In this and the next chapter, we turn to questions that require the use of all of our tools: differential equations, probability, computation, and a good deal of hard thinking about biological implications of the analysis. Do not be dissuaded: the material is accessible. However, accessing this material requires new kinds of thinking, because funny things happen when we enter the realm of dynamical systems with random components. These are generally called stochastic processes. Time can be measured either discretely or continuously and the state of the system can be measured either continuously or discretely. We will encounter all combinations, but will mainly focus on continuous time models. Much of the groundwork for what we will do was laid by physicists in the twentieth century and adopted in part or wholly by biologists as we moved into the twentyfirst century (see, for example, May (1974), Ludwig (1975), Voronka and Keller (1975), Costantino and Desharnais (1991), Lande et al. (2003)). Thus, as you read the text you may begin to think that I have physics envy; I don't, but I do believe that we should acknowledge the source of great ideas. Both in the text and in Connections, I will point towards biological applications, and the next chapter is all about them.

Thinking along sample paths

To begin, we need to learn to think about dynamic biological systems in a different way.