What was the idea behind Logistic growth model?

Question

The Malthus model is given by

$\frac{dP(t)}{dt}=rP(t)$, where $r$ is the growth rate. This model ignores the competition for resources among individuals. So, Verhulst came up with a model

$\frac{dP(t)}{dt}=rP(t) \left(1-\frac{P(t)}{K} \right)$, where $K$ is the carrying capacity of environment. My question is:

how he derived this model or what was the idea behind this model ?

In Strogatz book "Non linear dynamics and chaos", they give the following explanation:

Because $\frac{\dot P(t)}{P(t)}$, the per capita growth rate should decrease for the large population. A mathematical convenient way to incorporate these ideas is to assume that per capita growth rate decreases linearly with $P(t)$, which leads to logistic equation. Was this the original idea of Verhulst behind Logistic growth model ?

Answer 1

The Wikipedia page for Verhulst has links to his original writings. In the first paper from 1838, he basically only says (on p. 115) that since the growth rate of the population decreases as the number of inhabitants increases, we can subtract an unknown function $\varphi(p)$ from the right-hand side of the differential equation, $$ \frac{dp}{dt} = mp - \varphi(p) , $$ and that the simplest hypothesis that one can make about the form of this function is that $$ \varphi(p) = n p^2 . $$ (He also mentions some other possible forms, like $np^\alpha$ or $n \log p$.)

Answer 2

I think the most natural "intuition" for the logistic growth model is a combined proportion:

• $\frac{dP}{dt}$ is proportional to the population at time $t$: $\frac{dP}{dt} \sim P(t)$

$\bf{and}$

• $\frac{dP}{dt}$ is proportional to the remaining carrier capacity at time $t$: $\frac{dP}{dt} \sim (K-P(t))$

Combining these two proportions by multiplying leads to the logistic differential equation:

$$\frac{dP}{dt} \sim P(t)(K-P(t)) \Leftrightarrow \frac{dP}{dt} \sim P(t)\left(1-\frac{P(t)}{K}\right)$$

But I am not sure whether these were the thoughts of Verhulst.