Another differential equation that is often used to model population growth is the Gompertz equation, dn dt

= αn ln 

K n



, for some positive constants α and K. Some typical experimental data, together with a two fits to the data (using the Gompertz model and the logistic model), are shown in Fig. 25.16.

a. Use separation of variables (and then a computer) to show that the solution of the Gompertz equation is n(t) = KeCe−αt

, where C = ln 

n(0)

K



.

b. Plot some typical solutions of the Gompertz equation and describe their long-term behaviour.
Pick your favourite parameters.

c. What are the scientific interpretations of K and α? What (approximately) are the values of K and α in the Gompertz equation shown in Fig. 25.16?

d. Without solving the Gompertz equation (i.e., using a qualitative analysis) show that n = K is a stable steady state of the equation.