Logistic Growth and Model Limits

Definition A logistic function is a function of the form

with constants $L > 0$, $A > 0$ and $k > 0$. Its domain is all real numbers and its range is the open interval $(0, L)$. It is increasing, and its graph has the horizontal asymptote $y = 0$ as $t \to -\infty$ and $y = L$ as $t \to \infty$. The graph has the shape of a stretched letter S. Books use different letters for these constants, and the calculus lesson on logistic growth writes $K$ in place of $L$ and $r$ in place of $k$.

Definition In $P(t) = \frac{L}{1 + Ae^{-kt}}$ all three parameters must be positive. The starting value is

With $L > 0$, the condition $A > 0$ holds exactly when $0 < P(0) < L$, and $k > 0$ makes $P$ increasing. Under these conditions the denominator $1 + Ae^{-kt}$ is greater than $1$ for every $t$, so $P$ is defined everywhere and stays between $0$ and $L$. If instead $A < 0$ while $k > 0$, the denominator is zero at $t = \frac{\ln(-A)}{k}$ and the graph has a vertical asymptote there. If $A > 0$ but $k < 0$, the curve falls toward $0$ as $t$ grows.

Definition In a logistic model the parameter $L$ is the carrying capacity, the size the population approaches over time. Because $1 + Ae^{-kt} > 1$, every value satisfies $0 < P(t) < L$, and because $Ae^{-kt} \to 0$ as $t \to \infty$,

So the line $y = L$ is a horizontal asymptote of the graph. It is approached but never reached.

Definition Multiplying $P(t)$ by $1 + Ae^{-kt}$ and rearranging gives, for every $t$, the exact identity

While $P(t)$ is small compared with $L$, the factor in parentheses is close to $1$, so

Early logistic growth is therefore nearly exponential with continuous growth rate $k$. The exponential formula is too large by the factor $\frac{1}{1 - P(t)/L}$, which grows as $P(t)$ approaches $L$.

Definition Saturation is the late stage of logistic growth, when $P(t)$ is close to $L$ and growth has nearly stopped. The remaining gap is

which is positive for every $t$ and approaches $0$ as $t \to \infty$. Once $Ae^{-kt}$ is small the gap is approximately $LAe^{-kt}$, so it shrinks by about the factor $e^{-k}$ with each unit of time.

Definition The midpoint of a logistic curve is the point $(t_0, \frac{L}{2})$ with

Since $Ae^{-kt_0} = 1$, the function can be written as

and $P(t_0) = \frac{L}{2}$. The graph is symmetric about the midpoint, which means

To see this, put $u = e^{-ks}$. Then

which add to $L$. The curve is also steepest at the midpoint, where it changes from bending upward to bending downward. Calculus proves that fact, and here it is taken as known.

Definition In calculus the logistic function comes from an equation about its rate of change, a differential equation. Without that equation, a logistic model can still be read entirely from its formula. The carrying capacity is $L$, the starting value is $\frac{L}{1 + A}$, the early continuous growth rate is $k$, and $t_0 = \frac{\ln A}{k}$ is the time of half capacity and fastest growth. For any level $y$ with $0 < y < L$, solving $y = \frac{L}{1 + Ae^{-kt}}$ with logarithms gives the unique time at which the model reaches $y$,

The levels $0$ and $L$ are never reached. The model also rests on assumptions that real populations can break. The capacity stays fixed, growth is smooth with no seasons, harvests or disasters, and the rise is exactly symmetric about the midpoint.