Editing Population growth (section)

==Growth rate models==
The "population growth rate" is the rate at which the number of individuals in a population increases in a given time period, expressed as a fraction of the initial population. Specifically, population growth rate refers to the change in population over a unit time period, often expressed as a percentage of the number of individuals in the population at the beginning of that period. This can be written as the formula, valid for a sufficiently small time interval:

:<math>Population\ growth\ rate = \frac{ P(t_2)  -  P(t_1)} {P(t_1)(t_2-t_1)}</math>

A positive growth rate indicates that the population is increasing, while a negative growth rate indicates that the population is decreasing. A growth ratio of zero indicates that there were the same number of individuals at the beginning and end of the period—a growth rate may be zero even when there are significant changes in the [[birth rate]]s, [[death rate]]s, [[immigration rate]]s, and age distribution between the two times.<ref>[http://www.apheo.ca/index.php?pid=61 Association of Public Health Epidemiologists in Ontario] {{webarchive|url=https://web.archive.org/web/20080522092307/http://www.apheo.ca/index.php?pid=61 |date=22 May 2008 }}</ref>

A related measure is the [[net reproduction rate]]. In the absence of migration, a net reproduction rate of more than 1 indicates that the population of females is increasing, while a net reproduction rate less than one ([[sub-replacement fertility]]) indicates that the population of females is decreasing.

Most populations do not grow exponentially, rather they follow a [[Logistic function|logistic model]]. Once the population has reached its [[carrying capacity]], it will stabilize and the exponential curve will level off towards the carrying capacity, which is usually when a population has depleted most its [[natural resource]]s.<ref name=":0">{{Cite book|title=Campbell Biology|last1=Reece|first1=Jane|last2=Urry|first2=Lisa|last3=Cain|first3=Michael|last4=Wasserman|first4=Steven|last5=Minorsky|first5=Peter|last6=Jackson|first6=Robert|publisher=Pearson|year=2014}}</ref> In the world human population, growth may be said to have been following a [[linear trend]] throughout the last few decades.<ref name=":5" />
[[File:Logistic growth graph (population ecology).JPG|thumb|upright=1.4|The logistic growth of a population]]

===Logistic equation===

The growth of a population can often be modelled by the [[Population dynamics#Logistic population growth|logistic equation]]<ref>{{Cite book|title=Brief Applied Calculus|last1=Stewart|first1=James|last2=Clegg|first2=Daniel|publisher=Brooks/Cole Cengage Learning|year=2012}}</ref>

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

where
* <math>P(t)</math> = the population after time t;
* <math>t</math> = time a population grows;
* <math>r</math> = the relative growth rate coefficient;
* <math>K</math> = the carrying capacity of the population; defined by ecologists as the maximum population size that a particular environment can sustain.<ref name=":0" />

As it is a separable differential equation, the population may be solved explicitly, producing a [[logistic function]]:

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

where <math>A=\frac{K-P_0}{P_0}</math> and <math>P_0</math> is the initial population at time 0.