Many populations -- human, plant, and bacteria -- grow exponentially, at least at first. In time, these populations start to lose their resources (space, food, and so on).
Here's an example:
"Whale Numbers up 12% a Year" was a headline in a 1993 Australian newspaper. A 13-year study had found that the humpback whale population off the coast of Australia was increasing significantly. The actual data suggested the increase was closer to 14 percent!
When the study began in 1981, the humpback whale population was 350. Suppose the population has been increasing by about 14 percent each year since then. To find an increase of 14 percent, you could do either of the following (P stands for population):
Pthis year = Plast year + (0.14) * Plast year
Pthis year = (1.14) * Plast year
Each of these is a recursive rule. The first rule says that to know this year's population, start with last year's population (which is Plast year), then add the population growth. Since the growth is 14 percent of last year's population,
add (0.14) * Plast year.
The second rule is used more frequently because it's easier to calculate -- it incorporates the adding in one calculation. This equation shows that the second computation is equivalent to the first:
x + (0.14) * x = (1.14) * x
The second computation fits the format of an exponential function, because successive outputs have the same ratio (in this situation, the ratio is 1.14).