Differential Equations with Matlab 3rd Edition Matlab ProblemSet F Number 8:

Consider the predator-prey model dx/dt = x(4-3y) dy/dt = y(x-2)in which x >= 0 represents the population of the prey and y>= 0 represents the population of the predators.

a) Find all critical points of the system. At each criticalpoint, calculate the corresponding linear system and find theeigenvalues of the coefficient matrix; then identify the type andstability of the critical point.

b) Plot the vector field on a region small enough to distinguishthe critical points but large enough to judge the possible solutionbehaviors away from the critical points.

c) Use several initial data points (x0, y0) in the firstquadrant to draw a phase portrait for the system. Identify thedirection of increasing t on the trajectories you obtain. Use theinformation from parts (a) and (b) to choose a representativesample of initial conditions. Then combine the vector field andphase portrait on a single graph.

d) Explain from your phase portrait how the populations varyover time for initial data close to the unique critical pointinside the first quadrant. What happens for initial data far fromthis critical point?

e) Suppose the initial state of the population is given by x(0)= 1, y(0) = 1 Find the state of the population at t = 1, 2, 3, 4,5. f) Estimate to two decimal places the period of the solutioncurve that starts at (1, 1)