  We consider two species of particles performing random walks in a domain in
$\mathbb{R}^d$ with reflecting boundary conditions, which annihilate on
contact. In addition, there is a conservation law so that the total number of
particles of each type is preserved: When the two particles of different
species annihilate each other, particles of each species, chosen at random,
give birth. We assume initially equal numbers of each species and show that the
system has a diffusive scaling limit in which the densities of the two species
are well approximated by the positive and negative parts of the solution of the
heat equation normalized to have constant $L^1$ norm. In particular, the higher
Neumann eigenfunctions appear as asymptotically stable states at the diffusive
time scale.
