= Basally forced quadratic activator-inhibitor model
{title2=$(u_*,v_*)=(1+r,(1+r)^2)$}
A quadratic <activator> can have both basal production and activation inhibited by a second species. A normalized example has reaction rates $f(u,v)=1+ru^2/v-u$ and $g(u,v)=q(u^2-v)$, with $r,q>0$ and positive concentrations. At its <spatially homogeneous equilibrium>, $v=u^2$, so $u_*=1+r$ and $v_*=(1+r)^2$. Writing $a=(r-1)/(r+1)$, its reaction <Jacobian matrix> is $J=\begin{pmatrix}a&-r/(1+r)^2\\2q(1+r)&-q\end{pmatrix}$. Thus $\operatorname{tr}J=a-q$ and $\det J=q>0$, giving linear <asymptotic stability> when $q>a$ and instability when $q<a$. Basal production makes $a$ change sign at $r=1$; a positive activation feedback needed for a conventional <Turing instability> requires $r>1$.
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