Mutual repression stability criterion ($f^{\prime }(y_{\ast })g^{\prime }(x_{\ast })<1$)

At an equilibrium $x_{\ast }=f(y_{\ast })$, $y_{\ast }=g(x_{\ast })$, the Jacobian matrix has eigenvalues $-1\pm \sqrt{f^{\prime }(y_{\ast })g^{\prime }(x_{\ast })}$. Its product of repression gains is also $(\partial \log f/\partial \log y)(\partial \log g/\partial \log x)$. Product below one gives a stable node, above one a saddle, and equality the threshold for a zero eigenvalue. Robust multiple stable equilibria require an intervening unstable equilibrium; a gain exceeding one alone is not a sufficient global criterion for a particular pair of repression functions.

// Target: mathematical-biology.bigb