# Warning signs for wave speed transitions of noisy Fisher–KPP invasion fronts

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## Abstract

Invasion waves are a fundamental building block of theoretical ecology. In this study, we aim to take the first steps to link propagation failure and fast acceleration of traveling waves to critical transitions (or tipping points). The approach is based upon a detailed numerical study on various versions of the Fisher–Kolmogorov–Petrovskii–Piscounov equation. The main motivation of this work is to contribute to the following question: how much information do statistics, collected by a stationary observer, contain about the speed and bifurcations of traveling waves? We suggest warning signs based upon closeness to carrying capacity, second-order moments, and transients of localized initial invasions. However, we also show that these warning signs can be difficult to interpret if limited information is available and that the generalization of classical variance-based warning signs is problematic in the context of propagation failure.

## Keywords

Critical transitions Invasion waves Propagation failure Fisher–KPP FKPP SPDE## Notes

### Acknowledgments

I would like to thank the European Commission (EC/REA) for the support by a Marie-Curie International Reintegration grant. I also acknowledge the support via an APART fellowship of the Austrian Academy of Sciences (ÖAW). Furthermore, I would like to thank the two anonymous referees and the editor whose comments helped to improve the manuscript.

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