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

TL;DR

This paper analyzes a size-structured branching process modeling cell populations with asymmetry, demonstrating exponential growth, trait distribution stability, and conditions under which asymmetric division is evolutionarily optimal.

Contribution

It introduces a mathematical framework for asymmetric size-structured populations and proves exponential growth, trait convergence, and Darwinian optimality of asymmetry.

Findings

Population grows exponentially over time.

Trait distribution converges to a stable distribution.

Asymmetric division can be evolutionarily optimal under certain conditions.

Abstract

We study the long time behavior of an asymmetric size-structured measure-valued 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.