A Dynamical Model for the Origin of Anisogamy

The paper derives the anisogamous equilibrium size s* and its linear stability using the phenotype-flux ds/dt = ∂ϕ_ind/∂s with ϕ_ind(s|s̄) = c3 s^{-α}(1 + (s − s̄)/(w + |s − s̄|)), obtaining s* = w(1 − 3αx + √β)/(4αx^2), β = α^2x^2 − 6αx + 1, and Q < 0 for perturbations of the large-gamete mass, plus a leading-order decay for small-gamete perturbations; see Eqns. (8)–(11) , with model definitions in Eqns. (3)–(7) . The candidate solution reproduces the same derivations, including the quadratic reduction and branch selection for s*, and the linearization yielding Q(x,α) < 0, matching the paper’s formulas. Two minor differences: (i) the candidate explicitly highlights the positivity condition s* > 0, concluding αx < 3 − √8 (equivalently x < (3 − 2√2)/α), which aligns with Appendix A’s threshold though the paper emphasizes β > 0 for real s* ; and (ii) the candidate’s leading-order small-gamete perturbation omits a factor w in the coefficient, whereas the paper’s Eq. (11) includes −c3 α w/(w + s̄) ε^{−1−α} . Net: same core proof, correct stability conclusions; the model answer has a small algebraic slip in the ε → 0 asymptotic coefficient.