Thinking along sample paths

To begin, we need to learn to think about dynamic biological systems in a different way. The reason is this: when the dynamics are stochastic, even the simplest dynamics can have more than one possible outcome. (This has profound "real world'' applications. For example, it means that in a management context, we might do everything right and still not succeed in the goal.)

To illustrate this point, let us reconsider exponential population growth in discrete time:

which we know has the solution X(t) = (1 + l)tX(0). Now suppose that we wanted to make these dynamics stochastic. One possibility would be to assume that at each time the new population size is determined by the deterministic component given in Eq. (7.1) and a random, stochastic term Z(t) representing elements of the population that come from "somewhere else.'' Instead of Eq. (7.1), we would write

In order to iterate this equation forward in time, we need assumptions about the properties of Z(t). One assumption is that Z(t), the process uncertainty, is normally distributed with mean 0 and variance a2. In that case, there are an infinite number of possibilities for the sequence {Z(0), Z(1), Z(2), ...} and in order to understand the dynamics we should investigate the properties of a variety of the trajectories, or sample paths, that this equation generates. In Figure 7.1, I show ten such trajectories and the deterministic trajectory.

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