Wavelength Selection for Periodic Travelling Waves: An Unsolved Problem

Periodic travelling waves (PTWs) are a common solution type of models describing spatio-temporal patterns in biology and ecology. Particularly in ecology, pattern formation is regarded as a resilience mechanism and an ecosystem's ability to change its pattern wavelength is seen as a tool to adapt to environmental change. PTW solutions of corresponding mathematical models also possess this ability and typically undergo a cascade of wavelength changes in response to a gradual change in a bifurcation parameter. Extensive analysis has been conducted to develop a predictive understanding of parameter thresholds leading to wavelength changes. By contrast, theory on what determines PTW wavelength selection during a wavelength change is currently lacking and most conjectures stem from limited observations of specific simulations, or apply to special cases only. In this unsolved problems article, we first provide a review of how linear stability analysis and Busse balloon theory are used to predict parameter values at which PTW wavelength changes occur. On the topic of wavelength selection, we review the special case of PTWs in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>λ</mml:mi></mml:math> - <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ω</mml:mi></mml:math> systems, often used to predict wavelengths of predator-prey dynamics in the wake of an invasion front. For more general systems, we highlight that the Busse balloon theory that is so successful in determining parameter values of wavelength changes is unlikely able to provide information on PTW wavelength selection. Finally, we present new numerical trends of PTW wavelength selection during PTW-to-PTW transitions that highlight that some stable wavelengths are more frequently selected than others, and that cascades of wavelength changes can also result in extinction events despite bistability of the extinction state with PTWs. We conclude with a tentative list of potential approaches to unravel a deeper understanding of this topic. Combined, we aim to stimulate new approaches to gain more insights into the unsolved problem of PTW wavelength selection during PTW-to-PTW transitions.