= Turing threshold of a basally forced quadratic activator-inhibitor model
{c}
{title2=$k_c^2=\sqrt{q/p}$}
Add diffusivities $1,p$ to the <basally forced quadratic activator-inhibitor model>. A <Fourier mode> with $s=k^2$ has matrix $J-s\operatorname{diag}(1,p)$ and <determinant> $D(s)=ps^2+(q-pa)s+q$. When $q>a$, its <trace> is negative for every $s\geq0$. A positive-growth mode therefore occurs precisely when $D(s)<0$ for some positive $s$. The parabola has a positive minimizer only if $pa>q$, and its minimum is negative exactly when $(pa-q)^2>4pq$. Thus the ordinary <Turing instability> requires $a>0$, $q>a$ and $p>q/(\sqrt{1+a}-1)^2$. At onset, $k_c^2=(pa-q)/(2p)=\sqrt{q/p}$ and the critical <wavelength> is $2\pi(p/q)^{1/4}$. A bounded spatial domain further requires an allowed mode inside the unstable band.
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