Two-Sex Population Models: A Review of Marriage Functions and an Analysis of a Pair-Formation Model with Juveniles and Logistic Regulation

When a population consists of women and men and only couples can have children, how should fertility depend on the numbers of both sexes? We review the literature on this two-sex problem and analyse a pair-formation model with juveniles, single women, single men and couples, a marriage function, divorce, widowhood and logistic regulation. (1) No marriage function is agreed to be best. The axioms and Schoen’s consistency argument favour the harmonic mean, but it is only middling in ten-year projections of the United States, is supported in bean-beetle experiments only together with sex- and stage-specific density dependence, and implies implausibly early female marriage in projections for China and India. (2) Written as per-capita marriage hazards, the harmonic mean lets the hazard of the scarce sex at most double, while under the minimum function the surplus sex absorbs the whole squeeze. (3) Without regulation the model is homogeneous of degree one. Its two-sex exponential solution is unique up to scale; we give the exact existence condition and, for the harmonic mean, a closed form for the critical birth rate, above which the regulated model has a unique interior equilibrium. The equilibrium is locally stable in a wide parameter scan, but if density dependence counts adults only and maturation is concentrated, a Hopf bifurcation produces cycles of about 73 years. (4) The existence condition (1.6) in Theorem 1 of Hadeler (1993) should contain e−(μy−μx)τ; the printed e−2τμ_y contradicts the paper’s own eigenvalue equation and direct simulation. (5) Simulations of a rise in the sex ratio at birth reproduce Guilmoto’s findings that the marriage squeeze lags the sex ratio at birth by decades and outlasts it, and that for most of the century a female-dominance-type function predicts more lifelong single men than the harmonic mean. The paper is licensed under CC BY 4.0. The code in two-sex-pair-formation-code.zip is released under the MIT License (see LICENSE in the archive).