= Equal-strength power-law mutual repression
{title2=$f(y)=\lambda(1+y)^{-n},\quad g(x)=\lambda(1+x)^{-m}$}
For positive $m,n$, this <mutual repression> system can display <bistability> for some $\lambda$ exactly when both $m>1$ and $n>1$. If either exponent is at most one, it has a unique stable equilibrium for every $\lambda$. To prove the less obvious case $m\le1<n$, put $a=x/(1+x)$ and $b=y/(1+y)$ at equilibrium. Equality of the two expressions for $\lambda$ gives $a\le b(1-b)^{n-1}$. The loop gain is then bounded by $mn b^2(1-b)^{n-1}$, whose maximum is $4mn(n-1)^{n-1}/(n+1)^{n+1}<1$. Swap the two components for the other case. If both exponents exceed one, sufficiently large $\lambda$ yields two stable equilibria with $(x,y)\sim(\lambda,\lambda^{1-m})$ and $(x,y)\sim(\lambda^{1-n},\lambda)$. The weaker condition $mn\le1$ is sufficient to exclude multistability, but does not exhaust all exclusions when the same production prefactor is imposed.
// Target: mathematical-biology.bigb
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