## Coda

Why Initial Growth Rates?

One consideration behind the definition of invasion fitness is that the resulting quantities should link evolutionary features for arbitrary ecological scenarios with similar features for the classical simple viability selection model. In the latter model with viabilities vAA > vaA > vaa, the time taken for a gene substitution is largely determined by the initial and final exponential phases, which have as time constants the invasion fitnesses ln(vaA/vaa) > 0 and ln(vaA/vAA) < 0. Gene substitutions in more general ecologies generally follow the same pattern. Similarly, two alleles will coexist if they have positive invasion fitness in the environment set by the other allele, which for simple viability selection reduces to the classical vaA > vaa and vaA > vAA. Making the quantitative links requires (invasion) fitness to be defined as the asymptotic per capita growth rate.

For long-term evolution, invasion probabilities are the most important. However, for their full quantitative determination, less-accessible life-history details are needed, while the essential qualitative information, as well as a good quantitative estimate for small mutational steps, can be extracted (up to a proportionality constant) from invasion fitness.

Similar Phenotypes and Fisher's Fundamental Theorem

When the phenotypes under consideration are sufficiently similar, a case can even be made to drop the epithet 'invasion'. Under the similarity assumption, for fairly general classes of ecogenetic models, the change in the genetic make-up of the population approximately follows the differential equations derived customarily for simple viability selection, with log viabilities replaced by invasion fitnesses in the environment that would be generated by a clonally reproducing population with the average phenotype. This in retrospect vindicates Fisher's use of differential expressions mentioned in the introductory section, but only under the assumption of like phenotypes, and with a reference to the environment thrown in. (Note also that under the similarity assumption alleles on the different loci and h-states are all (almost) independent, so that there is no need to consider specifically reproductive-value-weighted allele frequencies; moreover, ln(R0)/Tb can replace the Malthusian parameters.) Under the same assumption, although the mean fitness of the population stays (approximately) zero, this lack of change can be decomposed into two opposing terms, the first corresponding to the expression brought to the fore in Fisher's fundamental theorem, and the second equal to the average change of the phenotypic fitnesses caused by the environmental change resulting from changes in the population composition, consonant with Fisher's verbal exegesis of his theorem.

### Not the Last Word

Apparently, invasion fitness, as defined in this article, does a good job. Only one assumption, infinite dilution, was needed for its definition. Perhaps the reach of the concept can be extended still a little further. However, there will certainly remain ecological scenarios where the extension fails. This does not mean that under those scenarios there never will be adaptive evolution, only that it is not possible to deduce its outcomes by means of the conceptual precision tool called invasion fitness.

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