Editing Carrying capacity (section)

==Mathematics==
The specific reason why a population stops growing is known as a limiting or [[regulating factor]].<ref>{{Cite web |title=Limiting Factors |url=https://education.nationalgeographic.org/resource/limiting-factors |access-date=2023-12-01 |website=education.nationalgeographic.org |language=en}}</ref>
[[File:Logistic_Carrying_Capacity.svg|thumb|Reaching carrying capacity through a logistic growth curve]]
The difference between the [[birth rate]] and the [[death rate]] is the [[Rate of natural increase|natural increase]]. If the population of a given organism is below the carrying capacity of a given environment, this environment could support a positive natural increase; should it find itself above that threshold the population typically decreases.<ref name="Storch-2019">{{cite journal |last1=Storch |first1=David |last2=Okie |first2=Jordan G. |title=The carrying capacity for species richness |journal=Global Ecology and Biogeography |date=October 2019 |volume=28 |issue=10 |pages=1519–1532 |doi=10.1111/geb.12987 |bibcode=2019GloEB..28.1519S |s2cid=202026304 }}</ref> Thus, the carrying capacity is the maximum number of individuals of a species that an environment can support in long run.<ref>{{cite journal |last1=Rees |first1=William E. |title=Ecological footprints and appropriated carrying capacity: what urban economics leaves out |journal=Environment and Urbanization |date=October 1992 |volume=4 |issue=2 |pages=121–130 |doi=10.1177/095624789200400212 |doi-access=free |bibcode=1992EnUrb...4..121R }}</ref>

[[Population decline|Population size decreases]] above carrying capacity due to a range of factors depending on the [[species]] concerned, but can include insufficient [[space]], [[Food security|food supply]], or [[sunlight]]. The carrying capacity of an [[Environment (biophysical)|environment]] varies for different species.{{citation needed|date=May 2020}}

In the standard ecological [[algebra]] as illustrated in the simplified [[Logistic function#In ecology: modeling population growth|Verhulst model]]<ref name="Verhulst18383">{{cite journal|last=Verhulst|first=Pierre-François|author-link=Pierre François Verhulst|year=1838|title=Notice sur la loi que la population poursuit dans son accroissement|url=https://books.google.com/books?id=8GsEAAAAYAAJ|format=PDF|journal=Correspondance Mathématique et Physique|volume=10|pages=113–121|access-date=3 December 2014}}</ref> of [[population dynamics]], carrying capacity is represented by the constant <math>K</math>:

: <math> {\mathrm{d}N\over\mathrm{d}t} = rN \left(1 - {N\over K}\right), </math>

where

* {{mvar|N}} is the [[population size]],
* {{mvar|r}} is the intrinsic [[rate of natural increase]]
* {{mvar|K}} is the carrying capacity of the local environment, and
* {{math|''dN/dt''}}, the [[derivative]] of {{mvar|N}} with respect to time {{mvar|t}}, is the rate of change in population with time.

Thus, the equation relates the growth rate of the population {{mvar|N}} to the current population size, incorporating the effect of the two [[Constant (mathematics)|constant parameters]] {{mvar|r}} and {{mvar|K}}. (Note that decrease is negative growth.) The choice of the letter {{mvar|K}} came from the [[German language|German]] ''Kapazitätsgrenze'' (capacity limit).<ref name="Verhulst18383">{{cite journal|last=Verhulst|first=Pierre-François|author-link=Pierre François Verhulst|year=1838|title=Notice sur la loi que la population poursuit dans son accroissement|url=https://books.google.com/books?id=8GsEAAAAYAAJ|format=PDF|journal=Correspondance Mathématique et Physique|volume=10|pages=113–121|access-date=3 December 2014}}</ref>

[[File:Logistic curve examples.png|thumb|This is a graph of population change utilizing the logistic curve model.  When the population is above the carrying capacity it decreases, and when it is below the carrying capacity it increases.]]

The [[Logistic function#In ecology: modeling population growth|Verhulst equation]] is a first-order [[ordinary differential equation]]. Combined with an [[initial value problem|initial value]] <math>N=N_0</math> for the population at time <math>t=0</math>, the [[initial value problem|solution]] takes the form of the [[logistic growth curve]]:

: <math>N(t) = {K\over 1 + A e^{-rt}},</math>

where

* {{mvar|e}} is [[e (mathematical constant)|Euler's number]],
* {{mvar|A}} is a constant determined by the initial population <math>N_0</math> via <math>A = {K\over N_0} - 1</math>.<ref>Murray, James D. "Mathematical biology: I. An introduction. Vol. 17." Springer Science & Business Media, 2007.</ref> When the initial population is below the carrying capacity, <math>A</math> is positive; when it is above, <math>A</math> is negative.
* {{mvar|r}} is the initial exponential growth rate, and
* {{mvar|K}} is the carrying capacity.

The logistic growth curve depicts how population growth rate and carrying capacity are inter-connected. As illustrated in the logistic growth curve model, when the population size is small, the population increases exponentially. However, as population size nears carrying capacity, the growth decreases and reaches zero at {{mvar|K}}.<ref>Swafford, Angela Lynn. "Logistic Population Growth: Equation, Definition & Graph." Study.com. N.p., 30 May 2015. Web. 21 May 2016. 
"Logistic Population Growth - Boundless Open Textbook." Boundless. N.p., n.d. Web. 21 May 2016.
</ref>

What determines a specific system's carrying capacity involves a [[Limiting factor#Ecology|limiting factor]]; this may be available supplies of [[food supply|food]] or [[water supply|water]], nesting areas, space, or the amount of [[waste]] that can be absorbed without degrading the environment and decreasing carrying capacity.