## Applications

In ESS calculations, including the calculation of population dynamical equilibria, any function of X and E that has the same sign as invasion fitness may be substituted for it without affecting the answer. Hence, for the calculation of evolutionary outcomes, ln(50) does as good a job as invasion fitness.

More quantitative precision is needed for dealing with evolutionary transients. The time needed for a gene substitution is largely determined by the initial fitness of the mutant allele and the final fitness of the allele it replaces, with the two fitnesses determined by the heterozygote's demographic behavior in the resident environments set by the two homozygotes. For mutants that are sufficiently similar to the resident, both quantities can be determined by means of [18]. The speed of long-term evolution is largely determined by the establishment probabilities of favorable mutants. For mutants

h with small effect these probabilities can be determined from [19].

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