  We establish an explicit rate of convergence for some systems of mean-field
interacting diffusions with logistic binary branching, towards the solutions of
non-linear evolution equations with non-local self-diffusion and logistic mass
growth, shown to describe their large population limits in arXiv:1303.3939. The
proof relies on a novel coupling argument for binary branching diffusion
processes, based on optimal transport, which allows us to sharply mimic the
trajectory of the interacting binary branching population by certain system of
independent particles with suitably distributed random space-time births. We
are thus able to derive an optimal convergence rate for the empirical measure
of the population in the dual bounded-Lipschitz distance on finite measures,
from the convergence rate in Wasserstein-2 distance of empirical distributions
of i.i.d. samples. Our approach and results extend techniques and ideas on
propagation of chaos from kinetic models to stochastic systems of interacting
branching populations, and appear to be new in this setting, even in the
simplest case of pure binary branching diffusions.
