Evolution on a smooth landscape

We study in detail a recently proposed simple discrete model for evolution on smooth landscapes. An asymptotic solution of this model for long times is constructed. We find that the dynamics of the population are governed by correlation functions that although being formally down by powers of $N$ (the population size) nonetheless control the evolution process after a very short transient. The long-time behavior can be found analytically since only one of these higher-order correlators (the two-point function) is relevant. We compare and contrast the exact findings derived herein with a previously proposed phenomenological treatment employing mean field theory supplemented with a cutoff at small population density. Finally, we relate our results to the recently studied case of mutation on a totally flat landscape.