Relative growth rate

{{Short description|Growth rate relative to size}} '''Relative growth rate''' ('''RGR''') is growth rate relative to size - that is, a rate of growth per unit time, as a proportion of its size at that moment in time. It is also called the exponential growth rate, or the continuous growth rate.

==Rationale== 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 <math>S</math> is the current size, and <math>\frac{dS}{dt}</math> its growth rate, then relative growth rate is :<math>RGR=\frac{1}{S}\frac{dS}{dt}</math>. If the RGR is constant, i.e., :<math>\frac{1}{S}\frac{dS}{dt} = k</math>, a solution to this equation is :<math>S(t) = S_0\exp(k\cdot t) </math>

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.

==Calculations==

In the simplest case of observations at two time points, RGR is calculated using the following equation:<ref>{{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 }}</ref>

:<math>RGR \ = \ {\operatorname{\ln(S_2) \ - \ \ln(S_1)}\over\operatorname{t_2 \ - \ t_1}\!}</math>,

where:

<math>\ln</math> = natural logarithm

<math>t_1</math> = time one (e.g. in days)

<math>t_2</math> = time two (e.g. in days)

<math>S_1</math> = size at time one

<math>S_2</math> = size at time two

When calculating or discussing relative growth rate, it is important to pay attention to the units of time being considered.<ref name="BriggsCochran2011">{{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}}</ref>

For example, if an initial population of S₀ bacteria doubles every twenty minutes, then at time interval <math>t</math> it is given by solving the equation: :<math>S(t) \ = \ S_0\exp(\ln(2)\cdot t) = S_0 2^t</math> where <math>t</math> 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 <math>S(3)=S_0 2^3</math>. 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, :<math>S(t) \ = \ S_0\exp(\ln(8)\cdot t) = S_0 8^t</math> where <math>t</math> is measured in hours, and the relative growth rate may be expressed as <math>\ln(2)</math> or approximately 69% per twenty minutes, and as <math>\ln(8)</math> or approximately 208% per hour.<ref name="BriggsCochran2011"/>

==RGR of plants== 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.

== See also == * Doubling time * Plant growth analysis

==References== {{reflist}}

Category:Plant physiology Category:Temporal rates