Abstract

This paper investigates a mathematical model for the growth of an age-structured population. The model includes the idea (due to Easterlin) that fertility is affected by the size of the cohort in which an individual is born. It is important to note that the model investigated represents only a reasonable first step in the direction of reality from the unrealistic assumption that mortality and fertility do not change with passing time. It is shown that this general model can lead to self-excited, persistent oscillations (called limit cycles in mathematical parlance) of the birth trajectory of the population. Using data for the United States from the twentieth century, it is shown that variations in the number of births are consistent with the model discussed.

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References

Arthur, W. B. (
1981
).
Why A Population Converges to Stability
.
American Mathematical Monthly
,
88
,
557
563
. 10.2307/2320504
Cohen, J. E. (
1976
).
Ergodicity of Age Structure in Populations with Markovian Vital Rates, I. Countable States
.
Journal of the American Statistical Association
,
71
,
335
339
. 10.2307/2285308
Cohen, J. E. (
1977
).
Ergodicity of Age Structure in Populations with Markovian Vital Rates, II. General States
.
Advances in Applied Probability
,
9
,
18
37
. 10.2307/1425814
Cohen, J. E. (
1979
).
Ergodic Theorems in Demography
.
Bulletin of the American Mathematical Society
,
1
,
275
295
. 10.1090/S0273-0979-1979-14594-4
Easterlin, R. A. (
1961
).
The American Baby Boom in Historical Perspective
.
American Economic Review
,
51
,
860
911
.
Easterlin, R. A. (
1968
).
The Current Fertility Decline and Projected Fertility Changes. Chapter 5 in Population, Labor Force, and Long Swings in Economic Growth: The American Experience
.
New York
:
Columbia University Press
.
Frauenthal, J. C. (
1975
).
A Dynamic Model for Human Population Growth
.
Theoretical Population Biology
,
8
,
64
73
. 10.1016/0040-5809(75)90039-8
Heuser, R. L. (
1976
).
Fertility Tables for Birth Cohorts by Color: United States, 1917–1973
.
Washington, D.C.
:
U. S. Department of Health, Education and Welfare
.
Keyfitz, N. (
1972
).
Population Waves
. In Greville, T. N. E. (Ed.),
Population Dynamics
(pp.
1
38
).
New York
:
Academic Press
.
Keyfitz, N. (
1977
).
Introduction to the Mathematics of Population: With Revisions
.
Reading, Mass.
:
Addison-Wesley
.
Lee, R. D. (
1974
).
The Formal Dynamics of Controlled Populations and the Echo, the Boom and the Bust
.
Demography
,
11
,
563
585
. 10.2307/2060471
Lee, R. D. (
1976
).
Demographic Forecasting and the Easterlin Hypothesis
.
Population and Development Review
,
2
,
459
468
. 10.2307/1971622
Leslie, P. H. (
1945
).
On the Use of Matrices in Certain Population Mathematics
.
Biometrika
,
33
,
183
212
. 10.1093/biomet/33.3.183
Lopez, A. (
1961
).
Problems in Stable Population Theory
.
Princeton, N.J.
:
Office of Population Research, Princeton University
.
Lotka, A. J. (
1939
). Théorie analytique des associations biologiques, Part II.
Analyse démographique avec application particulière à l’espèce humaine
.
Paris
:
Hermann & Cie
.
Miller, R. K. (
1971
).
Nonlinear Volterra Integral Equations
.
Menlo Park, Ca.
:
W. A. Benjamin
.
Vital Statistics ofthe United States. 1975,Volume 1— Natality. DHEW Pub. No. (PHS) 78–1113
. (
1978
).
Hyattsville, Md.
:
U. S. Department of Health, Education and Welfare
.
Samuelson, P. A. (
1976
).
An Economist’s Nonlinear Model of Self-generated Fertility Waves
.
Population Studies
,
30
,
243
247
. 10.2307/2173607
Smith, D. P. (
1981
).
The Problem of Persistence in Economic-Demographic Cycles
.
Theoretical Population Biology
,
19
,
125
146
. 10.1016/0040-5809(81)90013-7
Swick, K. E. (
1981
).
A Nonlinear Model for Human Population Dynamics
.
SIAM Journal of Applied Mathematics
,
40
,
266
278
. 10.1137/0140023
Swick, K. E. (
1981
).
Stability and Bifurcation in Age-Dependent Population Dynamics
.
Theoretical Population Biology
,
20
,
80
100
. 10.1016/0040-5809(81)90004-6