Adaptive dynamics
Adaptive dynamics studies long-term evolution of quantitative traits when ecological interactions determine the success of rare mutants.
Resident and mutant
Suppose a resident trait \(x\) establishes an ecological environment. A rare mutant with trait \(y\) has invasion fitness
\[s(y,x),\]which measures whether it can initially increase when rare.
If \(s(y,x)>0\), the mutant can invade; if \(s(y,x)<0\), it cannot under the model assumptions.
Selection gradient
A local direction of evolutionary change is determined by
\[\left.\frac{\partial s(y,x)}{\partial y}\right|_{y=x}.\]Evolutionary singular points
Points where this gradient vanishes can be analysed for evolutionary attraction, stability against invasion and possible branching.
Key idea. Adaptive dynamics links ecological population models to gradual evolution by asking whether nearby mutant traits can invade a resident population.