Abstract
We consider a diffusive 'replicator-mutator' equation to describe the adaptation of a large asexual population in a n-dimensional phenotypic space, under anisotropic mutation and selection effects. Though this equation has been studied previously when n = 1 or under isotropy assumptions, the n-dimensional anisotropic case remained unexplored. We prove here that the equation admits a unique solution, which is interpreted as the phenotype distribution, and we propose a new and general framework to the study of the quantitative behavior of this solution. When the initial distribution is Gaussian, the equation admits an explicit solution which remains normally distributed at all times, albeit with dynamic mean vector and variance-covariance matrix. In the general case, we derive a degenerate nonlocal parabolic equation satisfied by the distribution of the 'fitness components', and a nonlocal transport equation satisfied by the cumulant generating function of the joint distribution of these components. This last equation can be solved analytically and we then get a general formula for the trajectory of the mean fitness and all higher cumulants of the fitness distribution, over time. Such mean fitness trajectory is the typical outcome of empirical studies of adaptation by experimental evolution, and can thus be compared to empirical data. In sharp contrast with the known results based on isotropic models, our results show that the trajectory of mean fitness may exhibit (n−1) plateaus before it converges. It may thus appear 'non-saturating' for a transient but possibly long time, even though a phenotypic optimum exists. To illustrate the empirical relevance of these results, we show that the anisotropic model leads to a very good fit of Escherichia coli long-term evolution experiment, one of the most famous experimental dataset in experimental evolution. The two 'evolutionary epochs' that have been observed in this experiment have long puzzled the community: we propose that the pattern may simply stem form a climbing hill process, but in an anisotropic fitness landscape.