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 theexponential growth rate, or the continuous growth rate.

RGR is a concept relevant in cases where the increase in astate variable over time is proportional to the value of that state variable at the beginning of a time period. In terms ofdifferential equations, if${\displaystyle S}$ is the current size, and${\displaystyle \frac{dS}{dt}}$ its growth rate, then relative growth rate is

${\displaystyle RGR=\frac{1}{S}\frac{dS}{dt}}$.

If the RGR is constant, i.e.,

${\displaystyle \frac{1}{S}\frac{dS}{dt}=k}$,

a solution to this equation is

${\displaystyle S(t)=S_0\exp (k\cdot t)}$

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 isdoubling time.

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

${\displaystyle RGR=\frac{\ln ({\mathrm{S}}_2)\text{-}\ln ({\mathrm{S}}_1)}{{\mathrm{t}}_2\text{-}{\mathrm{t}}_1}}$,

where:

${\displaystyle \ln }$ =natural logarithm

${\displaystyle t_1}$ = time one (e.g. in days)

${\displaystyle t_2}$ = time two (e.g. in days)

${\displaystyle S_1}$ = size at time one

${\displaystyle S_2}$ = 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 S₀ bacteria doubles every twenty minutes, then at time interval${\displaystyle t}$ it is given by solving the equation:

${\displaystyle S(t)=S_0\exp (\ln (2)\cdot t)=S_0{2}^t}$

where${\displaystyle t}$ 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${\displaystyle S(3)=S_0{2}^3}$. 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,

${\displaystyle S(t)=S_0\exp (\ln (8)\cdot t)=S_0{8}^t}$

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

Inplant 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 asPlant growth analysis, and is further discussed in that section.

• Doubling time
• Plant growth analysis

• Plant physiology
• Temporal rates

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