  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 of various versions of
the Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) 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.
