Continuous stability and evolutionary convergence

A stochastic process of long-term evolution due to mutation and selection is defined over an asexually reproducing population, with selection according to a population game with a one-dimensional continuity of pure strategies. Limiting the analysis to mutations of small effect, it is shown that long-term dynamic stability in such a process is equivalent to continuous stability in the relevant population game. In the case of a one-dimensional strategy set (but not necessarily if the strategy set is multi-dimensional), this result is virtually independent of the distribution of mutations.