## 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:

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').

A more general version of this equation is:

P_{t2} = P_{t1}. exp(k.[t2-t1])

where P_{t1} and P_{t2} 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,

*, related to doubling time?*

**k**If we look at the population size at two times that are separated by the doubling time, t_{D}, we can rewrite the last equation:

t2-t1 = t_{D}

P_{t2} = P_{t1}. exp(k.tD)

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

_{Pt2} = 2.P_{t1}

So the growth equation can be rewritten in terms of P_{t1} only, and then re-arranged to give a relationship between k and t_{D}:

2.P_{t1} = P_{t1}. exp(k.t_{D})

exp(k.t_{D}) = 2.P_{t1}/P_{t1}= 2

k.t_{D} = 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)/ t_{D}

t_{D} = 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).