1 pt Any population P for which we can ignore immigration sa

(1 pt) 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 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 (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. Based on your solution curves, why is P = b/a called the threshold population? If P(0) > b/a , what happens to P in the long run? If P(0) = b/a , what happens to P in the long run? If P(0)

Solution

a) dP/dt <0 if aP²-bP<0

Or P(aP-b) <0

If P>0 and P<b/a or P<0 and P>b/a

i.e. (0<P<b/a) Or b/a<P<0

------------------------------------------------------------

dP/dt >0 if P>0 or P<b/a

-----------------------------------------------------

-----------------------------------------------------------------------------