## 5. Doubling time for populations

We started this document by looking at a geometric series as a simple model for population growth. However, we showed that this is not a robust model, as it does not allow us to make valid calculations for intermediate time-steps. From there, we moved to an exponential model of the form:

Pt = P0. exp(k.t)

The values in this equation are:

• P0 is the initial population, and Pt 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').

A more general version of this equation is:

Pt2 = Pt1. exp(k.[t2-t1])

where Pt1 and Pt2 are the population sizes at times t1 and t2 respectively, and [t2-t1] is the time interval separating these two observations.

For the initial example with the geometric series, we looked at a population whose population doubled each day. In this case, the period of one day is termed the doubling time. So how is the value of the growth constant, k, related to doubling time?

If we look at the population size at two times that are separated by the doubling time, tD, we can rewrite the last equation:

t2-t1 = tD

Pt2 = Pt1. exp(k.tD)

We also know that the population will have doubled over this period (by definition), so that:

Pt2 = 2.Pt1

So the growth equation can be rewritten in terms of Pt1 only, and then re-arranged to give a relationship between k and tD:

2.Pt1 = Pt1. exp(k.tD)

exp(k.tD) = 2.Pt1/Pt1= 2

k.tD = ln(2)

The symbol 'ln' indicates the natural logarithm (that is the logarithm expressed to the base e). So it is now possible to calculate the value of k for a given doubling time, or the doubling time if you know k:

k = ln(2)/ tD

tD = ln(2)/k

In the case of a doubling time of one day, the value of k is 0.693147 d-1, whilst for doubling times of two and five days the values are 0.346573 d-1 and 0.138629 d-1 respectively. Note that the constant is written with units 'per day' – it would have a different value if we had used hours as units but kept the doubling time as one day (ie 24 h).