7-5 Logistic Growth.

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Presentation transcript:

7-5 Logistic Growth

According to the model, what is the carrying capacity? 100 moose We are going to put together the two concepts we learned yesterday – partial fractions and logistic differential equations. Ex 5) In 1985 and 1987, the Michigan Department of Natural Resources airlifted 61 moose from Algonquin Park, Ontario to Marquette County in the Upper Peninsula. It was originally hoped that the population P would reach carrying capacity in about 2 years with a growth rate of According to the model, what is the carrying capacity? 100 moose b) This is what a slope field of the differential equation would look like.

(This involves quite a bit of work…stay organized!) Solve the differential equation with the initial condition P(0) = 61 and show that it conforms to the slope field. (This involves quite a bit of work…stay organized!) Partial fractions:

Ex 5c) cont… Set P = 61 at t = 0:

*Note: This algebra process is same every time The solution to is A is constant (by initial condition) M is carrying capacity k is growth constant

STAT  CALC  B: Logistic L1, L2, Store Y1 for graphing L1 L2 Ex 6) The table shows the population of Aurora, CO, for selected years between 1950 and 2003. Use logistic regression to find a logistic curve to model the data and superimpose it on a scatter plot of population against years after 1950. (Use your graphing calculator) Years after 1950 Population 11,421 20 74,974 30 158,588 40 222,103 50 275,923 53 290,418 Enter data in Lists STAT  CALC  B: Logistic L1, L2, Store Y1 for graphing L1 L2 Match!  Plot pts in scatterplot using STAT PLOT Window [–5, 60] x [–3600, 337000]

Numerator is carrying capacity = 316, 441 people Based on the regression equation, what will the Aurora population approach in the long run? Based on the regression equation, when would the population of Aurora first exceed 300,000 people? (Use the graph.) d) Write a logistic differential equation in the form Numerator is carrying capacity = 316, 441 people 59th year  2009 M Mk = –.1026 = 316440.7k k = 3.24 x 10–7 Mk

homework Pg. 373 # 4, 11, 15, 27, 30, 31, 37, 38, 39, 41, 42