## Modeling the dynamics of elephant populations

The dynamics of elephant populations have been modeled by several investigators. The type of model used and the demographic variables introduced into the models have depended on the aspect being investigatedâ€”population growth and regulation, productivity, viability of populations, elephant-vegetation dynamics, impacts of poaching on demography, tusk harvest or tusk inheritance, and so on.

The two major classes of models used in population dynamics are deterministic models and stochastic models. A deterministic model uses fixed values of demographic variables such as birth rates and death rates to track the fate of a population over time. Thus, an average birth rate and death rate may be applied to a population to determine its growth rate. These values could be varied, if needed, in relation to other variables, such as elephant density. A stochastic model, on the other hand, tracks the fate of individuals in the population by taking the birth and death rates to be probabilistic at the individual level. A degree of randomness or chance is introduced, with the result that the fate of a population can only be predicted with a certain probability. When a population is simulated using a computer, the outcomes are different for each run, and typically, an average of several hundred or thousand simulations is taken.

Results from deterministic and stochastic models are also different. It can be shown mathematically that population growth rates derived from a stochastic model are lower than those from a deterministic model. Actually, for a large population the differences are negligible but for smaller populations these are significant. Thus, a deterministic model of population dynamics provides, for practical purposes, an adequate representation of average trends in large populations.

These models are, however, grossly inadequate for application to small populations, for which a stochastic model is more realistic and appropriate. A deterministic model may indicate, for instance, a positive population growth rate for a small population, but a stochastic model may indicate a negative growth rate and high probability of extinction within a given period.

Stochastic models are mathematically complex, and analytical solutions are beyond the reach of most biologists (and many mathematicians). It is, however, much easier to use computer simulations to provide solutions with stochastic models. Concurrent with the spread of personal computers, there has naturally been an explosion in the use of stochastic simulation models in ecology. I first describe the application of deterministic models in elephant population dynamics before considering the more recent use of stochastic models for elephants.

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