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In [[Population biology]] and [[Demography]], the '''''Generation time''''' is a quantity that reflects the average time between two consecutive generations in the lineages of a [[population]]. However, there is no single definition of the generation time, and several quantities are each better suited for a particular use. In [[human population]], the [[generation time]] ranges from 20 to 30 years. Historians sometimes use this to date events, by converting generations into years to obtain rough estimates of time.

In [[Epidemiology]], the term has a different meaning and refers to the time
between receipt of [[infection]] by a host and maximal [[infectivity]] of that host.

== Definitions and corresponding formulas ==

The existing definitions of the generation time fall into two
categories: those that treat the generation time as a renewal time of the
population, and those that focus on the distance between individuals of one generation and the next. Below are the three most commonly used definitions:<ref name="Coale72" />

* '''The time its takes for the population to grow by a factor of its net reproductive rate:'''
The net reproductive rate <math>\textstyle R_0</math> is
the number of offsprings an individual is expected to produce during its
lifetime (a net reproductive rate of 1 means that the population is at its
demographic equilibrium). This definition envisions the generation time as a
renewal time of the population. It justifies the very simple definition used
in [[microbiology]] ("the time it takes for the population to double"), since one
can consider that during the exponential phase of bacterial growth, the
mortality is very low and as a result a [[bacterium]] is expected to be replaced by
two bacteria in the next generation (the mother cell and the daughter cell).
If the population dynamic is exponential with a growth rate
<math>\textstyle r</math> (i.e.<math>\textstyle n(t) \propto \exp(rt)</math>,
where <math>\textstyle n(t)</math> is the size of the population at time
<math>\textstyle t</math>), then this measure of the generation time is
given by:
::<math>T = \frac{\log R_0}{r}</math>.
Indeed, <math>\textstyle T</math> is such that <math>\textstyle n(t + T) =
R_0 \, n(t)</math>, i.e. <math>\textstyle \exp(rT) = R_0</math>.

* '''The average difference in age between parents and offsprings when the population is at the stable age distribution:'''
This definition is a measure of the
distance between generations rather than a renewal time of the population.
Since many demographical models are female-based (that is, they only take
females into account), this definition is often expressed as a mother-daughter
distance (the "average age of mothers at birth of their daugthers"). However,
it is also possible to define a father-son distance (average age of fathers at
the age of their sons) or not to take sex into account at all in the
definition. In age-structured [[population model]]s, an expression is given
by:<ref name="charlesworth" />
::<math>T = \int_0^{\infty} x e^{-rx} l(x) m(x) \, \mathrm{d}x</math>,
where <math>\textstyle r</math> is the growth rate of the
population, <math>\textstyle l(x)</math> is the survivorship function
(probability that an individual survives to age <math>\textstyle x</math>) and
<math>\textstyle m(x)</math> the maternity function (or birth function, or
age-specific fertility). For matrix population models, there is a more simple
and general formula:<ref name="BienvenuLegendre" />
::<math>T = \frac{ \lambda v w }{v F w} = \frac{1}{\sum e_{\lambda} ( f_{ij} )}</math>,
where <math>\textstyle \lambda = e^r</math> is the discrete-time growth rate of
the population, <math>\textstyle F = (f_{ij})</math> is its fertility
matrix, <math>\textstyle v</math> its reproductive value (row-vector)
and <math>\textstyle w</math> its stable stage distribution (column-vector);
the <math>\textstyle e_{\lambda}(f_{ij}) = \frac{f_{ij}}{\lambda}
\frac{\partial \lambda}{\partial f_{ij}}</math> are the
[[Elasticity (economics)|elasticities]] of <math>\textstyle \lambda</math>
to the fertilities.

* '''The age at which members of a given cohort are expected to reproduce''':
This definition is very similar to the previous one but the population need not
be at its stable age distribution. Moreover, it can be computed for different
cohorts and thus provide more information about the generation time in the
population. This measure is given by:<ref name="charlesworth" />
::<math>
T = \frac{\int_{x=0}^{\infty} x l(x) m(x) }{\int_{x=0}^{\infty} l(x) m(x) }
</math>.
Indeed, the numerator is the sum of the ages at which a member of the cohort
reproduces, and the denominator is <math>\textstyle R_0</math>, the average
number of offsprings it produces.

==References==
{{Reflist|refs=
<ref name="charlesworth">{{cite book|last=Charlesworth|first=Brian|title=Evolution in Age-structured Populations|year=1994|publisher=University of Cambridge Press|location=Cambridge|isbn=0-521-45967-2|pages=28–30}}</ref>
<ref name="Coale72">{{Cite book |last=Coale |first=A.J. |title=The Growth and Structure of Human Populations |publisher=Princeton University Press |year=1972 }}</ref>
<ref name="BienvenuLegendre">{{Cite journal |last1=Bienvenu |first1=F. |last2=Demetrius |first2=L. |last3=Legendre |first3=S. |title=arXiv:1307.6692 |url=http://arxiv.org/abs/1307.6692 |year=2013 }}</ref>
}}

[[Category:Demography]]
[[Category:Population]]
[[Category:Ecology]]

Sieving coefficient - Revision history

en>Foobarnix: Repairing links to disambiguation pages - You can help! equilibrium