Replace **rabbit** with **lemming** and **wolf** with **arctic fox**, and look at this analysis over in the [logistic thread](https://forum.azimuthproject.org/discussion/comment/21969/#Comment_21969)
The Lotka-Volterra in comment #1 is missing the death rate of the prey species
* \\(death(\rho): lemming\rightarrow \\)
The survival of the lemming appears to be more dependent on climate cycles since the cyclic lemming population period is always ~3.8 years, which happens to be the lunisolar tidal forcing period that I use in the [ENSO model](https://forum.azimuthproject.org/discussion/comment/21894/#Comment_21894). So \\( \rho \\) is actually \\( \rho(t) \\) and this follows a cyclic pattern as described in that thread. What this means is that the lemming population never crashes due to excess predation by the arctic fox (or snowy owl) but by the harshness of the climate, and so the predatory feedback factor in the Lotka-Volterra equation is likely not controlling. Those nonlinear limit cycles likely never occur for lemming/fox but may for rabbit/wolf.
Interesting and perfectly plausible how this missing factor completely changes the character of the solution.