Answer Happy

(iii) Simulate using MATLAB Consider the system z' = 3x - 4y, y=x-y. a) Use the Matlab command [P,D]=eig(A) (or any other software) to realize that this matrix is not diagonalizable (why?). b) Use instead the command [P,J]=jordan(A) to find the associated Jordan matrix J such that A = P.JP-1. Write the general solution of the system algebraically. c) What can we say about the stability of the origin in this case? Rock-paper-scissors. In the children's hand game of rock-paper-scissors, rock beats scissors (by smashing it); scissors beats paper (by cutting it); and paper beats rock (by covering it). In a biological setting, analogs of this non-transitive competition occur among certain types of bacteria (Kirkup and Riley, 2004) and lizard (Sinervo and Lively, 1996). Consider the following idealized model for three competing species locked in a life-and-death game of rock-paper-scissors: P= P(R-S) R = R(S-P) $ = S(P-R) where P, R, and S (all positive) are the sizes of the paper, rock, and scissors populations. (a) Write a few sentences explaining the various terms in these equations. Be sure to comment on why a given term has a plus or minus sign in front of it. Also, state some of the biological assumptions being made here. (b) Show that P+R+ S does not change in time. (c) Show that PRS also does not change in time. (d) How does the system behave as t→∞o? Prove by determining the stability of the system's steady states.