A note on bistability of a two-gene competitive system

Positive autoregulation together with mutual competition is one of the simplest mechanisms that can produce bistability in gene-regulatory models. We study a two-gene system in which each gene activates its own expression and the two genes compete through regulatory terms with Hill exponent one. We show first that the system has at least one and at most three equilibria in the positive quadrant. Exactly two positive equilibria can occur only at a degenerate nullcline tangency; thus, in the nondegenerate case, the number of positive equilibria is one or three. If there are exactly three distinct positive equilibria, then no nondegeneracy assumption is needed: all three equilibria are automatically hyperbolic, the two outer equilibria are asymptotically stable nodes, and the middle equilibrium is a saddle. Moreover, every positive solution converges to an equilibrium. Consequently, the positive quadrant is the disjoint union of the basins of attraction of the two stable nodes and the one-dimensional stable manifold of the saddle, yielding global bistability.