Relative growth rate

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

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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 is the current size, and its growth rate, then relative growth rate is

.

If the RGR is constant, i.e.,

,

a solution to this equation is

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

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

,

where:

= natural logarithm

= time one (e.g. in days)

= time two (e.g. in days)

= size at time one

= 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 it is given by solving the equation:

where 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 . 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,

where is measured in hours, and the relative growth rate may be expressed as or approximately 69% per twenty minutes, and as or approximately 208% per hour.[2]

RGR of plants

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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

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References

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  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.
  2. ^ 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.