The Verhulst model

This online calculator plots the Verhulst-Pearl equation, or logistic curve, using the given initial parameters.

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Created: 2023-11-13 11:41:42, Last updated: 2023-11-13 11:53:01

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You can read more about the parameters of the logistic equation and the logistic growth model below the calculator.

The Verhulst model

Population dynamics
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The logistic equation or the Verhulst equation

The Malthusian growth model assumes constant birth and death rates, i.e., a constant Malthusian parameter r. However, this assumes unlimited population growth, which is obviously unlikely for natural populations (including humans).

Pierre-François Verhulst also tried to find a "real" population law, but abandoned further research in this direction because the available data were insufficient to test the formulas and assess their accuracy. Instead, he decided to find an empirical formula that agreed well with the available data, which would allow him to find the maximum possible population size, on the assumption that over an infinitely large time interval this size should have stabilized due to limited resources1. Verhulst's work was published in 18382 and 18453.

Verhulst believed that with population growth, fertility should decrease and mortality should increase, that is, the rate of growth should not be a constant value, as in Malthusian model, but a function that decreases as the population grows.

In general terms, this could be expressed by the following equation:
$\frac{dP}{dt}=rP-f(P)$,
where P is the size of the population, r is the intrinsic growth rate in the absence of constraints, f(P) some dependence on P to be determined.

Verhulst checked at least four kinds of functions from P: $f(P)=sP^2, f(P)=sP^3,f(P)=sP^4,f(P)=s log(P)$, and decided that their results were quite close, so we can take the simplest of the hypotheses $f(P)=sP^2$, and get the well-known Verhulst Equation:
$\frac{dP}{dt}=rP-sP^2$.

The s parameter can be called the self-limitation coefficient or the coefficient of intraspecific competition, which may be due to competition for food resources, release of harmful metabolites into the environment (self-poisoning), and other reasons.

The appearance of such names is due to the fact that models of population dynamics, invented to describe population growth, were adopted by ecologists with the emergence and development of ecology. By the way, ecologists often use the designation N instead of P for population size in the formulas of the models.

By introducing the coefficient $K=\frac{r}{s}$, the equation can be written in the form of
$\frac{dP}{dt}=rP\frac{K-x}{K}$

The solution of this equation with initial condition $P(t_0)=P_0$ is as follows
$P(t)=\frac{KP_0}{P_0+(K-P_0)e^{-rt}}$
Ferhulst called this solution the logistic equation or logistic function. With time tending to infinity, P tends to K. The function tends to K from below if p₀ ＜K, with an inflection when K/2 is reached, or from above if p₀ ＞K. The parameter K was called the "upper limit of the population" by Verhulst. Ecologists call this parameter maximum population size, environmental capacity, or a measure of the capacity of the ecological niche of a population.

Interestingly, after Verhulst, the logistic equation was rediscovered by several researchers, most notably Raymond Pearl during a study of the rate of population growth in the United States in 19204. The equation is therefore sometimes referred to as the Ferhulst-Pearl equation. Pearl, as well as Malthus, believed that logistic curve growth was a universal law of growth of all living things, including population growth, but logistic curve growth was demonstrated mainly by laboratory experiments on animals and lower organisms, and attempts to predict population size did not match reality. Using Verhulst's model on what population data he had at the time for France gives a maximum population of 38.6 million1. France's population for 2020 is 67.39 million.

Nevertheless, modifications of the logistic equation with different types of f(P) are used by ecologists, for example, to describe population dynamics of commercial fish species. The model fairly well characterizes annual single-species populations with a short life span and individuals of approximately the same age.

1. Delmas, Bernard. (2004). Pierre-François Verhulst et la loi logistique de la population. Mathématiques et sciences humaines. 10.4000/msh.2893.

2. Verhulst, P. F., Notice sur la loi que la population poursuit dans son accroissement. Correspondance mathématique et physique, 10, 113—121, 1838.

3. Verhulst, P. F., Recherches Mathématiques sur La Loi D’Accroissement de la Population, Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres de Bruxelles, 18, Art. 1, 1—45, 1845 (Mathematical Researches into the Law of Population Growth Increase).

4. Pearl, Raymond and Lowell J. Reed. On the Rate of Growth of the Population of the United States since 1790 and its Mathematical Representation // Proceedings of the National Academy of Sciences of the United States of America (PNAS; USA). — 1920. — June 15 (v.6, №6). — p. 275–288.

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