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In the previous section we saw that the life tables and fecundity schedules drawn up for species with overlapping generations are at least superficially similar to those constructed for species with discrete generations. With discrete generations, we were able to compute the basic reproductive rate (R₀) as a summary term describing the overall outcome of the patterns of survivorship and fecundity. Can a comparable summary term be computed when generations overlap?

Note immediately that previously, for species with discrete generations, R₀ described two separate population parameters. It was the number of offspring produced on average by an individual over the course of its life; but it was also the multiplication factor that converted an original population size into a new population size, one generation hence. With overlapping generations, when a cohort life table is available, the basic reproductive rate can be calculated using the same formula:

(4.4)

and it still refers to the average number of offspring produced by an individual. But further manipulations of the data are necessary before we can talk about the rate at which a population increases or decreases in size, or, for that matter, about the length of a generation. The difficulties are much greater still when only a static life table (i.e. an age structure) is available (see later).

the fundamental net reproductive rate, R

We begin by deriving a general relationship that links population size, the rate of population increase, and time – but which is not limited to measuring time in terms of generations. Imagine a population that starts with 10 individuals, and which, after successive intervals of time, rises to 20, 40, 80, 160 individuals and so on. We refer to the initial population size as N₀ (meaning the population size when no time has elapsed). The population size after one time interval is N₁, after two time intervals it is N₂, and in general after t time intervals it is N_t. In the present case, N₀ = 10, N₁ = 20, and we can say that:

(4.5)

where R, which is 2 in the present case, is known as the fundamental net reproductive rate or the fundamental net per capita rate of increase. Clearly, populations will increase when R > 1, and decrease when R < 1. (Unfortunately, the ecological literature is somewhat divided between those who use ‘R’ and those who use the symbol λ for the same parameter. Here we stick with R, but we sometimes use λ in later chapters to conform to standard usage within the topic concerned.)

R combines the birth of new individuals with the survival of existing individuals. Thus, when R = 2, each individual could give rise to two offspring but die itself, or give rise to only one offspring and remain alive: in either case, R (birth plus survival) would be 2. Note too that in the present case R remains the same over the successive intervals of time, i.e. N₂ = 40 = N₁ R, N₃ = 80 = N₂ R, and so on. Thus:

(4.6)

and in general terms:

(4.7)

and:

(4.8)

R, R₀ and T

Equations 4.7 and 4.8 link together population size, rate of increase and time; and we can now link these in turn with R₀, the basic reproductive rate, and with the generation length (defined as lasting T intervals of time). In Section 4.6.1, we saw that R₀ is the multiplication factor that converts one population size to another population size, one generation later, i.e. T time intervals later. Thus:

(4.9)

But we can see from Equation 4.8 that:

(4.10)

Therefore:

(4.11)

or, if we take natural logarithms of both sides:

(4.12)

r, the intrinsic rate of natural increase

The term ln R is usually denoted by r, the intrinsic rate of natural increase. It is the rate at which the population increases in size – the change in population size per individual per unit time. Clearly, populations will increase in size for r > 0, and decrease for r < 0; and we can note from the preceding equation that:

(4.13)

Summarising so far, we have a relationship between the average number of offspring produced by an individual in its lifetime, R₀, the increase in population size per unit time, r (= ln R), and the generation time, T. Previously, with discrete generations (see Section 4.5.2), the unit of time was a generation. It was for this reason that R₀ was the same as R.