Limit theorems for supercritical age-dependent branching processes with neutral immigration

We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate theta, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have i.i.d. lifetimes durations (non necessarily exponential) during which they give birth independently at constant rate b. First, using spine decomposition, we relax previously known assumptions required for a.s. convergence of total population size. Then, we consider three models of structured populations: either all immigrants have a different type, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model, the vector (P_1,P_2,...) of relative abundances of surviving families converges a.s. In the first model, the limit is the GEM distribution with parameter theta/b.Comment: 24 pages, 3 figure

Speed of coming down from infinity for birth and death processes

We finely describe the speed of "coming down from infinity" for birth and death processes which eventually become extinct. Under general assumptions on the birth and death rates, we firstly determine the behavior of the successive hitting times of large integers. We put in light two different regimes depending on whether the mean time for the process to go from $n+1$ to $n$ is negligible or not compared to the mean time to reach $n$ from infinity. In the first regime, the coming down from infinity is very fast and the convergence is weak. In the second regime, the coming down from infinity is gradual and a law of large numbers and a central limit theorem for the hitting times sequence hold. By an inversion procedure, we deduce that the process is a.s. equivalent to a non-increasing function when the time goes to zero. Our results are illustrated by several examples including applications to population dynamics and population genetics. The particular case where the death rate varies regularly is studied in details.Comment: 30 pages. arXiv admin note: text overlap with arXiv:1310.740

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

Influence of barley malting operating parameters on T-2 and HT-2 toxinogenesis of Fusarium langsethiae, a worrying contaminant of malting barley in Europe.

The fungus Fusarium langsethiae, exclusively described in Europe at present, seems to have taken the place of other Fusarium species in barley fields over the last 5 years. It has proved to be a highly toxic type-A trichothecene producer (T-2 and HT-2 toxins). The aim of this work was to study the ecotoxinogenesis of this fungus the better to identify and manage the health risk it may pose during the beer manufacturing process. The influence of temperature and water activity on its growth rate and production of toxins are particularly assessed from a macroscopic point of view. Different cultures were grown on sterilized rehydrated barley with a water activity between 0.630 and 0.997 and a temperature ranging from 5 to 35°C. Biomass specific to F. langsethiae and T-2 and HT-2 toxins were quantified by real-time polymerase chain reaction and liquid chromatography-mass spectrometry, respectively. It appears that the optimal temperature and water activity for F. langsethiae toxinogenesis are 28°C and 0.997. This fungus was able to produce 2.22 g kg−1 of these toxins in 16 days on barley in optimal production conditions. The malting process seems to be a critical step because, in its temperature range, specific production was six times higher than under optimal temperatures for fungus growth. In the short-term, this work will help redefine the process conditions for malting. In the medium-term, the results will contribute to the development of a molecular tool to diagnose the presence of this contaminant and the detection of the toxins in barley, from fields to the end product