The [Lotka-Volterra equation](https://forum.azimuthproject.org/discussion/967/lotka-volterra-equation) is closely related to the logistic equation via a growth term, with a feedback term set so that a predatory species can further accelerate the prey toward a limiting value. But since the disappearance of prey will also cause the disappearance of the predator, a cyclic pattern can develop based on this coupled feedback. It's possible that that this is a real mechanism in actual ecological predator/prey relationships but it's difficult to verify. Any stochastic perturbation will likely knock the cycle off its current period.
One of the famous predator/prey behaviors in the Arctic latitudes is the Lemming/Arctic Fox cycle (or Snowy Owl). Over a long time interval this cycle has been estimated to have a period of 3.8 years. A wildlife ecologist working on the topic for **40+ years** finally seems to have pattern-matched to a plausible model -- published last year :
> Archibald, H. L. [Relating the 4-year lemming ( Lemmus spp. and Dicrostonyx spp.) population cycle to a 3.8-year lunar cycle and ENSO](https://tspace.library.utoronto.ca/bitstream/1807/97104/1/cjz-2018-0266.pdf). Can. J. Zool. 97, 1054–1063 (2019).
What he noted is that the lemming cycle happens to match a spring tide cycle, implying more of a climate related mechanism controlling the population. After coming across this paper I noted that the cycle appeared suspiciously close to the tidal forcing that I am using in the [ENSO model](https://forum.azimuthproject.org/discussion/comment/21894/#Comment_21894). The vertical dotted lines indicate the alignment (the inset is CC between ENSO and PDO )
Why it follows the more predictable tidal forcing rather than the more erratic ENSO response, I don't have an answer. But this does look more plausible than a predator-prey cycle. The environment is so harsh in the Arctic that climate factors likely control the health of the lemming population, and the predators then follow that cycle as well since that is their food supply. This may be a classic common-mode mechanism instead of a mutual resonance set by the eigenvalue or chaotic attractor of a differential equation.