 Turing threshold of the quadratic activator-inhibitor model

A mode with squared wavenumber $q$ in the quadratic activator-inhibitor model has negative trace $b-1-(1+d)q$ when $0<b<1$. Its determinant is $D(q)=d{q}^2+(1-db)q+b$. A positive minimum location and a negative minimum require $db>1$ and $(db-1{)}^2>4db$, equivalent to $db>3+2\sqrt{2}$. At onset, the double determinant root is $q_c=(1+\sqrt{2})/d$. In a bounded domain an admissible nonzero spatial mode must additionally lie in the open negative-determinant band.