1. HW: p. 369 #23 – 26 (all) #31, 38, 41, 42 Pick up one

2. Logistic growth is slowed by population- limiting factors M = Carrying capacity is the maximum population size that an environment can support Exponential growth is unlimited growth.

3. 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.

4. 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:

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

6. Partial Fractions Logistics Differential Equation

8. Logistics Growth ModelLogistics Differential Equation Logistics Growth Model

9. Logistic Growth Model Years Bears

10. 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?

11. 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? (round answers to nearest year)

12. At time zero, the population is 10.

13. After 10 years, the population is 23.

14. Years Bears We can graph this equation and use “intersect” to find the solutions. y=50 at 22 years y=75 at 33 years y=100 at 75 years

15. Years Bears What is the equation for the RATE of this logistic growth? When does the rate of population growth start to decrease? When the second derivative is equal to zero AND changes from positive to negative

16. When does the rate of population growth start to decrease? 50 Years Bears y=50 at 22 years Y1 Y2 Y3 50 Slopefield program Positive on (0, 100)