Fig 8. The probability of establishment of a new mutant allele as a function of distance from the edge of the region where it is beneficial, in an abrupt transition (left) and a gradual transition (right).

The allele is deleterious to the left and beneficial to the right (with selection coefficients ∓0.05 respectively); and the selection coefficient interpolates linearly between these in the central hatched region. The number of offspring is Poisson. The underlying landscape has demes every 15km and a migration rate of 0.5 between neighboring demes; the probability was found by numerically solving the equation for the probability of establishment of a multi-type branching process. Note that the approximation used in the text is here p _e ≈ 2s _p/ξ ² = 0.1, agreeing with the value in the center of the patch.