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:

Fertility Drugs

Medications used to enhance reproductive potential by stimulating ovulation or supporting other aspects of the reproductive process in individuals facing fertility challenges.

Pregnant

A condition where a female is carrying a developing embryo or fetus within her uterus.

Twins

Two offspring produced by the same pregnancy, which can be identical (monozygotic) or fraternal (dizygotic).

DNA

Deoxyribonucleic acid, a molecule that carries the genetic instructions used in the growth, development, functioning, and reproduction of all known living organisms and many viruses.