  We establish an explicit rate of convergence for some systems of mean-field
interacting diffusions with logistic binary branching towards the solutions of
nonlinear 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 diffusions 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, in the dual bounded-Lipschitz distance
on finite measures, for the empirical measure of the population, from the
convergence rate in 2-Wasserstein 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 simple case of
pure binary branching diffusions.
