Question

Consider the system of equations

dx/dt = x (1-x/3-y)
dy/dt = y (1-y/4-x)
taking (x, y) ≥ 0.

Definitions: a nullcline of this system is a line on which either dx/dt = 0 or dy/dt = 0. More precisely, a vertical nullcline of this system is a line on which dx/dy = 0, and likewise, a horizontal nullcline of this system is a line on which dy/dt = 0.

Write an equation for a vertical nullcline that is not a coordinate axis: ____ (enter your equation, e.g., y=x)

Answer 1

The vertical nullcline for the given system where \(dx/dt = 0\) and is not a coordinate axis is \(y = 1 - x/3\).

Step-by-step explanation:

The question is discussing systems of differential equations and is looking for the equation of a vertical nullcline that is not a coordinate axis. A vertical nullcline is where \(\frac{dx}{dt} = 0\), so using the given system \(dx/dt = x (1-x/3-y)\), we set this equation to zero and solve for y:

• \(0 = x (1 - x/3 - y)\)
• To have a nontrivial solution (aside from x=0, which is the y-axis), we set the factor in parentheses equal to zero \(1 - x/3 - y = 0\)
• This yields \(y = 1 - x/3\)

Therefore, the equation for a vertical nullcline that is not a coordinate axis for this system of differential equations is \(y = 1 - x/3\).