Section 4.4 The Logistic Equation (a very brief introduction)
References.
These notes just introduce this example of a scientifically interesting equation that can be solved by the methoids introuduced in Section 4.3, leaving further discussion and examples to the reference above.
Here I just state the equation itself: the Logistic Equation is
\begin{equation}
\frac{dP}{dt} = r P \left( 1 - \frac{P}{K} \right)\tag{4.4.1}
\end{equation}
where
- \(t\) is time,
- \(P = P(t)\) is the population size at that time,
- \(r\) is the natural growth rate: when the population is small, it behaves approximately as with \(dP/dt = rP\)), and
- \(K\) is the carrying capacity, which as will be seen is the natural equilibrium size of the polulation; the size at which resources are just sufficient to sustain the population.
This is separable, and indeed autonomous (there is no explicit dependence on the independent variable, \(t\)), so it can be solved by integration:
\begin{equation}
\int \frac{dP}{P \left( 1 - \frac{P}{K} \right)} = \int r dt, = r(t - t_0)\tag{4.4.2}
\end{equation}
but the integration takes some care, due to the zeros of the denominator for \(P=0\) and \(P=K\text{.}\)
openstax.org/books/calculus-volume-2/pages/4-4-the-logistic-equation
