This problem focuses on using Polymath, an ordinary differential equation (ODE) solver, and also a nonlinear equation (NLE) solver. These equation solvers will be used extensively in later chapters. Information on how to obtain and load the Polymath Software is given in Appendix D and on the CRE Web site.
a. Professor Sven Köttlov has a son-in-law, Štěpán Dolež, who has a farm near Riça, Jofostan where there are initially 500 rabbits (x) and 200 foxes (y). Use Polymath or MATLAB to plot the concentration of foxes and rabbits as a function of time for a period of up to 500 days. The predator–prey relationships are given by the following set of coupled ordinary differential equations:
dxdt=k1x−k2x⋅y
dxdt=k3x⋅y−k4y

Constant for growth of rabbits k₁ = 0.02 day–1
Constant for death of rabbits k₂ = 0.00004/(day × no. of foxes)
Constant for growth of foxes after eating rabbits k3 = 0.0004/(day × no. of rabbits)
Constant for death of foxes k₄ = 0.04 day–1
What do your results look like for the case of k₃ = 0.00004/(day × no. of rabbits) and t_final = 800 days? Also, plot the number of foxes versus the number of rabbits. Explain why the curves look the way they do. Polymath Tutorial (https://www.youtube.com/watch?v=nyJmt6cTiL4)
b. Using Wolfram and/or Python in the Chapter 1 LEP on the Web site, what parameters would you change to convert the foxes versus rabbits plot from an oval to a circle? Suggest reasons that could cause this shape change to occur.
c. We will now consider the situation in which the rabbits contracted a deadly virus also called rabbit measles (measlii). The death rate is rDeath = kDx with kD = 0.005 day–1. Now plot the fox and rabbit concentrations as a function of time and also plot the foxes versus rabbits. Describe, if possible, the minimum growth rate at which the death rate does not contribute to the net decrease in the total rabbit population.
d. Use Polymath or MATLAB to solve the following set of nonlinear algebraic equations
x³y – 4y² + 3x = 1
6y² – 9xy = 5
with initial guesses of x = 2, y = 2. Try to become familiar with the edit keys in Polymath and MATLAB. See the CRE Web site for instructions. You will need to know how to use this solver in later chapters involving CSTRs.