1 - Dynamic Modeling with Difference Equations

Summary

Whether we investigate the growth and interactions of an entire population, the evolution of DNA sequences, the inheritance of traits, or the spread of disease, biological systems are marked by change and adaptation. Even when they appear to be constant and stable, it is often the result of a balance of tendencies pushing the systems in different directions. A large number of interactions and competing tendencies can make it difficult to see the full picture at once.

How can we understand systems as complicated as those arising in the biological sciences? How can we test whether our supposed understanding of the key processes is sufficient to describe how a system behaves? Mathematical language is designed for precise description, and so describing complicated systems often requires a mathematical model.

In this text, we look at some ways mathematics is used to model dynamic processes in biology. Simple formulas relate, for instance, the population of a species in a certain year to that of the following year. We learn to understand the consequences an equation might have through mathematical analysis, so that our formulation can be checked against biological observation. Although many of the models we examine may at first seem to be gross simplifications, their very simplicity is a strength. Simple models show clearly the implications of our most basic assumptions.