The infinite dilution assumption allows proving the asymptotic exponential growth of the focal population. The definition of invasion fitness only requires this growth pattern. Such a pattern also occurs when individuals aggregate, with individuals in an aggregate interacting with each other for extended periods of time, provided the aggregates are infinitely diluted (so that, although the individuals appreciably influence their immediate environments, they only infinitesimally influence the environment outside the aggregate). Some examples of aggregates are (1) diploid individuals, or more generally genets, (2) family groups (with singles treated as families of size one), (3) patches in a structured metapopulation, and (4) pairs in so-called pair approximation calculations for a population supposedly living on the nodes of a graph. For the invasion problem only aggregates with at least one individual of the focal type count. Such aggregates will be referred to here as metaindividuals.
It is always possible to define an h-state of a metain-dividual in the form of a list of the types and h-states of all individuals in it, plus a scheme of their possibilities for interaction. In the case of a mutant, one other type of individuals are the residents, but in (1) they may in addition be alleles on additional polymorphic loci, and in (3) and (4) any other species in the metacommunity. The existence of at least one h-state representation, however impractical, allows invoking the multiplicative ergodic theorem. In special models more practical simplified representations may be possible. For instance, when the individuals in (3) are all the same but for a distinction between mutants and residents, a patch can be characterized with just the mutant and resident numbers, and when moreover the total number of individuals in a patch is constant, only the number of mutants is needed.
The disappearance of all focal types from an aggregate can be interpreted as a metadeath, the new appearance of aggregates with at least one individual of the focal type as metabirths. In (1), metabirths correspond to fertilizations; in (2), to the splitting of families; in (3), to the immigration of an individual of the focal type into a patch not yet containing such individuals (the infinite dilution assumption guarantees that a metaindividual once born does not experience further immigrations of the focal type); and in (4), to having a pair of adjacent nodes on the graph filled with non-focal-type individuals replaced by a pair with at least one focal type.
Since metaindividuals reproduce clonally and are infinitely diluted, all recipes for the calculation of invasion fitness and of proxies thereof equally apply to metaindividuals.
In addition, if the asymptotic per capita growth of the expected number of focal individuals in metaindividuals is less than the invasion fitness of the latter, the asymptotic per capita rate of initial increase of a type equals the invasion fitness of metaindividuals for that type, and hence can rightfully be called the invasion fitness of the type.
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