Calculator
A nullcline is where one time derivative becomes zero. In planar systems, x-nullclines satisfy dx/dt = 0 and y-nullclines satisfy dy/dt = 0. Their intersections are equilibrium points.
Formula used
For a planar dynamical system: dx/dt = f(x,y) and dy/dt = g(x,y). The x-nullcline is where f(x,y)=0. The y-nullcline is where g(x,y)=0. Intersections satisfy both equations and form equilibrium points.
- Linear: solve two lines; a nonzero determinant gives one intersection.
- Lotka–Volterra: nullclines factor into axes and constant lines.
- FitzHugh–Nagumo: intersection reduces to a cubic in v.
- Van der Pol: one nullcline is y=0; the other is rational in x.
- Rosenzweig–MacArthur: predator nullcline includes y=0 and often a vertical line.
How to use this calculator
- Select a model that matches your system.
- Enter parameters and keep units consistent.
- Set an x-range and step for sampling.
- Press Calculate to show results above the form.
- Export using the CSV or PDF buttons.
Example data table
These examples illustrate typical parameter choices and expected nullclines.
| Model | Example parameters | Expected nullclines | Example equilibrium |
|---|---|---|---|
| Lotka–Volterra | α=1.2, β=0.6, δ=0.4, γ=0.8 | y=0, x=0, y=α/β=2, x=γ/δ=2 | (2, 2) |
| Van der Pol | μ=2 | y=0 and y=x/(μ(1−x²)) | (0, 0) |
| Linear | a=1, b=1, c=−1; d=−1, e=2, f=−0.5 | ax+by+c=0 and dx+ey+f=0 | Computed by intersection |