View source for Carrying Capacity and Logistic Growth Rate

[[File:Logistic-curve.svg|thumb|320px|right|Standard logistic sigmoid function where <math>L=1,k=1,x_0=0</math>]]

A '''logistic function''' or '''logistic curve''' is a common S-shaped curve (sigmoid curve) with equation

: <math>f(x) = \frac{L}{1 + e^{-k(x-x_0)}},</math>

where
: <math>x_0</math>, the <math>x</math> value of the sigmoid's midpoint;
: <math>L</math>, the curve's maximum value;
: <math>k</math>, the logistic growth rate or steepness of the curve.

For values of <math>x</math> in the domain of real numbers from <math>-\infty</math> to <math>+\infty</math>, the S-curve shown on the right is obtained, with the graph of <math>f</math> approaching <math>L</math> as <math>x</math> approaches <math>+\infty</math> and approaching zero as <math>x</math> approaches <math>-\infty</math>.

==Modeling population growth==
[[File:Pierre Francois Verhulst.jpg|right|thumb|150px|Pierre-François Verhulst (1804–1849)]]
A typical application of the logistic equation is a common model of population growth, originally due to Pierre-François Verhulst in 1838, where the rate of reproduction is proportional to both the existing population and the amount of available resources, all else being equal. The Verhulst equation was published after Verhulst had read Thomas Malthus' ''An Essay on the Principle of Population'', which describes the Malthusian growth model of simple (unconstrained) exponential growth. Verhulst derived his logistic equation to describe the self-limiting growth of a biological population. The equation was rediscovered in 1911 by A. G. McKendrick for the growth of bacteria in broth and experimentally tested using a technique for nonlinear parameter estimation. The equation is also sometimes called the ''Verhulst-Pearl equation'' following its rediscovery in 1920 by Raymond Pearl (1879–1940) and Lowell Reed (1888–1966) of the Johns Hopkins University. Another scientist, Alfred J. Lotka derived the equation again in 1925, calling it the ''law of population growth''.

Letting <math>P</math> represent population size (<math>N</math> is often used in ecology instead) and <math>t</math> represent time, this model is formalized by the differential equation:

: <math>\frac{dP}{dt}=r P \left(1 - \frac{P}{K}\right),</math>

where the constant <math>r</math> defines the growth rate and <math>K</math> is the carrying capacity.

In the equation, the early, unimpeded growth rate is modeled by the first term <math>+rP</math>. The value of the rate <math>r</math> represents the proportional increase of the population <math>P</math> in one unit of time. Later, as the population grows, the modulus of the second term (which multiplied out is <math>-r P^2 / K</math>) becomes almost as large as the first, as some members of the population <math>P</math> interfere with each other by competing for some critical resource, such as food or living space. This antagonistic effect is called the ''bottleneck'', and is modeled by the value of the parameter <math>K</math>. The competition diminishes the combined growth rate, until the value of <math>P</math> ceases to grow (this is called ''maturity'' of the population).
The solution to the equation (with <math>P_0</math> being the initial population) is

:<math>P(t) = \frac{K P_0 e^{rt}}{K + P_0 \left( e^{rt} - 1\right)} = \frac{K}{1+\left(\frac{K-P_0}{P_0}\right)e^{-rt}}, </math>

where

:<math>\lim_{t\to\infty} P(t) = K.</math>

Which is to say that <math>K</math> is the limiting value of <math>P</math>: the highest value that the population can reach given infinite time (or come close to reaching in finite time). It is important to stress that the carrying capacity is asymptotically reached independently of the initial value <math>P(0) > 0</math>, and also in the case that <math>P(0) > K</math>.

In ecology, species are sometimes referred to as <math>r</math>-strategist or <math>K</math>-strategist depending upon the selective processes that have shaped their life history strategies.
Choosing the variable dimensions so that <math>n</math> measures the population in units of carrying capacity, and <math>\tau</math> measures time in units of <math>1/r</math>, gives the dimensionless differential equation

: <math>\frac{dn}{d\tau} =  n (1-n).</math>

=== Time-varying carrying capacity ===
Since the environmental conditions influence the carrying capacity, as a consequence it can be time-varying, with <math>K(t) > 0</math>, leading to the following mathematical model:

: <math>\frac{dP}{dt} = rP \cdot \left(1 - \frac{P}{K(t)}\right).</math>

A particularly important case is that of carrying capacity that varies periodically with period <math>T</math>:

: <math>K(t + T) = K(t).</math>

It can be shown that in such a case, independently from the initial value <math>P(0) > 0</math>, <math>P(t)</math> will tend to a unique periodic solution <math>P_*(t)</math>, whose period is <math>T</math>.

A typical value of <math>T</math> is one year: In such case <math>K(t)</math> may reflect periodical variations of weather conditions.

Another interesting generalization is to consider that the carrying capacity <math>K(t)</math> is a function of the population at an earlier time, capturing a delay in the way population modifies its environment. This leads to a logistic delay equation, which has a very rich behavior, with bistability in some parameter range, as well as a monotonic decay to zero, smooth exponential growth, punctuated unlimited growth (i.e., multiple S-shapes), punctuated growth or alternation to a stationary level, oscillatory approach to a stationary level, sustainable oscillations, finite-time singularities as well as finite-time death.

==Resources==
* [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(OpenStax)/08%3A_Introduction_to_Differential_Equations/8.4%3A_The_Logistic_Equation The Logistic Equation], Mathematics LibreTexts
* [https://courses.lumenlearning.com/boundless-biology/chapter/environmental-limits-to-population-growth/ Environmental Limits to Population Growth], Lumen Learning