Generation time

The existing definitions of generation time fall into two categories: those that treat 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:

The net reproductive rate $\backslash textstyle\; R\_0$ is the number of offspring an individual is expected to produce during its lifetime: $\backslash textstyle\; R\_0=1$ means demographic equilibrium. One may then define the generation time $T$ as the time it takes for the population to increase by a factor of $\backslash textstyle\; R\_0$. For example, in microbiology, a population of cells undergoing exponential growth by mitosis replaces each cell by two daughter cells, so that $\backslash textstyle\; R\_0=2$ and $T$ is the population doubling time.

If the population grows with exponential growth rate $\backslash textstyle\; r$, so the population size at time $t$ is given by

:$n(t)\; =\; \backslash alpha\; \backslash ,\; e^\{r\; t\}$,

then generation time is given by

:$T\; =\; \backslash frac\{\backslash log\; R\_0\}\{r\}$.

That is, $\backslash textstyle\; T$ is such that $n(t+T)=R\_0\backslash ,\; n(t)$, i.e. $e^\{r\; T\}=R\_0$.

This definition is a measure of the distance between generations rather than a renewal time of the population. Since many demographic 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 daughters"). However, it is also possible to define a father-son distance (average age of fathers at the birth of their sons) or not to take sex into account at all in the definition. In age-structured population models, an expression is given by:

:$T\; =\; \backslash int\_0^\{\backslash infty\}\; x\; e^\{-rx\}\; \backslash ell(x)\; m(x)\; \backslash ,\; \backslash mathrm\{d\}x$,

where $\backslash textstyle\; r$ is the growth rate of the population, $\backslash textstyle\; \backslash ell(x)$ is the survivorship function (probability that an individual survives to age $\backslash textstyle\; x$) and $\backslash textstyle\; m(x)$ the maternity function (birth function, age-specific fertility). For matrix population models, there is a general formula:

:$T\; =\; \backslash frac\{\; \backslash lambda\; \backslash mathbf\{v\; w\}\}\{\backslash mathbf\{v\; F\; w\}\}\; =\; \backslash frac\{1\}\{\backslash sum\; e\_\{\backslash lambda\}(f\_\{ij\})\}$,

where $\backslash textstyle\; \backslash lambda=e^r$ is the discrete-time growth rate of the population, $\backslash textstyle\; \backslash mathbf\{F\}=(f\_\{ij\})$ is its fertility matrix, $\backslash textstyle\; \backslash mathbf\{v\}$ its reproductive value (row-vector) and $\backslash textstyle\; \backslash mathbf\{w\}$ its stable stage distribution (column-vector); the $\backslash textstyle\; e\_\{\backslash lambda\}(f\_\{ij\})\; =\; \backslash frac\{f\_\{ij\}\}\{\backslash lambda\}\; \backslash frac\{\backslash partial\; \backslash lambda\}\{\backslash partial\; f\_\{ij\}\}$ are the elasticities of $\backslash textstyle\; \backslash lambda$ to the fertilities.

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 provides more information about the generation time in the population. This measure is given by:

:$T\; =\; \backslash frac\{\backslash int\_\{x=0\}^\{\backslash infty\}\; x\; \backslash ell(x)\; m(x)\; \backslash ,\; \backslash mathrm\{d\}x\}\{\backslash int\_\{x=0\}^\{\backslash infty\}\; \backslash ell(x)\; m(x)\; \backslash ,\; \backslash mathrm\{d\}x\}$.

Indeed, the numerator is the sum of the ages at which a member of the cohort reproduces, and the denominator is R₀, the average number of offspring it produces.

{{Reflist|refs=

{{Cite journal |last1=Bienvenu |first1=F. |last2=Legendre |first2=S. |title=A New Approach to the Generation Time in Matrix Population Models |journal=The American Naturalist |volume=185 |issue=6 |pages=834–843 |year=2015 |doi=10.1086/681104 |url=https://zenodo.org/record/49440 |pmid=25996867 |s2cid=3988634 |arxiv=1307.6692 |bibcode=2015ANat..185..834B }}

{{cite book |last=Charlesworth |first=B. |title=Evolution in Age-structured Populations |year=1994 |publisher=University of Cambridge Press |location=Cambridge |isbn=978-0-521-45967-9 |pages=28–30}}

{{Cite book |last=Coale |first=A.J. |title=The Growth and Structure of Human Populations |url=https://archive.org/details/growthstructureo0000coal |url-access=registration |publisher=Princeton University Press |year=1972 |pages=[https://archive.org/details/growthstructureo0000coal/page/18 18]–19|isbn=9780691093574 }}

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{{Time topics}}

Category:Ecology

Category:Population dynamics

Category:Time in life