n ln(n/K). Population size tends to increase more rapidly at low population sizes under the Gompertz equation than under the logistic equation.
This point is illustrated in the following graph with n[0]=1, K=100, and =0.1. Here, r for the logistic equation was chosen to have the same expected time until 50 individuals were present in the population (19 generations).
What are the equilibria of the Gompertz equation?
When is the equilibrium (with the species present) locally stable?
Challenge: A global solution to the Gompertz equation can be found by making a substitution of variables from n to y, using y=ln(n/K). First, what is n in terms of y? Second, what is dn/dt in terms of dy/dt (you will need to use the chain rule)? Make these substitutions into the Gompertz equation, simplify, and integrate the resulting equation to obtain the general solution for the population size at all future times.