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A nullcline is a point where a vector field changes direction . The name means zero (null) slope (cline).
Nullcline in Differential Equations
In an autonomous system of two differential equations
They can be defined as [2, 3]:
- x-nullcline: The set of points in the phase plane where dx/dt = 0. Geometrically, the vectors at these points are vertical (straight up and down). You can find the x-nullclines by solving f(x, y) = 0.
- y-nullcline: The set of points in the phase plane where dy/dt = 0. Geometrically, the vectors at these points are horizontal (moving to the left or right). You can find the x-nullclines by solving g(x, y) = 0.
If you have a different variable in your differential equations, the nullcline is called by that variable. For example, if you have dR/dt and dL/dt, then the curve where dR/dt = 0 is called the R null cline and the curve where dL/dt = 0 is called the L null cline.
Nullcline and Sketching Phase Planes
Nullclines are very useful for analyzing and sketching phase planes. The x-nullcline divides the phase plane into two regions where x moves in opposite directions. Similarly, the y-nullcline divides the phase plane into two areas where y either increases or decreases. Together, the x- and y-nullclines divide the phase plane into four regions :
- x and y increase
- x and y decrease
- x increases and y decreases
- y increases and x decreases.
In the 2D plane, every point where the nullclines intersect is an equilibrium point.
Example question: Find the x- and y- nullclines and the equilibrium point(s) for the following system of differential equations and sketch the result:
The nullclines happen when x′ = 0 or y′ = 0. Therefore, solve both equations for zero to get the nullclines:
The following graph shows the solution:
The vectors along the x-nullcline are vertical and along the y-nullcline they are horizontal; when a nullcline passes through the equilibrium point (-1, 1), the direction is reversed.
Although the nullclines are represented by lines in this example, they could technically be any curve.
 Izhikevich, E. (2007). Dynamical Systems in Neuroscience. MIT Press.
 Duke U. (2000). Systems of Differential Equations: Models of Species Interaction. Retrieved July 22, 2021 from: https://services.math.duke.edu/education/postcalc/predprey/pred3b.html
 A quick guide to sketching phase planes. Retrieved July 22, 2021 from: https://mcb.berkeley.edu/courses/mcb137/exercises/Nullclines.pdf
Stephanie Glen. "Nullcline: Definition, Examples & Graph" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/nullcline-definition-examples-graph/
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