Adaptive dynamics
theory
This page builds on the demographic machinery developed for the size-structured PDE to estimate the fitness of rare mutant phenotypes and so reason about evolutionary endpoints in a trait-, size-, and patch-structured metacommunity. The approach for calculating fitness implemented here follows Falster, Brännström, Westoby, & Dieckmann (2015).
Invasion fitness
We now consider how to estimate the fitness of a rare mutant individual with traits \(x^\prime\) growing in the light environment of a resident community with traits \(x\). We focus on phenotype-dependent components of fitness – describing the aggregate consequences of a given set of traits for growth, fecundity, and mortality – taking into account the non-linear effects of competition, but ignoring the underlying genetic basis for trait inheritance and expression. We also adhere to standard conventions in such analyses by assuming that the mutant phenotype is sufficiently rare to have a negligible effect on the light environment where it is growing (Geritz, Kisdi, Meszéna, & Metz, 1998). The approach for calculating fitness implemented here follows the approach described by Falster et al. (2015).
In general, invasion fitness is defined as the long-term per capita growth rate of a rare mutant phenotype in the environment determined by a resident phenotype (Metz, Nisbet, & Geritz, 1992). Calculating per capita growth rates, however, is particularly challenging in a structured metacommunity model (Gyllenberg & Metz, 2001; Metz & Gyllenberg, 2001). As an alternative measure of invasion fitness in metacommunities, we can use the basic reproduction ratio measuring the expected number of new dispersers arising from a single dispersal event. For metacommunities at demographic equilibrium, evolutionary inferences made using basic reproduction ratios are equivalent to those made using per capita growth rates (Gyllenberg & Metz, 2001; Metz & Gyllenberg, 2001).
We denote by \(R\left(x^\prime, x\right)\) the basic reproduction ratio of individuals with traits \(x^\prime\) growing in the competitive light environment of the resident traits \(x\). Recalling that patches of age \(a\) have frequency-density \(P(a)\) in the landscape (the equilibrium patch-age distribution derived on the size-structured PDE page), it follows that any seed with traits \(x^\prime\) has a probability of \(P(a)\) of landing in a patch of age \(a\). The basic reproduction ratio for individuals with traits \(x^\prime\) is then \[ R\left(x^\prime,x\right) = \int _0^{\infty} P\left(a\right) \, \tilde{R}\left(x^\prime, a, \infty \right) \, {\rm d}a , \tag{1}\] where \(\tilde{R}\left(x^\prime, a_0, a \right)\) is the expected number of dispersing offspring produced by a single dispersing seed with traits \(x^\prime\) arriving in a patch of age \(a_0\) up until age \(a\) (Gyllenberg & Metz, 2001; Metz & Gyllenberg, 2001). In turn, \(\tilde{R}\left(x^\prime, a,\infty\right)\) is calculated by integrating an individual’s fecundity over the expected lifetime of a patch, taking into account competitive shading from residents with traits \(x\), the individual’s probability of surviving, and its traits, \[ \tilde{R}(x^\prime, a_0, \infty) = \int_{a_0}^{\infty} S_{\rm D} \, f(x^\prime, H(x^\prime, a_0, a), E_{a}) \, S_{\rm I} (x^\prime, a_0, a) \, S_{\rm P} (a_0, a) \, {\rm d} a. \tag{2}\] Here \(S_{\rm I}\) is the individual survival probability and \(S_{\rm P}\) the patch survival probability, both defined on the size-structured PDE page.