some Thoughts on R₀ and r.

Generally, a population grows as (Krebs, p.140):

where: e is a constant (approx. 2.71828, the base of natural logarithms)

N is population density

t is time (0 at the start, and t some time-period “t” later)

r is the intrinsic rate of population increase under prevailing conditions

If you wish to know how quickly a population grows per generation, use G (generation time):

(this is given as “T” instead of “G” in Krebs’ notation)

Take the natural log of this equation, rearrange it, and you get:

The value in the brackets is essentially the multiplier by which population size has changed in one generation, or the relative change in number of breeders per generation. We also call this term the population net reproductive rate, and symbolize it as R₀.

[This value can also be derived from life-table data; see downloadable Demography document.]

This gives the approximate value only for r, since it assumes non-overlapping generations.

The theoretically ideal value for r – a population’s theoretic intrinsic maximum capacity for growth – can never be realized. An exponentially growing population may approach the theoretical maximum, or r_max, early in its growth, but never be actually “at” it. A population will exhibit a realized situation-specific value of r, called r_a or r_obs (“actual”, or “observed”, r), which varies from a little less than r_max, through positive values (increasing population), and zero (stable population size), to negative values (declining population size). See Box 11.1, page 162, in Krebs for a different explanation of this point. The units of r are individuals/individual/time, and it’s an instantaneous rate, saying how the population is changing “right now”.

If you want a finite rate (a prediction of what will happen a finite period into the future), use the calculation of l (lambda):

Krebs offers an example of this difference in usage at the end of Box 10.2, p.145.