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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 ?

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    $\begingroup$ I think this is an interesting question, but I'm not sure if you are looking for a historical account, or an explanation of the mathematics, which are nicely incorporated here, except for a critical issue: in the KA post the differential equations are not solved. $\endgroup$ – Antoni Parellada Nov 11 '19 at 13:48
  • $\begingroup$ @AntoniParellada Actually I am interested in the historical account. $\endgroup$ – Manoj Kumar Nov 11 '19 at 13:55
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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 $n \log p$.)

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  • $\begingroup$ Thanks I got the English translated version of this work. $\endgroup$ – Manoj Kumar Nov 11 '19 at 15:00
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    $\begingroup$ (+1) It is interesting to note how this "simplest hypothesis that one can make" as in the linked document by Verhulst comes together as $mp - np^2=(m-np)p$ with $m-np=r_\text{max}\frac{K- p}{K},$ with $K$ being the carrying capacity, and hence making the per capita rate $r$ linearly dependent on the population $p.$ $\endgroup$ – Antoni Parellada Nov 11 '19 at 15:36
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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.

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  • $\begingroup$ You want to say that $\frac{dP}{dt} \sim P(K-P)$ $\endgroup$ – Manoj Kumar Nov 11 '19 at 13:24
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    $\begingroup$ @ManojKumar: Multiplying the two proportions you get exactly this. Then you take out $K$ as a factor and put it into the factor of proportion and obtain the logistic equation you cited in your text. $\endgroup$ – trancelocation Nov 11 '19 at 13:28
  • $\begingroup$ @ManojKumar : I added this part to my answer. $\endgroup$ – trancelocation Nov 11 '19 at 13:32

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