Front propagation and global bifurcations in a multivariable reaction-diffusion model

Edgar Knobloch; Arik Yochelis

doi:10.1063/5.0147803

Front propagation and global bifurcations in a multivariable reaction-diffusion model

Chaos. 2023 May 1;33(5):053115. doi: 10.1063/5.0147803.

Authors

Edgar Knobloch¹, Arik Yochelis^{2

3}

Affiliations

¹ Department of Physics, University of California, Berkeley, California 94720, USA.
² Swiss Institute for Dryland Environmental and Energy Research, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 8499000, Israel.
³ Department of Physics, Ben-Gurion University of the Negev, Be'er Sheva 8410501, Israel.

PMID: 37192394
DOI: 10.1063/5.0147803

Abstract

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.