1. The number of bighorn sheep in a population increases at a rate that is proportional to the number of sheep present (at least for awhile.) So does any population of living creatures. Other things that increase or decrease at a rate proportional to the amount present include radioactive material and money in an interest-bearing account. If the rate of change is proportional to the amount present, the change can be modeled by:

2. Rate of change is proportional to the amount present. Divide both sides by y. Integrate both sides.

3. Exponentiate both sides. When multiplying like bases, add exponents. So added exponents can be written as multiplication.

4. Exponentiate both sides. When multiplying like bases, add exponents. So added exponents can be written as multiplication. Since is a constant, let.

5. At,. This is the solution to our original initial value problem.

7. 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.

8. 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 decreases as P approaches M and is negative if P exceeds M.

9. The equation then becomes: Logistics Differential Equation When P is small, how does behave? When P approaches M, how does behave?

10. Logistics Differential Equation We can solve this differential equation to find the logistics growth model.

11. Partial Fractions Logistics Differential Equation Let P = 0 Let P = M

12. Logistics Differential Equation Now solve for P

13. Where t is time, k is the relative growth rate, and M is the carrying capacity of the system. But what is A? To find A, let t = 0. Then P = P 0, the initial population. Logistics Growth Model

14. Suppose a flu-like virus is spreading through a population of 50,000 at a rate proportional to both the number of people already infected and to the number still uninfected. If 100 people were infected yesterday and 130 are infected today: 1.Write an expression for the number of people N(t) infected after t days. 2.Determine how many will be infected a week from today. 3.Indicate when the virus will be spreading the fastest.

16. 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?

17. 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?

18. At time zero, the population is 10.

19. After 10 years, the population is 23.

20. 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 