top line of this page); the file will be stored on your computer as yourmacid_A2. ipynb.

Question (total=10p)

Consider the following difference equation model:

Xt - 2x1_1 + X1-141-1- 5x1-1

(1)

yt = 2y _1 + X1-191-1 - 5y1-1

(2)

1. [2p] Find all its equilibrium points.

2. [2p] Calculate the Jacobian matrix at the equilibrium point (0,0) and at the equilibrium

point where x > 0 and y > 0.

3. [2p] Calculate the eigenvalues of each of the matrices obtained in part 2 of this question.

4. [2p] Based on these results, discuss the stability of the equilibrium point (0,0) and of the

equilibrium point where x > 0 and y > 0.

5. [2p] Develop a continuous-time model analogous to the discrete-time model considered in

this question.

Hints:

. Use the sample meshgrid code below to plot phase space trajectories for the continuous-time system

developed in part 5 of this question and to confirm your findings.

. The phase-space plot of the associated continuous-time system found in part 5 will have the same

equilibrium points and will show similar behaviour at the equilibrium point with x > 0 and y > 0,

but not at (0,0).