Assignment Problems with Solutions | Applied Linear Algebra | MATH 214, Assignments of Mathematics

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4. (22 Points) Consider the following system of clifferential equations governing the interaction between populations of rabbits, r(t), and foxes, f(t): tr =ar—bf and ay —=-cf+dr, where a.b.candd > 0. dt dt (a) (4 pts.) [f.X = iol define matrix A such that: di a = AX equation (1) Qa —b d -c (b) (8 pts) Given that a = 4, b = 2,c = 1 and d = 2 find, and clearly state, the eigenvalues and their corresponding eigenvectors of matrix A. ae (C A] ate \,=° > Ve | A > =O 0,3 \t (c) (2 pts) Write down the solution of equation (1) using your work from part (b). x = ¢, El. + cae [2 | (dl) (4 pts) Using the grid below, construct a phase diagram for the system. | ’ I 1 Lt See £ | Pe aoey ne (e) (4 pts.) What happens, in terms of the two species, if the initial conditions are as follows? (You can refer to your graph.) i. (2pt) r= 0 and f > 0? rabbits Shy deal, foxes clie ord. ii, (2Qpt) r= land f= 1? MAO HPO © blk yecien gras