Math: Applications for Living XIX: Population Decline with Exponential Models

Percent change is all around us. Sometimes, the percent change is based on the nature of the quantities involved (such as all those finance formulas). Other times, the data for a situation assumes an exponential property. Within our Math: Applications for Living course, we are looking at both half-life models and regular exponential models. The example today applies these models to data about African elephants.

Although the decline of African elephants is not consistently exponential, as it responds to economic and social forces, the general nature is still exponential. The current estimates used to predict future populations use differing rates of change, depending on assumptions. The best case scenario is a 3% reduction in population per year; some recent data shows a spike in poaching activities, which would result in reductions well over 3%.

For class, however, we gave a problem like this.

The population of African elephants is declining at a rate just under 4% per year. Estimate the percent of the current population which would remain in 100 years.

One way we approached this problem is to use the half-life model. In order to estimate the half-life, we used the banker’s rule of thumb for doubling-time: 70/P. In this case, 70/4 is 17.5 years; we will round that to 18 years. The calculation becomes:

0.5^(100/18)

The result is 0.02 (approximately), or 2%; 2% of the current population would remain in 100 years.

We also used the basic exponential model; with a 4% decrease per year, the multiplier is .96.

(0.96)^100

This result is 0.017 (approximately), also rounded to 2%.

Students were uniformly surprised by this result. They keep hearing about a half-million elephants declining at 4% per year, and this does not sound serious until we calculate the percent left after a longer period of time.

In our class, we do not explore the sociology of this problem, nor the political components (though it might be fun to combine mathematics and politics into one class). Our focus is on basic mathematical concepts. The most common theme in the class is percent change. This application got their attention!

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