A nullcline is a curve on a graph where one variable in a system of differential equations stops changing. More precisely, it’s the set of all points where the rate of change of a particular variable equals zero. Nullclines are one of the most practical tools for understanding how systems of two variables behave over time, because they let you sketch the big picture of a system’s dynamics without actually solving the equations.
The Basic Idea
Most real-world systems involve two or more quantities that change simultaneously and influence each other: predator and prey populations, voltage and recovery in a neuron, concentrations of two chemicals. You can describe these systems with a pair of equations that specify how fast each variable changes at any given moment. For a system with variables x and y, you’d write something like dx/dt = f(x, y) and dy/dt = g(x, y), where f and g are functions that depend on both variables.
The x-nullcline is the set of all points where dx/dt = 0, meaning x is momentarily not changing. The y-nullcline is the set of all points where dy/dt = 0, meaning y is momentarily not changing. You find them by taking each equation, setting it equal to zero, and solving. The result is typically a curve (or a set of curves) that you can draw on the x-y plane.
What Nullclines Tell You About Flow
The real power of nullclines is what they reveal about the direction of movement at every point in the plane. On the x-nullcline, x isn’t changing, so the system can only move straight up or straight down (purely in the y-direction). On the y-nullcline, y isn’t changing, so the system can only move straight left or right (purely in the x-direction). This gives you anchor points for sketching the flow of the entire system.
Each nullcline also divides the plane into regions. On one side of the x-nullcline, x is increasing; on the other side, x is decreasing. The same goes for the y-nullcline and y. To figure out which side is which, you just pick any single test point on one side and check the sign of the derivative there. That sign holds for the entire region on that side of the nullcline. By combining the information from both nullclines, you can determine the general direction of flow in every region of the plane: up-and-right, down-and-left, up-and-left, or down-and-right.
How to Find Them Algebraically
The procedure is straightforward. Take each equation in your system, set its right-hand side to zero, and solve for the relationship between x and y. Consider this example system:
- dx/dt = 2x(1 − x²) − xy
- dy/dt = 3y(1 − y³) − 2xy
For the x-nullcline, set 2x(1 − x²) − xy = 0. Factoring out x gives x(2 − 2x² − y) = 0, which produces two x-nullclines: the y-axis (where x = 0) and the curve x + y = 2. For the y-nullcline, set 3y(1 − y³) − 2xy = 0. Factoring out y gives y(3 − 3y² − 2x) = 0, producing two y-nullclines: the x-axis (where y = 0) and the line 2x + y = 3.
You then plot all of these curves on the same plane. The regions between them tell you the qualitative behavior of the system everywhere.
Intersections Are Equilibrium Points
Where an x-nullcline and a y-nullcline cross, both dx/dt and dy/dt are zero simultaneously. That means the system isn’t changing at all at that point. These intersections are the equilibrium points (also called fixed points or steady states) of the system. Finding all nullcline intersections is one of the most reliable ways to locate every equilibrium.
The geometry of the nullclines near an intersection also hints at stability. If the nullclines cross in a way that creates flow directed toward the intersection from all surrounding regions, the equilibrium is stable (the system will settle there). If flow is directed away, it’s unstable. In some cases, the relative slopes of the two nullclines at their crossing point can determine whether a fixed point is stable, unstable, or capable of producing oscillations. When the slopes are configured so that the system transitions from stable to unstable as a parameter changes, the system undergoes a Hopf bifurcation, which produces rhythmic, oscillatory behavior.
Nullclines vs. Isoclines
A nullcline is a special case of a more general concept called an isocline. An isocline is a curve along which the derivative equals some constant k. A nullcline is simply the isocline where k = 0. If you’ve encountered isoclines in the context of slope fields for a single differential equation, nullclines are the same idea applied to systems, focused specifically on the zero-growth curves.
Ecology: Predator-Prey and Competition
Nullclines appear constantly in population biology. In the classic Lotka-Volterra predator-prey model, with prey population x and predator population y, the equations are dx/dt = ax − bxy and dy/dt = −cy + dxy. The x-nullcline (prey stops changing) occurs when x = 0 or y = a/b, which is a horizontal line. The y-nullcline (predators stop changing) occurs when y = 0 or x = c/d, which is a vertical line. These two lines carve the plane into four regions with distinct flow directions, and their intersection at (c/d, a/b) is the equilibrium where both populations coexist. The flow around this point circulates, producing the famous boom-and-bust population cycles.
In competition models, where two species compete for the same resources, the nullclines are typically downward-sloping lines. The way these lines cross (or don’t cross) determines whether one species drives the other to extinction or whether coexistence is possible. A nullcline marks the boundary between population growth and decline for each species: above its nullcline, a species’ population is shrinking because resources are insufficient, and below it, the population is growing.
Neuroscience: The FitzHugh-Nagumo Model
One of the most elegant uses of nullclines is in models of how neurons fire. The FitzHugh-Nagumo model, a simplified description of nerve impulse generation, has two variables: a voltage-like variable V and a slower recovery variable W. The V-nullcline is an N-shaped (cubic) curve, and the W-nullcline is a straight line. Their intersection is the neuron’s resting state.
When a stimulus shifts the W-nullcline so that the intersection lands on the middle branch of the N-shaped V-nullcline, the equilibrium becomes unstable and the system begins to orbit around it, producing repetitive spiking that corresponds to a neuron firing over and over. The shape of the cubic nullcline is what creates the sharp, threshold-like behavior of nerve impulses: small perturbations return to rest, but a perturbation large enough to cross the middle branch of the V-nullcline triggers a large excursion that looks like an action potential.
How Parameter Changes Shift the Picture
Nullclines depend on the parameters in your equations, so changing a parameter reshapes one or both nullclines. This can move intersection points, create new ones, or destroy existing ones. When two equilibrium points merge and vanish as a parameter is tuned, that’s a saddle-node bifurcation. When a stable equilibrium becomes unstable and spawns oscillations, that’s the Hopf bifurcation mentioned earlier. In both cases, the qualitative shift in behavior is visible as a geometric change in how the nullclines relate to each other.
This is what makes nullcline analysis so useful in practice. You don’t need to solve the differential equations (which is often impossible analytically). Instead, you sketch the nullclines, identify their intersections, determine the flow direction in each region, and read off the system’s long-term behavior directly from the geometry. Changing a parameter and re-sketching the nullclines lets you see exactly how and why the system’s behavior changes.