Editing Logistic function (section)

=== In ecology: 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]] (see also [[population dynamics]]), originally due to [[Pierre François Verhulst|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 [[biology|biological]] population. The equation was rediscovered in 1911 by [[Anderson Gray McKendrick|A. G. McKendrick]] for the growth of bacteria in broth and experimentally tested using a technique for nonlinear parameter estimation.<ref name="McKendric Logistic">{{Cite journal | doi = 10.1017/S0370164600025426|journal=Proceedings of the Royal Society of Edinburgh|volume= 31 |date= January 1912|pages= 649–653 |title=XLV.—The Rate of Multiplication of Micro-organisms: A Mathematical Study|author= A. G. McKendricka|author2= M. Kesava Paia1|url=https://zenodo.org/record/1543653}}</ref> 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]].<ref>{{cite news|author=Raymond Pearl|author-link=Raymond Pearl|author2=Lowell Reed|author2-link=Lowell Reed|name-list-style=amp|title=On the Rate of Growth of the Population of the United States|url=http://math.bu.edu/people/mak/MA565/Pearl_Reed_PNAS_1920.pdf|date=June 1920|journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|page=275|number=6|volume=6}}</ref> 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 display="block">\frac{dP}{dt}=r P \left(1 - \frac{P}{K}\right),</math>

where the constant <math>r</math> defines the [[population growth rate|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 display="block">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 display="block">\lim_{t\to\infty} P(t) = K,</math>

where <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). 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 [[r/K selection theory|<math>r</math>-strategist or <math>K</math>-strategist]] depending upon the [[Natural selection|selective]] processes that have shaped their [[Biological life cycle|life history]] strategies.
[[Dimensional analysis|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 display="block">\frac{dn}{d\tau} =  n (1-n).</math>

==== Integral ====
The [[antiderivative]] of the ecological form of the logistic function can be computed by the [[Integration by substitution|substitution]] <math>u = K + P_0 \left( e^{rt} - 1\right)</math>, since <math>du = r P_0 e^{rt} dt</math>

<math display="block">\int \frac{K P_0 e^{rt}}{K + P_0 \left( e^{rt} - 1\right)}\,dt = \int \frac{K}{r} \frac{1}{u}\,du = \frac{K}{r} \ln u + C = \frac{K}{r} \ln \left(K + P_0 (e^{rt} - 1) \right) + C</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 display="block">\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 display="block">K(t + T) = K(t).</math>

It can be shown<ref>{{Cite journal |last1=Griffiths |first1=Graham |last2=Schiesser |first2=William |date=2009 |title=Linear and nonlinear waves |journal=Scholarpedia |language=en |volume=4 |issue=7 |page=4308 |doi=10.4249/scholarpedia.4308 |bibcode=2009SchpJ...4.4308G |issn=1941-6016|doi-access=free }}</ref> 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,<ref name="delay carrying">{{Cite journal | last1 = Yukalov | first1 = V. I. | last2 = Yukalova | first2 = E. P. | last3 = Sornette | first3 = D. | s2cid = 14456352 | doi = 10.1016/j.physd.2009.05.011 | title = Punctuated evolution due to delayed carrying capacity | journal = Physica D: Nonlinear Phenomena | volume = 238 | issue = 17 | pages = 1752–1767 | year = 2009 | arxiv = 0901.4714 | bibcode = 2009PhyD..238.1752Y }}</ref> 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.