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?

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Definitions:

Inelastic Supply

A situation where the quantity supplied of a good or service does not change significantly in response to price changes.

Health Care

The organized provision of medical services to individuals or communities.

Medical Services

Professional care or treatment provided by doctors and other health professionals to maintain or improve health.

Marginal Tax Rate

The additional tax liability a person faces divided by his or her additional taxable income. It is the percentage of an extra dollar of income earned that must be paid in taxes. It is the marginal tax rate that is relevant in personal decision-making.