Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of the differential equation dP/dt = c In (K/P) P where is a constant and K is the carrying capacity.
(a) Solve this differential equation.
(b) Compute lim t→∞ P(t).
(c) Graph the Gompertz growth function for K = 1000, P0 = 100, and c = 0.05, and compare it with the logistic function in Example 3. What are the similarities?
What are the differences?
(d) We know from Exercise 9 that the logistic function grows fastest when P = K/2. Use the Gompertz differential equation to show that the Gompertz function grows fastest when P = K/e.