The Population of Two Species Is Modeled by the System $$\left\{ \begin{array} { l } \frac { d x } { d t } = 2 x - x y \\\frac { d y } { d t } = - 4 y + x y\end{array} \right.$$

question 128

Essay

The population of two species is modeled by the system of equations $$\left\{ \begin{array} { l } \frac { d x } { d t } = 2 x - x y \\\frac { d y } { d t } = - 4 y + x y\end{array} \right.$$ .(a) Find an expression for $\frac { d y } { d x }$ .(b) A possible direction field for the differential equation in part (a) is given below: Use this graph to sketch a phase portrait with each of P, Q, R, and S as an initial condition. Describe the behavior of the trajectories near the nonzero equilibrium solutions.(c) Graph x and y as function of t. What happens to the population of the two species as the time t increases without bound?

---

Definitions:

Cancer

A group of diseases characterized by the uncontrolled growth and spread of abnormal cells in the body.

Heart Attacks

A medical condition caused by the blockage of blood flow to the heart muscle, often resulting in chest pain and potentially fatal if not treated promptly.

Stroke

A medical emergency involving the interruption of blood supply to a part of the brain, causing cell damage and possibly resulting in loss of functions.

Carbohydrates

Organic substances such as sugars, starches, and fibers, which are vital energy sources in our diet.