Axisymmetric ring solutions of the 2D Gray-Scott model and their destabilization into spots
The work addresses instability and nonlinear evolution of axially symmetric (ring-shaped) patterns in the Gray-Scott model of the reaction-diffusion type: \[ \begin{aligned} U_t & =D_U\nabla^2U - UV^2 +A(1-U),\\ V_t & =D_V\nabla^2V + UV^2 -BV,\end{aligned} \] for real fields \(U\) and \(V\) with real control parameters \(A\) and \(V\), which is known to demonstrate quite complex behavior (spontaneous spot multiplication, etc.) in two-dimensional simulations. In this work, solutions for ring-shaped patterns in the monostable sector of the model are constructed by means of an asymptotic method (starting from the one-dimensional solution in the radial direction as the zero-order approximation), and their stability is investigated against infinitesimal azimuthal perturbations, which tend to destroy the axial symmetry and split each ring into a set of localized spots. The stability problem is solved by reducing the one for the two coupled equations to an effective problem for a single nonlocal equation, which is achieved by eliminating the field \(U\). As a result, a type of the unstable eigenmode (the most unstable particular azimuthal harmonic) is found, thus making it possible to predict the number of spots into which the ring(s) are split. The predictions are confirmed by direct numerical simulations (which show the establishment of stationary patterns consisting of several layers of necklace-shaped arrays of spots). The analysis is also performed for the same model in the bistable section, where it reduces to the consideration of an instability of the Turing type.
scientific article; zbMATH DE number 2117592
scientific article