  We use traveling-wave theory to derive expressions for the rate of
accumulation of deleterious mutations under Muller's ratchet and the speed of
adaptation under positive selection in asexual populations. Traveling-wave
theory is a semi-deterministic description of an evolving population, where the
bulk of the population is modeled using deterministic equations, but the class
of the highest-fitness genotypes, whose evolution over time determines loss or
gain of fitness in the population, is given proper stochastic treatment. We
derive improved methods to model the highest-fitness class (the stochastic
edge) for both Muller's ratchet and adaptive evolution, and calculate analytic
correction terms that compensate for inaccuracies which arise when treating
discrete fitness classes as a continuum. We show that traveling wave theory
makes excellent predictions for the rate of mutation accumulation in the case
of Muller's ratchet, and makes good predictions for the speed of adaptation in
a very broad parameter range. We predict the adaptation rate to grow
logarithmically in the population size until the population size is extremely
large.
