Another example of exponential growth is the growth of an animal population.

Given two animals (male and female), we know how frequently they will reproduce on average, and how many offspring will be produced. These numbers are not precise, but with large numbers the errors average out. If the animals reproduce an average of three offspring per year, and on average two die per year, the end result is as if the number of animals grows by 50% annually.

Of course, the animals do not all reproduce at the same time. The process is more like continuous compounding. In the example, the appropriate model is continuous compounding with an APY of 50%.

This model is more accurate with shorter breeding periods. When studying microscopic creatures, that reproduce within hours, reasonable predictions can be made of the population growth over periods of shorter than a day. For insects,  a few days is often long enough for an accurate model. With humans, we need decades or even centuries. The “continuous compounding” model of a human population is used only for predicting the population movement in large cities, states or whole countries, because population fluctuations, caused by economic factors, the availability of highways, and so on, interfere with the model.

Sample Problem 1.1 A fish population doubles every year. At present it is 10,000. Approximately when will it reach 100,000? When will it reach 1,000,000?

Solution. After n years, the total population is 10,000×2n, so the questions are,  “when is 2n = 10?” and “when is 2ⁿ = 100?”

Now 2³ = 8,2⁴ = 16,2⁶ = 64,2⁷ = 128, so the answers are

100,000: during the 4th year;

1,000,000: during the 7th year.