25.1 The stochastic logistic model

In Section 24 we started to write down the format of a stochastic differential equation, which we will use the logistic equation for context:

\[\begin{equation} dx = rx \left(1 - \frac{x}{K} \right) \; dt + \mbox{ Noise } \; dt \tag{25.1} \end{equation}\]

It is helpful to identify the two different parts of Equation (25.1). The first part is called the deterministic part, and it does not involve the “Noise” term: \(\displaystyle rx \left(1 - \frac{x}{K} \right) \; dt\). The second part is called the stochastic part and is (I bet you guessed it!) the term that contains \(\mbox{ Noise } \; dt\) is the “stochastic part.”

While just writing \(\mbox{ Noise } \; dt\) doesn’t seem mathematical, its just a substitute for the type of stochastic process we have. For our purposes we are only going to consider random walks (a.k.a white noise a.k.a. Weiner processes), which we represent in shorthand with \(dW(t)\), so that \(dW(t)=\mbox{ Noise } \; dt\). This allows us to re-write Equation (25.1) as a more formal SDE (Equation (25.2)).

\[\begin{equation} dx = rx \left(1 - \frac{x}{K} \right) \; dt + dW(t) \tag{25.2} \end{equation}\]

Just to be clear the term \(dW(t)\) in Equation (25.2) is short hand to the differential equation \(\displaystyle \frac{dW}{dt} = \mbox{ Noise }\), where \(W(t)\) is the solution to a Weiner process. This white noise has the following characteristics:

• \(W(t)\) is continuous
• \(W(0)=0\)
• \(W(t)-W(s)\) independent (they call this independent increments)
• \(W(t)-W(s)\) is normally distributed with mean 0 and standard deviation \(\sqrt{t-s}\).

It does seem odd to write a differential equation in this form (i.e. \(dx = ...\) versus \(\displaystyle \frac{dx}{dt} = ...\)). but a good way to think of this stochastic differential equation is that a small change in the variable \(x\) (represented by the term \(dx\)) is computed in two ways:

\[\begin{equation*} \begin{split} \mbox{Deterministic part: } & rx \left(1 - \frac{x}{K} \right) \; dt \\ \mbox{Stochastic part: } & dW(t) \end{split} \end{equation*}\]

How does the stochastic part of this differential equation change the solution trajectory? It turns out that the “exact” solutions to problems like these are difficult (we will study a sample of them in 27). Rather than focus on an exact solution technique we are going to focus on how to apply a numerical method to simulate solution trajectories and then take the ensemble average each of the time points.