Adaptive dynamics
theory
This page builds on The big picture, especially the birth-rate loop, and on the demographic quantities defined in The size-structured PDE.
The idea first
Adaptive dynamics asks a simple question: if a new strategy appears at very low frequency, can it increase in the environment created by the strategies already present? The established strategies are called residents and the new strategy is called a mutant. Because the mutant is rare, we assume it experiences the residents’ competitive environment without noticeably changing that environment itself.
The calculation follows one mutant seed across all the patch ages at which it might arrive. In each possible patch, the seed may establish, grow, survive, and produce dispersing offspring. Averaging that lifetime return across the landscape gives a reproduction ratio: above one the mutant can invade, below one it cannot, and at one it exactly replaces itself. The equations below use the canopy-light environment developed on the size-structured PDE page. The fitness calculation follows Falster, Brännström, Westoby, & Dieckmann (2015).
Invasion fitness
Let \(x\) denote the traits of the resident community and \(x^\prime\) the traits of a rare mutant. We focus on phenotype-dependent fitness: the combined consequences of those traits for growth, fecundity, and mortality in the non-linear competitive environment created by the residents. We do not model the genetic basis of trait inheritance or expression. Following the standard rare-mutant assumption, the mutant has a negligible effect on the competitive environment—in this formulation, the canopy-light profile—in which it grows (Geritz, Kisdi, Meszéna, & Metz, 1998).
In general, invasion fitness is the long-term per-capita growth rate of a rare mutant in the environment determined by the residents (Metz, Nisbet, & Geritz, 1992). That rate is difficult to calculate directly in a structured metacommunity because plants differ in size and occupy patches of different ages (Gyllenberg & Metz, 2001; Metz & Gyllenberg, 2001).
At demographic equilibrium, we can instead use the basic reproduction ratio: the expected number of new dispersers ultimately produced by one dispersing seed. This ratio gives the same evolutionary ordering as the long-term per-capita growth rate, so it can be used to decide whether a mutant invades (Gyllenberg & Metz, 2001; Metz & Gyllenberg, 2001).
We denote by \(R\left(x^\prime, x\right)\) the mutant’s basic reproduction ratio in the competitive environment created by residents \(x\). Patches of age \(a\) have frequency-density \(P(a)\) in the landscape, as derived on the size-structured PDE page. More precisely, the probability that a randomly dispersed seed lands in a patch with age in a small interval \(\mathrm{d}a\) is \(P(a)\mathrm{d}a\).
The mutant may therefore land in a young, open patch or an old, shaded one. Its overall reproduction ratio averages its return over all those possible arrival ages: \[ R\left(x^\prime,x\right) = \int _0^{\infty} P\left(a\right) \, \tilde{R}\left(x^\prime, a, \infty \right) \, {\rm d}a , \tag{1}\]
Here \(\tilde{R}\left(x^\prime, a_0, a \right)\) is the expected number of dispersing offspring produced by one mutant seed that arrives when the patch is age \(a_0\), counted up to patch age \(a\) (Gyllenberg & Metz, 2001; Metz & Gyllenberg, 2001). In Equation 1 the endpoint is infinity, so the entire remaining lifetime of the patch is included.
That lifetime return is itself another integral: \[ \tilde{R}(x^\prime, a_0, \infty) = \int_{a_0}^{\infty} S_{\rm D} \, f(x^\prime, H(x^\prime, a_0, a), E_{a}) \, S_{\rm I} (x^\prime, a_0, a) \, S_{\rm P} (a_0, a) \, {\rm d} a. \tag{2}\]
The integrand is a product of four pieces:
- \(f\) is the mutant’s fecundity at its current height and light environment;
- \(S_{\rm I}\) is the probability that the mutant is still alive;
- \(S_{\rm P}\) is the probability that the patch has not yet been disturbed; and
- \(S_{\rm D}\) is the fraction of offspring that survive dispersal.
The individual and patch survival probabilities are defined on the size-structured PDE page. Multiplying the four terms gives expected viable offspring production at patch age \(a\); integrating from arrival age \(a_0\) onward gives the seed’s expected lifetime return.
Reading the result
The threshold is replacement:
- \(R(x^\prime, x) > 1\): one mutant seed produces more than one dispersing seed on average, so the mutant can invade;
- \(R(x^\prime, x) < 1\): it fails to replace itself and cannot invade; and
- \(R(x^\prime, x) = 1\): it exactly replaces itself in that resident environment.
Repeating this calculation across mutant trait values gives an invasion-fitness landscape. A resident that cannot be invaded by nearby alternatives is evolutionarily stable. That is distinct from convergence stability: a trait can be approached by gradual evolution yet still be invasible once reached, in which case it can be an evolutionary branching point rather than an endpoint. The adaptive-dynamics examples show both kinds of outcome.