1. 6. Section 9.4
Logistic Growth Model
Bears
Years

2. Essential Question: Why is logistic
growth model more realistic than the
exponential growth model
we have learned?

3.
What is the limiting factor?
What is it called?
Look at the graph of fish over time, describe some of its characteristics.

4. Mad Cow Disease

5. We have used the exponential growth equation
to represent population growth.
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 amount of the carrying capacity that remains:

6. The growth differential equation then becomes:
Logistics Differential Equation
M is maximum population, or carrying capacity, k is constant of proportionality, P is current population
We can solve this differential equation to find the logistics growth model.

7. Logistics Differential Equation
Partial
Fractions

8. Logistics Differential Equation

9. A is a constant determined by initial condition
Logistics Growth Model
A is a constant determined by initial condition
Do you need to memorize this?

10. Important points about Logistics
(the carrying capacity)
This is true no matter what the initial population is
The maximum rate of change will be at the point of inflection which will always be at ½ of M
If initial population is less than ½ of M, the curve is increasing, there will be 1 point of inflection, curve will be concave up until that point and concave down after it

11. Important points about Logistics
If initial population is more than ½ of M, the curve is always increasing, there is no point of inflection, and curve will always be concave down
If initial population is more than M, the curve is decreasing and always concave up

12. Example Given the equation dP/dt = 0.008P(100-P), answer the following
questions
For what values of P will the growth rate of dP/dt be close
to zero?
What is carrying capacity?
What is population when growth rate is maximized?
If initial population is 30, what does curve look like?
If initial population is 70, what does curve look like?
If initial population is 120, what does curve look like?

13. Example Given the equation dP/dt = 0.0085P(1-P/10) and P(0)=3,
answer the following questions
Find the equation for P
2. What is carrying capacity?
3. Use the answer to 1 to find P when t = 3?
4. Use the answer to 1 to find t when P = 7.

14. Example – Regression in calculator
The table shows the population of Aurora, CO for selected years between 1950 and 2003.
Years after 1950
Population
11,421 20
74,974 30
158,588 40
222,103 50
275,923 53
290,418

15. Example – Regression in calculator
Use logistic regression to find a logistic curve to model the data
Based on the regression equation, what will the Aurora population approach in the long run?
Based on the regression equation, when will the population of Aurora first exceed 300,000 people?
Write a logistic differential equation in the form dP/dt = kP(M – P) that models the growth of the Aurora data in the table.

16. Assignment
Worksheet