Evolutionary invasion analysis, also known as adaptive dynamics, is a set of mathematical modeling techniques that use differential equations to study the long-term evolution of traits in asexually and sexually reproducing populations.
This is a mathematical expression for the long-term exponential growth rate of the mutant subpopulation when it is introduced into the resident population in small numbers.
If the invasion fitness is positive (in continuous time), the mutant population can grow in the environment set by the resident phenotype.
[1] The basic principle of evolution via natural selection was outlined by Charles Darwin in his 1859 book, On the Origin of Species.
Though controversial at the time, the central ideas remain largely unchanged to this date, even though much more is now known about the biological basis of inheritance.
Darwin expressed his arguments verbally, but many attempts have since then been made to formalise the theory of evolution.
The best known are population genetics which models inheritance at the expense of ecological detail, quantitative genetics which incorporates quantitative traits influenced by genes at many loci, and evolutionary game theory which ignores genetic detail but incorporates a high degree of ecological realism, in particular that the success of any given strategy depends on the frequency at which strategies are played in the population, a concept known as frequency dependence.
Adaptive dynamics is a set of techniques developed during the 1990s for understanding the long-term consequences of small mutations in the traits expressing the phenotype.
Two fundamental ideas of adaptive dynamics are that the resident population is in a dynamical equilibrium when new mutants appear, and that the eventual fate of such mutants can be inferred from their initial growth rate when rare in the environment consisting of the resident.
This rate is known as the invasion exponent when measured as the initial exponential growth rate of mutants, and as the basic reproductive number when it measures the expected total number of offspring that a mutant individual produces in a lifetime.
To make use of these ideas, a mathematical model must explicitly incorporate the traits undergoing evolutionary change.
This can be difficult, but once determined, the adaptive dynamics techniques can be applied independent of the model structure.
is defined as the expected growth rate of an initially rare mutant in the environment set by the resident (r), which means the frequency of each phenotype (trait value) whenever this suffices to infer all other aspects of the equilibrium environment, such as the demographic composition and the availability of resources.
For each r, the invasion exponent can be thought of as the fitness landscape experienced by an initially rare mutant.
The landscape changes with each successful invasion, as is the case in evolutionary game theory, but in contrast with the classical view of evolution as an optimisation process towards ever higher fitness.
If the sign of the selection gradient is positive (negative) mutants with slightly higher (lower) trait values may successfully invade.
To determine the outcome of the resulting series of invasions pairwise-invasibility plots (PIPs) are often used.
In PIPs the fitness landscapes as experienced by a rare mutant correspond to the vertical lines where the resident trait value
If it is positive (negative) a mutant with a slightly higher (lower) trait-value will generically invade and replace the resident.
First, a degenerate case similar to the saddle point of a qubic function where finite evolutionary steps would lead past the local 'flatness'.
Second, a fitness maximum which is known as an evolutionarily stable strategy (ESS) and which, once established, cannot be invaded by nearby mutants.
Third, a fitness minimum where disruptive selection will occur and the population branch into two morphs.
we have If this does not hold the strategy is evolutionarily unstable and, provided that it is also convergence stable, evolutionary branching will eventually occur.
is negative, or equivalently The criterion for convergence stability given above can also be expressed using second derivatives of the invasion exponent, and the classification can be refined to span more than the simple cases considered here.
The emergence of protected dimorphism near singular points during the course of evolution is not unusual, but its significance depends on whether selection is stabilising or disruptive.
In the latter case, the traits of the two morphs will diverge in a process often referred to as evolutionary branching.
Geritz 1998 presents a compelling argument that disruptive selection only occurs near fitness minima.
By continuity and, since the fitness landscape for the dimorphic population must be a perturbation of that for a monomorphic resident near the singular strategy.
These show the region of coexistence, the direction of evolutionary change and whether points where the selection gradient vanishes are fitness maxima or minima.
One interesting finding to come out of this is that individual-level adaptation can sometimes result in the extinction of the whole population/species, a phenomenon known as evolutionary suicide.