## Agestructured models of population dynamics

The numerous studies of elephant demography in relation to habitat changes in eastern and southern Africa during the 1960s gave indications that population regulatory factors may operate under high densities. The model used by Richard Laws and colleagues to track changes in the North Bunyoro elephants was relatively crude and had several assumptions. For instance, calculations of mortality rates assumed a stable aged population, which may not have been true. None of the studies actually calculated the instantaneous rate of growth r of a population using appropriate demographic models. John Hanks, in col laboration with J. McIntosh, attempted to calculate the actual r and maximum possible growth rate rm using the well-known Euler equation.

They calculated the instantaneous growth rate r under different values of age-specific fecundity and mortality. The mean age of first conception in females was varied from 12 years to 22 years, while reproductive cessation was varied between 45 and 55 years in 5-year steps. Intercalving interval was changed in 1-year steps from 3 years to 9 years. In most field studies, the estimation of mortality rates poses the greatest problem because many carcasses are not discovered; further, there is underrepresentation of younger animals among the carcasses found. Hanks and McIntosh took three rates of mortality (low, medium, and high) across the age classes. Thus, juvenile (0-4 years) mortality was 5%, 10%, and 20% annually under low, medium, and high rates, respectively, while these were 1%, 1.5%, and 4% for ages 5-45 years. Rates were higher above 45 years of age.

The growth rate r calculated from various combinations of the above birth and death rates gave insights into which factors might regulate populations. Under ideal conditions, an elephant population experiencing low mortality and high reproductive rates (short intercalving interval and long reproductive life span) could grow at an intrinsic rate r of 0.047. This actually translates into an annual population growth of 4.8%. Considering that it is unlikely for a population to experience such ideal conditions for long, Hanks and McIntosh concluded that, in practice, the rm of an elephant population is more likely to be about 0.040 and the annual growth rate to be just under 4%. This result also corrected earlier estimates of "cropping rates" of 6% as the sustained yield quotas for elephant populations in Africa based on crude estimates of birth rates alone. The overall results were even more interesting from another perspective. The model clearly showed that, while intercalving interval influenced growth rates the most among various reproductive parameters, the mortality rates operating on a population were more important than reproductive rates in regulating the population. The population growth rate was especially sensitive to changes in juvenile mortality rate.

Such a sensitivity analysis is also important from a management perspective. If an elephant population is to be controlled, a mere reduction of the birth rate is usually not sufficient; it is more important to increase its death rate.

In the same year that Hanks and McIntosh published their results for simulated population growth in African elephants, a more sophisticated age-structured model using the "Leslie matrix" was used by Charles Fowler and Tim Smith to characterize the demographic conditions under which an elephant population attains equilibrium with its habitat. The Leslie matrix is an elegant method of projecting into the future an age-structured population under the influence of age-specific birth and survival rates. In the basic application of the Leslie model, the values for each element (age-specific fecundity and age-specific survivorship) of the matrix are constant with time. This is obviously unrealistic for real populations because both fecundity and mortality can be expected to vary with changes in habitat and population density. More-crowded populations could experience lower fecundity and higher mortality.

Fowler and Smith thus used a variable matrix in which fecundity and survivorship values were related to population density. Using demographic data from earlier studies in East Africa by Helmut Buechner, Irven Buss, George Petrides, Richard Laws, and their associates, they derived mathematical expressions for the relationships between elephant population density and fecundity (age at first calving, intercalving interval) and between density and survivorship (of elephants younger than 2 years old). These relationships were plugged into the basic Leslie model such that the age-specific fecundity and survivorship values of the matrix were determined by the density of the population. Survivorship values above 2 years old were assumed to be constant.

The Fowler-Smith model indicated that an elephant population would reach stability at a density of about 0.6 individuals per square kilometer. At this density, the age of sexual maturity in females is 12 years (the first calf being born 2 years later), and the intercalving interval is 4.6 years. Mortality rates during the first and second years are 5% and 3%, respectively.

When this model was applied to the estimated population structure and density data of Laws and Parker for the Murchison Falls elephants during 1945, it was seen that the population size changed very little. Fowler and Smith then increased the elephant density linearly from 0.8-1.2 individuals per square kilometer in 1945 to 4 per square kilometer in 1965. The resulting age distribution was not unlike that seen in the field during this year (fig. 7.6). Although the assumption of a linear increase in elephant density during 1945-1965 was almost certainly an error according to Laws and his associates, they accepted the contention of Fowler and Smith that such a variable matrix model incorporating density dependence was a useful predictive tool for management decisions. For this to happen, however, the relationships between density and fecundity or mortality will have to be based on much better empirical data.

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