Question

(1 point) Any population, P, for which we can ignore immigration, satisfies dP Birth rate – Death rate. dt 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 аP? — ЬР with a, b > 0. dt 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. Based on your solution curves, why is P = b/a called the threshold population? %3D 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) < b/a, what happens to Pin the long run?

Answer 1

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