J. Math. Biol.DOI 10.1007/s00285-007-0075-9 Mathematical Biology

Global models from the Canadian lynx cyclesas a direct evidence for chaos in real ecosystems

J. Maquet · C. Letellier · Luis A. Aguirre

Received: 18 May 2006 / Revised: 16 January 2007© Springer-Verlag 2007

Abstract Real food chains are very rarely investigated since long datasequences are required. Typically, if we consider that an ecosystem evolveswith a period corresponding to the time for maturation, possessing few dozenof cycles would require to count species over few centuries. One well knownexample of a long data set is the number of Canadian lynx furs caught by theHudson Bay company between 1821 and 1935 as reported by Elton and Nich-olson in 1942. In spite of the relative quality of the data set (10 undersampledcycles), two low-dimensional global models that settle to chaotic attractors wereobtained. They are compared with an ad hoc 3D model which was proposedas a possible model for this data set. The two global models, which were esti-mated with no prior knowledge about the dynamics, can be considered as directevidences of chaos in real ecosystems.

Keywords Ecosystems · Chaos · Global modelling · Data analysis

J. Maquet · C. Letellier (B)CORIA UMR 6614, Université de Rouen, BP 12,76801 Saint-Etienne du Rouvray cedex, Francee-mail: christophe.letellier@coria.fr

J. Maquete-mail: jean.maquet@coria.fr

L. A. AguirreDepartamento de Engenharia Eletrônica, Universidade Federeal de Minas Gerais,Av. Antônio Carlos 6627, 31270-901 Belo Horizonte, MG, Brazile-mail: aguirre@cpdee.ufmg.br

J. Maquet et al.

1 Introduction

Many ecological and biological populations oscillate with the remarkable prop-erty that their period length is quite constant while their amplitude experienceslarge fluctuations [14,26,29]. Among the rare data available, there are the cyclein populations of Lynx canadensis which has received wide attention on accountof the great amplitude of the rhythm it has produced in the fur catches of theHudson’s Bay Company over a long period. According to Elton and Nicholson’sconclusions [9], the fluctuations of the lynx are related to those of the snowshoerabbits (Lepus americanus) and aquatic species such as muskrats (OndatraZibethica). This is therefore an ecosystem involving at least three species andchaotic behaviors are a possibility as observed in many three species models(see [5,11,32] among others). Indeed, it is still a challenge to evidence chaos inreal data although its importance in ecology was reviewed in [15].

Unfortunately, all the models proposed are empirical, that is, the modelsare built to reproduced “chaotic” fluctuations which look like the observations.Indeed, almost all data sets available have few cycles and are too short for mosttechniques to directly identify a chaotic attractor. One of the very interestingmethods that avoid such difficulty consists in estimating a global model fromthe data [2,7,12]. Indeed, from a very limited number of cycles, it is possible toextract a global model—a set of three ordinary differential equations directlyestimated from the data without any constraint on the model structure—whichreproduces the main features of the dynamics [13,24]. It will be shown thatlow-dimensional global models can be estimated from the Canadian lynx furscaught (1821–1935) as reported by [9]. Moreover, such models happen to settleto a chaotic attractor. Using topological analysis, the chaotic attractor is foundto be topologically equivalent to the attractor solution of a standard three levelfood chain proposed by [5] as a good candidate for explaining the irregularfluctuations in the lynx populations. To our knowledge, this could be one of thevery first direct evidence of chaos in a real ecosystems.

2 The historical data

Ecological cycles are often characterized by a frequency of population whichremains relatively constant and erratic changes in abundance. Among exam-ples [14], the hare-lynx cycle is known since the investigations of Hudson’s BayCompany by [9]. Many studies were devoted to the dynamics underlying thesedata [29,30]. It has been shown that hare and lynx populations from differentregions of Canada synchronize in phase to a collective cycle that manifestsover millions of square kilometers [5]. There are thus interactions betweenthe different regions of Canada and using the total population would allow toreduce the “measurement errors”, even if some data are sometimes missing.We also believed that working with the whole population would help to have abetter observability as it will be discussed later. The total number of furs caught

Global models from the Canadian lynx cycles

Table 1 Hudson’s Bay Company lynx sale obtained as reported by [9]

Year # Year # Year # Year #

1821 4,849 1850 2,162 1879 9,948 1908 1,7411822 3,131 1851 1,386 1880 4,540 1909 1,9021823 3,130 1852 1,194 1881 3,218 1910 4,3241824 2,701 1853 1,540 1882 4,935 1911 10,0671825 2,610 1854 5,139 1883 15,651 1912 13,6811826 5,076 1855 13,062 1884 42,085 1913 19,0531827 7,527 1856 16,466 1885 63,651 1914 7,5831828 16,640 1857 25,392 1886 62,682 1915 7,5831829 24,103 1858 13,986 1887 31,958 1916 8,8511830 22,782 1859 8,470 1888 15,111 1917 2,4871831 9,422 1860 3,391 1889 6,657 1918 9561832 4,619 1861 1,663 1890 3,257 1919 1,0561833 3,116 1862 1,486 1891 3,898 1920 1,9181834 3,894 1863 2,921 1892 — 1921 3,5741835 6,311 1864 9,529 1893 — 1922 4,3801836 18,941 1865 27,051 1894 — 1923 7,6731837 34,744 1866 57,576 1895 — 1924 7,3181838 53,700 1867 49,508 1896 — 1925 9,2591839 42,256 1868 25,596 1897 14,919 1926 7,0781840 15,484 1869 8,600 1898 6,884 1927 4,3821841 3,912 1870 4,258 1899 2,070 1928 2,7241842 2,328 1871 1,712 1900 2,537 1929 2,3741843 2,935 1872 2,473 1901 4,548 1930 2,8881844 5,903 1873 5,858 1902 13,513 1931 3,3591845 14,113 1874 10,101 1903 26,741 1932 4,7761846 21,837 1875 25,748 1904 46,939 1933 6,8211847 35,942 1876 28,785 1905 53,827 1934 7,4531848 28,653 1877 20,733 1906 29,4371849 7,303 1878 10,550 1907 7,448

The total number of furs caught in the Northern department are here reported. Data between 1892and 1896 are missing

over the different regions of Canada will be thus investigated as reported in therecords of Hudson’s Bay Company [9] and here reported in Table 1.

The total number of furs is missing between 1892 and 1896. In order to havea continuous graphical representation, we introduced by hand “reconstructed”numbers as follows. Among the ten cycles available, we were looking for theclosest part of the available data to the cycle preceding the missing one. Wethen estimated the missing points by superposing this part of the data. Thewhole data set used for our analysis is shown in Fig. 1. Since the sampling rateis not sufficient for a safe dynamical analysis (less than 10 points per cycle areavailable), the data were interpolated by a factor of 10 using a standard Matlabprocedure preserving the Fourier spectrum as already used to analyze data froma thermionic diode experiment [23].

Among the conclusions (not necessarily correct) provided by Elton andNicholson, one can quote [9]:

• “The cycle in lynx furs is very violent and regular and has persistedunchanged for the whole period. Its average period is about 9.6 years”.

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1825 1850 1875 1900 1925

Year

0

10000

20000

30000

40000

50000

60000

70000

Lyn

x fu

rs c

augh

t Interpolated dataOriginal data points

Fig. 1 Evolution of the total number of lynx furs caught in the different regions of Canada between1821 and 1934. Data between 1892 and 1896 are inserted by hand as explained in the main text. Thedata were interpolated by a factor of 10 using a standard Matlab procedure preserving the Fourierspectrum

• “This cycle is a real one in lynx populations, which are dependent upon thesnowshoe rabbits (Lepus americanus) for food, and which starve when therabbits disappear periodically. It is therefore strong evidence of a similarcycle in snowshoe rabbits for the last 106 years”.

• “The wide synchronization of the cycle in different parts of Canada for atleast 100 years, its parallel occurrence both west and east of the Rockies, andits dependent occurrence in aquatic species such as the muskrat (Ondatrazibethica) and the salmon (Salmon salar), strongly suggest the existence ofa climatic factor partly controlling it”.

As clearly stated, more than two species interact in this ecosystem since lynx,rabbits and some aquatic species are involved. A food chain with at least threelevels is therefore under study. It is known since 1979 that three level foodchains can lead to chaotic oscillations [11]. In 1940s, the possibility for irregularfluctuations in populations was always correlated with climatic factors. Basedon the first three level food chain ecologists started to think in terms of speciesinteractions as a possible mechanism for such fluctuations [11]. In this latterinterpretation, climate fluctuations are not necessary for chaotic fluctuations inpopulations and an autonomous model can be expected.

By the late 1980s, it was shown that it is possible to obtain a global modelfrom a time series, that is, to estimate a set of differential or difference equationsdirectly from measurements [8]. Such models were qualified as “global” sincea single set of equations was sufficient to reproduce the whole phase space inopposition to “local” models which are valid only over a small domain of thephase space. Getting a global model directly from the data is the main objec-tive of this paper since, to our knowledge, there is no successful global modelreported in the literature.

Working in the paradigm of chaos, the first step in the dynamical analysis isto reconstruct a phase space from the time series available. This can be doneusing delay or derivative coordinates as proposed in [27]. Since our main taskis to obtain a differential model from these data, derivative coordinates willbe used. Delay coordinates will be used only for computing the embeddingdimension (see below). As usually done when experimental data are investi-gated, a slight smoothing is applied before computing the derivatives. This is

Global models from the Canadian lynx cycles

0 1 2 3 4 5 6 7 8 9 Dimension d

0

0,2

0,4

0,6

0,8

1

E2 (

d)

Fig. 2 Relative change in the average distance between neighbor points versus the dimension d ofthe phase space reconstructed from the smoothed interpolated data

done using a frequency cut in the Fourier spectrum [28]. The threshold hereused corresponds to a window of seven points of the interpolated data, that is, awindow with a length less than a tenth of the averaged time period. The num-ber of coordinates required to have a trajectory which does not intersect withitself—to avoid breaking the determinism—can be estimated using the falsenearest neighbors techniques as developed in [16] and improved in [6].

Cao’s algorithm is here used [6]. The idea is to increase the dimension d ofthe phase space up to the case where there is no longer self-intersections of thetrajectory. It was developed based on the fact that choosing too low an embed-ding dimension results in points that are far apart in the original phase spacebeing moved closer together in the reconstruction space. To avoid the choice ofa threshold to decide whether a neighbor is false or not, Cao used the relativechange in the average distance between two neighboring points in R

d when thedimension is increased from d to d + 1. When this index saturates around 1, theminimum dimension required to embed the trajectory without any self-crossingis reached. This minimal dimension is the so-called embedding dimension dE.

We made our computations for a time delay equal to 10 δt where δt is the timebetween two successive points in the interpolated data. Since the 10 availablecycles are clearly under the limit for having a proper estimation of the embed-ding dimension, we have to keep in mind that this is just a rough estimate. Fromour computations, the embedding dimension is low and could be equal to 3.One could object that the saturation value (E2 ≈ 1) is not exactly reached ford = 3 but only for d = 6 (Fig. 2). However we have to keep in mind that weused undersampled data which are definitely not noise free. By noise, we meanhere all factors—including climate factors—which are not strictly related to spe-cies interactions and which could result from high-dimensional dynamics. Thenoise thus affects the dimension estimation by inducing spurious false nearestneighbors and the saturation is obviously poorer when the data are noise con-taminated, as already pointed out [6]. This lack of a complete saturation doesnot hide the first clear change in the slope of the curve E2(d) around d = 3.This is a signature of a space with a sufficiently high dimension to embed quitesafely the data.

The reconstructed phase portrait (Fig. 3) appears like large loops arounda saddle-focus approximatively located at the left part of the attractor (whenthe first derivative Y < 0). Small oscillations around this saddle-focus are the

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0 10000 20000 30000 40000 50000 60000 70000

Number of lynx furs

-40000

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Fir

st d

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Fig. 3 Phase portrait reconstructed from the evolution of the total number of lynx furs caught inthe different regions of Canada between 1821 and 1934. The interpolated data are smoothed beforecomputing the derivatives

signature of the folding mechanism which should be responsible for the irregu-lar amplitude fluctuations. Such an organization of the reconstructed attractordoes not correspond to a very easy configuration to get a global model since itlooks similar to the portrait induced by the z-variable of the Rössler system, avariable reputed hard for a global modeling [17].

3 A three level food chain

A three level food chain with vegetation (x), herbivores (y) and predators (z)was introduced by Blasius et al. [4] as a good candidate to explain the irregularfluctuations in the time evolution of the number of lynx furs caught. This modelreads as

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

x = a(x − xf ) − α1xy1 + k1x

y = b(y − yf ) + α1xy1 + k1x

− α2yz

z = c(z − zf ) + α2yz

. (1)

The phase portrait is structured by a plane of fixed points (xf , yf = 0, zf ) andtwo other fixed points which have each one negative coordinate, x and y, respec-tively. The last two do therefore not play a relevant role in the structure of thephase portrait since a population cannot be negative. When b is chosen as thebifurcation parameter, the other being fixed at the values proposed by Blasiusand Stone [4], a bifurcation diagram reveals a period-doubling cascade as a routeto chaos. Beyond the accumulation point (b ≈ 1.025), there is a chaotic attractorcharacterized by a unimodal first-return map. Then, a very wide period-3 win-dow is easily identified (Fig. 4). A chaotic attractor is recovered after a second

Global models from the Canadian lynx cycles

Fig. 4 Bifurcation diagramcomputed for a = 1, c = 10,α1 = 0.1, α2 = 0.6, k1 = 0,xf = 1.5, yf = 0 and zf = 0.01

0,90,9511,051,1Bifurcation parameter b

10

20

30

40

50

60

xn

0.864

period-doubling cascade. This attractor is solution to the three level food chainup to a boundary crisis (b ≈ 0.864) which ejects the trajectory to infinity.

During the period-3 window, there is a significant change in the slope of thethree branches (b ≈ 0.91). This is the signature of a bifurcation which couldbe an inverse saddle-node bifurcation inducing a switch to another period-3limit cycle. This bifurcation is responsible for the topological changes observedin the chaotic behavior beyond the accumulation point of the period-doublingcascade. Thus, chaotic attractors observed after the accumulation point havea first-return map made of three different branches (Fig. 5b) and not two asusually observed. A possible scenario is actually that the period-3 limit cycleencoded by (100) loses its stability and that the period-3 limit cycle encodedby (201) becomes stable. This means that the period-3 orbit (100) with oneinteraction with the Poincaré section located in branch 1 and two in branch0 exchanges its stability with the orbit (201) which has one intersection withthe Poincaré section located in each branch of the first-return map (Fig. 5b).It is therefore not possible to obtain a b-value corresponding to a unimodalfirst-return map with a complete symbolic dynamics (all symbolic sequencesover {0, 1} have a corresponding orbit within the attractor). There is thereforeno parameter value b for which an homoclinic trajectory can be observed as forthe Rössler system [22], that is, when a first-return map to a Poincaré sectionlooks like a parabola characterized by two monotonic branches splitted by adifferentiable maximum. In the (nearly) homoclinic case, the increasing branchtouches the bisecting line. In the case of system (1) with a bifurcation parametercorresponding to a regime after the period-3 window (b = 0.864, for instance),the first increasing branch—encoded by “0”—is not even close to the bisectingline (Fig. 5b). Moreover, a third branch—encoded by 2—is observed, a branchalways observed after the period-3 window in system (1). Two extrema—oneminimum and one maximum—thus define three branches, a feature usuallyobserved only after a nearly homoclinic behavior. In the present case, such athree branch first-return map is observed up to the boundary crisis occurringfor b ≈ 0.864.

J. Maquet et al.

0 10 20 30 40 50 60 70

Vegetables x

0

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60 H

erbi

vore

s y

10203040506070

xn

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x n+1

0 1 2

(a) (b)

Fig. 5 2D projection of the chaotic attractor observed just before the boundary crisis which ejectsthe trajectory to the infinity. Parameter values: b = 0.864 and others as for Fig. 4. a Chaotic attractor;b First-return map

The chaotic attractor solution to the three level food chain is thus describedby a three branch template as shown in Fig. 6. Template identification from datais detailed in [10,22].

As observed with the Rössler system, there is a great difference betweenthe observability provided by each variable of the three level food chain. It isnow known that most—if not all—of nonlinear systems do present an observ-ability which strongly depends on the choice of the variable measured [17,19].Another realistic three level food chain was analyzed and it has been shownthat the generalist predator—at the end of the chain—actually provides thelowest observability of the ecosystem [20]. This means that counting the gener-alist predators was not the best strategy to observe the dynamics underlying theecosystem. This is particularly sensitive when global modeling is attempted (noglobal model was obtained from the time evolution of the generalist predatorwhile good models were obtained from the preys and the specialist predators).

In the present three level food chain (1), the lynx (generalist predator) arelocated at the end of the food chain and the time evolution of their populationprovides an observability of the dynamics lower than those offered when theherbivores (specialist predators) or the vegetation are counted. The observ-ability of the food chain from the different species can be estimated usingobservability indices [19] which are

δx = 0.030 > δy = 5.04 × 10−4 > δz = 4.96 × 10−6 ,

where x stands for the vegetation, y for the herbivores and z for the generalistpredator. The observability is related to the ability to recover the informationon the whole system from the measurements. This results from the fact thatit may be possible that some states from the original phase portrait cannot beseen in the reconstructed phase space. The observability indices quantify the

Global models from the Canadian lynx cycles

Fig. 6 Template of chaoticattractors solution to the threelevel food chain. Itcorresponds to the topologicalorganization of attractorscharacterized by a first-returnmap made of up to threemonotonic branches 2 1 0

quality of the measured variables (the observables) to provide all the requiredinformation. They are based on the eigenvalues of the Jacobian of the coor-dinate transformation between the original phase space and the reconstructedspace [19]. The three indices here computed estimate the relative quality ofobservable x, y and z to allow a good reconstruction of the underlying dynam-ics, respectively. These variables can thus be ranked from the best observabilityto the worst. In the present case, the dynamical variables are ranked as

x � y � z

This means that the reconstructed space induced by the x-variable has a bet-ter quality than those induced by the y or the z variable. According to sucha ranking, the best observable would be the amount of vegetable. Of course,it is hopeless to count the vegetation but counting rabbits would be a bettersolution for getting a global model for the underlying dynamics when comparedto counting lynx.

A first illustration of this lack of observability of the dynamics from the lynxpopulation can be given by a plane projection of the chaotic attractor recon-structed with derivative coordinates (Fig. 7). From the vegetation, the stretchingand the folding mechanisms responsible for the chaotic nature of the behav-ior are identifiable in the left upper part of the attractor. When herbivores

J. Maquet et al.

are counted, if the stretching mechanism is still obvious, it becomes hard toidentify the folding mechanism without a blow up of the attractor. This secondmechanism is located at the left of the saddle-focus but is squeezed by theway the y-variable sees the dynamics. Note that this variable is sensitive to theamplitude fluctuations (maxima) of herbivore population. These effects becomeworse when predators are counted. The reconstructed phase portrait appearslike few large loops related to the largest cycles and all dynamical mechanismsare restricted to a very small neighborhood of the saddle-focus. Since the non-linear mechanism are mostly located where variable z is around its minima, thenoise contamination—counting errors and influence of any external parameterslike climate changes—could quickly blurr the dynamics and prevent any modelidentification.

Thus, with phase portrait reconstructions from the different variables, it ispossible to have a first clue on how the choice of the observable may influencethe quality of the analysis and, in particular, the quality of the global model esti-mated. Note that the phase portrait reconstructed from the lynx furs caught bythe Hudson Bay Company (Fig. 3) is more similar to the attractor reconstructedfrom the y variable than to the attractor reconstructed from the z-variable. Thismay be justified by the fact that, men influence the lynx dynamics in the form ofa large, external mortality. They do not act as a superpredator since there is noobvious influence of the lynx population back on the human density. Huntingcould be considered as an undetermined forcing term. The lynx are thereforeno longer at the end of the food chain. The attractor induced by the lynx furscaught has thus a more identifiable folding mechanism—the small oscillations.The influence of this forcing term should be taken into account in an ad hocmodel for this ecosystem.

4 Global data-estimated models

From the ad hoc model proposed by Blasius et al., there are a few indications onthe possible nature of the dynamics underlying the ecosystem involving the lynxin the Hudson Bay. They can be sum up as follows. (i) The route to chaos couldbe a period-doubling cascade. (ii) There is a bifurcation around a wide period-3window which forbids a complete unimodal symbolic dynamics. (iii) The cha-otic attractor is destroyed through a boundary crisis. (iv) The lynx are locatedat the end of the food chain. The last point suggests that the lynx population isnot the best population to observe the underlying dynamics for analysis, suchas the estimation of a global model from data.

Basically, there are two types of models which can be obtained. (i) Con-tinuous models estimated from phase portraits reconstructed with derivativecoordinates and (ii) discrete models from a reconstruction with delay coor-dinates. From few different time series, these two techniques provided globalmodels with similar quality [20,25]. The technique used in this paper works withthe derivative coordinates to reconstruct a so-called differential embedding.The observable—the lynx population—is designated by X and the successive

Global models from the Canadian lynx cycles

Fig. 7 2D projections of thechaotic attractor solution tothe three food chainreconstructed from thedifferent species. Parametervalues: b = 0.864 and othersas for Fig. 4. a From thevegetation; b Fromthe herbivores; c Fromthe predators

0 10 20 30 40 50 60 70 Vegetation x

-200

-150

-100

-50

0

50

Fir

st d

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ativ

e

0 10 20 30 40 50 60 Herbivores y

-400

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0

100

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st d

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0 5 10 15 20 25

Predators z

-100

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(a)

(b)

(c)

J. Maquet et al.

Table 2 Structure of the 3D global model estimated from the lynx furs caught by the Hudson BayCompany

l Kp Term p Kp Term

1 −0.1400 × 10+4 11 −0.3340 × 10−9 X3

2 0.7800 × 10+0 X 12 −0.8400 × 10−9 X2Y3 −0.1730 × 10+1 Y 13 −0.1443 × 10−8 X2Z4 −0.9000 × 10−1 Z 14 0.1390 × 10−8 XY2

5 −0.2719 × 10−4 X2 15 −0.6000 × 10−8 XYZ6 0.5895 × 10−4 XY 16 0.1070 × 10−8 XZ2

7 0.1548 × 10−4 XZ 17 −0.5300 × 10−9 Y3

8 −0.1140 × 10−4 Y2 18 0.0 Y2Z9 0.2230 × 10−3 YZ 19 −0.4380 × 10−8 YZ2

10 −0.4800 × 10−4 Z2 20 0.1300 × 10−8 Z3

derivatives by Y, Z and Z, respectively. A set of (3) ordinary differential equa-tions thus takes a canonical form as

⎧⎨

⎩

X = YY = ZZ = F(X, Y, Z)

, (2)

where F(X, Y, Z) is the canonical function. All the procedure consists in approx-imating this canonical function using a multivariate polynomial basis as detailedin [12]. This function reads as follows

F(X, Y, Z) = K0 + K1X + K2Y + K3Z + K4X2 + · · · (3)

where the values of the coefficients Kp are returned by the estimation algorithm.After few trials, a successful model was estimated from 117 centers (roughly 13centers per cycle, that is, around 9 cycles). This “training” set has the disadvan-tage of using the cycle built by hand (1892–1896) but it does not use the last twocycles which have a very different underlying dynamics. The number of termsretained in the canonical functions is 20 (including the constant term K0). Thewhole global model is reported in Table 2.

Since the global model is very stable, that is, robust against parameterchanges, a bifurcation diagram versus one of the model parameter can becomputed. We choose to vary parameter K3 which corresponds to the Zterm in the canonical function. There is no significant departure between theso-obtained bifurcation diagram and others computed varying other parame-ters. The bifurcation diagram (Fig. 8) presents a period-doubling cascade as aroute to chaos. A wide period-3 window is also clearly identified. The chaoticattractor observed after the period-3 window is destroyed through a boundarycrisis (right part of the diagram). These three characteristics were found in thethree level food chain model proposed by Blasius et al.

As for the ad hoc model, it is not possible to observe a complete sym-bolic dynamics in the estimated global model, by varying parameter K3. The

Global models from the Canadian lynx cycles

Fig. 8 Bifurcation diagram ofthe data-estimated 3D globalmodel obtained from theinterpolated data. It iscomputed versus parameterK3

-0,25 -0,2 -0,15 -0,1

Parameter K3

0

5000

10000

15000

20000

xn

-0.175 -0.114

origin of this impossibility is also a bifurcation occurring near the period-3 win-dow. Nevertheless, the bifurcation is not the same as the one involved in themodel proposed by Blasius et al. Indeed, when a first-return map is computedjust before the period-3 window, the decreasing branch is layered (Fig. 9a).A remarkable characteristic is that, when parameter K3 is decreased, the sec-ond decreasing branch has a maximum with an ordinate which increases. Justbefore the period-3 window, its ordinate is equal to those of the critical pointbetween the increasing and the decreasing branches (differentiable maximum).Before the period-3 window, the dynamics can be described with two symbolsand the template (branches 0 and 1) is as the template proposed for the ad hocmodel (Fig. 6). This means that the ad hoc model (with b < 0.955) and the 3Dglobal model (K3 < −0.175) are topologically equivalent. They are thus dynam-ically equivalent (over the previous ranges of parameters). After the period-3window, the dynamics suddenly becomes much more complicated. Examplesof such “dynamical explosions” have been observed in various systems as thevan der Pol or Duffing equations, a 5D laser model [21] or in some of the Sprottsystems [31].

The gap between the upper boundary of the bifurcation diagram (Fig. 4)and the upper branch of the period-3 window is a signature of this bifurcation(Fig. 9b). In fact, as soon as the end of the second decreasing branch has anordinate equal to the maximum, the first-return map suddenly has new branchesas revealed after the period-3 window (Fig. 9b). This is a signature that thedata-estimated model has somewhat more complicated dynamics—as would beexpected because it is derived from real-life data—than the ad hoc model. Thepartition of such a map is not easy to define. This is the first significant depar-ture between the 3D global model estimated from the lynx furs caught andthe ad hoc model. Although both models display unusual bifurcations aroundthe period-3 windows—which is actually quite thrilling—the ways in which thedynamics developed are different in each model. In the ad hoc model, there isa large number of secondary oscillations as observed in a Rössler attractor of a

J. Maquet et al.

0 10000 20000 30000 40000

X

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0250050007500100001250015000

Xn

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1

0 10000 20000 30000 40000

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02500500075001000012500Xn

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7500

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Xn+

1

(a)

(b)

Fig. 9 Chaotic behavior solution to the data-estimated 3D model obtained from the lynx popula-tion as recorded by Hudson’s Bay Company. a Before the period-3 window: K3 = −0.175; b Afterthe period-3 window: K3 = −0.114

funnel type. This number is clearly greater in the 3D global model than in theoriginal data (compare Fig. 9b with Fig. 3).

Moreover, the time evolution of the number of lynx furs caught provided bythe 3D model does not look like the evolution reported by the Hudson BayCompany data (Fig. 1). Many reasons could be invoked. Briefly, there is anobvious observability problem which is similar to that presented by the Rösslersystem. It remains quite difficult to obtain a global model from the z-variableof the Rössler although the underlying dynamics is simple. This comes from thecoupling between the dynamical variables. Basically, the z-variable presentslethargies during which the evolution of the other variables is not seen [17].This is also the case of the z-variable of the three level food chain proposed byBlasius et al. Although this 3D ad hoc model is simple, the coupling betweenthe three levels is roughly correct. In particular, the lynx eat the rabbits andthe opposite is not true. This is sufficient to reach the conclusion that lynx

Global models from the Canadian lynx cycles

0 10000 20000 30000 40000 50000 60000

Number of Lynx furs caught X

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20000

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irst

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ive

Y

10000 20000 30000 40000 50000 60000

Xn

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30000

40000

50000

60000

Xn+

1

(a) (b)

Fig. 10 Chaotic behavior solution to the 4D global model estimated from the number of lynx furscaught by the Hudson Bay Company. a Chaotic attractor; b First-return map

population is certainly not the best observable to investigate this food chain.Indeed, taking into account the exact nature of the nonlinearities involved in thecoupling between the dynamical variables—the species—is not required sincethe observabilities from the variables can be ranked by using a simple graphdescription [18]. It is possible to balance the lack of observability by increasingthe embedding dimension as suggested by Takens’ theorem.

A 4D global model was therefore attempted. We thus need to compute thefourth derivative. We here reach the limit of the possibilities offered by thepoor data set investigated (around ten points per cycle before interpolation). Aglobal model was obtained with 93 points per cycle and 600 points retained forestimating the canonical function. The “training” set of this 4D global modelhas the advantage of not containing the cycle built by hand to replace the miss-ing data between 1892 and 1896. The best 4D global model obtained is notstable, that is, it cannot be integrated from a data point of the original dataset beyond 420 cycles (roughly corresponding to 4,000 years). The 4D model istherefore not robust against parameter change and a bifurcation diagram can-not be computed as for the 3D global model. This lack of stability comes fromthe relatively low quality of the data set. A projection of the chaotic attractorobserved (Fig. 10a) looks very similar to those reconstructed from the originaldata (Fig. 3). As expected, increasing the embedding dimension helps to balancethe lack of observability and the dynamics captured by the 4D global model isbetter than those captured by the 3D global model previously discussed [1].

A first-return map to a Poincaré section reveals three main monotonicbranches with a layered structure (Fig. 10b). The first increasing branch touchesthe first bisecting line, signature of the existence of lethargies in the time series.In fact, such lethargies are related to the existence of an inner fixed point ofthe saddle-focus type whose neighborhood is visited. The attractor could thuscontain a homoclinic orbit (at least a secondary one). The dynamics of thismodel thus captures one of the very obvious features of the original data. Theselethargies, also observed in the z-variable of the Rössler system, are responsible

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0 200 400 600 800

Years

0

10000

20000

30000

40000

50000

60000

Lyn

x po

pula

tion

Fig. 11 Evolution the lynx population obtained by numerically integrating the 4D global model.Lethargies are closely reproduced as in the original data

for the lack of observability [17] and are therefore difficult to reproduce. It maybe checked that the 4D global model generates a times series (Fig. 11) whichlooks very similar to the original time series (Fig. 1). In particular, lethargiesnear zero are observed, a property not observed with the 3D global model.

The second increasing branch (right part of the map) also reaches the firstbisecting line. Its end has an ordinate which reaches those of the maximumof the map. A dynamical explosion is thus expected as discussed for the 3Dglobal model. The fact that both global models present this feature stronglysuggest that the dynamics underlying the data from the Hudson Bay Companyhave something in common with the ad hoc model postulated by Blasius andcolleagues. The 4D global model also suggests the ecosystem is close to suchan explosion. This could be the origin of the lack of stability of the 4D globalmodel.

This first-return map (Fig. 10b) could explain another origin of the diffi-culty to capture the original dynamics. It is known that overparametrizationof estimated models produces layered first-return maps [3]. Thus a first anal-ysis of Fig. 10b could indicate that the model is overparameterized. However,an important remark is that, unlike the models in [3], the model under analy-sis is 4D. Consequently, a Poincaré section of such a 4D model would be a 3Dsurface. Therefore, a 2D projection (as in Fig. 10b) of the first return map to sucha section could well be layered without the model being overparameterized.

The remark in the previous paragraph could suggest that the system dynamicsunderlying the data is, in fact, more complicated than an attractor characterizedby a one dimensional map. Hence the layered structure of the map (Fig. 10b)would point to true complexity rather than overparametrization. This wouldsuggest that the dynamics is better described by a 4D model (4D dynamics aremore often characterized by layered map than 3D models). Maybe this is a firstclue to include the human activity (hunting) in this ecosystem since this modelwas estimated directly from the data.

Unfortunately, to the best of our knowledge, there is no guaranteed wayof testing such hypotheses using such a limited data set nor is there a rigor-ous (dynamical) way of validating the estimated models. The global modelshere proposed have the advantage that no prior knowledge has been used in

Global models from the Canadian lynx cycles

their estimation. Therefore all dynamical features revealed by the models wereestimated from the data although we cannot provide accuracy levels.

5 Conclusion

Investigating data sets as the number of lynx furs caught by the Hudson BayCompany is always very challenging since there are only ten undersampledcycles. This is not sufficient to perform topological analysis of the originaldynamics with confidence and, consequently, no model can be rigorously vali-dated. In the past, many papers were devoted to this data set (since it remainsone of the very rare examples of a real food chain followed for over a century).Unfortunately, the amount of data available is not sufficient for any technique(statistical or dynamical). For these reasons, ad hoc models are welcome sincethey provide possible mechanisms for explaining the fluctuations observed inthe data. Nevertheless, with global modeling techniques it is possible to extracta set of governing equations from a quite limited amount of data. This wassuccessfully performed with various techniques, most of time from data setsarising from numerical models.

This paper has presented and discussed two global models estimated fromthe available data set without any prior knowledge on the underlying dynam-ics. This means that the coupling between the different species—the dynamicalvariables—are not imposed and only results from the numerical estimation ofthe canonical function projected on a multivariate polynomial basis. In spite ofthe poor quality of the data set, the models here obtained capture some dynam-ical features contained in the data which are also found in ad hoc models. Inparticular, the period-doubling cascade is identified in the 3D global model aswell as a long period-3 window. However, there are also differences betweenthe estimated and the ad hoc models. In both estimated global models, a crisisis identified: it involves a third branch in the first-return map with an end hav-ing an ordinate equal to those of the maxima of the map. This is a dynamicalcharacteristic which is not observed in the ad hoc model proposed by Blasiuset al.

Obtaining global models from the data set collected by Elton and Nicholsonhelps to support the idea that actual ecosystems could evolve in a chaotic way—both models are chaotic—without any significant environmental changes. Ourmodels thus suggest that the underlying dynamics is determinist or, at least,has a deterministic component which is chaotic. These models do support theconclusion that there is a deterministic, chaotic component in the dynamicsunderlying the lynx population. Indeed, it has been checked in many differentsituations that it is typically impossible to obtain a global model with a chaoticattractor for solution from any surrogate data set. On the other hand, the simi-larities observed between the 4D global model and the original data suggest thatthe ecosystem involving the lynx could correspond at least to a four level foodchain. Many possibilities could be invoked to justify a fourth variable. Thus, the

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Shannon activity (hunting) but also an alternative prey as suggested by Blasiuset al. [4] could be considered for future ad hoc models of this ecosystem.

Acknowledgments We wish to thank Dair José de Oliveira who interpolated the data for us. Theauthors acknowledge partial funding by CNRS and CNPq.

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