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.