Natural Resource Mathematics
This assignment is due on Friday, September 17,
10:00 am. Make sure that you show clearly the reasoning you use to solve the
problems. It is also advisable to keep a photocopy of the assignment you hand
in. You are free to use any reference you wish as long as you cite your source.
However, you must define and explain all notation or concepts used that were not
covered in the lecture or a prerequisite course. Each question is worth 25 points.
Q1. Consider the Rosenzweig-MacArthur predator-prey model presented in class,
(a) What are nullclines of N and P?
(b) What are equilibria of N and P?
(c) Show the parameter condition in which an internal equilibrium is stable.
(d) Describe in words how eutrophication and harvesting in this model affect the stability of the internal equilibrium.
(e) Derive an analytical solution of dN dt when a = 0.
(f) Consider the Lotka-Volterra predator-prey model. Using the model and R, plot trajectories given parameter values r = 1, a = 2, c = 0.5, and d = 0.1 with the initial condition N(0) = 0.2 and P(0) = 0.5, from t = 0 to t = 200.
Repeat for N(0) = 0.5 and N(0) = 1.
(g) Derive an analytical solution of dP dN of the Lotka-Volterra predator-prey model.
How does the analytical solution help explain the behaviour in the graphs of (f)?
