There are many important classes of species whose birthrate is not proportional to the population size N(t). Suppose, for example, that each member of the population requires a partner for reproduction and that each member relies on chance encounters to meet a mate. If the expected number of encounters is proportional to the product of the numbers of males and females, and if these are equally distributed in the population, then the number of encounters, and hence the birthrate, is proportional to N(t)2. The death rate is still proportional to N(t). Consequently, the population size N(t) satisfies the differential equation:
dN/dt = bN² - aN,
where a and b are positive constants.
(a) Solve for N(t) given N(0).
(b) Find the long term behavior by taking t→[infinity] in your solution in (a). Do this for the case N(0) < a/b.