The forest population of a species of possum can be modelled by the following equation.
P(n+1) = r(P – P(n)) (P(n)/P)
Here P(n) (between 0 and P) represents the population at the start of year n, P is the maximum population the forest can sustain and r is a positive constant.
(i) Find P(1) and P(2) if r = 1, P = 625, P(0) = 125.
(ii) What happens to the population from year to year when r = 13/3, P = 130, P(0) = 40?
(iii) Show that when r = 3 and P(0) is two thirds of P, the population remains constant year to year according to the model. More generally, what is P(n)/P in terms of r for which the population is constant?
(iv) Find P(n) in terms of P(0) when r = 2 (ignore rounding to the nearest integer).
(v) Using a spreadsheet or computer program, investigate the behaviour of P(n) for different positive values of P(0) and r, describing any observations you make. The relationship above is known as a logistic map which is known to exhibit chaotic behaviour.
Question created by Chaitanya Rao, Daniel Mathews, Norman Do