These extreme peaks - spikes in effective population size (Ne) immediately followed by an even more extreme collapse are really common! Here are a few examples from the literature:

Is there any new result about the origins of old (i.e. pre-european) Native American DNA in Rapanui genomes? Also, have we ever found traces of Rapanui DNA in Native American genomes? Asking for a friend :)

PIV models are available in BEAST (beast.community) and PhyREX (github.com/stephaneguin...).

We applied this technique to the West Nile virus in the USA and showed that PIV models were indeed able to predict the arrival of that virus on the West Coast in 2002 using data collected before the end of 2001, for instance.

Also, since PIV models enable the estimation of speed at the tips of a phylogeny, we used them to predict future dispersal patterns.

Simulations show that "Phylogenetic Integrated Models" (PIV) provide estimates of dispersal speed that are more accurate than that obtained with previous approaches.

We devised a new model which, instead of describing the evolution of coordinates, focuses on the evolution of velocity. The coordinates are then obtained by integrating out velocities, which can be done at no extra computational cost.

-> Brownian trajectories have infinite variation: no matter how much you zoom in a trajectory, it will never be smooth... The concept of instantaneous speed simply does not exist here! This leads to difficulties for assessing the speed of dispersal under models such as the "Relaxed Random Walk".

Inferring speed of dispersal/migration in phylogeography is not trivial. For instance, assuming a Brownian process to model the evolution of spatial coordinates is a sensible approach overall but does not provide adequate means to measure speed. Why?