Theory
f(x',x) = r(x')[1 - N*(x)/K(x',x)]
K(x,x') = K₀·exp(-(x-x')²/2σ_K²)
Branching: ∂²f/∂x'²|_{x'=x*} > 0
The Pairwise Invasibility Plot (PIP): if f(x',x) > 0 (green), a rare mutant x' invades a resident x. Evolutionary singular strategies are fixed points of the canonical equation. Branching occurs when the singular strategy is convergent stable but invasible.
Legend
■ Green: mutant can invade
■ Red: mutant cannot invade
— Diagonal: resident = mutant
White dots: current trait values
Branching: resident moves to singular point, then splits into two lineages