1. This problem examines Discrete Malthusian and Logistic growth models, which are appropriate for studying simple organisms over limited time periods. The Malthusian growth model is given by the equation:
Bn+1 = Bn + rBn = (1 + r)Bn,
where n is the time in minutes and r is the rate of growth. The Logistic growth equation is given by
Bn+1 = Bn + rBn(1 - Bn/M),
where M is the carrying capacity of the population.

a. Begin with a simulation of the Malthusian growth model starting with 1000 bacteria (or B0 = 1000). Assume that the growth rate r = 0.024/min. Write an expression for the number of bacteria at each min. Simulate this dynamical system, then create a table with the number of individuals at n = 1, 3, and 5 hr (60, 180, and 300 min). How long does it take for this population to double?

b. Next we examine a population of bacteria that satisfies the Logistic growth law. Start again with B0 = 1000 bacteria, but use a growth rate of r = 0.029/min. Assume that M = 1,000,000. Simulate this model for 300 min, then create a table with the number of individuals after n = 1, 3, and 5 hr (60, 180, and 300 min). How long does it take for these bacteria to double? (Note that in this case since you do not have a formula to find the doubling time, you will have to use your output from the simulation.)

c. On a single graph plot the populations of both bacterial cultures (Malthusian and Logistic) for n from 0 to 300. (Be sure to use lines to represent these simulations and not points, labeling which line represents which model.) Use your data to determine the first time that the population from Malthusian growth model exceeds the one growing according to the Logistic growth model.