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.

Apply proper techniques in managing patients with chest tubes and tracheostomies to prevent complications.

Develop short-term care plans for patients with breathing difficulties or other conditions that address achievable goals.

Understand how clinical information systems support management and decision-making processes in healthcare organizations.

Distinguish between data, information, knowledge, and wisdom in the context of healthcare.

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Definitions:

Supply Curve

A graphical representation showing the relationship between the price of a good or service and the quantity of that good or service suppliers are willing to offer for sale at various prices.

Price of Sugar

The cost at which sugar is sold or bought in the market.

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Equilibrium Price

The pricing point where the volume of a good or service provided equals the volume demanded, achieving market harmony.