Frequency-dependent selection drives traits to an evolutionary branching point where the population splits into two morphs
Consider a trait x (e.g., beak size). Fitness of a rare mutant x' in resident x population is determined by the invasion fitness: s(x',x) = r[1 − C(x',x)/K(x)], where K(x)∝exp(−x²/2σ_K²) is the carrying capacity and C(x',x)∝exp(−(x'−x)²/2σ_C²) is competition. The PIP shows regions where mutants can invade (green) or cannot (dark). When σ_C < σ_K, the singular strategy x*=0 is a convergence stable attractor but evolutionarily unstable — selection becomes disruptive and the population branches into two ecologically distinct morphs (evolutionary branching → dimorphism → speciation).