ChE 433: Process Modeling and Systems Theory In class, we studied Robert Mays iterative population growth model, and explored the variety of solutions it generates. Here, let us focus on a similar kind of model for a different type of system. The model proposed calculates the population densities of a prey (Xn) and a predator (Yn) in successive years n. The population densities Xn+1 and Yn+1 in any given year are calculated from the population densities Xn and Yn in the previous year according to the following equations: Xn+1 = a*Xn*(1-Xn) - b*Xn*Yn Yn+1 = -c*Yn + d*Xn*Yn Where a, b, c, and d, are parameters. We would like to understand the dynamic behavior resulting from the use of this map as these parameters are varied. You are to perform the following steps (some steps need to be done in a computational environment, like Excel, or MATLAB, or other): 1. Determine the fixed point solutions. This system has 3 possible types of solutions (A: the trivial solution where both do not survive, B: a solution where The prey survives but not the predator, and C: a solution where both survive). Develop expressions for each one of these solutions. Focusing on solutions of type B (i.e. the predator does not exist), obtained when b=0, c=0, and d=0. 2. Develop a diagram showing the change in solutions as a is varied from 1 to 4. This is similar to the one in the Aris paper and also developed in class. 3. Select values of a that lead to: (i) a fixed point solution, (ii) a solution of period 2 (P2), (iii) A P4 solution, and (iv) a chaotic solution. 4. Plot the evolution series (Xn vs. n) for each of the cases selected above. 5. In part 2, we will focus on solutions of type C.