Long-time behavior and darwinian optimality for an asymmetric size-structured branching process

Abstract

We study the long time behavior of an asymmetric size-structured measurevalued growth-fragmentation branching process that models the dynamics of a population of cells taking into account physiological and morphological asymmetry at division. We show that the process exhibits a Malthusian behavior; that is that the global population size grows exponentially fast and that the trait distribution of individuals converges to some stable distribution. The proof is based on a generalization of Lyapunov function techniques for non-conservative semi-groups. We then investigate the fluctuations of the growth rate with respect to the parameters guiding asymmetry. In particular, we exhibit that, under some special assumptions, asymmetric division is optimal in a Darwinian sense.

Ergodic Behaviour of a Multi-Type Growth-Fragmentation Process Modelling the Mycelial Network of a Filamentous Fungus

“…This approach was also used in the recent paper [18] in which individuals (bacteria) can be of two types with dierent growth parameters: for both types of individuals, the trait considered grows exponentially fast but at two dierent rates α 0 , α 1 > 0, and the branching/fragmentation rate is common to all individuals and is trait-dependent. During each fragmentation event, the length of the parent is split between the two ospring in xed proportions θ 0 , θ 1 = 1 − θ 0 .…”

“…During each fragmentation event, the length of the parent is split between the two ospring in xed proportions θ 0 , θ 1 = 1 − θ 0 . Because of the very quick elongation of both types of individuals, together with the assumption that the (positive) branching rate tends to innity as the individual length goes to innity, it is natural (although not at all easy to prove) that the same form of convergence (1.8) as in similar systems with only one type of individuals should occur in their framework, and indeed this constitutes the main results of [18]. In contrast, in our case closed individuals do not elongate and the length of open individuals increases rather slowly (linearly with time).…”