Criteria for the (in)stability of planar interfaces in singularly perturbed 2-component reaction–diffusion equations

Paul Carter, Arjen Doelman, Kaitlynn Lilly, Erin Obermayer, Shreyas Rao

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We consider a class of singularly perturbed 2-component reaction–diffusion equations which admit bistable traveling front solutions, manifesting as sharp, slow–fast–slow, interfaces between stable homogeneous rest states. In many example systems, such as models of desertification fronts in dryland ecosystems, such fronts can exhibit an instability by which the interface destabilizes into fingering patterns. Motivated by the appearance of such patterns, we propose two versions of a 2D stability criterion for (transversal) long wavelength perturbations along the interface of these traveling slow–fast–slow fronts. The fronts are constructed using geometric singular perturbation techniques by connecting slow orbits on two distinct normally hyperbolic slow manifolds through a heteroclinic orbit in the fast problem. The associated stability criteria are expressed in terms of the nonlinearities of the system and the slow–fast–slow structure of the fronts. We illustrate and further elaborate on the general set-up by explicitly working out the existence and transversal (in)stability of traveling fronts in a number of example systems/models. We analytically establish the instability of invading bare soil/vegetation interfaces against transversal long wavelength perturbations in several dryland ecosystem models and numerically recover fingering vegetation patterns counter-invading an invading desertification front.

Original languageEnglish (US)
Article number133596
JournalPhysica D: Nonlinear Phenomena
StatePublished - Feb 2023

Bibliographical note

Funding Information:
The authors gratefully acknowledge the anonymous referees for their comments that helped improve the manuscript, and Ehud Meron for his input and feedback, both on the analysis of ecosystem model (4.11) and on the impact of (the magnitude of) on the dynamics of (1.1) . PC was supported by the NSF through grants DMS-2204758 and DMS-2105816 . KL, EO, and SR were supported by the NSF REU program through the grant DMS-2204758 . AD acknowledges the hospitality of Arnd Scheel and the School of Mathematics during his stay at the University of Minnesota as Ordway Visiting Professor.

Publisher Copyright:
© 2022 The Author(s)


  • Ecosystem dynamics
  • Geometric singular perturbation theory
  • Interface
  • Reaction–diffusion equations
  • Sideband instability


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