Delayed loss of stability of periodic travelling waves: Insights from the analysis of essential spectra

• Published in: Journal of theoretical biology Vol. 595; p. 111945
• Main Authors: Eigentler, Lukas, Sensi, Mattia
• Format: Journal Article
• Language: English
• Published: England Elsevier Ltd 07.12.2024
• Subjects: Delayed loss of stability; Models, Biological; Periodic travelling waves; Periodicity; Plane waves; Time Factors; Vegetation patterns; Wavetrains; Plane waves; Periodic travelling waves; Wavetrains; Delayed loss of stability; Vegetation patterns
• ISSN: 0022-5193; 1095-8541; 1095-8541
• DOI: 10.1016/j.jtbi.2024.111945

Periodic travelling waves (PTW) are a common solution type of partial differential equations. Such models exhibit multistability of PTWs, typically visualised through the Busse balloon, and parameter changes typically lead to a cascade of wavelength changes through the Busse balloon. In the past, the stability boundaries of the Busse balloon have been used to predict such wavelength changes. Here, motivated by anecdotal evidence from previous work, we provide compelling evidence that the Busse balloon provides insufficient information to predict wavelength changes due to a delayed loss of stability phenomenon. Using two different reaction–advection–diffusion systems, we relate the delay that occurs between the crossing of a stability boundary in the Busse balloon and the occurrence of a wavelength change to features of the essential spectrum of the destabilised PTW. This leads to a predictive framework that can estimate the order of magnitude of such a time delay, which provides a novel “early warning sign” for pattern destabilisation. We illustrate the implementation of the predictive framework to predict under what conditions a wavelength change of a PTW occurs. •Wavelength changes of periodic travelling waves after destabilisation are subject to a delay.•Busse balloons provide insufficient information to predict wavelength changes.•We can predict the order of magnitude of delays through essential spectra.•Our predictor applies to a wide class of models admitting periodic travelling waves.