Bifur ations and Chaos in E ology:Lynx Returns RevisitedJAVIER G. P. GAMARRA1;2 and RICARD V. SOL�E1;31Complex Systems Resear h GroupDepartment of Physi s FEN,Universitat Polit�e ni a de CatalunyaCampus Nord B5, 08034 Bar elonaE.mail: javier� omplex.up .es, ri ard� omplex.up .esPhone: 0-34-3-4017056 ; Fax: 0-34-3-40171002Centre Te nol�ogi Forestal de CatalunyaPujada del Seminari s/n, 25280, Solsona3Santa Fe Institute1399 Hyde Park RoadSanta Fe, New Mexi oUSABiosket hResear h interests of the autors in lude Theoreti al E ology, Non-linear Spatio-temporalPopulation Dynami s (and Non-linear Time Series Analysis), Sto hasti Pro esses andComplex Adaptive Systems

Submitted to E ology LettersRunning Title: Bifur ations, trapping and lynx1

Abstra tOne of the most popular data sets in e ology, that of lynx fur returns, is analyzedin order to look for eviden e for a bifur ation pro ess. This bifur ation seemsto be present from the observation of a shift in the amplitude of os illations ofthe lynx time series. The possibility for su h a bifur ation was �rst proposed byS ha�er in 1985. This author suggested that a possible sour e for the qualitative hange of lynx's u tuations was an in reased trapping e�ort, whi h eventually leadto high-amplitude, haoti dynami s. By studying the available information fromthe Hudson Bay Company re ords, we have found eviden e for su h an in reasedtrapping pressure whi h rapidly rose lose to the shift from low-amplitude to large-amplitude u tuations. Although an in rease in the top-predator mortality in athree-spe ies food web typi ally leads to simpler dynami s and eventual predatorextin tion, here we show that a re ent model involving a minimum bound in the lynxpopulation, due to the presen e of alternative preys in the lynx diet, onsistentlysupports the presen e of a bifur ation phenomenon.Keywords haos, lynx fur returns, population y les, strange attra tors,trapping e�ort, three-trophi models

2

1 Introdu tionThe most famous ompilation of population data along time omes from the well-knowntwo enturies statisti s of Canadian lynx (Lynx anadensis Kerr) fur trade ompiled fromthe Hudson's Bay Company ar hives (Elton & Ni holson 1942). Its meaning and inter-pretation have been a matter of ontroversy over the last �fty years. Based mainly on lassi al time series analysis and the presen e of spurious orrelations, most of the theoriesfound a ten-year y le in lynx returns. Early onje tured auses for this y le (for a briefreview see Royama 1992) ome from: a) external events, su h as the sunspot y le (Elton1924) or weather and forest-�re y le (Grange 1965); b) internal auses, su h as prey pop-ulation stresses (Ma Luli h 1937, Green & Evans 1940), food supplies for its main prey,the snowshoe hare (Lepus ameri anus Erxleben) (La k 1954), predation (Moran 1953) orthe prey-predator ratio (Ak� akaya 1992).As far as the presen e of a ten year y le was so remarkable, none of them dis ussedthe possibility of in luding non-linear dynami s leading to the presen e of bifur ationsand haos in the system. The onset of haoti dynami s in the e ologi al literature inthe mid-70's (May 1976; May & Oster 1977) opened a di�erent road for new thoughts inpopulation e ology, and in fa t, numerous studies have been reported involving the use ofnon-linear time series statisti al analysis on lynx data (see Tong 1990). In this ontext, analternative theoreti al framework for the lynx ase was developed by Gilpin (1973, 1979).Based on ontinuous Lotka-Volterra 3D equations, the model was able to display haos.Further work, using the te hnique of attra tor re onstru tion (Takens 1981) showed that athree-dimensional dynami al system ould be responsible of lynx u tuations, suggestingthat three variables (lynx, hare and vegetation) might be at play (S ha�er 1984, 1985).Experimental eviden e for the three-dimensional stru ture of lynx dynami s ame ontos ienti� knowledge in 1995 (Krebs et al. 1995). By manipulating food supply for hareand the number of predators (lynx), they found non-additive e�e ts in the populationdensity of hares. This further suggested the importan e not only of dire t, but also ofindire t intera tions in the vegetation-hare-lynx system.The time series of lynx u tuations has three phases. In the �rst one, from 1735to 1820, the number of skins harvested showed small os illations with a mean of over3

5000 skins for all Canada (Fig. 1A). From 1821 to 1904 the series is the one with largeramplitudes and is the most typi ally studied in the literature. The third series, startingaround 1905, is the less learly de�ned, displaying again small-s ale u tuations. Inspite of the di�erent trust that may onfere the olle tion of data (depending of the years onsidered, the data were ompiled from di�erent sour es and there was an amalgamationof two trading ompanies in 1821), the statisti s show strong shifts revealing three learphases.1735 1785 1835 1885 1935

Time (year)

0

20000

40000

60000

Num

ber o

f skin

s

10−2

10−1

100

frequency

0

5e+02

1e+03

2e+03

Four

ier sp

ectru

m

10−2

10−1

100

frequency

0

5e+03

1e+04

2e+04

1774 1790 1806 1822 18380

500

1000

1500

2000Nu

mbe

r of e

mplo

yees

1735−1820 1820−1935

A

B C

Figure 1: (A) Series of lynx fur returns re orded by the Hudson's Bay Company for two enturiesand (inset) Total salaried worfor e from the Hudson's Bay Company (open ir les) and the sumof Hudson's Bay and NorthWest ompanies (bla k ir les). Both ompanies amalgamated in1821, under the name of HBC. (data from Goldring 1979 and Moodie et al. 1992). (B-C) Powerspe trum orresponding to lynx returns before and after 1821.Many developments have been done sin e S ha�er's ontribution, but most of themhave fo used on the large- u tuations phase. The question of what ould have a tedover the system that resulted in those shifts was tentatively answered by S ha�er himself(S ha�er 1985). In his paper he laid two onje tures: one, the shift over 1820 was dueto an in rease in hunting pressure due mainly to the amalgamation of the Hudson's Bay4

Company and the Northwest Company (HBC and NWC hereafter); se ond, it ould bedue to some event in the early 19th entury that resulted in a displa ement from onebasin of attra tion to another.The �rst onje ture of S ha�er is not a pure spe ulation. In fa t, we have found ev-iden e for an in reased trapping e�ort in fur trade histori al ar hives, on erning that ould have a ted from the beginning of the 19th entury. The early fur trade, at thebeginning of the 18th entury, was hara terized by an initial division of Canada in twotrading areas: one, the Hudson's Bay watershed, was regulated by the British HBC; theother, omprising the territories around Lake Huron and Lake Superior and the Athabas a ountry, was the trading territory of the St. Lawren e's Fren h traders. Little trade withnatives was done, sin e the area exploited was not too large. From 1774, the two tradesystems started ompetition, and a ra e for the fur trade with Indians took pla e. Thenumber of trading posts in reased exponentially and the areas exploited overlapped. Forthe Indians, ompetition raised fur pri es. In 1784 the NWC got to manage the fur tradein the area formerly exploited by the St. Lawren e small traders. Dire t ompetitionbetween two equally sized ompanies further in reased. Competition in reased the pri esthat European traders paid for furs. Sin e the pri es rose, fewer furs were traded and In-dian traders and trappers had more time for other a tivities. Competition also in reasedinvestment in posts and personnel (Fig. 1A), whi h in reased still more the pri es of furs.This situation proved to be ruinous, with expenses soaring and pro�ts falling. Mean-while, Indian dependen e on trapping for trade was getting a relevant role in their so ialand demographi organization (Bishop 1972). In 1815 the HBC �nally penetrated theAthabas a Country, destroying NWC's monopoly in the region, and ompetition be ameopen warfare. Eventually, the British government pressed the two ompanies to amalga-mate in 1821 in bene�t of the fur trade. Afterwards, under the domain of HBC, monopolyof fur trade be omes a fa t, fur pri es and administrative employment de rease and In-dians, now totally dependent of the fur trade (traders an then introdu e new poli ieswithout restri tions due to ompetition) and with an in reasing population, are obligedto in rease trapping e�ort.In this letter we want to explore S ha�er's onje ture, in the sense that, if haoti 5

dynami s is at play we should be able to reprodu e the so- alled route to haos by slightly hanging one parameter des ribing hunting pressure in a simple non-linear model.2 Material, Methods and ResultsThe three-dimensional stru ture in lynx intera tions was sharplintrodu ed in the UPCA(Uniform Phase evolution and Chaoti Amplitude) model (Blasius et al. 1999). Theauthors developed a 3D ontinuous model in whi h the intera tions Lotka-Volterra orHolling-type: dvdt = av � �1f1(v; h)dhdt = �bh + �1f1(v; h)� �2f2(h; l)dldt = � (l � l�) + �2f2(h; l) (1)where a is the vegetation growth rate, �1; �2 are intera tion strengths between vegetation-hare and hare-lynx, and b is a natural mortality rate for hares. As a fa t, this model isvery similar to others formerly developed as generalizations of three-spe ies lineal foodwebs (Hastings & Powell 1991) or planktoni populations (S he�er 1991). One of themain di�eren es of the UPCA model is the addition of l�, a parameter des ribing aminimum bound in the lynx population due to the possible existen e of alternative preysin the lynx diet (Bergerud 1983), expressing the degree of spe ialization of the lynx asa predator. This parameter avoids the rash or extin tion of the lynx population underextreme dynami s. The spatial ounterpart of the model is able to reprodu e the hugevariability of the y le's amplitude and the relative syn hronization in its frequen y underlow dispersion values, a result formerly found in the lynx series, at least for the two �rstphases des ribed, where populations seem to u tuate with 10 years intervals (see Fig.1B-C).A omplete des ription of the parameters involved in the UPCA model an revealthat there is one of them spe ially intuitive for our purposes. Parameter represents amortality rate for lynx population. It may a ount for natural mortality and mortalitydue to external auses, su h as trapping or hunting.6

Thus the UPCA model an be easily modi�ed in order to introdu e the e�e t ofin reasing trapping pressure. We may be able to de�ne a new variable (t), that, in thesimplest ase may be de�ned as (t) = 0 + Æ where 0 is the initial trapping and/ornatural mortality rate and Æ is some annual in rease in trapping pressure. resemblesthe mortality rate of the superpredator in a three-spe ies food hain (Hastings & Powell1991), and in the absen e of a minimum bound su h as l�, a linear in rease in (t) drivesthe population to extin tion in both models (see inset in Fig. 2 for S he�er's model).The ombined assumption on the existen e of a minimum bound and a linear in rease intrapping pressure is not irrelevant: �rst, be ause a predator su h as the lynx is not a fullygeneralist spe ies (Bergerud 1983), and se ond, be ause a linear in rease in the numberof Indian trappers ould have a ted from the late 1700's to 1850, approximately. Besides,this linearity an result in ounterintuitive, nonlinear re ords in lynx dynami s.

10 15 20 25

Death (trapping) rate, c

0

2

4

6

Maxim

um ly

nx po

pulat

ion

0 2 4 6

algae

0

0.1

0.2

0.3

0.4

0.5

Carn

ivoro

us zo

oplan

kton

Figure 2: Bifur ation diagram for the modi�ed version of the UPCA model, where is thebifur ation parameter. vmax = 150. The rest of parameters are the same as in Blasius et al.(1999). Inset Attra tor resulted from applying a linear in rease in (t) for S he�er's (1991)planktoni three-trophi model. r = 0:265. The rest of parameters are the same as in theoriginal manus ript. Observe that, after drawing some orbits, the top-predator's traje tory rash to extin tion. 7

We have performed simulations on the UPCA model with two other assumptions:� Blasius et al. (1999) assume exponential growth (dvdt = av) for vegetation in theabsen e of hares. We have assumed, more realisti ally, logisti growth:dvdt = av(1� vvmax ) (2)where vmax is the arrying apa ity for vegetation. In our simulations we have setvmax = 150� The intera tion terms areHolling type II f1(v; h) = vh(1 + k1v) (3)Lotka-Volterra f2(h; l) = hl (4)� We assume that the natural mortality rate is not a�e ted by hanges in trappinge�ort (i.e., there are no indire t intera tions involving mortality su h as a redu tionin the mortality rate due to a redu tion of the population due to an in rease in trap-ping). Thus, our assumption does not involve any multipli ative e�e t in mortalitydue to an in rease in (t).The rest of parameters are the same as in Blasius et al. (1999).As shown in Fig. 2, the in orporation of (t) as a linear in reasing fun tion is learlyfollowed by a period-doubling route to haos from a more or less stable dynami s forlynx populations. The temporal evolution of the simulation (Fig. 3A) also resembles theoriginal data. The amplitude of the os illations also grows with the in reasing "trappinge�ort" or mortality term.Thus, we ould at a �rst sight support the idea that a slow growth of trapping pressure ould have taken pla e following su essive olonization of Canadian forests by fur traders,so the idea of haoti dynami s behind the data ould be supported through a ontinuousroute to haos, but none of the important events that o urred ould have played any role8

in it, so S ha�er's initial onje ture would not have any sense, sin e the amalgamation ofNWC and HBC would not be signi� ant at all.0

5

10

15

20

Hunti

ng pr

essu

re

0

0.5

1

1.5

2

2.5

Lynx

0 1000 2000 3000Time(iterations)

0

5

10

15

20

Hunti

ng pr

essu

re

0 1000 2000 30000

0.5

1

1.5

2

2.5

Lynx

A

B

Figure 3: Time series and re onstru ted attra tors produ ed by the modi�ed UPCA model for:top a linear in rease in (t) (hunting or trapping pressure); bottom (t) as a \step" fun tion. (t)is represented in dashed lines.If we re all the return map from the lynx fur returns (by plotting su essive maxima)as in Fig. 4(A), we an observe that its stru ture reminds that of a unimodal map,similar to the logisti map for populations with non-overlapping generations (May 1976).For the lynx data, there an be seen two di�erent sets of points: the one with loweramplitudes orresponds to the years before the amalgamation, that an be viewed as aperiodi attra tor (S ha�er 1985). The other set of points, though somewhat noisy, maydes ribe a two-point y le or even haoti dynami s. This di�eren e between these twosets an not be adequately des ribed in terms of a linear in reasing trapping e�ort. Insu h a ase, the two sets would overlap, due to the slow displa ement over the "tuning"parameter (t). Instead, we made simulations where = � 0 if t � t 1 otherwise9

0 20000 40000 60000 80000Lynx maximum (i)

0

20000

40000

60000

80000

Lynx

max

imum

(i+1)

A

1743−1821

1829−1905

0 0.2 0.4 0.6 0.8 1Lynx maximum (i)

0

0.2

0.4

0.6

0.8

1

Lynx

max

imum

(i+1)

B

Figure 4: Time series and re onstru ted attra tors produ ed by the modi�ed UPCA model for:top a linear in rease in (t) (hunting or trapping pressure); bottom (t) as a \step" fun tion. (t)is represented in dashed lines.In this "step" fun tion we assume 2 >> 1, and t is the iteration number. Thus,now the trapping e�ort "jumps" along the bifur ation diagram due to some signi� antevent. The return map orresponding to this simulation is represented in Fig. 4(B),and des ribes two hara teristi sets de�ning a stable point ( orresponding to periodi dynami s in the original series) and a more omplex attra tor in a unimodal map. Thismap resembles more a Ri ker map than a logisti one, but a more similar return map an be obtained by introdu ing random u tuations in the l� parameter. Moreover,the attra tor re onstru ted from the simulation reprodu es a sudden shift in the orbitsfollowed by the dynami s (Fig. 3B).3 Dis ussionA onsiderable number of auses has been proposed in the existing literature for explain-ing the aparent ten year y le for the series of lynx returns taken from the Hudson's BayCompany ar hives. Re ently, however, some studies have shown: �rst, that the aparent10

y li ity is not so obvious, se ond, that although there is some dominant frequen y in-trinsi in the series (Fig. 1) (see Tong 1990), the amplitude of the series shows a errati , haoti route (S ha�er 1985). Blasius et al. (1999) demostrate trough a simple tritrophi model similar to that of Arneodo et al. (1980) that uniform frequen y and haoti am-plitude an be easily a hieved.The utilization of tritrophi webs for hara terizing lynx dynami s is the simplest and learest example of haoti dynami s for a natural e osystem. But the histori al series oflynx returns, starting from 1735, shows tremendous qualitative and quantitative hangesthat an not be reprodu ed without some kind of \intervention analysis". A simple modelsu h as the UPCA model (Blasius et al. 1999) is a good one for this purpose; however,others ould have been used (Bas ompte et al. 1997).The introdu tion of the trapping e�ort as the \intervention variable" shows, a ordingto S ha�er's �rst onje ture, that a period-doubling route to haos might have a ted overthe system. Thus, be omes the bifur ation parameter, and onstitutes an additionaleviden e for the haoti ity of the lynx series, and ould even represent a fourth trophi levelinsert into the system (Weinstein 1977). Moreover, histori al re ords point learly towardthis possibility. The monopoly reated over the pelts market sin e the amalgamation ofNWC and HBC ompanies in 1821 seems to be the key event. At that time, Indians hadbeen driven into a new mer antile e onomy based thoroughly on the fur trade. Thus,spe ulation and monopoly by the HBC, and the fa t that the pri es paid for pelts to thetrappers remained �xed for one entury (Newman 1985), drew a system of exploitationin whi h a higher trapping pressure was its main hara teristi .In this ontext, we have supported the idea that a sudden shift, rather that a smoothin rease, ould have o urred in the trapping pressure, in response to the amalgamation.The sudden hange in the attra tor re onstru ted from the original series (see S ha�er1984) and the orresponding return map of lynx maxima (Fig 4(A)) support this idea. Theamalgamation was followed by a sudden redu tion in the number of permanent salariedworkfor e and the number of posts dispersed over the territory in a short time step (only�ve years), so the Indians had been for ed rapidly towards a greater exploitation.There still remain some open questions. S ha�er's se ond onje ture, whi h states11

that a shift between two di�erent basins of attra tions ould have happened, an not benegle ted yet. In any ase, the deterministi nature of the u tuations and the existen eof some internal event playing a role at some point at the beginning of the 19th enturyrepresents a strong eviden e for explaining this shift.Besides, the in lusion of spa e in the system may have ounterintuitive responsesin lynx dynami s, and an support the deterministi nature of it, giving raise to theappearan e of spatial waves (Bas ompte et al. 1997; Blasius et al. 1999). Su h spatialsyn hrony may indu e the growth of stable stru tures not only in lynx, but also on haresand vegetation.Nowadays, there are re ommendations for lynx harvesting from governmental poli ies,su h as the suspension of harvests for the 3-4 year low of hare y le (Brand & Keith 1979),so potentially more lynx available for harvesting in peak years. This orresponds to somekind of periodi perturbations in the trapping pressure, a ting as a form of haos ontrol(see Sol�e et al. 1999 and referen es therein). The onsequen es of this type of perturbationregime under spatial domain still need a theoreti al ba kground and will be regarded infuture work.A knowledgementsAnne Morton and the Hudson's Bay Company kindly gave invaluable histori al informa-tion on trapping and workfor e in Canada. This work has been partially supported bythe Santa Fe Institute (RVS) and grant DGYCIT PB97-0693.

12

4 Referen esAk� akaya H. R. (1992) Population y les of mammals: eviden e for a ratio-dependentpredation hypothesis. E ol. Monogr., 62(I), 119-142.Arneodo A. P., Collet P. & Tresser C. (1980) O urren e of strange attra tors in three-dimensional Volterra equations. Phys. Lett., 79a, 259-263.Bas ompte J., Sol�e R. V. & Mart�inez N. (1997) Population y les and spatial patterns insnowshoe hares: an individual-oriented simulation. J. Theor. Biol., 187, 213-222.Bergerud A. (1983) Prey swit hing in a simple e osystem. S i. Am., 249(6), 130-141.Blasius B., Huppert A. & Stone L. (1999) Complex dynami s and phase syn hronizationin spatially extended e osystems. Nature, 399, 354-359.Brand C. J. & Keith L. B. (1979) Lynx demography during a snowshoe hare de line inAlberta. J. Wildl. Manage., 43, 827-849.Elton, C. (1924) Periodi u tuations in the number of animals: their auses and e�e ts.Brit. J. Exper. Biol., 2, 119-163.Elton C. & Ni holson M. (1942) The ten year y le in numbers of the Lynx in Canada.J. Anim. E ol., 11, 215-244.Gilpin M. E. (1973) Do hares eat lynx? Am. Nat., 107, 727-730.Gilpin M. E. (1979) Spiral haos in a predator-prey model. Am. Nat., 113, 306-308.Goldring P. (1979) Papers on the labour system of the Hudson's Bay Company, 1821-1900:Vol. I HBC Ar hives, 362 (I).Grange W. B. (1965) Fire and tree growth relationships to snowshoe rabbits. Pro .Annual Tall Timbers Fire E ol., Conf 4, 110-125.Green R. G. & Evans C. A. (1940) Studies on a population y le of snowshoe hares onthe lake Alexander area. J. Wildl. Manage., 4, 220-238.Hastings A. & Powell T. (1991) Chaos in a three-spe ies food hain. E ology, 72, 896-903.Krebs C. J., Boutin S., Boonstra R., Sin lair A. R. E., Smith J. N. M., Dale M. R. T.,13

Martin K. & Turkington R. (1994) Impa t of food and predation on the snowshoe hare y le. S ien e, 269, 1112-1115.La k D. (1954) The Natural Regulation of Animal Numbers. Oxford Univ. Press, Oxford.Ma Luli h D. A. (1937) Flu tuations in the Number of the Varying Hare (Lepus ameri- anus). University of Toronto Press, Toronto.May R. M. (1976) Simple mathemati al models with very ompli ated dynami s. Nature,261, 459-457.May R. M. & Oster G. F. (1976) Bifur ations and dynami omplexity in simple e ologi almodels. Am. Nat., 110, 573-599.Moodie D. W., Kaye B., Lytwyn V. P. & Ray A. J. (1992) Peoples of the boreal forestand parkland. In: Histori al Atlas of Canada R. Cole Harris Vol. I (ed. Matthews G.J.). University of Toronto Press, pl. 65.Moran P. A. P. (1953) The statisti al analysis of the Canadian lynx y le. Aust. J. Zool.,1, 163-173.Newman P. C. (1985) Company of Adventurers Vol. 1. Viking, Markham.Royama T. (1992) Analyti al Population Dynami s. Chapman & Hall, London.S ha�er W. M. (1984) Stret hing and folding in lynx fur returns: eviden e for a strangeattra tor in nature? Am. Nat., 124, 798-820.S ha�er W. M. (1985) Order and haos in e ologi al systems. E ology, 66, 93-106.S he�er M. (1991) Should we expe t strange attra tors behind plankton dynami s-and ifso, should we bother? J. Plankton. Res., 13, 1291-1305.Sol�e R. V., Gamarra J. G. P., Ginovart M. & L�opez D. (1999) Controlling haos in e ology:from deterministi to individual-based models. Bull. Math. Biol.Takens F. (1981) Dete ting strange attra tors in turbulen e. Le ture Notes in Math., 898.Tong, H. (1990) Non-linear Time Series: A Dynami al Systems Approa h. Oxford Univ.Press, Oxford. 14

Weinstein M. S. (1977). Hares, lynx and trappers. Am. Nat., 111, 806-808.

15