Adaptive Dynamics & Evolutionary Branching

Invasion Fitness · Pairwise Invasibility Plot · ESS · Branching

Controls

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Competition Model

Logistic: dN/dt = N(r − αN) — competitive effect
α(x,y) = exp(−(x−y)²/2σ_c²) — Gaussian competition
K(x) = K₀·exp(−(x−x_opt)²/2σ_K²) — resource kernel
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Invasion Fitness

Trait Evolution

Theory

f(y,x) = r(1 − N*·α(y,x)/K(y)) — invasion fitness
Resident x survives if f(x,x) = 0 (equilibrium)
Mutant y invades if f(y,x) > 0 (PIP: green region)
ESS: ∂f/∂y = 0 at y=x (singular strategy)
CSS (convergent stable): trait evolves to ESS
Branching point: ∂²f/∂y² > 0 at ESS (min selection!)
→ disruptive selection → speciation
σ_c < σ_K: ESS is branching (evolutionary branching)