Any population, P, for which we can ignore immigration, satisfies dP/dt=Birth rate “ Death rate
For organisms which need a partner for reproduction but rely on a chance encounter for meeting a mate, the birth rate is proportional to the square of the population. Thus, the population of such a type of organism satisfies a differential equation of the form dP/dt=(aP^2) “ (bP) with a,b > 0 This problem investigates the solutions to such an equation.
(a) Sketch a graph of dP/dt against P. Note when dP/dt is positive and negative. dP/dt<0 when P is in _________ dP/dt>0 when P is in _________ (Your answers may involve a and b. Give your answers as an interval or list of intervals: thus, if dP/dt is less than zero for P between 1 and 3 and P greater than 4, enter (1,3),(4,infinity).)
(b) Use this graph to sketch the shape of solution curves with various initial values: use your answers in part (a), and where dP/dt is increasing and decreasing to decide what the shape of the curves has to be.
If P(0)>b/a, what happens to P in the long run? P ->_______
If P(0)=b/a, what happens to P in the long run? P -> _________
If P(0)<b/a, what happens to P in the long run? P ->_________