2. Biology 354 (Ecology and Evolution) uses the discrete logistic growth model

Pn+1 = f(Pn) = Pn + rPn(1 - Pn/M).

This problem explores some of the complications that can arise as the parameter r varies.

a. Let M = 5,000. The first step in studying this model is to find all equilibria (where the population stays the same). Determine the equilibria by solving

Pe = f(Pe) for Pe.

b. Let r = 1.89 with P0 = 2,000. Simulate the discrete dynamical system for 50 generations. Make a table listing the population for every fifth generation ( P0, P5, P10, ..., P50 ). Graph the solution of the dynamical system and write a brief description of what you observe in your solution.

c. Repeat the process in Part b. with r = 2.1 and r = 2.62. (You can make a separate table for these simulations or simply add these to your table in Part b. with appropriate labeling.) Don't forget to write a description of these solutions and how they compare to each other and your solution in Part b. What behavior do you observe for the solution in relation to the larger of the two equilibria?

d. Bonus: Find a parameter value that gives you an oscillation with period 3. This always implies that the dynamical system has gone through chaos. Show your simulation that gives the period 3 oscillation.