Refined stability thresholds for localized spot patterns for the Brusselator model in $\mathbb{R}^2$

In the singular perturbation limit ε → 0, we analyse the linear stability of multi-spot patterns on a bounded 2-D domain, with Neumann boundary conditions, as well as periodic patterns of spots centred at the lattice points of a Bravais lattice in $\mathbb{R}^2$ , for the Brusselator reaction–diffusion model $$ \begin{equation*} v_t = \epsilon^2 \Delta v + \epsilon^2 - v + fu v^2 \,, \qquad \tau u_t = D \Delta u + \frac{1}{\epsilon^2}\left(v - u v^2\right) \,, \end{equation*} $$ where the parameters satisfy 0 < f < 1, τ > 0 and D > 0. A previous leading-order linear stability theory characterizing the onset of spot amplitude instabilities for the parameter regime D = ${\mathcal O}$ (ν −1 ), where ν = −1/log ϵ , based on a rigorous analysis of a non-local eigenvalue problem (NLEP), predicts that zero-eigenvalue crossings are degenerate. To unfold this degeneracy, the conventional leading-order-in-ν NLEP linear stability theory for spot amplitude instabilities is extended to one higher order in the logarithmic gauge ν. For a multi-spot pattern on a finite domain under a certain symmetry condition on the spot configuration, or for a periodic pattern of spots centred at the lattice points of a Bravais lattice in $\mathbb{R}^2$ , our extended NLEP theory provides explicit and improved analytical predictions for the critical value of the inhibitor diffusivity D at which a competition instability, due to a zero-eigenvalue crossing, will occur. Finally, when D is below the competition stability threshold, a different extension of conventional NLEP theory is used to determine an explicit scaling law, with anomalous dependence on ϵ , for the Hopf bifurcation threshold value of τ that characterizes temporal oscillations in the spot amplitudes.

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