Nullclines
Nullclines are curves in the phase plane along which one component of the system has zero instantaneous rate of change.
Definitions
For
\[\frac{dx}{dt}=f(x,y),\qquad \frac{dy}{dt}=g(x,y),\]the \(x\)-nullcline is defined by
\[f(x,y)=0,\]and the \(y\)-nullcline by
\[g(x,y)=0.\]Equilibria
An equilibrium occurs at an intersection of the nullclines because both derivatives are zero there.
Qualitative use
Nullclines divide the phase plane into regions in which \(x\) and \(y\) increase or decrease. Combining these signs provides a useful picture of the vector field.
Key idea. Nullclines identify where individual components stop changing instantaneously and help organise phase-plane analysis.