Relative growth rate explained

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

S is the current size, and (dS)/(dt) its growth rate, then relative growth rate isRGR = (1)/(S) (dS)/(dt) .If the RGR is constant, i.e.,(1)/(S) (dS)/(dt) = k ,a solution to this equation isS(t) = S _{0\exp(k ⋅} 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.

Calculations

In the simplest case of observations at two time points, RGR is calculated using the following equation:^[1]

RGR = {\operatorname{ ln (S _{2)} - ln (S _{1)}\over\operatorname{t} _{2} - t _{1}}} ,

where:

ln = natural logarithmt _{1} = time one (e.g. in days)t _{2} = time two (e.g. in days)S _{1} = size at time oneS _{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 S₀ bacteria doubles every twenty minutes, then at time interval

t it is given by solving the equation:S(t) = S _{0\exp(ln(2) ⋅} 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) = S _{0\exp(ln(8) ⋅} t) = S _{0} 8 ^{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.

See also

• Doubling time
• Plant growth analysis

Notes and References

1. Hoffmann . W.A. . Poorter . H. . Avoiding bias in calculations of Relative Growth Rate . Annals of Botany . 2002 . 90 . 1 . 37–42 . 10.1093/aob/mcf140. 12125771 . 4233846 .
2. Book: William L. Briggs. Lyle Cochran. Bernard Gillett. Calculus: Early Transcendentals. 24 September 2012. 2011. Pearson Education, Limited. 441. 978-0-321-57056-7.