Note 2 – Initial growth rate

An instantaneous growth rate of slightly less than ${\displaystyle 2}$% is necessary if the average annual rate is to be ${\displaystyle 2}$%, because the instantaneous rate is being applied later in the year to a bigger population than was present at the start of the year. Thus, if ${\displaystyle k=0.02}$ when ${\displaystyle \Delta t=1}$, then ${\displaystyle k}$ shrinks a little as ${\displaystyle \Delta t}$ gets smaller, which means ${\displaystyle K}$ will be a little smaller than ${\displaystyle 0.02}$ for the same population. In fact, to have a ${\displaystyle 2}$% yearly growth, K should be approximately ${\displaystyle 0.0198\text{.}}$