Consider the general logistic growth model Pn+1 Pn = rPn ( 1 Pn /M ) where 0 < r < 3 and M > 0. While this is often a reasonable model of population growth in a restricted environment, unusual behavior can occur in some parameter regimes due to the overshooting nature of difference equation models.

(a) Determine the fixed points and their corresponding linear stabilities as a function of r and M.

(b) It is possible to determine whether a difference equation has a 2-cycles (e.g. a repeating sequence like {1, 3, 1, 3, . . .}) by computing xn+2 = f(xn+1) = f(f(xn)) = g(xn). The fixed points of g(xn) correspond to points on a 2-cycle. For the equation (19), show that there is a 2-cycle for r > 2 by computing g(Pn) = f(f(Pn)) and solving g(P) = P. (Hint: use a computer to do the algebra!)

(c) For the following parameter values, simulate a few steps forward using P0 = 5 and M = 50, then explain what you see and whether it is consistent with the results of part (a) and (b): (i) r = 1 (ii) r = 2 (iii) r = 2.25 (iv) r = 2.5 (v) r = 2.75 (vi) r = 2.9