telech

Canadian Lynx counted by the Hudson Bay Company

Christophe LETELLIER
29/02/2008
JPG - 18.6 kb
Hudson Bay Company

Many ecological and biological populations oscillate with the remarkable property that their period length is quite constant while their abundance levels have large amplitude fluctuations. The earliest evidence for fluctuations comes from records of the Hudson Bay Company regarding the populations of lynx and hares. This company bought and sold lynx pelts. Their records (Fig. 1) - studied by Charles Sutherland Elton and Mary Nicholson [1] - show that the number of animals fluctuates with a period of about nine years, so that about a dozen cycles were observed in the data during the century. The amplitude of these cycles varies significantly from year to year, as would be expected for a chaotic system as observed in many three species models (see [2], [3], [4] among others).

PNG - 20.7 kb
Fig. 1 Lynx furs caught
texte - 17.9 kb
Interpolated Elton’s data

The counts from the Hudson Bay Company are undersampled since only around ten points are available per cycle (one point per year for a cycle which has an average period equal to 9.2 year). These data have been interpolated to allow a global modelling [5]. A 4D model producing transient chaos was obtained (Fig. 2).

PNG - 55.8 kb
The 4D model phase portrait

[1] C. Elton & M. Nicholson, The Ten-Year Cycle in numbers of the Lynx in Canada, Journal of Animal Ecology, 11, 215-244, 1942.

[2] M. Gilpin, Spiral chaos in a predator prey model, The American Naturalist, 113, 306-308, 1979.

[3] R. K. Upadhyay, S. R. K. Jyengar & V. Rai, Chaos: an ecological reality?, International Journal of Bifurcation & Chaos, 8 (6), 1325-1333, 1998.

[4] B. Blasius, A. Huppert & L. Stone, Complex dynamics and phase synchronization in spatially extended ecological systems, Nature, 399, 354-359, 1999.

[5] J. Maquet, C. Letellier & L. A. Aguirre, Global models from the Canadian Lynx cycles as a first evidence for chaos in real ecosystems, Journal of Mathematical Biology, 55 (1), 21-39, 2007.

Documents

Interpolated Elton’s data
texte · 17.9 kb
3333 - 23/04/17

ATOMOSYD © 2007-2017 |  Follow-up of the site's activity  |  SPIP  |  scoty  |  MàJ . 16/04/2017