Initially identical populations in identical environments may subsequently diverge from one another not only via the effects of genetic drift on neutral alleles, but also by selection on beneficial alleles that arise stochastically by mutation. In the simple case of one locus with two alleles in a haploid organism, a full range of combinations of population sizes, selection pressures, mutation rates and fixation probabilities reveals two qualitatively distinct dynamics of divergence among such initially identical populations. We define a non-dimensional parameter k that describes conditions for the occurrence of these different dynamics. One dynamic (k > 1) occurs when beneficial mutations are sufficiently common that substitutions within the populations are essentially simultaneous; the other dynamic (k < 1) occurs when beneficial mutations are so rare that substitutions are likely to occur as isolated events. If there are more than two alleles, or multiple loci, divergence among the populations can be sustained indefinitely if k < 1. The parameter k pertains to the nature of biological evolution and its tendency to be gradual or punctuated.