  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.
