Suppose that a population develops according to the logistic equation dP/dt = 0.

Question

Suppose that a population develops according to the logistic equation dP/dt = 0.03P - 0.0003P^2 where t is measured in weeks. What is the carrying capacity? M = What is the value of k? k = A direction field for this equation is shown. Where are the slopes close to 0? (Enter your answers as a comma-separated list.) P = Where are they largest? (Enter your answers as a comma-separated list.) P = Which solutions are increasing? (Enter your answer using interval notation.) P_0 elementof Which solutions are decreasing? (Enter your answer using interval notation.) P_0 elementof Use the direction field to sketch solutions for initial populations of 20, 40, 60, 80, 120, and 140. What do these solutions have in common? How do they differ? All of the solutions approach P = 100 as t increases. Also all the solutions are increasing. The solutions differ since some have an inflection point and some don't. All of the solutions approach P = 100 as t increases. Also all the solutions are decreasing. The solutions differ since some have an inflection point and some don't. None of the solutions have an inflection point. The solutions differ since for 0

Explanation / Answer

Solution(a)

dP/dt= 0.3P-0.0003P^2

logistic equation: dP/dt = kP(t)(1-(P(t)/M)

dP/dt = 0.3P(1 - P/100)

So from logistic equation

K = 0.3

And Carrying capacity M = 100

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