Quadratic activator-inhibitor model ($u_t=u_{xx}+u^2/v-bu,v_t=d{v}_{xx}+u^2-v$)

For positive concentrations and positive parameters, this reaction–diffusion system has homogeneous equilibrium $(u_{\ast },v_{\ast })=(1/b,1/b^2)$ and reaction Jacobian matrix $\left(\begin{array}{cc}b& -b^2\\ 2/b& -1\end{array}\right)$. Its trace is $b-1$ and determinant is $b$, making the equilibrium reaction-stable exactly when $0<b<1$. Stable complex eigenvalues permit damped oscillations, which must be distinguished from sustained periodic oscillations.

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.

For the diffusionless quadratic activator-inhibitor model, choose the Dulac function $u^{-2}$. Then $u^{-2}f=1/v-b/u$ and $u^{-2}g=1-v/u^2$, giving divergence $(b-1)/u^2$. For $0<b<1$ it is strictly negative throughout the simply connected positive quadrant. Green's theorem rules out a periodic orbit: its boundary flux would be zero because the rescaled vector field is tangent to the orbit, while its interior divergence integral would be negative.