Abstract
Turing's theory of pattern formation has been used to describe the formation of self-organized periodic patterns in many biological, chemical, and physical systems. However, the use of such models is hindered by our inability to predict, in general, which pattern is obtained from a given set of model parameters. While much is known near the onset of the spatial instability, the mechanisms underlying pattern selection and dynamics away from onset are much less understood. Here, we provide physical insight into the dynamics of these systems. We find that peaks in a Turing pattern behave as point sinks, the dynamics of which is determined by the diffusive fluxes into them. As a result, peaks move toward a periodic steady-state configuration that minimizes the mass of the diffusive species. We also show that the preferred number of peaks at the final steady state is such that this mass is minimized. Our work presents mass minimization as a potential general principle for understanding pattern formation in reaction diffusion systems far from onset.
Highlights
Pattern formation occurs in a huge variety of natural and living systems [1], from chemical reactions [2,3] to living cells [4,5,6] to environmental patterns [7]
In systems described by reaction diffusion (RD) equations, the formation of spatially periodic patterns can be explained by the Turing instability, in which patterns emerge due to the presence of two or more interacting components that diffuse at different rates [8,9,10,11]
Sufficient conditions for pattern formation can be determined in the so-called Turing or linear regime, in which a spatially uniform stable steady state becomes linearly unstable to spatial perturbations in the presence of diffusion [9]
Summary
Pattern formation occurs in a huge variety of natural and living systems [1], from chemical reactions [2,3] to living cells [4,5,6] to environmental patterns [7]. While multiple peaks often form initially, consistent with the linear prediction, coarsening rapidly occurs, leaving mispositioned peaks that move slowly towards opposite quarter positions, while maintaining their shape [Fig. 1(c)] Note that this movement is only observed because of the competition instability. Since peaks do not move, the position of this final peak is determined by whichever peak of the transient state remains after coarsening, i.e., the steady-state solutions are in general not symmetric as might naively be expected by the boundary conditions We tested these conclusions by initializing the system with a single preformed peak (constructed as a translation of the non-mass-conserved steady-state solution). The mass-conserved case b = 0 with reflective boundary conditions has a continuum of single-peak stable states, whereas there is at most one unique single-peak solution for b > 0 This implies that regular positioning is not an intrinsic property of the system but rather depends on b. These results demonstrate a connection between peak movement towards the regular positioned configuration and the flow (turnover) of mass through the system
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