6.5 Logistic Growth Model Years Bears Greg Kelly, Hanford High School, Richland, Washington

We have used the exponential growth equation to represent population growth. The exponential growth equation occurs when the rate of growth is proportional to the amount present. If we use P to represent the population, the differential equation becomes: The constant k is called the relative growth rate.

The population growth model becomes: However, real-life populations do not increase forever. There is some limiting factor such as food, living space or waste disposal. There is a maximum population, or carrying capacity, M. A more realistic model is the logistic growth model where growth rate is proportional to both the amount present ( P ) and the fraction of the carrying capacity that remains:

The equation then becomes: Our book writes it this way: Logistics Differential Equation We can solve this differential equation to find the logistics growth model.

Partial Fractions Logistics Differential Equation

Logistics Growth ModelLogistics Differential Equation So the solution for: is:

Example: Logistic Growth Model Ten grizzly bears were introduced to a national park 10 years ago. There are 23 bears in the park at the present time. The park can support a maximum of 100 bears. Assuming a logistic growth model, when will the bear population reach 50? 75? 100?

Ten grizzly bears were introduced to a national park 10 years ago. There are 23 bears in the park at the present time. The park can support a maximum of 100 bears. Assuming a logistic growth model, when will the bear population reach 50? 75? 100?

At time zero, the population is 10.

After 10 years, the population is 23. Let’s leave k as is here…

Years Bears We can graph this equation and use “trace” to find the solutions. y=50 at 22 years y=75 at 33 years y=100 at 75 years Before graphing, use the store function on your calculator to assign We’d rather not round k when plugging into the calculator so that our graph can be as accurate as possible

Years Bears It appears too that the graph is steepest around 22 years. What does this mean in terms of y=50 at 22 years Before graphing, use the store function on your calculator to assign

To find the maximum rate of change, we need to find when the second derivative equals zero.

Which happens when P = 0 M – P = 0 which happens when P = M or when the population has reached its carrying capacity M – 2P = 0 which happens when 2P = M or when the population has reached half of its carrying capacity So in any logistic growth model, the rate of increase of a population is always a maximum when 