The following discrete-time dynamical systems describe the populations of two competing strains of bacteria

at+1 = sat

bt+1 = rbt.

For the following values of the initial conditions a0 and b0, and the capita production s and r,

a. Find the number of each type as a function of time.

b. Find the fraction of type a as a function of time.

c. Use leading behavior or L'Hopital's rule to find the limit of the fraction as t → ∞.

d. Compute the fraction at t = 0, 10, 20, and 50 and compare with your limit.

1. a0 = l04, b0 = 106, s = 2.0,r = 1.5.

2. a0 = 104, b0 = 106, s = 1.5, r = 2.0.

3. a0 = l04, b0 = 105, s = 0.8,r = 1.2.

4. a0 = 104, b0 = 105, s = 0.5, r = 0.3.