In the paradigm we considered the problem dx/dt = 4x - 2xy dy/dt = xy - 3y This

Question

In the paradigm we considered the problem dx/dt = 4x - 2xy dy/dt = xy - 3y This is an example of a "predator-prey" population model, x represents the population of a prey species that naturally grows exponentially but is controlled by predation by the predator species (hence the -2xy term). y represents the population of a predator species that needs to find sufficient prey to survive (hence the -2xy term). As observed in the text, this system has an equilibrium point at (3,2), but linearization failed to classify it. However, using the fact that dy/dx = dy/dt/dx/dt we can rewrite this system as a first-order equation in x and y. Solve this equation. Note that this tells you how x and y are related, but doesn't tell you how they evolve with respect to t.

Explanation / Answer

dx/dt = 4x - 2xy

dy/dt = xy - 3y

dy/dx = [dy/dt]/[dx/dt]

dy/dx = [xy-3y]/[4x-2xy]

[4x-2xy]dy - [xy-3y]dx = 0

integrating,

4xy - xy2 - x2y/2 + 3xy = C

7xy - xy2 - x2y/2 = C

at (3,2)

42 - 12 - 9 = C

C = 21

then equation becomes,

7xy - xy2 - x2y/2 = 21

14xy - 2xy2 - x2y = 42

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