Adaptive dynamics
Adaptive dynamics studies gradual evolution of quantitative traits when ecological interactions determine whether rare mutants can invade a resident population.
From fixed strategies to continuous traits
Replicator dynamics usually begin with a fixed collection of strategies. Adaptive dynamics instead lets a trait vary continuously.
Examples include body size, virulence, resource-use traits or dispersal tendency. Denote the resident trait by
\[x\]and a mutant trait by
\[y.\]The resident ecological environment
The resident population first changes according to an ecological model. For example, suppose its density \(N\) satisfies
\[\frac{dN}{dt}=F(N,x).\]For a fixed resident trait \(x\), assume the ecology approaches an equilibrium
\[N^*(x).\]The mutant is then introduced at very low density into this resident-created environment.
Invasion fitness
The mutant's initial per-capita growth rate is called its invasion fitness:
\[\boxed{s(y,x)}.\]The first argument is the mutant trait and the second is the resident trait.
If
\[s(y,x)>0,\]the rare mutant can initially increase. If
\[s(y,x)<0,\]it declines when rare.
Why the resident has zero invasion fitness against itself
At its ecological equilibrium, a resident type has zero net per-capita growth. Therefore a consistent invasion-fitness function normally satisfies
\[\boxed{s(x,x)=0}.\]This identity is important when analysing nearby mutants.
Nearby mutants
Adaptive dynamics commonly assumes mutations are sufficiently small that
\[y=x+\delta,\qquad |\delta|\ll1.\]A Taylor expansion around \(y=x\) gives
\[s(x+\delta,x)\approx s(x,x)+\delta\left.\frac{\partial s(y,x)}{\partial y}\right|_{y=x}.\]Since \(s(x,x)=0\),
\[s(x+\delta,x)\approx\delta\left.\frac{\partial s(y,x)}{\partial y}\right|_{y=x}.\]The selection gradient
Define
\[\boxed{g(x)=\left.\frac{\partial s(y,x)}{\partial y}\right|_{y=x}}.\]This is the local selection gradient.
If \(g(x)>0\), nearby mutants with slightly larger trait values are locally favoured. If \(g(x)<0\), slightly smaller values are favoured.
Evolutionary singular points
A singular trait \(x^*\) satisfies
\[\boxed{g(x^*)=0}.\]At this point there is no first-order directional advantage for an infinitesimally nearby mutant.
However, this condition alone does not tell us what happens evolutionarily.
Two different stability questions
Adaptive dynamics distinguishes two ideas that are easy to confuse:
| Question | Meaning |
|---|---|
| Convergence stability | Do nearby resident traits evolve toward the singular point? |
| Evolutionary stability | Once the resident is at the singular point, can nearby mutants invade it? |
A singular point can satisfy one property without satisfying the other.
Convergence stability
For a one-dimensional trait, local evolutionary movement follows the sign of \(g(x)\). A simple local condition for attraction is
\[\boxed{g'(x^*)<0}.\]Then traits just below \(x^*\) tend to move upward and traits just above it tend to move downward.
Evolutionary stability
At a singular point, examine the curvature of mutant invasion fitness with respect to the mutant trait:
\[\left.\frac{\partial^2s(y,x)}{\partial y^2}\right|_{y=x=x^*}.\]If
\[\boxed{\left.\frac{\partial^2s}{\partial y^2}\right|_{y=x=x^*}<0},\]then \(y=x^*\) is locally a maximum of invasion fitness. Nearby mutants have negative invasion fitness, so the resident is locally resistant to invasion.
Evolutionary branching
An especially interesting case occurs when a singular point is convergence stable but not evolutionarily stable.
If nearby residents evolve toward \(x^*\), but at \(x^*\) sufficiently nearby mutants can invade, selection can become disruptive. Under additional coexistence conditions, the population may split into diverging trait lineages.
This phenomenon is called evolutionary branching.
Canonical equation
Under assumptions of rare mutations, small mutational steps and separation between ecological and evolutionary time scales, evolutionary change can often be approximated by a canonical equation of the form
\[\boxed{\frac{dx}{dt}=K(x)g(x)},\]where \(K(x)>0\) collects factors such as mutation rate, resident population size and mutational variance.
The selection gradient determines the local direction, while \(K(x)\) controls the evolutionary speed.
A simple ecological example
Suppose a resident with trait \(x\) creates equilibrium density \(N^*(x)\), and a rare mutant with trait \(y\) has per-capita growth
\[s(y,x)=r(y)-\alpha(y,x)N^*(x).\]Here \(r(y)\) is the mutant's intrinsic growth contribution and \(\alpha(y,x)\) measures the competitive effect of the resident environment on the mutant.
The selection gradient is
\[g(x)=r'(x)-\left.\frac{\partial\alpha(y,x)}{\partial y}\right|_{y=x}N^*(x).\]This shows explicitly how ecological competition shapes evolutionary direction.
Pairwise invasibility
The sign of
\[s(y,x)\]can be examined across pairs of resident and mutant traits. Regions where \(s(y,x)>0\) identify mutants capable of invading particular residents.
A pairwise invasibility plot provides a graphical way to locate singular points and understand local evolutionary directions.
Trait substitution sequence
Under the rare-mutation assumption, a typical conceptual sequence is:
a resident reaches ecological equilibrium; a rare mutant appears; invasion fitness determines whether it initially grows; a successful mutant may replace the resident; ecology relaxes again before the next mutation.
Repeated substitutions generate gradual evolutionary movement through trait space.
Key assumptions
Classical adaptive-dynamics arguments commonly assume mutations are rare, mutational steps are small, ecological dynamics are fast relative to evolutionary change, and the resident environment is sufficiently well defined before a new mutant appears.
These assumptions should be checked against the biological system rather than treated as universal truths.
Adaptive dynamics and replicator dynamics
Replicator dynamics track changes among strategies already present in the population. Adaptive dynamics focuses on invasion by new nearby mutant traits and the resulting long-term movement through trait space.
The two frameworks answer related but different evolutionary questions.
Adaptive dynamics and finite populations
Classical adaptive dynamics often uses deterministic ecological equilibria and deterministic invasion criteria. In finite populations, a mutant with positive invasion fitness can still disappear by chance when initially rare.
Stochastic birth–death or branching-process models can therefore complement invasion-fitness analysis when establishment probabilities matter.
Eco-evolutionary feedback
The central feedback can be summarised as
\[\boxed{\text{trait}\to\text{ecological environment}\to\text{invasion fitness}\to\text{trait change}}.\]This makes adaptive dynamics a natural bridge between ecological modelling and evolutionary theory.
End of Evolutionary Mathematics
This section began with allele and genotype frequencies, then developed mutation, selection and genetic drift through Wright–Fisher and Moran models. Evolutionary game theory introduced frequency-dependent fitness, replicator dynamics converted fitness differences into deterministic frequency change, and adaptive dynamics extended the framework to continuously evolving traits.
The next major section changes the mathematical representation again: instead of assuming that every individual interacts equally with every other individual, it represents biological relationships explicitly as a network.