Relative growth rate

Relative growth rate (RGR) is growth rate relative to size. 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 formula_1 is the current size, and formula_2 its growth rate, then relative growth rate is
If the relative growth rate is constant, i.e.,
a solution to this equation is
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:
where:
formula_7 = natural logarithm
formula_8 = time one (e.g. in days)
formula_9 = time two (e.g. in days)
formula_10 = size at time one
formula_11 = size at time two
When calculating or discussing relative growth rate, it is important to pay attention to the units of time being considered.
For example, if an initial population of formula_12bacteria doubles every twenty minutes, then at time interval formula_13 it is given by the equation
where formula_13 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 formula_16. 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 formula_18 is measured in hours, and the relative growth rate may be expressed as formula_19 or approximately 69% per twenty minutes, and as formula_20 or approximately 208% per hour.
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. 