Lemming/Fox Dynamics not Lotka-Volterra

Appendix E of the book contains information on compartmental models, of which resource depletion models, contagion growth models, drug delivery models, and population growth models belong to.

undefinedOne compartmental population growth model, that specified by the Lotka-Volterra-type predator-prey equations, can be manipulated to match a cyclic wildlife population in a fashion approximating that of observations. The cyclic variation is typically explained as a nonlinear resonance period arising from the competition between the predators and their prey. However, a more realistic model may take into account seasonal and climate variations that control populations directly. The following is a recent paper by wildlife ecologist H. L. Archibald who has long been working on the thesis that seasonal/tidal cycles play a role (one paper that he wrote on the topic dates back to 1977! ).

Archibald, H. L. Relating the 4-year lemming ( Lemmus spp. and Dicrostonyx spp.) population cycle to a 3.8-year lunar cycle and ENSO. Can. J. Zool. 97, 1054–1063 (2019).

These are his main figures:

fig1
fig2

This 3.8 year cycle directly agrees with the ENSO model driven by the fortnightly tropical cycle (13.66 days) interacting with an annual cycle, which is indicated in the middle right pane in the figure below:

This image has been resized to fit in the page. Click to enlarge.

Below is an expanded view of the tidal forcing used in the climate index model in comparison to the peak years in lemming population. The tidal forcing meandering square wave aligns with the cyclic peak lemming populations. The 3.8 year cycle derives directly from the aliased beat cycle 1/(27-365.242/13.6608) = 3.794 years. The dotted lines are a guide to the eye

This image has been resized to fit in the page. Click to enlarge.

This does not validate the ENSO model but like the cyclic anchovy and sardine populations off the coast of Peru and Chile, it substantiates the view that climate variations impact wildlife populations. What the Lemming/Arctic Fox cycle does is show the ENSO-to-lunisolar-forcing connection more directly. In other words, the lemming population may be more sensitive to the precursor tidal forcing than to the resulting (and more erratic) ENSO cycles. Archibald discusses this aspect:

“One big question raised but not answered by this study: Is the ENSO cycle driven by or somehow connected with the 3.8-year lunar cycle? The correlation between predicted lemming/Arctic fox peak years and January-March SOI peak years (Table 1) suggests that ENSO might be driven by the 3.8-year lunar cycle. Zhang (2001) surmised, “There must exist some behind and deep mechanism of linking rodent outbreaks and ENSO together”. However, there is not much evidence for such a linkage. The only reference found to a relationship between a 3.8-year lunar cycle and atmospheric circulation is Wilson’s (2012) finding of a significant 3.78-year (+- 0.06) periodicity in the peak latitude anomaly of the subtropical high-pressure ridge over eastern Australia from 1860 to 2010. However, some worldwide climate oscillations contain a periodicity of ~3.8 years (selected examples in Table 4), and it is clear that these oscillations are interconnected. For example, Li and Lau (2012) examined the late winter teleconnection between the ENSO and the NAO and found that the average NAO index values in 19 El Niño winters and 20 La Niña winters were -0.35 and 0.51, respectively: positive NAO index values are more frequent during La Niña winters, and negative winter NAO index values are more frequent during El Niño events.”

Archibald (2019)

The solution to the Laplace Tidal Equations provide the direct linkage from 3.8 years to the observed ENSO pattern. The 3.8 year pattern remains embedded in this pattern along with the nonlinear superharmonics that are generated from the forcing input.


Will leave this recent citation here:

Image
Baker, L. E. & Sutherland, B. R. The evolution of superharmonics excited by internal tides in non-uniform stratification. Journal of Fluid Mechanics 891, (2020).

This may help explain why the LTE wave solution aligned precisely along the equator sustains a standing wave mode with characteristic Mach-Zehnder superharmonics. A Google Scholar search for “superharmonic cascade” only gives 4 hits.

4 thoughts on “Lemming/Fox Dynamics not Lotka-Volterra

  1. In contrast, the number of hits for “subharmonic cascade” is like 334.

    Without context, a harmonic is typically a superharmonic as creating multiples of a frequency is easier than creating a fraction of a frequency level, the latter being a subharmonic.

    https://scholar.google.com/scholar?q=%22subharmonic+cascade%22

    In a sense, a superharmonic cascade is the same as creating overtones or simply generating harmonics, which is simple to do.

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  2. Regarding the superharmonics paper of Baker and Sutherland, another by Sutherland and Jefferson

    When they talk of a triad it is a triplet of waves such that the wave-numbers add up as such : k = k_a + k_b

    https://www.wolframalpha.com/input/?i=plot+sin%2827t%29+%2Bsin%2830t%29+%2B+cos%283t%29

    This could be a pair of rapidly oscillating TIW waves with a beat frequency difference that equals the main ENSO dipole. This may have the advantage of emulating the behavior of TIW waves such that they are really only apparent during the La Nina phase!
    IOW, during El Nino the beat of the TIW pair cancels

    Triad LTE modulation:

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  3. Pingback: Triad Waves | GeoEnergy Math

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