We consider a discrete model that describes locally regulated spatial population with mortality selection. This was studied in parallel by Bolker and Pacala Dieckmann, Law Murrell. first generalize this adding dependence. Then we give pathwise description terms of Poisson point measures. show different normalizations may lead to macroscopic approximations model. The approximation is deterministic gives rigorous sense the number density. second superprocess previously Etheridge. Finally,...

This survey concerns the study of quasi-stationary distributions with a specific focus on models derived from ecology and population dynamics. We are concerned long time behavior different stochastic size processes when 0 is an absorbing point almost surely attained by process. The hitting this point, namely extinction time, can be large compared to physical fluctuate for amount before actually occurs. phenomenon understood quasi-limiting distributions. In paper, general results...

We are interested in modelling Darwinian evolution, resulting from the interplay of phenotypic variation and natural selection through ecological interactions. Our models rooted microscopic, stochastic description a population discrete individuals characterized by one or several adaptive traits. The is modelled as point process whose generator captures probabilistic dynamics over continuous time birth, mutation, death, influenced each individual's trait values, interactions between...

In this paper, we are interested in a stochastic differential equation which is nonlinear the following sense: both diffusion and drift coefficients depend locally on density of time marginal solution. When law initial data has smooth with respect to Lebesgue measure, prove existence uniqueness for equation. Under more restrictive assumptions density, approximate solution by system n moderately interacting processes obtain trajectorial propagation chaos result. Finally, study fluctuations...

In this paper we study quasi-stationarity for a large class of Kolmogorov diffusions. The main novelty here is that allow the drift to go −∞ at origin, and diffusion have an entrance boundary +∞. These diffusions arise as images, by deterministic map, generalized Feller diffusions, which themselves are obtained limits rescaled birth–death processes. Generalized take nonnegative values absorbed zero in finite time with probability 1. An important example logistic diffusion. We give sufficient...

Mathematical modeling offers the opportunity to test hypothesis concerning Myeloproliferative emergence and development. We tested different mathematical models based on a training cohort (n=264 patients) (Registre de la c\^ote d'Or) determine evolution times before JAK2V617F classical disorders (respectively Polycythemia Vera Essential Thrombocytemia) are diagnosed. dissected time diagnosis as two main periods: from embryonic development for mutation occur, not disappear enter in...

Signs of ageing become apparent only late in life, after organismal development is finalized. Ageing, most notably, decreases an individual’s fitness. As such, it commonly perceived as a non-adaptive force evolution and considered by-product natural selection. Building upon the evolutionarily conserved age-related Smurf phenotype, we propose simple mathematical life-history trait model which organism characterized by two core abilities: reproduction homeostasis. Through simulation this...

Abstract We perform an asymptotic analysis of models population dynamics with a fractional Laplacian and local or nonlocal reaction terms. The first part the paper is devoted to long time/long range rescaling Fisher-KPP equation. This based on exponential speed propagation population. In particular we show that only role in determining this at initial layer where it determines thickness tails solutions. Next, such also possible for non-local terms, as selection-mutation models. However,...

Abstract We describe in detail the speed of `coming down from infinity' for birth-and-death processes which eventually become extinct. Under general assumptions on rates, we firstly determine behavior successive hitting times large integers. identify two different regimes depending whether mean time process to go n +1 is negligible or not compared reach ∞. In first regime, coming infinity very fast and convergence weak. second gradual a law numbers central limit theorem sequence hold. By an...

Nous considérons une classe de processus naissance-et-mort décrivant population constituée $d$ sous-populations types différents qui intéragissent entre elles. L'espace d'état est $\mathbb{Z}^{d}_{+}$ (il donc non borné). supposons que la s'éteint presque sûrement, sorte l'unique distribution probabilité stationnaire masse Dirac à l'origine. faisons dépendre ces d'un paramètre d'échelle $K$ qu'on peut interpréter comme l'ordre grandeur taille totale au temps $0$. Etant donné un intervalle...

We consider the Navier–Stokes equation in dimension 2 and more precisely vortex satisfied by curl of velocity field.We show relation between this a nonlinear stochastic differential equation. Next we use probabilistic interpretation to construct approximating interacting particle systems which satisfy propagation chaos property: laws empirical measures tend, as number particles tends $\infty$, deterministic law for marginals are solutions equation.This pathwise result justifies completely...

We introduce two stochastic chemostat models consisting in a coupled population-nutrient process reflecting the interaction between nutrient and bacterias with finite volume. The concentration evolves continuously but depending on population size, while size is birth death coefficients time through concentration. shared by bacteria creates regulation of bacterial size. latter fluctuations due to random births deaths individuals make go almost surely extinction. Therefore, we are interested...