Evolutionary invasion analysis, also known as adaptive dynamics, is a set of techniques for studying long-term phenotypical evolution developed during the 1990s. It incorporates the concept of frequency dependence from game theory but allows for more realistic ecological descriptions, as the traits vary continuously and gives rise to a non-linear invasion fitness (the classical fitness concept is not directly applicable to situations with frequency dependence).
Contents
- Introduction and background
- Fundamental ideas
- Monomorphic evolution
- Invasion exponent and selection gradient
- Pairwise invasibility plots
- Evolutionarily singular strategies
- Polymorphic evolution
- Invasion exponent and selection gradients in polymorphic populations
- Evolutionary branching
- Trait evolution plots
- Other uses
- References
Introduction and background
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. They link population dynamics to evolutionary dynamics and incorporate and generalise the fundamental idea of frequency-dependent selection from game theory.
Fundamental ideas
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. It is sometimes called the invasion fitness of mutants.
To make use of these ideas, a mathematical model must explicitly incorporate the traits undergoing evolutionary change. The model should describe both the environment and the population dynamics given the environment, even if the variable part of the environment consists only of the demography of the current population. The invasion exponent can then be determined. This can be difficult, but once determined, the adaptive dynamics techniques can be applied independent of the model structure.
Monomorphic evolution
A population consisting of individuals with the same trait is called monomorphic. If not explicitly stated otherwise, the trait is assumed to be a real number, and r and m are the trait value of the monomorphic resident population and that of an invading mutant, respectively.
Invasion exponent and selection gradient
The invasion exponent
We will always assume that the resident is at its demographic attractor, and as a consequence
The selection gradient is defined as the slope of the invasion exponent at
which holds whenever
Pairwise-invasibility plots
The invasion exponent represents the fitness landscape as experienced by a rare mutant. In a large (infinite) population only mutants with trait values
Evolutionarily singular strategies
The selection gradient
Singular strategies can be located and classified once the selection gradient is known. To locate singular strategies, it is sufficient to find the points for which the selection gradient vanishes, i.e. to find
If this does not hold the strategy is evolutionarily unstable and, provided that it is also convergence stable, evolutionary branching will eventually occur. For a singular strategy
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.
Polymorphic evolution
The normal outcome of a successful invasion is that the mutant replaces the resident. However, other outcomes are also possible; in particular both the resident and the mutant may persist, and the population then becomes dimorphic. Assuming that a trait persists in the population if and only if its expected growth-rate when rare is positive, the condition for coexistence among two traits
and
where
Invasion exponent and selection gradients in polymorphic populations
The invasion exponent is generalised to dimorphic populations straightforwardly, as the expected growth rate
and
In practise, it is often difficult to determine the dimorphic selection gradient and invasion exponent analytically, and one often has to resort to numerical computations.
Evolutionary branching
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. To understand this heuristically, consider a dimorphic population
and, since
the fitness landscape for the dimorphic population must be a perturbation of that for a monomorphic resident near the singular strategy.
Trait evolution plots
Evolution after branching is illustrated using trait evolution plots. These show the region of coexistence, the direction of evolutionary change and whether points where the selection gradient vanishes are fitness maxima or minima. Evolution may well lead the dimorphic population outside the region of coexistence, in which case one morph is extinct and the population once again becomes monomorphic.
Other uses
Adaptive dynamics effectively combines game theory and population dynamics. As such, it can be very useful in investigating how evolution affects the dynamics of populations. 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.