Unit 7 Test Tuesday Feb 11th

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Presentation transcript:

Unit 7 Test Tuesday Feb 11th AP #2 Friday Feb 7th Computer Lab (room 253) Monday Feb 10th HW: p. 357 #23-26, 31, 38, 41, 42

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

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: Logistic Differential Equation We can solve this differential equation to find the logistic growth model.

Logistic Differential Equation Partial Fractions

Logistic Differential Equation

Logistic Growth Model

Logistic Growth Model Bears Years

Logistic Growth Model Example: 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 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.

We can graph this equation and use “trace” to find the solutions. 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

Logistic Growth diff eq  solution

If you are told Logistic Growth you can go directly from diff eq to Carrying Capacity Population “Room to grow” constant rate time