Before revision:  Here , we require perturbation strengths w such that 1/N\ll w\ll 1 for typical expanding populations of size N \infty; this range is in contrast to w=\mathcal O(1/N) for the diffusion approximations of the model. The first main result proves that the density processes of strategy converge to the replicator equation in the limit of a large finite population subject to appropriate, but general, conditions.
After revision:  Here , we require perturbation strengths w such that 1/N\ll w\ll 1 for typical expanding populations of size N 1. Under appropriate conditions on the space, the limiting density processes of strategy converge to the replicator equation in the limit of a large finite population subject to appropriate, but general, conditions.