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

question 105

Essay

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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Definitions:

Overhead Volume Variance

The difference between the budgeted and actual volume of overhead costs.

Direct Labor Hours

The total hours worked by employees directly involved in the manufacturing of products or in providing services.

Budgeted Overhead

Budgeted overhead is the estimated total cost of all overhead expenses planned for a specific period in the future.

Controllable Overhead Variance

The difference between the actual overhead costs incurred that are directly manageable by a manager and the budgeted or standard overhead costs.