## Relationships between the variables

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: Figure 4.13 A reconstructed static life table for the modules (tillers) of a Carex bigelowii population. The densities per m2 of tillers are shown in rectangular boxes, and those of seeds in diamond-shaped boxes. Rows represent tiller types, whilst columns depict size classes of tillers. Thin-walled boxes represent dead tiller (or seed) compartments, and arrows denote pathways between size classes, death or reproduction. (After Callaghan, 1976.)

0-5 6-10 11-15 18-20 21-25 26-30 31-35 36-40 Number of leaves per tiller

0-5 6-10 11-15 18-20 21-25 26-30 31-35 36-40 Number of leaves per tiller

Figure 4.13 A reconstructed static life table for the modules (tillers) of a Carex bigelowii population. The densities per m2 of tillers are shown in rectangular boxes, and those of seeds in diamond-shaped boxes. Rows represent tiller types, whilst columns depict size classes of tillers. Thin-walled boxes represent dead tiller (or seed) compartments, and arrows denote pathways between size classes, death or reproduction. (After Callaghan, 1976.)

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 below).

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 Nt. In the present case, N0 = 10, N1 = 20, and we can say that:

use X 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 = N1R, N3 = 80 = N2R, and so on. Thus:

N3 = N1R X R = N0R X R X R = N0R3, and in general terms:

the fundamental net reproductive rate, R

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 X for the same parameter. Here we stick with R, but we sometimes

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 4.5.2, 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:

Nt = NR But we can see from Equation 4.8 that:

Therefore:

Ro = RT, or, if we take natural logarithms of both sides: ln R0 = T ln R.

. . , . . ^ , The term ln R is usually denoted r, the intrinsic rate of

. . by r, the intrinsic rate of natural increase.

natural increase

It is the rate at which the population increases in size, i.e. 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:

Summarizing 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.

The most precise way to calculate r is from the equation:

where the lx and mx values are taken from a cohort life table, and e is the base of natural logarithms. However, this is a so-called 'implicit' equation, which cannot be solved directly (only by iteration, usually on a computer), and it is an equation without any clear biological meaning. It is therefore customary to use instead an approximation to Equation 4.13, namely:

where Tc is the cohort generation time (see below). This equation shares with Equation 4.13 the advantage of making explicit the dependence of r on the reproductive output of individuals (R0) and the length of a generation (T). Equation 4.15 is a good approximation when R0 ~ 1 (i.e. population size stays approximately constant), or when there is little variation in generation length, or for some combination of these two things (May, 1976).

We can estimate r from Equation 4.15 if we know the value of the cohort generation time Tc, which is the average length of time between the birth of an individual and the birth of one of its own offspring. This, being an average, is the sum of all these birth-to-birth times, divided by the total number of offspring, i.e.: 