This Web Extension shows how to derive the geometric growth equations (Equations 9.1 and 9.2 in the textbook) and how to convert each equation to the other.
We’ll begin by working through an example where λ is constant and equals 2. Let Nt represent the number of individuals in the population at time t. If the population has 10 individuals at time t = 0, the starting population size can be written as
N0 = 10
With λ = 2, the population size after one time period (i.e., when t = 1) is
N1 = 2N0 = 2 × 10 = 20
The population doubles every time period, so when t = 2, the population size is
N2 = 2N1 = 2 × 20 = 40
because N1 = 2N0 and N0 = 10, N2 can also be written as
N2 = 2N1 = 2(2N0) = 22N0 = 4 × 10 = 40
Similarly, because we have just found that N2 = 22N0, when t = 3, we have N3 = 2N2 = 2(22N0) = 23N0 = 80
When t = 4, to what power should 2 be raised in the equation N4 = 2?N0 ? The answer is 4, since when t = 4, we have N4 = 2N3 = 160 = 24N0. What we see here is that the size of this population can be predicted for any time t by raising 2 to the tth power and multiplying that number by N0:
Nt = 2t N0
We can generalize from the results we found for this particular case where λ = 2. Whatever its value, λ is a multiplier that allows us to predict the size of the population in the next time period:
Nt +1 = λNt
which is Equation 9.1. Thus, as we saw when λ = 2, the population size can be predicted for any time t by raising λ to the tth power and multiplying that number by N0:
Nt = λt N0
which is Equation 9.2.
We have just used Equation 9.1 to derive Equation 9.2. We can also use Equation 9.2 to derive Equation 9.1. If we replace the “t” in Equation 9.2 by a “t+1” we have:
Nt +1 = λt +1N0 = λ [λ tN0] = λNt
Thus,
Nt +1 = λNt
which is Equation 9.1.
