Refined stability thresholds for localized spot patterns for the Brusselator model in R

In the singular perturbation limit ǫ → 0, we analyze the linear stability of multi-spot patterns on a bounded 2-D domain, with Neumann boundary conditions, as well as periodic patterns of spots centered at the lattice points of a Bravais lattice in R, for the Brusselator reaction-diffusion model vt = ǫ ∆v + ǫ − v + fuv , τut = D∆u+ 1 ǫ ( v − uv ) , where the parameters satisfy 0 < f < 1, τ > 0, andD > 0. A previous leading-order linear stability theory characterizing the onset of spot amplitude instabilities for the parameter regime D = O(ν), where ν = −1/ log ε, based on a rigorous analysis of a nonlocal 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 centered at the lattice points of a Bravais lattice in R, 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. Our higher-order analysis also provides a detailed characterization of the spectrum of the linearization of the spot pattern within the small ball |λ| = O(ν) ≪ 1 of the spectral plane whenever D is sufficiently close to this competition stability threshold. For the finite-domain problem the second term in the asymptotic expansion of this critical value of D is shown to depend on the matrix spectrum of the Neumann Green’s matrix. For the periodic spot problem, this second term is shown to depend on the regular part of the Bloch Green’s function for the Laplacian. 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.