The Mathematics of Populations: Demographics

Abstract

What are population problems? Here are some examples:

• 1. What will the population of the U.S. be in 10 years? It is reasonable to look at the record of population growth in the past and extrapolate this information into the future to predict population size. In this mathematical procedure, parameters such as the birth and death rates are estimated using past records, and the population of the future is projected by solving a mathematical model.

• 2. How will this population be distributed among age groups? The same extrapolation methods are used, but the model becomes more complicated when we account for age structure. Now the parameters to be identified from available data are age-specific birth and death rates. The solution of an appropriate mathematical model determines the population’s future age distribution.

• 3. How are exhaustible natural resources best managed? A resource, such as a bacterial culture or a fishery or a forest, must be managed to provide optimal production while at the same time protecting the resource from extinction. Legal, social, financial and moral questions almost always arise with management of natural resources. Here we avoid all but the relatively simple questions of calculating yields from exhaustible resources, and we describe consequences of various optimal harvesting policies. Even with this severe restriction, there are difficult problems since each resource is part of a complicated interdependent network. The mathematical tools introduced here give some guides to study the economics of ecological systems, and they illustrate potential glaring pitfalls in management policies. In particular, the resource population might be driven into chaotic dynamics that can lead to its collapse.

• 4. Do the results of genetic engineering pose a serious threat to our environment? To answer this question, we must understand the ways genetic information can be inherited and how pathogenic agents can spread throughout a population. Genetic inheritance has been studied by philosophers, theologians, biologists, and physicians for more than a century. Even now, this research is controversial. We restrict attention here to genetic inheritance of relatively simple traits in bacteria and humans Mathematical techniques are useful to project what gene distributions will be in future generations. We discuss genetics problems in small populations (small plasmid pools) and in large ones where there are many participants.

• 5. Can the spread of a contagious disease be predicted? Questions about the occurrence and control of epidemics are raised by the spread of bacteria and viruses. Disease dynamics are traditionally described in terms of the numbers of susceptibles and infectives. We first discuss disease propagation in small populations such as families, and then in large ones such as schools or cities, to gain useful insight into the mechanisms of disease spread and control. Economic considerations also come up in disease control; for example, if a vaccine is available for a particular disease, how would the population best be inoculated within the constraints imposed by time, staff, medical facilities, age-specific host susceptibility, and the presence of competing diseases?

• 6. Are insect infestations in crops predictable? Study of the geographical distribution of populations is the key to answering this kind of question. There are many possible ways populations move through regions, and mathematical analysis has led to methods for describing them. We study a random walk model and eventually derive a diffusion approximation to it. The derivation of the diffusion equation from a random process can be applied to the genetics and epidemics examples as well, and with this we see how the small population and large population models are related.