Relative growth rate

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 ${\displaystyle S}$  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 is doubling time.

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

${\displaystyle RGR\ =\ {\operatorname {\ln(S_{2})\ -\ \ln(S_{1})} \over \operatorname {t_{2}\ -\ t_{1}} \!}}$ ,

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]

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.

• Doubling time
• Plant growth analysis