3. In class we studied the discrete dynamical system for Malthusian growth. This problem extends this growth law to include immigration or emigration of the population.

a. Suppose a population of organisms satisfies the Malthusian growth law with immigration

where n is the number of years, r = 1.15 is the annual growth rate (15% per year), and m_a = 200 is the yearly number of immigrants. Suppose the initial population A₀ = 100,000. Simulate this model for 10 generations, n = 1,..10. List the populations at n = 1, 2, 5, and 10, and graph your solution.

b. This equation would be difficult for you to solve exactly. However, with the help of MAPLE's rsolve command, we can solve this discrete dynamical system. The solution satisfies:

Verify this solution agrees with your results in Part a. at n = 5 and 10. Use this solution to determine how long it takes for the population to double., i.e., find n such that A_n = 2A₀.

c. Suppose another population of organisms satisfies the Malthusian growth law with emigration

where n is the number of years, q = 1.13 is the annual growth rate, and m_b = 200 is the yearly number leaving the region. Suppose the initial population B₀ = 200,000. Again simulate this model for 10 generations, n = 1,..10. List the populations at n = 1, 2, 5, and 10.

d. Take into account the sign change for emigration (or use MAPLE) and find the solution to the Malthusian growth model with emigration. How long does it take for this population to double?

e. Use the solutions from Parts b. and d. to graph the populations A and B on a single graph for n = 0,..,50. Find how long it takes until Population A is equal to Population B ( i.e., find the value of n when the populations are equal) and give the population at that time.