  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.
