Consider the Following Predator-Prey System Where X and Y Are $\begin{array} { l } \frac { d x } { d t } = 2 x - x y \\\frac { d y } { d t } = - 4 y + x y\end{array}$

question 49

Essay

Consider the following predator-prey system where x and y are in millions of creatures and t represents time in years: $\begin{array} { l } \frac { d x } { d t } = 2 x - x y \\\frac { d y } { d t } = - 4 y + x y\end{array}$
(a) Show that (4, 2) is the nonzero equilibrium solution.
(b) Find an expression for $\frac { d y } { d x }$ .
(c) The direction field for the differential equation is given below: (i) Locate (4, 2) on the graph.(ii) Sketch a rough phase trajectory through P indicated in the graph.(d) With the aid of the phase trajectory, answer the following questions:
(i) For the region $0 < x < 4$ and 0 < y < 2, is x (t) increasing or decreasing? Is y (t) increasing or decreasing? Describe in words how the two species interact with one another.(ii) For the region x > 4 and 0 < y < 2, is x (t) increasing or decreasing? Is y (t) increasing or decreasing? Describe in words how the two species interact with one another.(iii) For the region x > 4 and y > 2, is x (t) increasing or decreasing? Is y (t) increasing or decreasing? Describe in words how the two species interact with one another.(iv) For the region 0 < x < 4 and y > 2, is x (t) increasing or decreasing? Is y (t) increasing or decreasing? Describe in words how the two species interact with one another.(e) Suggest a pair of species which might interact in the manner described by this system.

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