24.2 Concluding Thoughts

You may be wondering how this discussion of random walks connects back into stochastic differential equations. With the ideas of a random walk developed here and in Section 23, we can now understand how small changes in a variable or parameter affect the solutions to a differential equation.

For example if we consider the following logistic differential equation \(\displaystyle \frac{dx}{dt} = rx \left(1-\frac{x}{K} \right)\), a naive way to add stochasticity is to add an additional term (which we call “Noise”)

\[\begin{equation} \frac{dx}{dt} = rx \left(1-\frac{x}{K} \right) + \mbox{ Noise }. \end{equation}\]

One way that we examine this is by multiplying the \(dt\) term over to the right hand side:

\[\begin{equation} dx = rx \left(1-\frac{x}{K} \right) \; dt + \mbox{ Noise } \; dt \end{equation}\]

The first term (\(\displaystyle rx \left(1-\frac{x}{K} \right) \; dt\)) is called the “deterministic part” of the equation. The second term (\(\mbox{ Noise } \; dt\)) is the “stochastic part.” If this “Noise” term represents a random walk, this will affect the solution trajectory. However, we would expect that the average ensemble of solutions to behave similarly to the deterministic solution.

A model for this Noise process ties directly into Equation (24.3). and how to build this into our simulation models will be the focus of the final sections.