  The diffusion approximation (DA) is widely used in the analysis of stochastic
population dynamics, from population genetics to ecology and evolution. DA is
an uncontrolled approximation that assumes the smoothness of the calculated
quantity over the relevant state space and fails when this property is not
satisfied. This failure becomes severe in situations where the direction of
selection switches sign. Here we employ the WKB (large-deviations) method,
which requires only the logarithm of a given quantity to be smooth over its
state space. Combining the WKB scheme with asymptotic matching techniques, we
show how to derive the diffusion approximation in a controlled manner and how
to produce better approximations, applicable for much wider regimes of
parameters. We also introduce a scalable (independent of population size)
WKB-based numerical technique. The method is applied to a central problem in
population genetics and evolution, finding the chance of ultimate fixation in a
zero-sum, two-types competition.
