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Understanding Logistic Growth Through a Grizzly Bear Population Model

This resource explores the concept of logistic growth, as demonstrated by a real-world scenario involving grizzly bears introduced to a national park. The growth of the bear population is modeled using a differential equation, illustrating proportionality to both the current population and the park's carrying capacity. By analyzing the initial population of 10 bears introduced a decade ago, we determine the timeline for reaching various population milestones (50, 75, and 100 bears). This exercise reinforces the principles of logistic growth in ecology.

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Understanding Logistic Growth Through a Grizzly Bear Population Model

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  1. Logistic Growth 10.4

  2. A question on the 2008 AP exam:

  3. A question on the 2008 AP exam:

  4. If the growth rate (or decay rate) of a population, P, is proportional to the population itself, we say : In other words, the larger the population, the faster it grows. The smaller the population, the slower it grows. Solving this differential equation results in the population growth model :

  5. 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, L. A more realistic model is the logistic growth model where growth rate is proportional to both the amount present and the carrying capacity.

  6. Logistics Differential Equation The equation then becomes: We can solve this differential equation to find the logistics growth model.

  7. Logistics Growth Model Logistics Differential Equation

  8. 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?

  9. 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?

  10. Note: The value of A can also be found algebraically by substituting P(0)=10.

  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?

  12. Bears Years We can graph this equation and use “trace” to find the solutions. y=50 at 22 years We can also solve algebraically. y=75 at 33 years y=100 at 75 years p

  13. Logistics Growth Model Solve the initial value problem. Factor out the 100. Logistics Differential Equation Find A.

  14. Match the differential equation with the corresponding slope field. The window is [-20,100]x[-10,200]. a. 1. b. 2. c. 3.

  15. Homework Page 456 #1-3,7-21 odd, 14,16

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