= Mutual repression stability criterion
{title2=$f'(y_*)g'(x_*)<1$}
At an equilibrium $x_*=f(y_*)$, $y_*=g(x_*)$, the <Jacobian matrix> has eigenvalues $-1\pm\sqrt{f'(y_*)g'(x_*)}$. 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
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