Section 1.7 Verhulst ("logistic") and Gompertz population models
Consider a population that has a maximum possible size, called the carrying capacity of the population. We will use a constant \(K\) to denote the carrying capacity. However, carrying capacity of a population can change in many ways, such as from a lack of resources, the spread of disease, or the development of a new technology, but we will only consider populations having a constant carrying capacity here. The Malthusian model of Section 1.2 is a crude population model that does allow carrying capacity to be considered. We consider below two population models that take carrying capcity into account.
Subsection 1.7.1 Verhulst ("logistic") model
The Verhult model (also called "logistic model") says that the rate of change of the size of the population is proportional to the size of the population multiplied by the proportion that the environment can handle
\begin{equation*}
\dfrac{\mathrm{d}P}{\mathrm{d}t} = rP\left(1-\dfrac{P}{K}\right),
\end{equation*}
where the proportionality constant \(r\) is the growth rate and \(K\) is the carrying capacity.
If the population at time \(t\text{,}\) \(P(t)\text{,}\) is less than \(K\text{,}\) then we observe that the proportion of population that the environment can handle is \(\dfrac{P(t)}{K} < 1\text{.}\) From this we conclude that the remaining proportion of the population the environment can handle is \(1-\dfrac{P(t)}{K} > 0\text{.}\) In other words, the population can still grow whenever \(P(t) < K\text{.}\) Similarly, if the population is larger than \(K\) at time \(t\text{,}\) then we conclude that \(1-\dfrac{P(t)}{K} < 0\text{,}\) meaning the population must shrink since its size is beyond the carrying capacity \(K\) of the environment. We summarize this in the following table. Table 1.7.1. Signs of \(\dfrac{\mathrm{d}P}{\mathrm{d}t}\) in the Verhulst model
| Relation between \(P\) and \(K\) | Sign of \(1-\dfrac{P(t)}{K}\) | Sign of \(\dfrac{\mathrm{d}P}{\mathrm{d}t}\) | Effect on solution curve |
|---|---|---|---|
| \(P(t)< K\) | \(+\) | \(+\) | Increasing at time \(t\) |
| \(P(t)=K\) | \(0\) | \(0\) | Unchanging at time \(t\) |
| \(P(t) > K\) | \(-\) | \(-\) | Decreasing at time \(t\) |
Subsection 1.7.2 Gompertz model
The Gompertz model works similarly to the Verhulst model, except instead of multiplying by \(1-\dfrac{P(t)}{K}\text{,}\) we instead multiply by \(\ln\left(\dfrac{K}{P(t)}\right)\)
\begin{equation*}
\dfrac{\mathrm{d}P}{\mathrm{d}t}=rP\ln\left(\dfrac{K}{P(t)}\right).
\end{equation*}
This model captures the carrying capacity idea because if \(P(t) < K\text{,}\) then \(\dfrac{K}{P(t)} > 1\text{.}\) Since the logarithm function is positive for any input larger than \(1\text{,}\) we obtain that \(\ln\left(\dfrac{K}{P(t)}\right)\) is a positive quantity, so the right-hand side of the differential equation will also be positive.
