A Dynamical Model for Human Population

Abstract

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. Sqlutions 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. This is a preliminary report of the results of an investigation into the stability properties of reproductive systems whose dynamics are independent of overall scale, and in which agespecific fertility and mortality rates are (provisionally) assumed to be non-age-specific, and constant through time. Special cases of this general class have previously been partially analyzed by D. G. Kendall (1), L. A. Goodman (2), J. H. Pollard (3), and others. Scale-independent modelOur population model is defined by the following assumptions:1. There are two sexes with total numbers TI(t) and T2(t).2. Monogamy prevails, with T3(t) marriages existing at time t.The number of single individuals of each sex is therefore NAT(t) = T1(t) -T3(t); N2(t) = T2(t) -T3(t).[1]S. The population is replenished only through births to married couples. The fertility rates, per marriage, for female and male births are fi and f2 respectively, and are independent of parental ages. 4. The (instantaneous) rate at which marriages occur depends nonlinearly on the then-existing stocks of single individuals N1(t) and N2(t), vanishing when either Nf vanishes, and increasing without limit as either Ni increases. We will write this (nonnegative) rate as MI[Nj(t),N2(t)].5. The mortality rates (X1,X2) of single individuals are constants, independent of age, as is the mortality rate of couples, X3, due to death (or divorce, or permanent sterilization, etc.). 6. The dynamics of the system is independent of its overall scale, and of the common scale of its components. Dynamical equationsGiven the assumptions above, the behav...

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