This is essentially Checkpoint Project 2 Part 2, which related Chapters 5 and 6, but with a different set of equations --- and, for extra credit, a stochastic parameter, to bring in material from Chapter 7. Consider the following population model. dx/dt = a;x(1 x/K) pxy dyldt = apy + 3fxy 1. (by hand) Identify the variables and parameters. Explain each piece of these equations: Which expressions account for exponential growth? Logistic growth? Coupling between species? Which variable represents the prey? Which variable represents the predator? What does each parameter represent? (See Section 6.1 of Buzby and Lee, equations 6.1 and 6.2.) Choose a particular species for x and y, e.g. rats and owls. 2. (by hand) Find the Jacobian matrix. (See Section 6.5 of Buzby and Lee.) 3. (by hand) Find the equilibria. (These will be in terms of ay, a,, K, and f. You do not need to plug in the equilibria to the Jacobian yet.) Written as a system of difference equations, this model becomes Xp+] = Xp + alxn(l _ X,,/K) - xnyn Yo+l = n T a2y + Bxnyn 4. Write code that simulates this model. (Modify your code from Example 5.5.4.) Play with the parameters and initial conditions. You can start with: a=.2,00=-2,K=10, x5 =.5,y =.2,and g = .04555. (This is approximately the value of g that leads to perfectly oscillatory behavior, at least for these choices of ay, a;. and K; make it slightly smaller for stable behavior, or slightly larger for unstable behavior.) Graph the populations of the two species against time. Also graph a phase plane diagram. Settle on parameters and initial conditions that feel reasonable. 5. (by hand, or in MATLAB) Evaluate your expressions for the equilibria using the parameter values you chose. Classify the equilibria by finding eigenvalues of the Jacobian. (See Theorem 6.5.1 of Buzby and Lee.) 6. What are the limitations of this model? How could it be improved