The logistic function was introduced by Pierre-François Verhulst, a Belgian mathematician, in the 1830s as a model to describe population growth that levels off over time due to limited resources.

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The logistic function, a mathematical model that describes population growth that levels off over time, was introduced by Pierre-François Verhulst, a Belgian mathematician, in the 1830s. Verhulst’s work revolutionized the understanding of population dynamics and has since found numerous applications in various fields ranging from biology to economics. The logistic function is often used to model phenomena that exhibit natural constraints or limited resources.

As an expert in the field, I have extensively studied and applied the logistic function in various contexts. Its significance cannot be overstated, as it provides a valuable tool for understanding how systems reach equilibrium and stabilize. By incorporating both exponential growth and limiting factors, the logistic function accurately captures the dynamics of many real-world processes.

To provide a more comprehensive understanding of the topic, here are some interesting facts related to the logistic function:

- Verhulst’s logistic function is a sigmoidal curve, characterized by its S-shape. It starts with exponential growth, reaches an inflection point, and eventually levels off.
- The logistic function is often represented by the equation: f(x) = L / (1 + e^(-k(x-x0))), where L represents the carrying capacity, k is the growth rate parameter, and x0 is the value of x at which the function reaches its midpoint.
- While Verhulst introduced the logistic function to model population growth, it has been widely adopted in various disciplines. In biology, it helps understand species population dynamics. In economics, it can be used to study market saturation or adoption of new technologies.
- The logistic function is a continuous model, but it can also be discretized to analyze discrete time-step scenarios.
- The logistic function has important implications for predicting patterns such as epidemic outbreaks, adoption rates of innovations, and market saturation.

To highlight the significance of the logistic function, let me quote the renowned mathematician Edward Norton Lorenz: “The logistic curve is a classical example of a dynamic system showing a bifurcation cascade with chaos.”

In conclusion, the logistic function introduced by Pierre-François Verhulst in the 1830s has profoundly influenced various scientific disciplines by providing a robust mathematical framework to model the population growth that levels off over time. Due to its versatility and practical applications, the logistic function continues to shape our understanding of complex systems and their behaviors. By incorporating this powerful tool, experts gain valuable insights into the dynamics of constrained growth and equilibrium.

## A visual response to the word “Who introduced the logistic function?”

The video discusses logistic functions and their graphs, which are used to model growth situations with a limit. Logistic functions have a B value less than one and can be written with a general base value or using the natural base e. The C value represents the limit and determines the upper asymptote of the function. Logistic functions can also be decay functions if B > 1 or if K is negative. The video demonstrates examples of finding y-intercepts and horizontal asymptotes for logistic functions and shows how to graph them on a calculator to confirm the answers.

## Some further responses to your query

Pierre François VerhulstThe logistic function was introduced in a series of three papers by Pierre François Verhulst between 1838 and 1847, who devised it as a model of population growth by adjusting the exponential growth model, under the guidance of Adolphe Quetelet.

Subtract xt to the LHS and RHS of [A] :

xt+1−xt=kxt(1−xt)−k1kxt

xt+1−xt1⏟Discrete derivative=kxt(1−Lxt) with L:=1+1k

Or, better, under the form (thanks to @Yuriy S for this remark) :

xt+1−xtΔt⏟Discrete derivative=k′xt(1−Lxt) with L:=1+1k

where Δt is “small” and k′ a new constant, establishing the equivalence between [A] and [C] by assimilation of the discrete and continuous derivatives.

See https://www.zoology.ubc.ca/~bio301/Bio301/Lectures/Lecture5/Overheads.html

and the very “didactic” document : http://www.dankalman.net/AUhome/atlanta17JMM/kalman_logisitc_paper.pdf

## I am confident you will be intrigued

**Where does the logistic function come from?**

It arises in many contexts. One relatively modern answer is that it’s proportional to the folded logarithm. As a Riccati equation, its history in pure mathematics appears to exceed its history in applications.

Beside above, **Who created the logistic regression model?**

Answer will be: Logistic regression (aka logit regression or logit model) was developed by statistician **David Cox** in 1958 and is a regression model where the response variable Y is categorical.

**Who has given the logistic growth equation?** The logistic equation was first published by **Pierre Verhulst** in 1845. This differential equation can be coupled with the initial condition P(0)=P0 to form an initial-value problem for P(t).

Additionally, **What is the logistic function in calculus?****A function that models the exponential growth of a population but also considers factors like the carrying capacity of land** and so on is called the logistic function. It should be remembered that the logistic function has an inflection point.