ChE 433: Process Modeling and Systems Theory

Continuing with the Predator-Prey model given by the following equations:

Xn+1 = a*Xn*(1-Xn) - b*Xn*Yn

Yn+1 = -c*Yn + d*Xn*Yn

it when b=1, c=0.2, and d=3.3. We would like to study the solution behavior as a changes. The attached figures show the solution bifurcation diagrams when plotted versus a for both Xn and Yn . Solutions of type B are plotted in blue, and those of type C are plotted in red. (Hint: Even when b=0, the predator could be absent). You are to perform the following steps:

1. Determine the range of a necessary for the Predator to be able to survive.

2. Explain the plots that are attached to the best of your ability.

3. Focus on the following values of a= 3.25, 3.45, 3.65, 3.88, and 3.95. [Hint: Note that for solution type C a window appears containing a period doubling cascade and based on the strange solution of type P5 (period 5)].

4. For each of the values of a, plot (i) the evolution series (Xn vs. n), (ii) the evolution series (Yn vs. n), and (iii) the state space plot of (Yn vs. Xn). [Hint: eliminate transients. i.e. do not plot the first few points in the series to get rid of the initial condition effect.]

5. Discuss each case. Make use of the state space plot to gain more understanding of each case.

6. Make plots for the case a=3.49, b=1, c=0.2, and d=3.1. The state space plot should look like a donut. This is a solution of periodicity we have not encountered yet. Search the internet for the nature of this solution and what it is regularly called.