26.3 Generalizing the approach.

As mentioned previously, another way to think about the logistic differential equation as what as known as a “birth-death process.”

Let’s call the part of the differential equation that contributes to a positive rate as a “birth process” and parts that contribute to a negative rate as a “death process.” If we have the differential equation

\[\begin{equation*} \frac{dx}{dt} = \alpha(x)-\delta(x) \end{equation*}\]

Then we would simulate the birth death process with \(\mu = \alpha(x)-\delta(x)\) and \(\sigma^{2} = \alpha(x)+\delta(x)\).

For a multivariable system of equations the process is the same, however because we have a system of equations the calculations for the expected value and variance require more knowledge of matrix algebra which is beyond the scope here. The provided code does take this into account.