Self-replication of mesa patterns in reaction–diffusion systems

Abstract

Certain two-component reaction-diffusion systems on a finite interval are known to possess mesa (box-like) steadystate patterns in the singularly perturbed limit of small diffusivity for one of the two solution components. As the diffusivity D of the second component is decreased below some critical value Dc, with Dc = O(1), the existence of a steady-state mesa pattern is lost, triggering the onset of a mesa self-replication event that ultimately leads to the creation of additional mesas. The initiation of this phenomena is studied in detail for a particular scaling limit of the Brusselator model. Near the existence threshold Dc of a single steady-state mesa, it is shown that an internal layer forms in the center of the mesa. The structure of the solution within this internal layer is shown to be governed by a certain core problem, comprised of a single non-autonomous second-order ODE. By analyzing this core problem using rigorous and formal asymptotic methods, and by using the Singular Limit Eigenvalue Problem (SLEP) method to asymptotically calculate small eigenvalues, an analytical verification of the conditions of Nishiura and Ueyema [Physica D, 130, No. 1, (1999), pp. 73-104], believed to be responsible for self-replication, is given. These conditions include: (1) The existence of a saddle-node threshold at which the steady-state mesa pattern disappears; (2) the dimple-shaped eigenfunction at the threshold, believed to be responsible for the initiation of the replication process; and (3) the stability of the mesa pattern above the existence threshold. Finally, we show that the core problem is universal in the sense that it pertains to a class of reaction-diffusion systems, including the Gierer-Meinhardt model with saturation, where mesa self-replication also occurs.

Diffuse interface surface tension models in an expanding flow

“…1. Similar phenomenon have been observed in numerous reaction-diusion systems, see for example [24], [20], [17] and references therein. In this section we perform a detailed study of the breakup phenomenon for the advective Cahn-Hilliard model in one dimension.…”

Spatially Localized Structures in Diblock Copolymer Mixtures

“…For pure diblock copolymers, on the other hand, the region between micelletype structures must be filled with a homogeneous mixture, which may or may not be unstable. This paper's results have some qualitative similarity to a variety of systems that arise in the reaction-diffusion literature [13,21,27,45]. Pattern formation in variational models similar to ours has been studied by Muratov [26].…”

“…Their structure suggests a dynamical pathway that leads to growth of peaks on either side of an existing peak, consistent with numerical obser- vation. Note this is different from the even-symmetry "dimple"-shaped eigenfunction that leads to pulse splitting in other self-replication problems [8,21].…”

“…Self-replication dynamics are an "aftereffect" of this bifurcation, whereby the remnants of the unstable manifolds at the bifurcation point form a pathway leading from N -peak states to (N + 1)-peak states. Many of these ingredients have been verified for particular systems, such as the Brusselator model [21].…”

Simple PDE Model of Spot Replication in Any Dimension

“…Many other reaction-diffusion systems have been found to exhibit self-replication behaviour. These include the ferrocyanideiodide-sulfite system ( [11]), the Belousov-Zhabotinsky reaction ( [12], [18] ), the Gierer-Meinhardt model ( [16], [9], [15]), the Bonhoffer van-der-Pol-type system ( [6], [7]) and the Brusselator [13].…”