nullcline

Nullcline is a curve in the phase plane of a dynamical system along which one component of the velocity is zero. For a two-dimensional system written as dx/dt = F(x,y), dy/dt = G(x,y), the x-nullcline is the set of points where F(x,y) = 0, and the y-nullcline is the set where G(x,y) = 0. On the x-nullcline, the horizontal component of the motion is zero, so the trajectory moves vertically (changes in y only). On the y-nullcline, the vertical component is zero, so motion is horizontal (changes in x only). Where both nullclines intersect, both derivatives vanish, yielding equilibria.

Nullclines are useful for qualitative analysis and phase portrait construction. The slope of trajectories satisfies dy/dx = G(x,y)/F(x,y) wherever F ≠ 0, so nullclines indicate where the slope is vertical or horizontal (dy/dx infinite or zero). The plane is partitioned into regions by the nullclines, and the signs of F and G in each region determine the direction of motion. Equilibria are found at the intersections of the x- and y-nullclines.

Example: in the Lotka–Volterra predator-prey model dx/dt = x(a − by), dy/dt = −y(c − dx) with positive parameters a,b,c,d, the x-nullcline consists of x = 0 and y = a/b, while the y-nullcline consists of y = 0 and x = c/d. Intersections yield equilibria at (0,0) and (c/d, a/b).

In higher dimensions, nullclines generalize to sets where a particular component of the vector field is zero, still helping locate equilibria and sketch the qualitative dynamics.