Relative growth rate

Relative growth rate (RGR) is growth rate relative to size. 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 $S$ is the current size, and ${\frac {dS}{dt}}$ its growth rate, then relative growth rate is

{\frac {1}{S}}{\frac {dS}{dt}}

.

If the relative growth rate is constant, i.e.,

{\frac {1}{S}}{\frac {dS}{dt}}=k

,

a solution to this equation is

S_{t}=\exp ^{(kt)}

.

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:^[1]

RGR\ =\ {\frac {(\ln S_{2}-\ln S_{1})}{(t_{2}-t_{1})}}

,

where:

$\ln$ = 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.^[2]

For example, if an initial population of $S_{0}$ bacteria doubles every twenty minutes, then at time interval $T$ it is given by the equation

S_{T}\ =\ S_{0}2^{T}=S_{0}\exp ^{(\ln(2)\cdot 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}=S_{0}8^{t}=S_{0}\exp ^{(\ln(8)\cdot T)}

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

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.

References

^ Hoffmann, W.A.; Poorter, H. (2002). "Avoiding bias in calculations of Relative Growth Rate". Annals of Botany. 90 (1): 37–42. doi:10.1093/aob/mcf140. PMC 4233846. PMID 12125771.
^ ^a ^b William L. Briggs; Lyle Cochran; Bernard Gillett (2011). Calculus: Early Transcendentals. Pearson Education, Limited. p. 441. ISBN 978-0-321-57056-7. Retrieved 24 September 2012.

[1] Hoffmann, W.A.; Poorter, H. (2002). "Avoiding bias in calculations of Relative Growth Rate". Annals of Botany. 90 (1): 37–42. doi:10.1093/aob/mcf140. PMC 4233846. PMID 12125771.

[BriggsCochran2011-2] William L. Briggs; Lyle Cochran; Bernard Gillett (2011). Calculus: Early Transcendentals. Pearson Education, Limited. p. 441. ISBN 978-0-321-57056-7. Retrieved 24 September 2012.

[1]

[2]

Relative growth rate

Rationale

Calculations

RGR of plants

See also

References