Question: A shrimp population P in a bay grows according to the logistic equation
(dP/dt) = 0.8P(1 - 0.01P),
with P in tons of shrimp and t in years.
(a) Sketch a graph of dP/dt against P.
(b) What is the predicted long-term population of shrimp in the bay, given any positive initial condition? (c) If shrimp are harvested out of the bay by fishermen at a rate of 10 tons per year, what is the new differential equation showing both the natural logistic growth and the constant harvesting?
(d) Sketch a graph of dP/dt against P for the differential equation given in part (c).
(e) What are the equilibrium values for the differential equation given in part (c)?
(f) Use the graph in part (d) to determine whether the shrimp population increases or decreases from each of the following populations:
(i) P = 12;
(ii) P = 25;
(iii) P = 75.