Suppose that a population develops according to the logistic equation dP/dt = 0.05P 0.0005P2 where t is

Question:

Suppose that a population develops according to the logistic equation dP/dt = 0.05P €“ 0.0005P2 where t is measured in weeks.
(a) What is the carrying capacity? What is the value of k?
(b) A direction field for this equation is shown at the right. Where are the slopes close to 0? Where are they largest? Which solutions are increasing? Which solutions are decreasing?
(c) 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? Which solutions have inflection points? At what population levels do they occur?
(d) What are the equilibrium solutions? How are the other solutions related to these solutions?