Self-replication of mesa patterns in reaction–diffusion systems

Certain two-component reaction–diffusion systems on a finite interval are known to possess mesa (box-like) steady-state patterns in the singularly perturbed limit of small diffusivity for one of the two solution components. As the diffusivity DD of the second component is decreased below some critical value DcDc, with Dc=O(1)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 DcDc of a single steady-state mesa, it is shown that an internal layer forms in the centre 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 nonautonomous second-order ODE. By analysing 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 Ueyama [Y. Nishiura, D. Ueyama, A skeleton structure of self-replicating dynamics, Physica D 130 (1) (1999) 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.