From Self-Limiting Growth to the Logit Function

Mathematics is often taught as if formulas arrive fully formed. We are shown the final expression, perhaps a short derivation, and then asked to remember the result. What usually gets lost in that process is the original motivation: what concrete problem made someone invent the model in the first place?

I find mathematical ideas much easier to hold onto when they are tied back to the reasoning that produced them. The logistic and logit functions are a good example. Today they show up in statistics, machine learning, and forecasting, but they did not begin as generic tools for classification. They grew out of a much older question about population, scarcity, and what happens when growth runs into limits.

This post follows that chain of ideas from the beginning. I focus on one historical path into the logistic function: how it emerged from attempts to understand population pressure and human misery, and how it later became a much more general tool. The short version is:

1. Malthus modeled population with constant per-capita growth.
2. Verhulst asked how to correct that model once growth starts to slow itself down.
3. The resulting solution is the logistic function.
4. Later statisticians reused the same curve because it maps unrestricted values into valid probabilities.
5. The associated transformation from probabilities to log-odds became the logit.

Why Population Became a Mathematical Problem

In the late eighteenth and early nineteenth centuries, population was not an abstract academic topic. It sat directly inside political economy. Governments and scholars were worried about food supply, wages, poverty, poor relief, taxation, military strength, and crowding. Context The word “statistics” was still closely tied to the state. Counting births, deaths, households, and productive capacity was part of understanding how a polity functioned and what it could sustain. The core question was not simply whether the population was large, but whether population growth could outrun what a country was able to support^1,2.

There were competing explanations for the causes of human misery. Term In this debate, misery meant material suffering: poverty, hunger, disease, overcrowding, dependence, and early death. It was not just a mood; it referred to concrete social and biological hardship. William Godwin represented a more optimistic view. He argued that people are highly improvable through reason, education, and freer social arrangements. On that view, many social problems came less from natural limits than from bad institutions and coercive social organization^3,1.

Thomas Malthus took a harder line. He thought this optimism underestimated a basic constraint: population pressure on subsistence. Food is necessary for life, people continue to reproduce when conditions allow, and agriculture cannot expand without limit. For Malthus, many forms of suffering were therefore not only institutional, but also the result of a population pressing against the resources available to sustain it¹.

Malthus and the Problem of Exponential Growth

Malthus’s starting point can be written as

This says the per-capita growth rate is constant. If every individual contributes, on average, the same amount to future growth, then larger populations grow faster in absolute terms.

The solution is the familiar exponential law

This is a clean and sensible first approximation. When population is small relative to the available land and food supply, exponential growth is exactly the kind of behavior one might expect. But it also reveals the weakness of the model: if the per-capita growth rate remains constant forever, then the total population grows without bound¹.

Malthus used this argument to push back against the more optimistic view that social improvement alone could remove misery. Even if institutions become better, unchecked population growth still runs into material limits. Verhulst accepted that basic concern, but asked a more precise mathematical question: what happens once growth begins to slow because the size of the population itself starts to matter?

Verhulst’s Correction: Growth That Slows Itself Down

Pierre-Francois Verhulst accepted the Malthusian picture as a useful approximation at low population density, but argued that it could not remain true indefinitely. As population rises, crowding, competition, and limited resources feed back into the growth process itself^4,6.

So instead of assuming a constant per-capita growth rate, he introduced a retardation term:

The key modeling question is then: what shouldbe? Math Verhulst’s linear choice became the standard one, but he explicitly noted that other retardation terms were conceivable as well, for example.

Verhulst’s simplest useful choice was to let the reduction in per-capita growth be proportional to the size of the population itself:

That gives

This is where the quadratic term comes from. It is not a mysterious extra ingredient. It appears because total growth is population times per-capita growth, and the per-capita growth rate has been assumed to decline linearly with population size Math Once you write the per-capita growth rate as, multiplying bygives. So the quadratic term is a consequence of a linear crowding correction on the per-capita scale. ⁴.

Whenis small, theterm is negligible compared with the linear term, so we recover the Malthusian model as a first approximation. But asgrows, the quadratic term matters more and more, and growth begins to slow.

The interactive figure makes the modeling contrast visible. The exponential model never asks whether there is still room to grow. The logistic model does. As a result, it bends and eventually saturates.

Solving the Logistic Equation

The logistic differential equation can be written as

A useful substitution is

Differentiating gives

so

Substituting into the original equation yields

Multiplying through bygives a first-order linear equation:

This is much easier to solve. First solve the homogeneous equation

If we try an exponential ansatz, then

So the homogeneous solution is

Now find a particular solution. Since the forcing term on the right-hand side is the constant, the simplest guess is a constant solution. Substituting gives

So

Combining both pieces,

Since, we have

If we now define

then

and the solution becomes

Renaming the constantgives the standard form

This is the logistic function. In the original growth setting it maps time to a bounded interval betweenand the carrying capacity.

Why Is It Called the Logistic Function?

The name is older than the statistical use. Verhulst introduced it for the growth curve itself, not for a probability transformation. What is less satisfying is that the exact etymology is not fully settled Name In his 1845 memoir Verhulst says he will give the name “logistique” to the curve, but he does not clearly justify the term. He also wrote the integrated result first with time as a logarithmic function of population, rather than in the later explicit form. That original logarithmic presentation is one reason historians connect the name to the curve’s log form, not to modern “logistics” in the sense of supply chains. ^5,9.

My own reading is that the name may be connected to the way Verhulst first wrote the solution:

In that form, time appears as a logarithmic function of population. That makes a log-related name plausible, but it is still only an inference; Verhulst did not spell out the etymology. The important point is simply that the name belongs to the nineteenth-century growth model, not to the later statistical use.

From Growth Curves to Probability Models

Much later, statisticians faced a different problem. In dose-response studies and binary outcome models, the quantity of interest is often a probability. A straight-line model is awkward in that setting because it can predict values outside the interval fromto^7,8.

What they needed instead was a smooth monotone curve that takes an unrestricted input and turns it into a valid probability. The probit model was one early answer. History The term “probit” predates “logit” in twentieth-century statistics. Joseph Berkson later coined “logit” in analogy to “probit” when advocating the logistic formulation for bioassay work. Berkson’s move was important because the logistic curve had simple algebra and a direct interpretation in terms of odds^7,8.

The standard logistic function in statistics is

This is the same S-shaped functional form as the normalized growth curve, but now the output lives inand is read as a probability rather than a population level.

The Logit and the Inverse Logit

The logit function maps a probability to the real line:

The quantity

is the odds, so the logit is the log-odds. Its inverse is exactly the standard logistic function:

That is why the same curve appears in two rather different settings:

• in population models, it comes from self-limiting growth;
• in statistics, it comes from mapping unrestricted scores into valid probabilities.

Why the Logit Is Useful

The practical strength of the logit is that probabilities are constrained while linear predictors are not. A regression model often wants to work on an unrestricted scale:

But if the outcome is a probability, then we need

The logit provides exactly the bridge needed to move between those two domains:

This is why logistic regression is built the way it is. Effects are modeled additively on the log-odds scale, and then transformed back into probabilities through the inverse logit.

In that sense, the modern use of the logistic function is conceptually different from Verhulst’s original problem, but mathematically it reuses the same form because the same shape solves a similar structural requirement: bounded output with smooth, monotone behavior.

Closing Thoughts

What I like about this story is that it makes the mathematics feel less arbitrary. The logistic function was not invented because somebody wanted an S-shaped curve for its own sake. It emerged from a specific attempt to correct exponential growth by building resource limits into the model itself. Later, the same curve reappeared because statisticians needed a disciplined way to turn real-valued predictors into probabilities.

That kind of migration across disciplines is one of the most interesting things about mathematics. An idea begins inside one concrete problem, shaped by one historical setting, and then later becomes part of a much wider conceptual toolkit. The logistic function began as a model of self-limiting growth. Today it helps express risk, probability, and decision-making across medicine, ecology, and statistical forecasting.

Remembering that history makes the formulas easier to hold onto. More importantly, it gives them back some of their original logic. The equations stop looking like isolated symbolic tricks and start looking like answers to real questions people once had to take seriously.