Problem

This problem focuses on using Polymath, an ordinary differential equation (ODE) solver, an...

This problem focuses on using Polymath, an ordinary differential equation (ODE) solver, and also a non-linear 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 E and on the DVD-ROM.

(a) There are initially 500 rabbits (x) and 200 foxes (y) on Farmer Oat’s property. 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:
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 tfinal = 800 days? Also plot the number of foxes versus the number of rabbits. Explain why the curves look the way they do.
Vary the parameters k1, k2, k3, and k4. Discuss which parameters can or cannot be larger than others. Write a paragraph describing what you find.
(b) Use Polymath or MATLAB to solve the following set of nonlinear algebraic equations:
with initial guesses of x = 2, y = 2. Try to become familiar with the edit keys in Polymath and MATLAB. See the DVD-ROM for instructions.