Transcritical Bifurcation

Two equilibria pass through each other and exchange stability — neither is created nor destroyed, they simply swap roles

Controls

Parameter μ: -0.50

μ < 0: x*=0 stable, x*=μ unstable

System: ẋ = μx − x²

Fixed points: x* = 0 and x* = μ (always exist!)

Stability (linearize):
At x*=0: f'(0) = μ → stable if μ<0, unstable if μ>0
At x*=μ: f'(μ) = −μ → stable if μ>0, unstable if μ<0

At μ=0: The two fixed points coincide at x*=0. Stability is exchanged.

Examples: Laser threshold (below: off; above: lasing), epidemic model (below R₀=1: disease-free; above: endemic), logistic growth with harvesting.