the transcritical bifurcation
/ tranz-KRIT-i-kal /
Sometimes a fixed point is forced to exist no matter what — most naturally the 'zero' state, like a population of zero or a switched-off device, which is always an equilibrium because nothing breeds from nothing. As you turn a knob, a second fixed point comes sweeping in, the two cross paths, and at the crossing they swap their stability: the formerly stable one becomes unstable and vice versa, but neither is created or destroyed. That trading of stability as two persistent fixed points pass through each other is the transcritical bifurcation.
Its normal form is x' = r x - x^2 = x(r - x). There are always exactly two fixed points, x = 0 and x = r, and they exist on both sides of the transition — unlike the saddle-node, nothing vanishes. Checking stability with f'(x) = r - 2x: at x = 0 the slope is r, and at x = r the slope is -r. So for r < 0 the origin is stable and x = r is unstable; for r > 0 they have switched, the origin now unstable and x = r stable. At r = 0 the two collide and exchange roles. In the bifurcation diagram this is two straight lines crossing transversally, with stability swapping across the intersection.
The transcritical bifurcation is the canonical picture of a threshold for invasion or extinction. The 'zero' state being always present is exactly the situation in which a new species, a disease, or a laser mode either persists or dies out depending on whether a control parameter exceeds a critical value: below threshold the zero state is the stable destiny (extinction), above threshold the zero state goes unstable and a positive state takes over (establishment). The basic reproduction number crossing 1 in an epidemic model is precisely a transcritical bifurcation.
In an epidemic, let the infected fraction obey roughly x' = (R0 - 1) x - x^2 near zero, where R0 is the basic reproduction number. The disease-free state x = 0 is stable when R0 < 1 (outbreaks die out) and unstable when R0 > 1 (the disease establishes). R0 = 1 is the transcritical bifurcation — the epidemic threshold.
The always-present zero state trades stability with a positive state as R0 crosses 1 — the mathematical face of an epidemic threshold.
Unlike the saddle-node, no fixed point is born or destroyed here — both exist throughout, they only swap stability. The transcritical form depends on the zero state being structurally fixed; a generic perturbation that breaks that constraint turns it into a pair of saddle-nodes.