Bifurcations and Chaos inEcology: Lynx ReturnsRevisitedJavier G. P. GamarraRicard V. Solé

SFI WORKING PAPER: 1999-10-067

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SANTA FE INSTITUTE

Bifurcations and Chaos in Ecology:Lynx Returns Revisited

JAVIER G. P. GAMARRA1;2 and RICARD V. SOL�E1;3

1Complex Systems Research GroupDepartment of Physics FEN,

Universitat Polit�ecnica de CatalunyaCampus Nord B5, 08034 Barcelona

E.mail: javier@complex.upc.es, ricard@complex.upc.es

Phone: 0-34-3-4017056 ; Fax: 0-34-3-4017100

2Centre Tecnol�ogic Forestal de CatalunyaPujada del Seminari s/n, 25280, Solsona

3Santa Fe Institute1399 Hyde Park RoadSanta Fe, New Mexico

USA

Biosketch

Research interests of the autors include Theoretical Ecology, Non-linear Spatio-temporal

Population Dynamics (and Non-linear Time Series Analysis), Stochastic Processes and

Complex Adaptive Systems

Submitted to Ecology Letters

Running Title: Bifurcations, trapping and lynx

1

Abstract

One of the most popular data sets in ecology, that of lynx fur returns, is analyzed

in order to look for evidence for a bifurcation process. This bifurcation seems

to be present from the observation of a shift in the amplitude of oscillations of

the lynx time series. The possibility for such a bifurcation was �rst proposed by

Scha�er in 1985. This author suggested that a possible source for the qualitative

change of lynx's uctuations was an increased trapping e�ort, which eventually lead

to high-amplitude, chaotic dynamics. By studying the available information from

the Hudson Bay Company records, we have found evidence for such an increasedtrapping pressure which rapidly rose close to the shift from low-amplitude to large-

amplitude uctuations. Although an increase in the top-predator mortality in a

three-species food web typically leads to simpler dynamics and eventual predator

extinction, here we show that a recent model involving a minimum bound in the lynx

population, due to the presence of alternative preys in the lynx diet, consistently

supports the presence of a bifurcation phenomenon.

Keywords

chaos, lynx fur returns, population cycles, strange attractors,

trapping e�ort, three-trophic models

2

1 Introduction

The most famous compilation of population data along time comes from the well-known

two centuries statistics of Canadian lynx (Lynx canadensis Kerr) fur trade compiled from

the Hudson's Bay Company archives (Elton & Nicholson 1942). Its meaning and inter-

pretation have been a matter of controversy over the last �fty years. Based mainly on

classical time series analysis and the presence of spurious correlations, most of the theories

found a ten-year cycle in lynx returns. Early conjectured causes for this cycle (for a brief

review see Royama 1992) come from: a) external events, such as the sunspot cycle (Elton

1924) or weather and forest-�re cycle (Grange 1965); b) internal causes, such as prey pop-

ulation stresses (MacLulich 1937, Green & Evans 1940), food supplies for its main prey,

the snowshoe hare (Lepus americanus Erxleben) (Lack 1954), predation (Moran 1953) or

the prey-predator ratio (Ak�cakaya 1992).

As far as the presence of a ten year cycle was so remarkable, none of them discussed

the possibility of including non-linear dynamics leading to the presence of bifurcations

and chaos in the system. The onset of chaotic dynamics in the ecological literature in

the mid-70's (May 1976; May & Oster 1977) opened a di�erent road for new thoughts in

population ecology, and in fact, numerous studies have been reported involving the use of

non-linear time series statistical analysis on lynx data (see Tong 1990). In this context, an

alternative theoretical framework for the lynx case was developed by Gilpin (1973, 1979).

Based on continuous Lotka-Volterra 3D equations, the model was able to display chaos.

Further work, using the technique of attractor reconstruction (Takens 1981) showed that a

three-dimensional dynamical system could be responsible of lynx uctuations, suggesting

that three variables (lynx, hare and vegetation) might be at play (Scha�er 1984, 1985).

Experimental evidence for the three-dimensional structure of lynx dynamics came onto

scienti�c knowledge in 1995 (Krebs et al. 1995). By manipulating food supply for hare

and the number of predators (lynx), they found non-additive e�ects in the population

density of hares. This further suggested the importance not only of direct, but also of

indirect interactions in the vegetation-hare-lynx system.

The time series of lynx uctuations has three phases. In the �rst one, from 1735

to 1820, the number of skins harvested showed small oscillations with a mean of over

3

5000 skins for all Canada (Fig. 1A). From 1821 to 1904 the series is the one with larger

amplitudes and is the most typically studied in the literature. The third series, starting

around 1905, is the less clearly de�ned, displaying again small-scale uctuations. In

spite of the di�erent trust that may confere the collection of data (depending of the years

considered, the data were compiled from di�erent sources and there was an amalgamation

of two trading companies in 1821), the statistics show strong shifts revealing three clear

phases.

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 recorded by the Hudson's Bay Company for two centuries

and (inset) Total salaried worforce from the Hudson's Bay Company (open circles) and the sum

of Hudson's Bay and NorthWest companies (black circles). Both companies amalgamated in

1821, under the name of HBC. (data from Goldring 1979 and Moodie et al. 1992). (B-C) Power

spectrum corresponding to lynx returns before and after 1821.

Many developments have been done since Scha�er's contribution, but most of them

have focused on the large- uctuations phase. The question of what could have acted

over the system that resulted in those shifts was tentatively answered by Scha�er himself

(Scha�er 1985). In his paper he laid two conjectures: one, the shift over 1820 was due

to an increase in hunting pressure due mainly to the amalgamation of the Hudson's Bay

4

Company and the Northwest Company (HBC and NWC hereafter); second, it could be

due to some event in the early 19th century that resulted in a displacement from one

basin of attraction to another.

The �rst conjecture of Scha�er is not a pure speculation. In fact, we have found ev-

idence for an increased trapping e�ort in fur trade historical archives, concerning that

could have acted from the beginning of the 19th century. The early fur trade, at the

beginning of the 18th century, was characterized by an initial division of Canada in two

trading areas: one, the Hudson's Bay watershed, was regulated by the British HBC; the

other, comprising the territories around Lake Huron and Lake Superior and the Athabasca

country, was the trading territory of the St. Lawrence's French traders. Little trade with

natives was done, since the area exploited was not too large. From 1774, the two trade

systems started competition, and a race for the fur trade with Indians took place. The

number of trading posts increased exponentially and the areas exploited overlapped. For

the Indians, competition raised fur prices. In 1784 the NWC got to manage the fur trade

in the area formerly exploited by the St. Lawrence small traders. Direct competition

between two equally sized companies further increased. Competition increased the prices

that European traders paid for furs. Since the prices rose, fewer furs were traded and In-

dian traders and trappers had more time for other activities. Competition also increased

investment in posts and personnel (Fig. 1A), which increased still more the prices of furs.

This situation proved to be ruinous, with expenses soaring and pro�ts falling. Mean-

while, Indian dependence on trapping for trade was getting a relevant role in their social

and demographic organization (Bishop 1972). In 1815 the HBC �nally penetrated the

Athabasca Country, destroying NWC's monopoly in the region, and competition became

open warfare. Eventually, the British government pressed the two companies to amalga-

mate in 1821 in bene�t of the fur trade. Afterwards, under the domain of HBC, monopoly

of fur trade becomes a fact, fur prices and administrative employment decrease and In-

dians, now totally dependent of the fur trade (traders can then introduce new policies

without restrictions due to competition) and with an increasing population, are obliged

to increase trapping e�ort.

In this letter we want to explore Scha�er's conjecture, in the sense that, if chaotic

5

dynamics is at play we should be able to reproduce the so-called route to chaos by slightly

changing one parameter describing hunting pressure in a simple non-linear model.

2 Material, Methods and Results

The three-dimensional structure in lynx interactions was sharplintroduced in the UPCA

(Uniform Phase evolution and Chaotic Amplitude) model (Blasius et al. 1999). The

authors developed a 3D continuous model in which the interactions Lotka-Volterra or

Holling-type:dv

dt= av � �1f1(v; h)

dh

dt= �bh + �1f1(v; h)� �2f2(h; l)

dl

dt= �c(l � l�) + �2f2(h; l) (1)

where a is the vegetation growth rate, �1; �2 are interaction strengths between vegetation-

hare and hare-lynx, and b is a natural mortality rate for hares. As a fact, this model is

very similar to others formerly developed as generalizations of three-species lineal food

webs (Hastings & Powell 1991) or planktonic populations (Sche�er 1991). One of the

main di�erences of the UPCA model is the addition of l�, a parameter describing a

minimum bound in the lynx population due to the possible existence of alternative preys

in the lynx diet (Bergerud 1983), expressing the degree of specialization of the lynx as

a predator. This parameter avoids the crash or extinction of the lynx population under

extreme dynamics. The spatial counterpart of the model is able to reproduce the huge

variability of the cycle's amplitude and the relative synchronization in its frequency under

low dispersion values, a result formerly found in the lynx series, at least for the two �rst

phases described, where populations seem to uctuate with 10 years intervals (see Fig.

1B-C).

A complete description of the parameters involved in the UPCA model can reveal

that there is one of them specially intuitive for our purposes. Parameter c represents a

mortality rate for lynx population. It may account for natural mortality and mortality

due to external causes, such as trapping or hunting.

6

Thus the UPCA model can be easily modi�ed in order to introduce the e�ect of

increasing trapping pressure. We may be able to de�ne a new variable c(t), that, in the

simplest case may be de�ned as c(t) = c0 + Æ where c0 is the initial trapping and/or

natural mortality rate and Æ is some annual increase in trapping pressure. c resembles

the mortality rate of the superpredator in a three-species food chain (Hastings & Powell

1991), and in the absence of a minimum bound such as l�, a linear increase in c(t) drives

the population to extinction in both models (see inset in Fig. 2 for Sche�er's model).

The combined assumption on the existence of a minimum bound and a linear increase in

trapping pressure is not irrelevant: �rst, because a predator such as the lynx is not a fully

generalist species (Bergerud 1983), and second, because a linear increase in the number

of Indian trappers could have acted from the late 1700's to 1850, approximately. Besides,

this linearity can result in counterintuitive, nonlinear records in lynx dynamics.

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: Bifurcation diagram for the modi�ed version of the UPCA model, where c is the

bifurcation parameter. vmax = 150. The rest of parameters are the same as in Blasius et al.

(1999). Inset Attractor resulted from applying a linear increase in c(t) for Sche�er's (1991)

planktonic three-trophic model. r = 0:265. The rest of parameters are the same as in the

original manuscript. Observe that, after drawing some orbits, the top-predator's trajectory

crash to extinction.

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 the

absence of hares. We have assumed, more realistically, logistic growth:

dv

dt= av(1�

v

vmax

) (2)

where vmax is the carrying capacity for vegetation. In our simulations we have set

vmax = 150

� The interaction terms are

Holling 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�ected by changes in trapping

e�ort (i.e., there are no indirect interactions involving mortality such as a reduction

in the mortality rate due to a reduction of the population due to an increase in trap-

ping). Thus, our assumption does not involve any multiplicative e�ect in mortality

due to an increase in c(t).

The rest of parameters are the same as in Blasius et al. (1999).

As shown in Fig. 2, the incorporation of c(t) as a linear increasing function is clearly

followed by a period-doubling route to chaos from a more or less stable dynamics for

lynx populations. The temporal evolution of the simulation (Fig. 3A) also resembles the

original data. The amplitude of the oscillations also grows with the increasing "trapping

e�ort" or mortality term.

Thus, we could at a �rst sight support the idea that a slow growth of trapping pressure

could have taken place following successive colonization of Canadian forests by fur traders,

so the idea of chaotic dynamics behind the data could be supported through a continuous

route to chaos, but none of the important events that occurred could have played any role

8

in it, so Scha�er's initial conjecture would not have any sense, since the amalgamation of

NWC and HBC would not be signi�cant 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 reconstructed attractors produced by the modi�ed UPCA model for:

top a linear increase in c(t) (hunting or trapping pressure); bottom c(t) as a \step" function. c(t)

is represented in dashed lines.

If we recall the return map from the lynx fur returns (by plotting successive maxima)

as in Fig. 4(A), we can observe that its structure reminds that of a unimodal map,

similar to the logistic map for populations with non-overlapping generations (May 1976).

For the lynx data, there can be seen two di�erent sets of points: the one with lower

amplitudes corresponds to the years before the amalgamation, that can be viewed as a

periodic attractor (Scha�er 1985). The other set of points, though somewhat noisy, may

describe a two-point cycle or even chaotic dynamics. This di�erence between these two

sets can not be adequately described in terms of a linear increasing trapping e�ort. In

such a case, the two sets would overlap, due to the slow displacement over the "tuning"

parameter c(t). Instead, we made simulations where

c =�c0 if t � tcc1 otherwise

9

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 reconstructed attractors produced by the modi�ed UPCA model for:

top a linear increase in c(t) (hunting or trapping pressure); bottom c(t) as a \step" function. c(t)

is represented in dashed lines.

In this "step" function we assume c2 >> c1, and t is the iteration number. Thus,

now the trapping e�ort "jumps" along the bifurcation diagram due to some signi�cant

event. The return map corresponding to this simulation is represented in Fig. 4(B),

and describes two characteristic sets de�ning a stable point (corresponding to periodic

dynamics in the original series) and a more complex attractor in a unimodal map. This

map resembles more a Ricker map than a logistic one, but a more similar return map

can be obtained by introducing random uctuations in the l� parameter. Moreover,

the attractor reconstructed from the simulation reproduces a sudden shift in the orbits

followed by the dynamics (Fig. 3B).

3 Discussion

A considerable number of causes has been proposed in the existing literature for explain-

ing the aparent ten year cycle for the series of lynx returns taken from the Hudson's Bay

Company archives. Recently, however, some studies have shown: �rst, that the aparent

10

cyclicity is not so obvious, second, that although there is some dominant frequency in-

trinsic in the series (Fig. 1) (see Tong 1990), the amplitude of the series shows a erratic,

chaotic route (Scha�er 1985). Blasius et al. (1999) demostrate trough a simple tritrophic

model similar to that of Arneodo et al. (1980) that uniform frequency and chaotic am-

plitude can be easily achieved.

The utilization of tritrophic webs for characterizing lynx dynamics is the simplest and

clearest example of chaotic dynamics for a natural ecosystem. But the historical series of

lynx returns, starting from 1735, shows tremendous qualitative and quantitative changes

that can not be reproduced without some kind of \intervention analysis". A simple model

such as the UPCA model (Blasius et al. 1999) is a good one for this purpose; however,

others could have been used (Bascompte et al. 1997).

The introduction of the trapping e�ort as the \intervention variable" shows, according

to Scha�er's �rst conjecture, that a period-doubling route to chaos might have acted over

the system. Thus, c becomes the bifurcation parameter, and constitutes an additional

evidence for the chaoticity of the lynx series, and could even represent a fourth trophic level

insert into the system (Weinstein 1977). Moreover, historical records point clearly toward

this possibility. The monopoly created over the pelts market since the amalgamation of

NWC and HBC companies in 1821 seems to be the key event. At that time, Indians had

been driven into a new mercantile economy based thoroughly on the fur trade. Thus,

speculation and monopoly by the HBC, and the fact that the prices paid for pelts to the

trappers remained �xed for one century (Newman 1985), drew a system of exploitation

in which a higher trapping pressure was its main characteristic.

In this context, we have supported the idea that a sudden shift, rather that a smooth

increase, could have occurred in the trapping pressure, in response to the amalgamation.

The sudden change in the attractor reconstructed from the original series (see Scha�er

1984) and the corresponding return map of lynx maxima (Fig 4(A)) support this idea. The

amalgamation was followed by a sudden reduction in the number of permanent salaried

workforce and the number of posts dispersed over the territory in a short time step (only

�ve years), so the Indians had been forced rapidly towards a greater exploitation.

There still remain some open questions. Scha�er's second conjecture, which states

11

that a shift between two di�erent basins of attractions could have happened, can not be

neglected yet. In any case, the deterministic nature of the uctuations and the existence

of some internal event playing a role at some point at the beginning of the 19th century

represents a strong evidence for explaining this shift.

Besides, the inclusion of space in the system may have counterintuitive responses

in lynx dynamics, and can support the deterministic nature of it, giving raise to the

appearance of spatial waves (Bascompte et al. 1997; Blasius et al. 1999). Such spatial

synchrony may induce the growth of stable structures not only in lynx, but also on hares

and vegetation.

Nowadays, there are recommendations for lynx harvesting from governmental policies,

such as the suspension of harvests for the 3-4 year low of hare cycle (Brand & Keith 1979),

so potentially more lynx available for harvesting in peak years. This corresponds to some

kind of periodic perturbations in the trapping pressure, acting as a form of chaos control

(see Sol�e et al. 1999 and references therein). The consequences of this type of perturbation

regime under spatial domain still need a theoretical background and will be regarded in

future work.

Acknowledgements

Anne Morton and the Hudson's Bay Company kindly gave invaluable historical informa-

tion on trapping and workforce in Canada. This work has been partially supported by

the Santa Fe Institute (RVS) and grant DGYCIT PB97-0693.

12

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