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We study a continuous time model for the frequency distribution of aninfinitely large asexual population in which both beneficial and deleteriousmutations occur and the fitness is additive. When beneficial mutations areignored, the exact solution for the frequency distribution is known to be aPoisson distribution. Here we include beneficial mutations and obtain exactexpressions for the frequency distribution at all times using an eigenfunctionexpansion method. We find that the stationary distribution is non-Poissonianand related to the Bessel function of the first kind. We also provide suitableapproximations for the stationary distribution and the time to relax to thesteady state. Our exact results, especially at mutation-selection equilibrium,can be useful in developing semi-deterministic approaches to understandstochastic evolution.

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