  The class of epistatic fitness landscapes is much more diverse than the class
of non-epistatic landscapes, and so it stands to reason that there exist
dynamical phenomena that can only be realized in the presence of epistasis.
Here, we compare evolutionary dynamics on all finite epistatic landscapes
versus all finite non-epistatic landscapes, under weak mutation. We first
analyze the mean fitness trajectory - that is, the time course of the expected
fitness of a population. We show that for any epistatic fitness landscape and
starting genotype, there always exists a non-epistatic fitness landscape and
starting genotype that produces the exact same mean fitness trajectory. Thus,
surprisingly, the space of mean fitness trajectories that can be realized by
epistatic landscapes is no more diverse than the space of mean fitness
trajectories that can be realized by non-epistatic landscapes. On the other
hand, we show that epistatic fitness landscapes can produce dynamics in the
time-evolution of the variance in fitness across replicate populations and in
the time-evolution of the expected number of substitutions that cannot be
produced by any non-epistatic landscape. These results on identifiability have
implications for efforts to infer epistasis from the types of data often
measured in experimental populations.
