and quality control of drinking water in Niort town in western France. The municipal drinking water supply mainly comes from a karstic resurgence, the Vivier spring, which is registered as a " Grenelle " priority water supply since 2013. There is a strong pressure from agriculture, which is illustrated since the 90' by nitrate concentration that exceed the European drinking water standards. There is also an increasing pressure on water quantity, mainly due to irrigation and drinking water demand, particularly in low water periods when the karst can be subject to collapses due to the low pressure in the karstic galleries. Modelling the hydrogeology of the area will help to optimize the effective quantitative and qualitative water resource management. Hydrogeological and agronomical modelling is done using the BICHE-MARTHE software chain, developed at the BRGM. Comparing observed and simulated groundwater levels, stream flows, springs flows and overflow at the Vivier spring gives satisfactory results considering the limited knowledge on the area. This part of the modelling has been strengthened by a comparison with a sensibility approach with GARDENIA regarding irrigation and with an approach using neural networks. The model integrates Agricultural practices observed in the catchments area to simulate nitrate transfers. The resulting nitrate concentrations are correct for the Vivier spring and its associated catchments (Gachet I and III) and stays within a reasonable range for other observation points on the catchments area. Modelling, together with the learnings of the measurements campaign analyses, allows us to better understand how the Vivier hydrosystem works. The spring has two supply methods: a short one-year-cycle, during which meteoric waters get through the karstic system and join the Vivier spring, and a multi-year cycle during which the effective rainfall slowly percolates through non-karstic rocks. Basic simulations are conducted to better identify the impact of agricultural and quantitative pressures on the water supply. They outline the karstic system sensibility to any * Intervenan

Wavefront Aberrations in Subjects Wearing Soft Aspheric Contact Lenses and Those Wearing Spherical Ones

Purpose: To measure the level of higher order aberrations (HOA) when wearing a soft aspheric contact lens (CL), compared to a spherical CL, in myopic subjects. Method: Fifteen myopic subjects aged 20-30 years were tested for the presence of dry eye. Aberrometry measurements were done without a contact lens as well as with a spherical CL and an aspheric CL. Root mean square error (RMS) of HOA, spherical aberration (SA) and coma were measured five times in an interval of 15 seconds without blinking for each of the 3 conditions. Results: Wearing a spherical CL produced a significant increase of SA and horizontal coma compared to an eye without a contact lens. When wearing an aspheric CL, there was a trend towards a smaller increase of these aberrations. However, the difference between both types of lens was not statistically significant. In terms of total HOA, these were higher when wearing the spherical CL, while they tended to be less with the aspheric CL. As for the variations between blinks, there was a similar increase in total HOA and individual modes with time for the three conditions. Conclusion : Wearers of aspheric CL seem to show a tendency towards smaller amounts of SA, horizontal coma and HOA in general in comparison with wearers of SCL. However, total HOA increases during a long interval between blinks, no matter the condition