## 3. An exponential model for population growth

We introduced the geometric series in a 'word equation' like:

Population (tomorrow) = Daily rate of increase × Population (today)

We have then shown that we can decrease the time interval over which we evaluate the series, and gradually approached a population growth curve that does not change significantly for further decrease in interval. We can write a new word equation for this curve, which involves an exponent:

Population (tomorrow) = Population (today) × exponent (growth constant × time difference)

This gives us a way to measure the population change reliably at any time, or for any time difference. Written in a mathematical format:

P_{t} = P_{0}. exp(k.t)

The values in this equation are:

- P
_{0}is the initial population, and P_{t}is the population at time t. - The expression exp(k.t) is termed the 'exponent' of the time (t) multiplied by a growth constant (k), which has the units of time
^{-1}(that is, 'per unit time').

This can be set up using a spreadsheet, and a plot of an exponential growth model looks like this: