and therefore the exponential rate of increase is:

r = loge(Nt+1/Nt) = logeA We will try this out on a doubling and halving. With a doubling:

and so:

Thus a halving and a doubling both provide the same exponential rate of increase, 0.693, which in the case of a halving has the sign reversed (i.e. — 0.693). It makes the point again that a rate of decrease is simply a negative rate of increase.

The finite rate of increase (i.e. the growth multiplier A) and the exponential rate of increase r must each have a unit attached to them. In our example the unit was a year, and so we can say that the population is multiplied by A per year. The exponential rate r is actually the growth multiplier of loge numbers per year. That is something of a mouthful and so we say that the population increased at an exponential rate r on a yearly basis. Note that A and r are simply different ways of presenting the same rate of change. They do not contain independent information.

Unlike the finite rate of increase, the exponential rate of increase can be changed from one unit of time to another by simple multiplication and division. If r = — 0.693 on a yearly basis then r = — 0.693/365 = — 0.0019 on a daily basis. That simplicity is not available for A.

The equations given above were simplified to embrace only one unit of time. They can be generalized to:

Nt = N0ert where N0 is population size at the beginning of the period of interest and Nt is the population size t units of time later. The average exponential rate of increase over the period is:

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