Deterministic evolution of an asexual population under the action of beneficial and deleterious mutations on additive fitness landscapes

We study a continuous time model for the frequency distribution of an infinitely large asexual population in which both beneficial and deleterious mutations occur and the fitness is additive. When beneficial mutations are ignored, the exact solution for the frequency distribution is known to be a Poisson distribution. Here we include beneficial mutations and obtain exact expressions for the frequency distribution at all times using an eigenfunction expansion method. We find that the stationary distribution is non-Poissonian and related to the Bessel function of the first kind. We also provide suitable approximations for the stationary distribution and the time to relax to the steady state. Our exact results, especially at mutation-selection equilibrium, can be useful in developing semi-deterministic approaches to understand stochastic evolution.