A population of individuals with a heritable trait x. Adaptive dynamics: mutants invade when f(x_mut, x_res) > 0. Watch evolutionary branching at fitness minima — the origin of diversity.
Adaptive dynamics (Dieckmann & Law 1996): the trait evolves along the fitness gradient ∂f/∂x_mut|=0. Evolutionary branching occurs at singular strategies where the fitness gradient vanishes AND the second derivative is negative (minimum of fitness landscape) — a generic route to speciation.