Dynamics of adaptation in an anisotropic phenotype-fitness landscape

Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): F. Hamel, F. Lavigne, G. Martin, L. RoquesAbstractWe study the dynamics of adaptation of a large asexual population in a n-dimensional phenotypic space, under anisotropic mutation and selection effects. When n=1 or under isotropy assumptions, the ‘replicator-mutator’ equation is a standard model to describe these dynamics. However, the n-dimensional anisotropic case remained largely 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 behaviour of this solution. Our method builds upon 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 appea...
Source: Nonlinear Analysis: Real World Applications - Category: Research Source Type: research