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The Main Ideas

The growth of a population depends on many factors, and often depends on the way that one population interacts with other populations. For example, a sophisticated model may include predator-prey interaction, parasitic interactions, mutualistic interactions, and more. In this module, however, we will focus on simpler models of a population.

Intuitively, the rate at which a population changes depends on at least three factors. For the purpose of this lab, we assume a specific mathematical form for these factors. For a real-life model, the way that a population depends on each factor would need to be determined by field observations and detailed knowledge of the population's social and reproductive characteristics. The factors we will consider in this module are:

Net Birth Rate for the Population

We assume that a population changes linearly in proportion to the current value of the population. The constant of proportionality represents the net birth rate for the population, and must account for a variety of factors such as

The proportion of the population that will mate;
The number of offspring for each mating pair;
The proportion of the population that will die during the next period of time.

Overcrowding and Scarcity of Resources

We assume that a population is limited in size by resources such as the availability of food and land. We lump all of these factors into a single overcrowding term that will serve to decrease the population when it grows too large to be supported by the available resources.

Harvesting

The removal of a constant number of individuals from a population during each time period is known as harvesting (or sometimes fishing). As its name implies, this harvesting is often accomplished by humans. Advocates of harvesting point to stable populations of deer, fish, and other game animals, as evidence that harvesting can be used to reduce the number of animals who needlessly die from starvation or other natural causes. On the other hand, unregulated harvesting can lead a population to the brink of extinction, as is evidenced by well-known examples such as the North American Bison (Bison bison) and several populations of whales. We will build up our understanding and intuition about each of these factors by considering them in turn. We will first consider population models that change according to the net birth rate of the current population, and will find that this leads to exponential growth or decay of the population. We then introduce an overcrowding term, and discover that the population may stabilize at a certain number of animals. We then study the effect of harvesting on populations. Along the way, we will learn new geometric techniques to help us qualitatively analyze solutions of differential equations.

The period of time during which a population changes depends on whether the population has a mating season (like many mammals) or whether the population reproduces continuously (like bacteria, some insects, and domesticated rabbits). Thus, depending on the species being studied, it may be convenient to measure time in decades (e.g., a census of human population), years, or even hours (for some petri dish experiments).

Next: Unbounded Populations
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