 Quadratic activator-inhibitor model

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.