Consider the system of equations dx | dt = = x(3 - x - 4y) dy dt = y(1 - 3x), taking (x, y) > 0. Recall that a nullcline of this system is a line on which t dy = = 0. Likewise, a vertical nullcline of this dt dx dy system is a line on which = 0, and a horizontal nullcline of this system is a line on which = 0. dt dt (a) Write an equation for the (non-zero) vertical (x-)nullcline of this system: y=(3-X)/4 (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal (y-)nullcline: y=0 (Enter your equation, e.g., y=x.) (Note that there are also nullclines lying along the axes.) (b) What are the equilibrium points for the system? Equilibria = (0,0),(3,0), (1/3,2/3) (Enter the points as comma-separated (x,y) pairs, e.g., (1,2), (3,4).) (c) Use your nullclines to estimate trajectories in the phase plane, completing the following sentence: If we start at the initial position (1/3, 1), trajectories converge to the point (-4/9,0) (Enter the point as an (x,y) pair, e.g., (1,2).)
Consider the system of equations dx | dt = = x(3 - x - 4y) dy dt = y(1 - 3x), taking (x, y) > 0. Recall that a nullcline of this system is a line on which t dy = = 0. Likewise, a vertical nullcline of this dt dx dy system is a line on which = 0, and a horizontal nullcline of this system is a line on which = 0. dt dt (a) Write an equation for the (non-zero) vertical (x-)nullcline of this system: y=(3-X)/4 (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal (y-)nullcline: y=0 (Enter your equation, e.g., y=x.) (Note that there are also nullclines lying along the axes.) (b) What are the equilibrium points for the system? Equilibria = (0,0),(3,0), (1/3,2/3) (Enter the points as comma-separated (x,y) pairs, e.g., (1,2), (3,4).) (c) Use your nullclines to estimate trajectories in the phase plane, completing the following sentence: If we start at the initial position (1/3, 1), trajectories converge to the point (-4/9,0) (Enter the point as an (x,y) pair, e.g., (1,2).)
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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Transcribed Image Text:Consider the system of equations
dx
|
dt
=
= x(3 - x - 4y)
dy
dt
= y(1 - 3x),
taking (x, y) > 0.
Recall that a nullcline of this system is a line on which t
dy
= = 0. Likewise, a vertical nullcline of this
dt
dx
dy
system is a line on which = 0, and a horizontal nullcline of this system is a line on which = 0.
dt
dt
(a) Write an equation for the (non-zero) vertical (x-)nullcline of this system:
y=(3-X)/4
(Enter your equation, e.g., y=x.)
And for the (non-zero) horizontal (y-)nullcline:
y=0
(Enter your equation, e.g., y=x.)
(Note that there are also nullclines lying along the axes.)
(b) What are the equilibrium points for the system?
Equilibria (0,0),(3,0), (1/3,2/3)
(Enter the points as comma-separated (x,y) pairs, e.g., (1,2), (3,4).)
(c) Use your nullclines to estimate trajectories in the phase plane, completing the following sentence:
If we start at the initial position (1/3, 1), trajectories converge to the point (-4/9,0)
(Enter the point as an (x,y) pair, e.g., (1,2).)
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