How to use the logistic growth model for a population
A population may grow quickly when resources are plentiful and slow down as it gets crowded. The logistic model builds that slowdown into the growth rate. Let’s use the starting population and one later count to predict what happens next.
Read the logistic equation
Here is the population, is the carrying capacity, and sets the growth rate. Below capacity, the population grows. As approaches , the factor approaches zero and growth slows. Above capacity, the model gives a decrease toward .
For a starting population with , separating and integrating the equation gives this form.
Example: Fit the model to a duck population
Suppose the population starts at , the capacity is , and after two years there are ducks. The starting count gives . Use the two-year count to find .
Keep the exact value for until the last step. Since , the five-year prediction is
The model predicts about ducks. Substituting and gives and , as required.
Check the example from the video
In the video, , , and . That gives and . After four years the model gives
Compare with exponential growth
When the population is small relative to capacity, is close to , so growth is approximately exponential. If a bacteria population grows exponentially and increases tenfold in eight hours, then . Its doubling time is hours.