Relative growth rate

RGR is a concept relevant in cases where the increase in a state variable over time is proportional to the value of that state variable at the beginning of a time period. In terms of differential equations, if $S$ is the current size, and $\backslash frac\{dS\}\{dt\}$ its growth rate, then relative growth rate is

:$RGR=\backslash frac\{1\}\{S\}\backslash frac\{dS\}\{dt\}$.

If the RGR is constant, i.e.,

:$\backslash frac\{1\}\{S\}\backslash frac\{dS\}\{dt\}\; =\; k$,

a solution to this equation is

:$S(t)\; =\; S\_0\backslash exp(k\backslash cdot\; t)$

Where:

• S(t) is the final size at time (t).
• S₀ is the initial size.
• k is the relative growth rate.

A closely related concept is doubling time.

In the simplest case of observations at two time points, RGR is calculated using the following equation:{{cite journal |last1=Hoffmann |first1=W.A. |last2=Poorter |first2=H. |title=Avoiding bias in calculations of Relative Growth Rate |journal=Annals of Botany |date=2002 |volume=90 |issue=1 |pages=37–42 |doi=10.1093/aob/mcf140|pmid=12125771 |pmc=4233846 }}

:$RGR\; \backslash \; =\; \backslash \; \{\backslash operatorname\{\backslash ln(S\_2)\; \backslash \; -\; \backslash \; \backslash ln(S\_1)\}\backslash over\backslash operatorname\{t\_2\; \backslash \; -\; \backslash \; t\_1\}\backslash !\}$,

where:

$\backslash ln$ = natural logarithm

$t\_1$ = time one (e.g. in days)

$t\_2$ = time two (e.g. in days)

$S\_1$ = size at time one

$S\_2$ = size at time two

When calculating or discussing relative growth rate, it is important to pay attention to the units of time being considered.{{cite book|author1=William L. Briggs|author2=Lyle Cochran|author3=Bernard Gillett|title=Calculus: Early Transcendentals|url=https://books.google.com/books?id=_cMLQgAACAAJ|accessdate=24 September 2012|year=2011|publisher=Pearson Education, Limited|page= 441|isbn=978-0-321-57056-7}}

For example, if an initial population of S₀ bacteria doubles every twenty minutes, then at time interval $t$ it is given by solving the equation:

:$S(t)\; \backslash \; =\; \backslash \; S\_0\backslash exp(\backslash ln(2)\backslash cdot\; t)\; =\; S\_0\; 2^t$

where $t$ is the number of twenty-minute intervals that have passed. However, we usually prefer to measure time in hours or minutes, and it is not difficult to change the units of time. For example, since 1 hour is 3 twenty-minute intervals, the population in one hour is $S(3)=S\_0\; 2^3$. The hourly growth factor is 8, which means that for every 1 at the beginning of the hour, there are 8 by the end. Indeed,

:$S(t)\; \backslash \; =\; \backslash \; S\_0\backslash exp(\backslash ln(8)\backslash cdot\; t)\; =\; S\_0\; 8^t$

where $t$ is measured in hours, and the relative growth rate may be expressed as $\backslash ln(2)$ or approximately 69% per twenty minutes, and as $\backslash ln(8)$ or approximately 208% per hour.

In plant physiology, RGR is widely used to quantify the speed of plant growth. It is part of a set of equations and conceptual models that are commonly referred to as Plant growth analysis, and is further discussed in that section.

• Doubling time
• Plant growth analysis

{{reflist}}

Category:Plant physiology

Category:Temporal rates