2. This problem has a very significant role in the evolution of bacteria cultures. One finds that even a slight advantage in growth rate can significantly effect the balance of bacteria in a particular environment in a relatively short period of time. Contaminated cultures by these opportunistic cultures have more than once skewed scientific findings, so need your understanding. Consider two strains of bacteria growing in a culture according to the discrete Malthusian growth model. Suppose these cultures satisfy the equations:

Assume that the initial populations are A₀ = 20,000 and B₀ = 80,000 with the growth parameters given by r = 0.034 and s = 0.029, and n in minutes.

a. Find the general solutions for each strain A_n and B_n, then determine the doubling times for each of these strains of bacteria. Compute when the populations of A and B are equal, and find their populations at that time.

b. Let P_n be the fraction of the population that is strain A. Then

 

It can be shown that the discrete dynamical system for P_n is

 

(Bonus 7 points if you show how this is true from the expressions above.) Find the populations for A and B and the value of P for times 0, 5, 10, 20, and 60 min., then make a table of these values.

c. Find all equilibria for the discrete dynamical system for P_n. Compute the derivative for the updating function F(P_n) (write F '(P_n) in your lab report), and determine the stability of these equilibria. Discuss biologically what the equilibria mean and what is the significance of the stable equilibrium. Graph the updating function for P_n along with the identity map for the domain of P_n, including this in your report.