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We study the existence and stability of propagating fronts in Meinhardt’s multivariable reaction-diffusion model of branching in one spatial dimension. We identify a saddle-node-infinite-period bifurcation of fronts that leads to episodic front propagation in the parameter region below propagation failure and show that this state is stable. Stable constant speed fronts exist only above this parameter value. We use numerical continuation to show that propagation failure is a consequence of the presence of a T-point corresponding to the formation of a heteroclinic cycle in a spatial dynamics description. Additional T-points are identified that are responsible for a large multiplicity of different unstable traveling front-peak states. The results indicate that multivariable models may support new types of behavior that are absent from typical two-variable models but may nevertheless be important in developmental processes such as branching and somitogenesis.
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Instability mechanisms of repelling peak solutions in a multi-variable activator–inhibitor system
We study the linear stability properties of spatially localized single- and multi-peak states generated in a subcritical Turing bifurcation in the Meinhardt model of branching. In one spatial dimension, these states are organized in a foliated snaking structure owing to peak–peak repulsion but are shown to be all linearly unstable, with the number of unstable modes increasing with the number of peaks present. Despite this, in two spatial dimensions, direct numerical simulations reveal the presence of stable single- and multi-spot states whose properties depend on the repulsion from nearby spots as well as the shape of the domain and the boundary conditions imposed thereon. Front propagation is shown to trigger the growth of new spots while destabilizing others. The results indicate that multi-variable models may support new types of behavior that are absent from typical two-variable models.
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- Award ID(s):
- 1908891
- PAR ID:
- 10464782
- Date Published:
- Journal Name:
- Chaos: An Interdisciplinary Journal of Nonlinear Science
- Volume:
- 32
- Issue:
- 12
- ISSN:
- 1054-1500
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0125535
