Dynamics of adaptation: extreme value domains, distance to fitness optimum and fitness correlations

We study the properties of adaptive walk performed by a maladapted asexual population in which beneficial mutations fix sequentially until a local fitness peak is reached. Here we consider three factors that govern the adaptation dynamics: the extreme value domain of beneficial mutations, initial distance to the local fitness optimum and the correlations amongst the fitnesses. We show that there is a transition in the behaviour of the walk length and average fitness fixed during adaptation when the mean and variance of the fitness distribution respectively become infinite. When the mean is finite, walk length decreases logarithmically with initial fitness but is a constant otherwise. We also find that the walks are longer for faster decaying fitness distributions and correlated fitnesses. For fitness distributions with finite variance, the fitness fixed during initial steps does not depend on the fitness of the local optimum but increases with the local peak fitness otherwise. Interestingly, the fitness difference between successive steps shows a pattern of diminishing returns for bounded distributions and accelerating returns for fat-tailed distributions. These trends are found to be robust with respect to fitness correlations.