Vincent Bansaye

Branching Feller diffusion for cell division with parasite infection

We now deal with a continuous time model for dividing cells which are infected by parasites. We assume that parasites proliferate in the cells and that their lifetimes are much shorter than the cell lifetimes. The quantity of parasites (X t : t ≥ 0) in a cell is modeled by a Feller diffusion (see Chapter 3 and Definition 4.1). The cells divide in continuous time at rate τ(x) which may depend on the quantity of parasites x that they contain. When a cell divides, a random fraction F of the parasites goes in the first daughter cell and a fraction (1 − F) in the second one. More generally, splitting Feller diffusion may model the quantity of some biological content which grows (without resource limitation) in the cells and is shared randomly when the cells divide (for example, proteins, nutriments, energy or extrachromosomal rDNA circles in yeast).

Lower large deviations for supercritical branching processes in random environment

Branching processes in random environment (Zn: n ≥ 0) are the generalization of Galton-Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical regime, the process survives with a positive probability and grows exponentially on the non-extinction event. We focus on rare events when the process takes positive values but lower than expected. More precisely, we are interested in the lower large deviations of Z, which means the asymptotic behavior of the probability {1 ≤ Zn ≤ exp(nθ)} as n → ∞. We provide an expression for the rate of decrease of this probability under some moment assumptions, which yields the rate function. With this result we generalize the lower large deviation theorem of Bansaye and Berestycki (2009) by considering processes where ℙ(Z1 = 0 | Z0 = 1) > 0 and also much weaker moment assumptions.

CELL CONTAMINATION AND BRANCHING PROCESSES IN A RANDOM ENVIRONMENT WITH IMMIGRATION

We consider a branching model for a population of dividing cells infected by parasites. Each cell receives parasites by inheritance from its mother cell and independent contamination from outside the cell population. Parasites multiply randomly inside the cell and are shared randomly between the two daughter cells when the cell divides. The law governing the number of parasites which contaminate a given cell depends only on whether the cell is already infected or not. We first determine the asymptotic behavior of branching processes in a random environment with state-dependent immigration, which gives the convergence in distribution of the number of parasites in a cell line. We then derive a law of large numbers for the asymptotic proportions of cells with a given number of parasites. The main tools are branching processes in a random environment and laws of large numbers for a Markov tree.

Small positive values for supercritical branching processes in random environment

Branching Processes in Random Environment (BPREs)

Tightness for processes with fixed points of discontinuities and applications in varying environment

(Z_n:n\geq0)

Phenotypic diversity and population growth in a fluctuating environment

are the generalization of Galton-Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of

Queueing for an infinite bus line and aging branching process

\mathbb{P}(1 \leq Z_n \leq k|Z_0=i)

Feller Diffusion with Random Catastrophes

,

Birth and Death Processes

k,i\in\mathbb{N}

Markov Processes along Continuous Time Galton-Watson Trees

as