What is logistic growth?
It is the simplest model of a population that cannot grow forever. With unlimited resources a population would grow exponentially, doubling again and again. But no real environment is infinite: food, space and water run out. The logistic model adds that brake with three ingredients: the population size (N), its intrinsic growth rate (r) —how fast it reproduces while still small— and the carrying capacity (K), the ceiling the environment can sustain.
It was formulated by the Belgian mathematician Pierre-François Verhulst in 1838, who wanted a law of population growth that did not shoot off to infinity. The result is the characteristic S-shaped curve: nearly exponential growth at first, a slowdown in the middle, and a gentle plateau as the population approaches K.
Why does it slow down? The carrying capacity
The heart of the model is the braking factor (1 − N/K). When the population is small compared with K, this factor is close to 1 and growth is almost exponential; as N approaches K, the factor tends to 0 and growth stops. The differential equation that describes it is:
Unlike other population models, this equation has an exact solution as a function of time, so the curve can be computed directly —no numerical integration, no accumulated error:
The system has two equilibria: extinction at N = 0 (unstable) and the carrying capacity at N = K (stable). Any population starting with N₀ > 0 converges to K —growing if it started below, declining if it started above.
The S-curve and its inflection point
The most useful feature of the logistic curve is its inflection point: the moment when growth stops accelerating and starts braking. It happens exactly when the population reaches half of the carrying capacity, N = K/2, at the instant:
At that point the population grows at its maximum speed, whose value is remarkably simple:
In resource management this quantity is known as the maximum sustainable yield (MSY): the rate at which a population —fish, timber, game— could be harvested while keeping it right at half of its capacity, where it replenishes individuals as fast as possible. An inflection point only exists if the population starts below K/2; if it starts higher, it is born already in the slowdown phase and the curve is concave from the outset.
A real case: Gause’s cultures
The classic empirical example comes from the ecologist Georgii Gause, who in The Struggle for Existence (1934) grew the protozoan Paramecium in tubes with a steady supply of bacteria as food. Counting the individuals day by day, his populations traced a textbook logistic curve: an exponential start, a slowdown, and a stable plateau where density no longer rose —the carrying capacity set by the available food.
These experiments turned a century-old mathematical idea into a phenomenon you could measure on the bench, and they are why the logistic model is today the gateway to population ecology. We present the case qualitatively: the value of K depends on each experimental setup and should not be read as a universal constant.