Chapter 9.4, Problem 22E

Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of the differential equation

$\frac{dP}{dt}=c\ln \left(\frac{M}{P}\right)P$

where c is a constant and M is the carrying capacity.

(a) Solve this differential equation.

(b) Compute lim_t→∞P(t).

(c) Graph the Gompertz growth function for M = 1000, P₀ = 100, and c = 0.05, and compare it with the logistic function in Example 2. What are the similarities? What are the differences?

(d) We know from Exercise 13 that the logistic function grows fastest when P = M/2. Use the Gompertz differential equation to show that the Gompertz function grows fastest when P = M/e.