We want the average number of particles that are inside the small red strike circle. That’s simple proportionality. Since the particles are randomly distributed, the expected proportion of particles inside the red circle will be the same as the ratio of the area a of the red circle to the whole area A: 

Here we have used  for the expected number of new offspring at a lightning strike.

That gives us a simple equation for ΔN:

ΔN = 0.01N

This is a formula for the growth rate of the population.  Since there is one lightning strike every second, the equation says that the population grows at the average rate of 0.01N per second. That’s a 1% growth rate.

The take-home message is that the growth rate is proportional to the size.  This type of growth is called exponential.