As a population grows, its additional members produce virile offspring, such that the more a population grows, the faster it continues to grow. This exponential growth follows a continuous compounding rate that can lead to alarming growth patterns, such as a single bacterium growing to several million bacteria within 24 hours.
Calculating Growth Rate
Before calculating a future population size, you need to calculate the growth rate by considering known sizes at two time periods. Divide the later population size by the initial size and then take the natural log of the result. The result is the "k" growth rate between the two times. Divide the growth rate by the number of time units to calculate the periodic growth rate. As an example, if a colony of bacteria grew from 2,000 to 6,000 in two hours, divide 6,000 by 2,000 and then take the natural log of the result (3) to calculate the k-value of 1.099 for the two hours. Divide by 2 to calculate the hourly k-value of 0.5495.
Estimating Future Size
Armed with the k-value, you can estimate a future population size under the assumption that the growth rate remains constant. Multiply the growth rate by the number of time periods, apply your calculator's exponential function and then multiply by the initial population size. If your calculator doesn't have an "e [to the power of x]" function, raise 2.71828 to the power of the growth rate times the time periods and then multiply by the initial population size. To continue the example, to calculate the colony size after 10 hours, raise 2.71828 to the power of 10 times 0.5495 and then multiply by the initial population size of 2,000. This estimates the population size after 10 hours as 486,943 bacteria.
Calculating Time to a Specific Size
You can reverse the population size calculation to discover when a population will reach a specific size. Divide the target population size by the initial size, take the natural log and then divide by the k-value. To continue the example, to calculate when the colony will double in size to 4,000 bacteria, take the natural log of 2 (4,000 divided by 2,000) and then multiply by 0.5495 to calculate 0.38 hours, or roughly 23 minutes.
Nothing can experience constant exponential growth forever. Resources, such as food and space, are limited and impose a maximum population size. Disease, environmental disasters and changes to predator populations can also influence populations. Furthermore, human populations are subject to behavioral changes, such as contraception usage, and social or governmental influences, such as China's recently loosened one-child policy.