Last Updated on Tue, 10 May 2022 |Population Growth
Once we have the basic life table, we are in a position to calculate the same types of growth-rate statistics we discussed in the first two chapters. The first of these is the net reproductive rate, R0. This is an equivalent to the R (Eqn. 1.4) we developed for populations with discrete generations. The net reproductive rate represents the increase in the population per generation, and is defined as the mean number of female offspring produced per female in the population per generation. This value is found by incorporating both the fertility and survivorship functions of the life table. For each age class the product of I* x m* is found. This product is the contribution a particular age class is making toward population growth per generation. The net reproductive rate for the population as a whole is the sum of these products for all age classes:
In Table 4.3 the calculated net reproductive rate for the gray squirrel population is 1.119. This means that the average female squirrel replaces herself with 1.119 female squirrels per generation. As in the case of the net reproductive rate for non-overlapping populations, an R0 > 1 means that the population, according to the life table, has the potential to increase every generation. The opposite is also true: an R0 < 1 means that the population is decreasing every generation.
Although the net reproductive rate is an important statistic, we usually want to know the growth rate per year (or some other defined period). When we compare growth rates among different types of populations, the usual currency is r, the intrinsic rate of increase, or the finite rate of increase (X), since both are measured for a specific unit of time. The intrinsic rate of increase can be extracted from life history data using an equation developed by Euler, although some authors give credit to Lotka (see Mertz 1970, or Case 2000 for its derivation). It is most often known as the Euler equation; but in any event, it is considered to be a "characteristic equation" of demography (Dingle 1990).
This equation is useful because it allows us to determine the intrinsic rate of increase from the life table. However, since r is an exponent in a summation, it cannot be explicitly solved for if there are more than two age classes. Values of r must be estimated and tried in the Euler equation until a value is found that satisfies it. However, Laughlin (1965) and May (1976a) showed that there exists an excellent approximation for r. Assuming a stable age distribution, the approximation is based on the following:
If G = generation time, we can write: — = R0.
It is also true (Eqn. 1.8) that —G = er<
Therefore, we can set R0 = erG.
Taking natural logs of both sides of the equation gives us ln R0 = rG and therefore:
This tells us that the intrinsic rate of increase can be found by dividing the natural log of the increase per generation by the generation time. We now have an approximation for r, but we must calculate G, the mean generation time. Mean generation time is actually a somewhat slippery concept, and can be defined in various ways. Here we will use the definition, the mean age of the mothers at the time of their daughter's birth. This is the same definition as, "the average interval between reproductive onset in two successive generations" (Dingle 1990). Generation time is estimated according to the following equations, in which the age, x, is weighted by its realized fecundity, lxmx. In the second equation, discrete age intervals are used:
G = xlxmxdx (4.11a)
These equations, however, can only be used if the population is not growing. In order to account for the number of offspring being produced per individual female, the right side of the equation must be divided by the net reproductive rate, R0. Therefore we use Equation 4.12 in estimating G:
Once we have approximated the value of G, r can be estimated using Equation 4.10. Note that all approximations of r gained using Equation 4.10 must be verified by the Euler equation! Since Equation 4.10 simply approximates r, the value of r must be verified, or the approximation adjusted using Equation 4.9, the Euler equation. The way r is estimated and confirmed is illustrated in Example 4.1.
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