A dynamical model for human population.

We analyze a simple, deterministic model of the dynamics of population changes in a bisexual, reproductive system based on marriage. Our model is one of a general class, special cases of which have been previously discussed within the framework of population biology by D. G. Kendall, L. A. Goodman, J. H. Pollard, and others. Here, we extend and complete previous analyses of systems characterized by first-degree homogeneous, unbounded marriage functions, allowing for arbitrary birth and death rates. The dynamics of the model is determined by three coupled first-order, nonlinear differential equations, similar to those used in the description of chemical reactions and of radioactive decay chains. Solutions of the differential equation system are classified according to the associated patterns of birth and death rates of the two sexes, and growth and stability properties are discussed. This preliminary report gives conditions sufficient to insure the existence of a unique, exponential mode of population growth or decay, with a finite ratio of the sexes. We also exhibit other conditions which, in contrast to the standard, linear demographic analysis of Lotka, guarantee that the sex ratio asymptotically becomes infinite. The model manifests a delicate balance between the vital parameters that alerts one to the possibility of selfaggravating distortions of the sex ratio, once a monogamous society's fertility falls below the replacement value.