A population P(t) has births occuring at a rate proportional to P2, and a deaths occuring at a rate proportional to P. What will happen to the population?
This problem is described on pages 74-76 of Edwards & Penney (1996).
The population grows according to the differential equation
dP/dt = k P2 - h P = k P[P - a]where a = h/k < 0.
If the initial population is P0 = a then dP/dt = 0 and the population does not change.
If P0 > a the population increases without bound. The following animation shows the fate of a population with a = 20 and P0 = 21.
If P0 < a the population decreases to zero.
