# Population Growth Formula

## Formula

### Summary

Expression Description
The population at time .
The initial population, sometimes modeled as so the formula returns the percentage of growth.
Euler’s number is shorthand for the exponential function where
The relative growth rate constant. If , the population is exponentially growing, if the population stays the same, and if the population is decaying or decreasing.
The time elapsed.

## Usage

The formula for population growth, shown below, is a straightforward application of the function. Given an initial population size and a growth rate constant , the formula returns the population size after some time has elapsed.

Note, as mentioned above, this formula does not explicitly have to use the exponential function. For example, the same exponential growth curve can be defined in the form or as another exponential expression with a different base. However, by using the exponential function, the formula inherits a bunch of useful properties that make performing calculus a lot easier.

### Growth Rate Constant

In practice, the growth rate constant is calculated from data. For example, say we have two population size measurements and taken at time and . It’s tempting to say that the growth rate is , since the population doubled in unit of time, however this linear way of thinking is a trap. Instead, let’s solve the formula for and calculate the growth rate constant.

Given the growth constant, the exponential growth curve is now fitted to our original data points as shown in the figure below.

More abstractly, the growth rate constant changes how fast the population grows.

### Formula Variations

Other useful variations of this formula are: 1) The logistic growth formula which models bounded population growth. 2) Given an initial population and growth rate, how long does it take for the population to double? 2) How to calculate the half-life of a population in decay.