On the Dynamics, Equilibria, and Economics of Gender Balance at Naturist Venues

We propose a dynamical systems approach to model the demographic evolution of naturist venues. A coupled ODE system yields a characteristic polynomial P(r) whose root structure determines whether a venue is stable, exhibits a tipping point, or is irreversibly doomed. Under male indifference to gender imbalance (Case A), a critical ratio r₁ separates demographic collapse from recovery; restoring male sensitivity (Case B) relaxes viability conditions and provides an endogenous recovery mechanism. Exact phase-space trajectories are obtained by partial-fraction decomposition, and a Sundman temporal desingularisation reduces the ratio dynamics to a polynomial flow in dual time. The "no single males" entry rule is modelled as a binding constraint; Karush-Kuhn-Tucker conditions show that the global revenue optimum is explicitly identifiable, with the shadow price of female attendance equal to her own access fee plus the male slot her presence unlocks, providing a rigorous economic justification for cross-subsidisation and showing that revenue and welfare maximisation converge on the same strategy. A Voronoi percolation analogy motivates a female-fraction connectivity benchmark of Pc = 1/2. Microfoundations derive the ODE as the mean-field limit of individually rational discrete-choice decisions; a scalar best-response map identifies its fixed points with the roots of P(r) and its self-correction condition with ODE stability. A theorem establishes that κ*=1 (one admitted male per admitted female) is the unique revenue-maximising proportional quota subject to the connectivity benchmark, with the KKT shadow prices and the constrained ODE substitution following as corollaries.