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2. Consider the predator-prey system (r for rabbits and y for foxes) x' = 2x - 4xy, y = 4xy - 2y. (a) Show that for x, y > 0, E(x, y) = 2x + 2y - ln x-lny is a conserved quantity, that is, the time derivative of E(x(t), y(t)) is zero for any solution (r(t), y(t)) in the first quadrant. (b) Find the critical points and the Jacobian matrices of the system. Analyze the critical points.

2. Consider the predator-prey system (r for rabbits and y for foxes) x' = 2x - 4xy, y = 4xy - 2y. (a) Show that for x, y > 0, E(x, y) = 2x + 2y - ln x-lny is a conserved quantity, that is, the time derivative of E(x(t), y(t)) is zero for any solution (r(t), y(t)) in the first quadrant. (b) Find the critical points and the Jacobian matrices of the system. Analyze the critical points.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. Consider the predator-prey system (r for rabbits and y for foxes) x' = 2x - 4xy, y = 4xy - 2y. (a) Show that for x,y> 0, E(x, y) = 2x + 2y - ln x-lny is a conserved quantity, that is, the time derivative of E(r(t), y(t)) is zero for any solution (r(t), y(t)) in the first quadrant. (b) Find the critical points and the Jacobian matrices of the system. Analyze the critical points. Hint. Although the system is not of the form, with a conserved function E(x, y), a linear centre is also a stable nonlinear centre.
Transcribed Image Text:2. Consider the predator-prey system (r for rabbits and y for foxes) x' = 2x - 4xy, y = 4xy - 2y. (a) Show that for x,y> 0, E(x, y) = 2x + 2y - ln x-lny is a conserved quantity, that is, the time derivative of E(r(t), y(t)) is zero for any solution (r(t), y(t)) in the first quadrant. (b) Find the critical points and the Jacobian matrices of the system. Analyze the critical points. Hint. Although the system is not of the form, with a conserved function E(x, y), a linear centre is also a stable nonlinear centre.
Hint: A DE of the form " + f(x) = 0 gives the system x' = y₁ y' = -f(x). It is conservative with the energy function E(x, y) = y² + F(x) where F(x) = f(y) dy.
Transcribed Image Text:Hint: A DE of the form " + f(x) = 0 gives the system x' = y₁ y' = -f(x). It is conservative with the energy function E(x, y) = y² + F(x) where F(x) = f(y) dy.
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