Consider the Predator-Prey System $\frac { d x } { d t } = 2 x - x y , \frac { d y } { d t } = - 4 y + x y$

Consider the predator-prey system $\frac { d x } { d t } = 2 x - x y , \frac { d y } { d t } = - 4 y + x y$ , where x and y are in millions of creatures and t represents time in years.(a) Find equilibrium solutions for this system.(b) Explain why it is reasonable to approximate this predator-prey system as $\frac { d x } { d t } \approx 2 x , \frac { d y } { d t } \approx - 4 y$ , if the initial conditions are x(0) = 0.001 and y(0) = 0.002.(c) Describe what this approximate system tells about the rate of change of each of the specie populations x(t) and y(t) near (0, 0).(d) Find the solution for the approximate system given in part (b).(e) Sketch x (t) and y (t) as determined in part (d) on the same coordinate plane.(f) Sketch a phase trajectory through (0.001; 0.002) for the predator-prey system. Describe in words what happens to each population of species and the interaction between them.

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