Autocatalytic bistability arises when a species X catalyzes its own production.
Model: ẋ = S + k₁x²/(1+x²) − k₂x
ẋ = S + k₁x²/(1+x²) − k₂x
The nullcline f(x) = k₁x²/(k₂(1+x²)) can intersect the input line S/k₂ at three points —
giving two stable fixed points (LOW, HIGH) separated by an unstable saddle.
Hysteresis: the system follows different paths as S increases vs. decreases.
This motif underlies cell differentiation, memory, and genetic switches.