Levins’ metapopulation model from Exercise 7.2.15 describes a population consisting of patches that can be either occupied or vacant. Occupied patches create more occupied patches by sending individuals to unoccupied patches. If the frequency of occupied patches is p, we would therefore expect that the rate at which new patches become occupied is proportional to both p and 1 2 p.

Patches also become unoccupied through mortality at a constant rate. The differential equation for p is

where c and m are constants.

(a) Find the solution to this initial-value problem.

(b) Under what conditions on the constants will the frequency of occupied patches go to zero as t l `?

dp dt = cp(1 p) mp = p(0) - Po