Chapter 9.1, Problem 28E

GOMPERTZ GROWTH CURVE Suppose a quantity Q(t) does not exceed some number C; that is, Q(t) ≤ C for all t. Suppose further that the rate of growth of Q(t) is jointly proportional to its current size and the difference between C and the natural logarithm of its current size. What is the size of the quantity Q(t) at any time t? Show that the mathematical formulation of this problem leads to the differential equation

$\frac{dQ}{dt}=kQ\left(C-\ln Q\right)Q\left(0\right)=Q_0$

where Q₀ denotes the size of the quantity present initially. The graph of Q(t) is called the Gompertz growth curve. This model, like the ones leading to the learning curve and the logistic curve, describes restricted growth.