Nullclines
Nullclines are curves that help us understand two-dimensional dynamical systems without first solving the differential equations. They show where one variable temporarily stops changing.
Start with two variables
Consider
\[\frac{dx}{dt}=f(x,y),\qquad \frac{dy}{dt}=g(x,y).\]The state of the system is a point \((x,y)\) in the phase plane. As time passes, this point moves because \(x\) and \(y\) change.
The x-nullcline
The \(x\)-nullcline is found by setting
\[\boxed{f(x,y)=0}.\]At every point on this curve,
\[\frac{dx}{dt}=0.\]So at that instant there is no horizontal component of motion. The system may still move vertically because \(dy/dt\) need not be zero.
The y-nullcline
The \(y\)-nullcline is found from
\[\boxed{g(x,y)=0}.\]Along this curve,
\[\frac{dy}{dt}=0.\]There is therefore no vertical component of motion at that instant, although \(x\) may still change.
Why nullcline intersections are equilibria
At an intersection of the two nullclines,
\[f(x,y)=0\qquad\text{and}\qquad g(x,y)=0.\]Therefore
\[\frac{dx}{dt}=0,\qquad\frac{dy}{dt}=0.\]The state has no instantaneous motion in either direction. Hence the intersection is an equilibrium.
A geometric picture
Nullclines do more than locate equilibria
The curves divide the phase plane into regions. Inside each region, the signs of \(dx/dt\) and \(dy/dt\) are usually fixed.
These signs tell us the qualitative direction in which trajectories move.
Four basic directions
| Signs | Motion |
|---|---|
| \(dx/dt\gt0,\ dy/dt\gt0\) | right and upward |
| \(dx/dt\gt0,\ dy/dt\lt0\) | right and downward |
| \(dx/dt\lt0,\ dy/dt\gt0\) | left and upward |
| \(dx/dt\lt0,\ dy/dt\lt0\) | left and downward |
How to determine the signs
Choose one test point in a region and substitute it into \(f(x,y)\) and \(g(x,y)\). Their signs determine the local horizontal and vertical directions.
If the nullclines are the only places where the corresponding derivatives change sign, the same sign pattern applies throughout that connected region.
A simple worked system
Consider
\[\frac{dx}{dt}=x(3-x-y),\] \[\frac{dy}{dt}=y(2-x-y).\]This form is similar to a simple competition model, with both populations affected by total crowding.
Find the x-nullclines
Set \(dx/dt=0\):
\[x(3-x-y)=0.\]Therefore
\[\boxed{x=0\qquad\text{or}\qquad y=3-x}.\]Find the y-nullclines
Set \(dy/dt=0\):
\[y(2-x-y)=0.\]Therefore
\[\boxed{y=0\qquad\text{or}\qquad y=2-x}.\]Find the equilibria
Equilibria are simultaneous solutions. In the non-negative phase plane, the nullcline intersections give
\[(0,0),\qquad(3,0),\qquad(0,2).\]The two sloping nullclines are parallel in this example, so they do not create an interior coexistence equilibrium.
Worked nullcline diagram
Why axes often become nullclines in biology
Population equations frequently contain the population itself as a factor:
\[\frac{dx}{dt}=xF(x,y).\]Then \(x=0\) automatically gives \(dx/dt=0\). Biologically, if a population is absent and there is no immigration or spontaneous appearance, it remains absent.
Nullclines in predator–prey models
For a Lotka–Volterra predator–prey system
\[\frac{dN}{dt}=rN-aNP,\] \[\frac{dP}{dt}=eaNP-mP,\]the prey nullclines are
\[N=0\qquad\text{or}\qquad P=\frac{r}{a},\]and the predator nullclines are
\[P=0\qquad\text{or}\qquad N=\frac{m}{ea}.\]The positive nullclines are horizontal and vertical lines. Their intersection gives the positive coexistence equilibrium.
Predator–prey interpretation
Below the prey nullcline \(P=r/a\), predator abundance is low enough that prey increase. Above it, predation is strong enough that prey decrease.
To the right of the predator nullcline \(N=m/(ea)\), there is enough prey for predators to increase. To the left, predators decline.
Predator–prey nullclines and directions
What happens exactly on a nullcline?
Suppose a trajectory crosses the \(x\)-nullcline. At the crossing instant, \(dx/dt=0\), so its motion is purely vertical if \(dy/dt\ne0\).
Likewise, on the \(y\)-nullcline the instantaneous motion is horizontal if \(dx/dt\ne0\).
Crossing a nullcline can mark a turning point
When a trajectory crosses an \(x\)-nullcline, the sign of \(dx/dt\) may change. Then \(x(t)\) changes from increasing to decreasing, or vice versa.
So nullcline crossings often correspond to maxima or minima of one variable along a trajectory.
Nullclines and phase portraits
A phase portrait contains trajectories or a vector field. Nullclines provide the organising framework for that portrait.
By drawing the nullclines first and determining signs in each region, we can often anticipate the broad flow before calculating any numerical solution.
Nullclines and stability
Nullclines can suggest whether trajectories appear to move toward or away from an equilibrium, but they do not always provide a complete stability test.
For rigorous local classification in a two-dimensional nonlinear system, the Jacobian and its eigenvalues are usually used.
Nullclines versus phase lines
A phase line is used for one state variable. Nullclines are the natural extension to systems with two state variables.
In one dimension, equilibria divide a line into sign intervals. In two dimensions, nullclines divide a plane into sign regions.
Nullclines versus trajectories
A nullcline is generally not a trajectory. It is a set of points where one derivative vanishes.
A trajectory can cross a nullcline because the other variable can still be changing.
Biological feasibility
For population models, we usually interpret only
\[x\ge0,\qquad y\ge0.\]Parts of mathematically defined nullclines lying at negative population values may therefore be irrelevant biologically.
A practical workflow
For a two-variable system, set \(dx/dt=0\) and draw every resulting \(x\)-nullcline. Then set \(dy/dt=0\) and draw every \(y\)-nullcline. Find their intersections, check biological feasibility, test the signs of both derivatives in each region, add directional information, and then use the resulting structure to interpret trajectories and equilibria.