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
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]\displaystyle{ S }[/math] is the current size, and [math]\displaystyle{ \frac{dS}{dt} }[/math] its growth rate, then relative growth rate is
- [math]\displaystyle{ RGR=\frac{1}{S}\frac{dS}{dt} }[/math].
If the RGR is constant, i.e.,
- [math]\displaystyle{ \frac{1}{S}\frac{dS}{dt} = k }[/math],
a solution to this equation is
- [math]\displaystyle{ 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:[1]
- [math]\displaystyle{ RGR \ = \ {\operatorname{\ln(S_2) \ - \ \ln(S_1)}\over\operatorname{t_2 \ - \ t_1}\!} }[/math],
where:
[math]\displaystyle{ \ln }[/math] = natural logarithm
[math]\displaystyle{ t_1 }[/math] = time one (e.g. in days)
[math]\displaystyle{ t_2 }[/math] = time two (e.g. in days)
[math]\displaystyle{ S_1 }[/math] = size at time one
[math]\displaystyle{ 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.[2]
For example, if an initial population of S0 bacteria doubles every twenty minutes, then at time interval [math]\displaystyle{ t }[/math] it is given by solving the equation:
- [math]\displaystyle{ S(t) \ = \ S_0\exp(\ln(2)\cdot t) = S_0 2^t }[/math]
where [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ S(t) \ = \ S_0\exp(\ln(8)\cdot t) = S_0 8^t }[/math]
where [math]\displaystyle{ t }[/math] is measured in hours, and the relative growth rate may be expressed as [math]\displaystyle{ \ln(2) }[/math] or approximately 69% per twenty minutes, and as [math]\displaystyle{ \ln(8) }[/math] 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
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. PMID 12125771.
- ↑ 2.0 2.1 William L. Briggs; Lyle Cochran; Bernard Gillett (2011). Calculus: Early Transcendentals. Pearson Education, Limited. p. 441. ISBN 978-0-321-57056-7. https://books.google.com/books?id=_cMLQgAACAAJ. Retrieved 24 September 2012.
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