Demographic responses of a site-faithful and territorial predator to its fluctuating prey:...

Demographic responses of a site-faithful and territorial predator

to its fluctuating prey: Long-tailed skuas and arctic lemmings

Frederic Barraquand∗,1, Toke T. Høye2,3, John-Andre Henden1, Nigel G. Yoccoz1,

Olivier Gilg5,7, Niels M. Schmidt2,4, Benoit Sittler6,7, and Rolf A. Ims1

1Department of Arctic and Marine Biology, University of Tromsø, 9037 Tromsø,

Norway

2Arctic Research Centre, Aarhus University, DK-8000 Aarhus, Denmark

3Department of Bioscience, Aarhus University, DK-8410 Rønde, Denmark

4Department of Bioscience, Aarhus University, DK-4000 Roskilde, Denmark

5Laboratoire Biogeosciences, UMR CNRS 5561, Universite de Bourgogne, 21000

Dijon, France

6Institut fur Landespflege, University of Freiburg, 79106 Freiburg, Germany

7Groupe de Recherche en Ecologie Arctique, 21440 Francheville, France

∗ Corresponding author: [email protected]

Summary

1. Environmental variability, through interannual variation in food availability or climatic variables,

is usually detrimental to population growth. It can even select for constancy in key life-history traits,

though some exceptions are known. Changes in the level of environmental variability are therefore

important to predict population growth or life-history evolution. Recently, several cyclic vole and

lemming populations have shown large dynamical changes, that might affect the demography or life

histories of rodent predators.

2. Skuas constitute an important case study among rodent predators, because of their strongly

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saturating breeding productivity (they lay only two eggs) and high degree of site fidelity, in which

they differ from nomadic predators raising large broods in good rodent years. This suggests that they

cannot capitalize on lemming peaks to the same extent as nomadic predators, and might be more

vulnerable to collapses of rodent cycles.

3. We develop a model for the population dynamics of long-tailed skuas feeding on lemmings to

assess the demographic consequences of such variable and nonstationary prey dynamics, based on data

collected in NE Greenland. The model shows that populations of long-tailed skua sustain well changes

in lemming dynamics, including temporary collapses (e.g. 10 years). A high floater-to-breeder ratio

emerges from rigid territorial behaviour and a long life expectancy, which buffers the impact of adult

abundance’s decrease on the population reproductive output.

4. The size of the floater compartment is affected by changes in both mean and coefficient of

variation of lemming densities (but not cycle amplitude and periodicity per se). In Greenland, the

average lemming density is below the threshold density required for successful breeding (including

during normally cyclic periods). Due to Jensen’s inequality, skuas therefore benefit from lemming

variability; a positive effect of environmental variation.

5. Long-tailed skua populations are strongly adapted to fluctuating lemming populations, an

instance of demographic lability in the reproduction rate. They are also little affected by poor lemming

periods, if there are enough floaters, or juveniles disperse to neighbouring populations. The status of

Greenland skua populations therefore strongly depends upon floater numbers and juvenile movements,

which are not known. This reveals a need to intensify colour-ringing efforts on the long-tailed skua at

a circumpolar scale.

Text: c. 7900 words (Main text: c. 7100 + Legends: c. 800)

Key-words environmental variance; floaters; population cycles; territoriality; demographic buffer-

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Introduction

Classic ecological theory demonstrates that variability in vital rates is inherently detrimental to pop-

ulation growth (Lewontin & Cohen, 1969), which suggests that environmental variability negatively

influences population growth rate and density. However, recent theoretical developments (e.g. Drake,

2005; Boyce et al., 2006) have shown that some positive effects of environmental variability are possi-

ble when the relationships between vital rates and the environmental variables are nonlinear, due to

Jensen’s inequality. This spawned life-history theory considering the possibility of selection for convex

reaction norms, or demographic lability (Koons et al., 2009), which might happen in systems that are

subjected to strong environmental variability. In many cases, nonetheless, the effect of environmental

variation on population growth is overall negative (van de Pol et al., 2010; Jonzen, et al., 2010), and

even more so when density-dependence is not at work (Barraquand & Yoccoz, 2013). Demographic and

life-history theory on nonlinear reaction norms can be further complexified with temporal autocorrela-

tion in environmental variables. Non-linearities can indeed transform the colour of the environmental

noise (Laakso, Kaitala & Ranta, 2001, 2003; Garcia-Carreras & Reuman, 2011), and it has been

shown that temporal autocorrelation can seriously affect population growth (Tuljapurkar & Haridas,

2006). Yet, nonlinearities and noise temporal autocorrelation may combine in non-intuitive ways in

empirically-derived population dynamics models; whether temporal autocorrelation matters in such

empirically grounded models is still unclear (van de Pol et al., 2011). Predicting how populations react

to changes in environmental variability therefore requires population dynamics models with explicit

functional relationships to environmental variables. Conceiving and analysing such a model is what

we attempt here, in the case of an arctic-breeding seabird, the long-tailed skua, whose demography is

strongly forced by the cyclic and nonstationary nature of its lemming prey population dynamics. The

model is parametrised with long-term data from Greenland.

In arctic ecosystems where vole and lemming populations are often strongly oscillating, specialist

rodent predators have evolved various solutions to cope with such environmental variability (Andersson

& Erlinge, 1977). Nomadic specialists such as snowy owls or arctic foxes track their main prey over

vast distances, and trade the costs of dispersal for the odds of finding prey-rich breeding grounds

(Andersson & Erlinge, 1977). However, other predator species preying on cyclic rodents adopt an

opposite strategy, and display a strong site tenacity (i.e. both site fidelity and territoriality), which is

thought to be adaptive for bird species with small clutches and high adult survival (Andersson, 1980).

The long-tailed skua is a good example of that life history strategy. This peculiar long-lived seabird



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specializes on a terrestrial food resource just for the breeding season: voles and lemmings (Andersson,

1976). Most rodent predators can respond strongly to rodent outbreaks; foxes and snowy owls can

have more than a dozen young in good rodent years. In contrast, long-tailed skuas do not lay more

than two eggs. This strongly saturating breeding capacity suggests that they would benefit from a less

variable food supply, with mostly intermediate values of lemming densities. Yet these birds should be

adapted to the large multiannual fluctuations of their prey, as theory predicts (Andersson, 1980). The

basic life-history theory developed for such animals contrasts fully cyclic versus random environmental

variation (Andersson, 1980). Theory is therefore missing to connect knowledge of predator demography

to more realistic prey dynamics through nonlinear functional forms, especially in the case where prey

dynamics is changing.

Cycles of northern voles and lemmings have recently been reported to fade in a number of arctic

and boreal ecosystems (e.g. Kausrud et al., 2008; Ims, Henden & Killengreen, 2008; Gilg, Sittler &

Hanski, 2009; Schmidt et al., 2012; Cornulier et al., 2013), and in general, it is well-known that rodent

population dynamics can alternate between periods of cyclic and noncyclic dynamics (Steen, Yoccoz &

Ims, 1990; Angerbjorn et al., 2001; Henden, Ims & Yoccoz, 2009). The explanations for such temporal

(as well as spatial) variation in cyclic tendency of northern rodent populations generally invoke changes

in snow cover and quality (Hanski, Hansson & Henttonen, 1991; Ims, Henden & Killengreen, 2008;

Kausrud et al., 2008), although changes in local community composition have been put forward as a

possible cause of changing dynamics in populations of boreal voles (Hanski & Henttonen, 1996; Sundell

& Ylonen, 2008; Brommer et al., 2010). In lemmings, snow frost-melt events have been shown in a

mountain-tundra ecosystem (Finse, southern Norway) to be influential in stopping lemming outbreaks,

and maintaining a prolonged lemming-poor period that extends from 1995 up to 2012 (Kausrud et al.,

2008 and E. Framstad, pers. comm.). The site-tenacious strategy described above depends on the

rodent cycle ‘kicking back’ at some point - but when exactly? A question of interest, in this context

of nonstationary lemming dynamics, is how, and for how long, populations of site-tenacious predators

such as long-tailed skuas can withstand such changes in prey dynamics? Moreover, it is yet unclear

how variability in the ‘normal’ 3-5 year lemming cycle affects site-tenacious predators such as skuas.

A previous study on the Arctic fox (Henden et al., 2008) examined the consequences of environ-

mental variability for a predator with stronger reproductive responses, and found positive effects of

higher variability in the rodent cycle, at low average rodent density (though see Discussion for a re-

evaluation). Given the strongly saturating breeding success of long-tailed skuas, we could expect the



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opposite trend, a negative effect of variability. Our model shows that Greenlandic long-tailed skua

populations would actually benefit from more variability in the lemming cycle, confirming theoretical

possibilities. The model also highlights the importance of the floater compartment (currently unob-

served) for population persistence, and shows skua populations could include quite large numbers of

floaters, which contributes to their ability to withstand lemming-poor periods. This echoes recent

concerns of conservation biologists realizing that breeding birds might sometimes be only the ‘tip of

the iceberg’ (Penteriani, Ferrer & Delgado, 2011; Katzner et al., 2011), and large floater compartments

of bird populations might be missed due to focus on territory holders. Finally, our results suggest that

some skua populations might act as sources and others as sinks, stressing the need for more monitoring

of skua survival and movements (e.g. through colour-ringing) so that survival and dispersal rates can

be evaluated.

Methods

SPECIES ECOLOGY AND STUDY SITE

The ecology of the long-tailed skua has been thoroughly described in Andersson (1976). Outside of the

breeding season, long-tailed skuas are kleptoparasitic, migratory seabirds (Sittler, Aebischer & Gilg,

2011; Wiley & Lee, 1998), and during the breeding season, they specialize on rodents (in the study

site, lemmings). They can consume large quantities of lemmings, up to 5 per day per individual in

peak rodent years (Gilg et al., 2006). Breeding skuas are territorial, and fight to access territories

of constant size across years, which keeps the breeder compartment of the population fairly stable

(Andersson, 1976; Gilg et al., 2006; Meltofte & Høye, 2007), including in the absence of reproduction.

We use data from two study sites in NE Greenland, at Karupelv valley, Traill island, and Zackenberg

research station (c. 300 km North).

Lemming time series from both sites are shown in Fig. 1. High amplitude fluctuations were present

in Karupelv (Traill island), but have now collapsed into more dampened fluctuations. The dynamics at

Zackenberg seems synchronous to that of Traill island, though the amplitude of fluctuations is smaller

(Fig. 1).



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MODEL STRUCTURE

In order to highlight the most important model components, the model is constructed from the bottom

up, first with a relatively detailed description of population structure and lemming dynamics (to avoid

missing important ecological processes). During its analysis, we progressively simplify the model,

which allows for analytical solutions that confirm and extend the simulation results. Demographic

stochasticity and other sources of environmental stochasticity than lemming fluctuations are ignored

in the model. This is a one-sex model for a large closed population in an environment forced only by

a fluctuating prey density, and with very strong density regulation.

Lemming population dynamics

We used two different annual models for lemming dynamics, both phenomenological. The first model,

usually called the Maynard Smith model (Maynard Smith & Slatkin, 1973; Maynard Smith, 1974;

Grenfell et al., 1992), is quite useful to model cyclic/noncyclic alternance, and it produces cycles with

very skewed distributions, as often observed in lemmings. In contrast, the second model, called log-

linear AR(2) model (Royama, 1992), produces cycles with less asymmetric distributions, but is more

helpful to model smoother changes in variance, for a constant median (constant mean log-density, as in

Henden et al., 2008). Finally, we also reduced the lemming dynamics to a simple lognormal probability

distribution (without temporal autocorrelation), at very little loss of generality.

The simple Maynard-Smith model differs from other discrete-time models by its sigmoid-shaped

density dependence, that allows for long cycles despite the absence of delayed density-dependence

(Getz, 1996). It is commonly written

Nt+1 = NtRN

1 + (Nt/K)γ(1)

with K the threshold density marking the onset of density-dependence, RN a maximal population

growth rate, and γ the abruptness of density-dependence. This model has the desirable property that

when RN is large, γ almost only affects periodicity and RN mostly amplitude (max-min densities), see

Appendix S3 for more detail.

The second model is a second-order autoregressive model, on a logarithmic scale (Royama, 1992)

that exhibits quasi-cycles when under the influence of environmental stochasticity, usually written with

logarithms (xt = ln(Nt)) in its centered form



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xt+1 −m = (1 + Ω1)(xt −m) + Ω2(xt−1 −m) + ǫt (2)

where ǫt ∼ N(0, σ2).

This model is helpful to separate the effects of periodicity and variance (Henden et al., 2008).

Skua population structure

Adult breeders do not necessarily breed every year in the model, but return each year to their breeding

site to defend their territory (Gilg et al., 2006; Meltofte & Høye, 2007). The number of breeders is

denoted Bt. Productivity (breeding output) π(Nt) depends on the density of lemmings Nt in year

t. The fraction of breeders that survive each year is sA, and we assume for simplicity that once

they acquire a territory, breeders do not lose it (consistent with the fact they come back each year

irrespective of whether there are enough lemmings to reproduce).

The yearly production then enters the juvenile stage (see Figure 2). We use a stage-based framework

for simplicity, with Jt the number of juveniles. We assume that the annual survival probability of

juveniles is a constant sJ , while the annual probability of leaving the juvenile stage is another constant

ϕ, whose inverse 1/ϕ is the average duration of the juvenile period. Once individuals leave the juvenile

stage they become floaters (numbers Ft).

Floaters stay floaters until they can finally enter the breeding population. This is where the

territoriality of skuas and the resulting density-dependent recruitment comes in. We use a form of

strong density-dependence previously applied by Brommer et al. (2000) to a model for territorial owls.

Until there are less floaters than available territories, all territories freed by breeder death are taken

over by floaters (according to the data available, Meltofte & Høye, 2007). We define KB as the total

number of available territories.

This model is akin to a ‘musical chair’ or ‘lottery’ contest: there are KB − Bt ‘seats’ available at

the ‘skua breeding table’, and these seats are always filled when there are enough floaters available

around. This leads to the formula for the recruitment rate to the breeder population, R(Bt) =

min(KB −Bt, Ft)/Ft.

The above assumptions on skua life-history lead to the life-cycle graph of Fig. 2, and the following

projection matrix representation:



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Xt =

Jt

Ft

Bt

, Xt+1 =M(Nt)Xt, M(Nt) =

(1 − ϕ)sJ 0 π(Nt)

ϕsJ (1−R(Bt))sA 0

0 R(Bt)sA sA

(3)

We assume that skua breeding success depends on lemming density, but that skuas have a negligible

effect on lemming densities. This is clearly an approximation. We know skuas in Greenland can have

an important effect on lemming populations, notably by reducing lemming densities to levels where

regulation by stoats is possible (Gilg, Hanski & Sittler, 2003; Gilg et al., 2006). They are responsible

for keeping the lemming cycle within bounds in Gilg et al.’s model, being present even in low-lemming

years (Gilg, Hanski & Sittler, 2003, J.A. Henden & F. Barraquand, unpublished data). However, here

we are mostly concerned with the effect of lemmings on skua populations, which warrants the use of

such a bottom-up approximation.

PARAMETRIZATION

The following sigmoid skua productivity function proved suitable to represent the empirical data (Fig.

2b and Gilg, Hanski & Sittler, 2003; Gilg et al., 2006):

π(N) = πm

(

1−1

1 + (N/Nthresh)η

)

(4)

Other functional forms are possible, provided the function is sigmoid. The importance of a sigmoid

shape comes from two facts: (1) we know empirically the curve accelerates at low densities from the

data (Fig. 2b), and (2) it has to decelerate at large densities, because the maximum number of eggs is

two.

We chose an asymptote πm = 1.75 because the maximum number of eggs layed is 2, and there is

always some nest predation (Meltofte & Høye, 2007). Nthresh = 6 and η = 3 have been chosen so

that the function matches that estimated on the Traill island skua population (Gilg et al., 2006, and

Fig. 2b). Even with the combined dataset (data from both sites), there is a large margin for error in

the parametrisation, given the data scarcity around the inflection point. Accordingly we consider two

additional values for the threshold, i.e. Nthresh = 6± 2 (Fig 2b).

While the total number of territories, KB, is approximately 20 in in the extended Zackenberg area

based on maximum observed data, we chose 25 to get a conservative estimate of the maximum number



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of breeder territories in the population.

The adult annual survival probability sA is directly related to the average duration of the adult

stage lA = 1/(1− sA) which is itself closely related to longevity. For instance, sA = 0.9 corresponds to

an average adult life span of 10 years and sA = 0.95 an adult life span of 20 years. Hence, 0.9 - 0.95

seems a range of acceptable values for that species, in line with the estimates of Andersson (1976). We

assume floaters and breeders have the same survival probabilities for parsimony.

We considered a range of juvenile survival probabilities sJ between 0.5 and 0.8. Andersson (1976)

suggests 0.75-0.8, but if there is some juvenile emigration, which is likely, apparent survival could be

lower. We assumed a transition rate ϕ = 0.25, which implies individuals attaining maturity spend on

average 4 years in the juvenile compartment.

Results

NUMBER OF FLOATERS, ASSUMING HABITAT SATURATION

When floaters are available, R(Bt) = KB − Bt/Ft because the number of floaters (Ft) is large when

compared to the number of free territories KB − Bt. In that case, the habitat is saturated, and the

matrix multiplication for the breeder compartment of the model yields Bt = KBsA (Appendix S1).

Hence the number of breeders is fixed to B = KBsA at all times (provided floater abundance is

large), and is dependent only on their survival rate. Combining this result with equation 3, we can

then calculate the number of juveniles as

Jt+1 = (1− ϕ)sJJt + 0.5πtBt = (1− ϕ)sJJt + πt0.5KBsA (5)

where πt = π(Nt) is the productivity, hereby obtaining the number of juveniles as a simple recur-

rence equation. Unfortunately, because the sequence (πt) is externally driven by lemming dynamics,

this is not possible to solve right away for equilibrium values. We assume in the following, as a first

step, that the productivity is constant, i.e. lemmings are not fluctuating (an assumption later relaxed).

Given equation 5, we obtain

J∗ =0.5πKBsA

1− sJ + ϕsJ(6)

J∗ increases with KB, sA, π, and decreases with ϕ (less juveniles if they mature faster). ∂J∗

∂sJ> 0



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as well.

Including constant numbers of juveniles and breeders into the remaining floater equation (Appendix

S1), and solving for equilibrium yields a floater-to-breeder ratio (for π fixed)

ρ = F ∗/B∗ = lA0.5ϕsJπ

1− sJ + ϕsJ− 1 (7)

where lA = 11−sA

is the average duration of the adult stage. The factor 0.5ϕsJπ/(1− sJ + ϕsJ) =

ϕsJJ∗/B∗ is the number of juveniles recruiting annually into the floater compartment per breeder.

Let us call this the ‘effective adult production’, say πA. Then we arrive at a simple expression for the

floater-to-breeder ratio

ρ = lAπA − 1 (8)

This is the lifetime production of adults (both breeders and floaters) by a breeder individual,

minus one. So the floater-to-breeder ratio is the net contribution of the average breeder to the adult

pool. Importantly, this explains why there should be so many floaters in long-lived territorial bird

populations; as long as one breeder produces at least two adults during its life, there should be as

many breeders as floaters (ρ = 1). This last result is quite remarkable. Of course, in real populations

floater numbers will probably be much smaller because floaters settle also in suboptimal habitats

(though see Katzner et al., 2011). However, the model still suggests a very large floater compartment

emerging from the type of recruitement we assumed from field observations.

FLOATERS BUFFER LEMMING LOWS

Floaters delay the decline of the breeder population (Fig. 3), because as long as there are floaters they

can replace the breeders, and new breeders can reproduce as soon as the lemming cycle restart. A key

element for this to work is a high adult survival rate.

The process by which the floater pool is emptied can be analysed mathematically, provided a few

simplifications. Assuming breeder numbers are constant as above (B = KBsA) and production of

juveniles has already stopped, we have Ft+1 = sAFt −L(sA), where L(sA) = KBsA(1− sA) is the loss

of floaters to the breeder compartment (demonstration in Appendix S2). L(sA) decreases with sA for

the observed values, and sA is also the common ratio of this arithmetico-geometric sequence. Thus,

the larger adult survival sA, the slower the decline in floater numbers.



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We performed a sensitivity analysis of the model to the duration of the lemming low-density period

(from 0 to 50 years). Figure 3b reveals that longevity promotes population persistence. Again, the

mathematical approximation presented in Appendix S2 allows one to verify the numerical findings. We

computed the time period separating the stop of juvenile production from the decline of the breeder

pool, and it is shown to be approximatively T ≈ −1 + lA × ln(1 + ρ0) (see Appendix S2) where ρ0

is the initial floater-to-breeder ratio and lA the average duration of the adult stage, closely related

to longevity. Thus, the time for the breeder pool to decline scales proportionally with the average

duration of the adult stage (minus one year), and the coefficient of proportionality is ln(1+ ρ0), which

means that T increases but decelerates with ρ0. In situations where the initial floater-to-breeder ratio

is small (e.g. ρ0 = 0.2) breeder numbers can decline fast (0-2 years), while in situations where ρ0 is

close to 1 or more, it will take between 5 and 10 years for reasonable values of lA. See section “Effect

of periods without lemmings on skua population persistence” for more results on how T is influenced

by ρ0.

THE EFFECT OF VARIABILITY IN LEMMING DENSITIES DEPENDS ON THE MEAN

Let us assume that (Nt) can be represented as a simple random sequence, characterised by a mean and

variance (we show in the following sections that temporal autocorrelation does not matter). Starting

from the juvenile equation 5, we can write the following expectation (assuming the process is ergodic,

i.e. averaging over time and realizations yields the same result)

E(Jt+1) = (1− ϕ)sJE(Jt) +KBsA0.5E(πt) (9)

Defining J∗ = E(Jt) = E(Jt+1) the long run average of juvenile abundance, we obtain

J∗ =KBsA0.5πe1− sJ + ϕsJ

(10)

where πe is E(π(Nt)) (the expected productivity). So we recover the same expression as in the

constant-productivity case (section “Number of floaters, assuming habitat saturation”), except here

J∗, and by extension F ∗ which is linearly related to J∗, include an expected instead of constant

productivity. How πe depends on lemming interannual variability can be seen with a second-order

Taylor development (Appendix S1, using primes for derivatives), which eventually leads to:



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E(π(Nt)) = πe ≈ π(N ) +1

2π′′(N)σ2

N (11)

Because of the usual scaling between variance and mean, it is desirable to rewrite that formula

with the coefficient of variation CV. There is a good correlation between CV and the commonly

used S-index for rodent cycles (e.g. Ugland & Stenseth, 1985, and our results). The expression of

the expected productivity (on which both juvenile and floater abundances depend) becomes πe =

π(N) + 12π

′′(N)N2CV2. Thus, depending on the value of average lemming abundance N

1. A large negative effect of increased lemming variability (CV) is expected at high N (N >>

Nthresh), because π′′(N) < 0 and this is multiplied by N2 which is large.

2. A small positive effect of increased lemming variability is expected at low N (N << Nthresh),

because π′′(N) > 0 but this is multiplied by N2 which is small.

Consequently, we can expect that the effect of variability in lemming density (without changes

in the mean) on the average numbers of juveniles and floaters will, in general, be quite negative,

unless average lemming density is low (below 4 lemmings/ha, which is actually the case here). This

approximation works only for moderate amplitude fluctuations; if instead both concave and convex

portions of the productivity π(N) function are used frequently on a (Nt) sequence, the approximation

is likely to break down. A more general method is presented in section “Expected productivity: A

general expression for large and skewed rodent variability”.

COEFFICIENT OF VARIATION IS MORE RELEVANT THAN CYCLE AMPLITUDE

The effect of process standard devation σ in the log-linear AR(2) model (on a log-scale, so this is a

measure correlated to CV) on the quantity of floaters and juveniles (Fig. 4a) depends on whether

Nmean < Nthresh (positive effect) or Nmean > Nthresh (negative effect). For the populations studied

in NE Greenland (both Zackenberg and Traill Island), averages of lemming density suggest that more

variability would actually be beneficial (Fig. 4b).

The Maynard Smith model provides a different story than the loglinear AR(2) model, and shows the

difference between the effect of cycle amplitude (max-min densities) and the effect of cycle variability

(i.e. CV or S-index, Stenseth, 1999). Increasing the maximum growth rate RN in the Maynard

Smith model leads to oscillations of higher amplitude (Appendix S3). However, despite the increase

in amplitude, the coefficient of variation saturates (Appendix S3). Increasing RN increases cycle



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amplitude but not variability in a statistical sense. Interestingly, the effect of increasing RN on the

floater and juvenile compartment are negligible when CV saturates (not shown) and therefore, cycle

amplitude per se is not important.

Using again the loglinear AR(2) model, we found no discernable effects on the numbers of floaters

and juveniles of cycle periodicity when variance and mean of lemming densities were constant. This is

in line with previous modelling results (Henden et al., 2008). However, this should not be interpreted

as an absence of an effect of the period of the lemming cycle in general. Changes in periodicity are

often correlated to change in mean and variance (Henden et al., 2008). Therefore, the period of the

lemming cycle, as illustrated in the following sections, can influence the skua population; but it does

so mostly through its indirect effect on the mean lemming density.

EXPECTED PRODUCTIVITY: A GENERAL EXPRESSION FOR LARGE AND SKEWED RO-

DENT VARIABILITY

As shown above, a key quantity in the model is the temporal average of skua productivity π(N) =

1T

∑Ti=1 π(Ni), that converges in the long term limit (T → ∞) to the expectation of π(N) with respect

to all possible N values, denoted E(π(N)). The Taylor development of π(N) presented in section

“The effect of variability in lemming densities depends on the mean” shows that increased lemming

variability (i.e. increased CV) has small positive effects at low N and large negative effects at large N .

However, such an approximation is limited to small lemming variability (e.g. CV < 0.25); we provide

here an expression for large lemming variability.

For any continuous random variable X with probability density ψ(x), the relation E(f(X)) =´

f(x)ψ(x)dx is valid. Formally, this is even true for any ergodic stochastic process, which is a reason-

able assumption for lemming densities (Nt) in the cyclic regime, and expected productivity is therefore

obtained with the formula

E(π(N)) =

ˆ

∞

0

π(x)ψ(x)dx (12)

where ψ(x) is the marginal probability distribution of the lemming values. We can therefore

compute the expression without resorting to stochastic simulations, through numerical integration or

analytical derivations. It is difficult to obtain a closed form solution for E(π(N)), but greater analytical

insight can be obtained if we replace the sigmoid productivity function by a step function, i.e. π(N) = 0

for N < Nthresh and πm above Nthresh. The equation then becomes



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E(π(N)) =

ˆ Nthresh

0

0× ψ(x)dx +

ˆ

∞

Nthresh

πm × ψ(x)dx = πm(1−Ψ(Nthresh)) (13)

where 1−Ψ(Nthresh) =´

∞

Nthreshψ(x)dx is the probability that N is above the reproductive thresh-

old and Ψ the cumulative distribution function of N . The expected productivity, in the case of an

extremely steep sigmoid (threshold function), is therefore the maximal productivity times the frequency

of lemming densities above the threshold.

Figures 5 show how lemming variability affects the expected productivity (see also Appendix S4).

In each case, the expected productivity was evaluated by numerical integration of the deterministic

integral (as opposed to stochastic simulation). We are varying jointly the mean and variability (coef-

ficient of variation), looking at how variability affects expected productivity when keeping the mean

(and in Appendix S4 the median) constant.

The mean/CV decomposition is a somewhat theoretical way of looking at variability effects: in

real datasets, a less variable lemming cycle might correspond to both lower mean and lower CV. Or,

taking an example from modelling, the MS population model (Appendix S3) suggests that very high

maximum growth rates, leading to high amplitude cycles, always correspond to higher mean but CV

saturates.

Therefore, a real trajectory of change in rodent dynamics might correspond to many possible curves

in the (mean, CV) plane. In the case of the Traill island series (before and after cycle loss, i.e. pre-

or post-2000), we see decreases in both mean and coefficient of variation (Fig. 5). These changes can

however also be represented by a constant median (constant mean on logarithmic scale) and decreasing

S-index (Appendix S4).

EFFECT OF PERIODS WITHOUT LEMMINGS ON SKUA POPULATION PERSISTENCE

When lemming peaks are really well-delineated, because lemming density is almost zero outside of

peaks; or equivalently the reproductive threshold Nthresh is large with respect to average lemming

density (such as on Greenland), it becomes appropriate to think of a binary sequence of skua repro-

ductive events, Erepro = (00001000100010001). We show in the preceding section (Fig. 5 and Appendix

S4) that such usual simplification (Andersson, 1980; Brommer et al., 2000) yields qualitatively similar

results to a sigmoid function for reproduction.

Let us consider an expected skua productivity πe = πm/p, where p is the period between peaks

(and 1/p the frequency of above-threshold years). Note this does not necessarily assume a regular time



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series, since in a sequence of Bernoulli variables with parameter 1/p, we would have the same mean.

A key question in the context of long-term population persistence in such a poor environment is:

what is the critical value p for which there are no floaters anymore? As we have shown previously, the

presence of floaters postpones the decline of the breeder pool during a lemming shortage. Thus in a

territorial population with few breeders, the real criterion for long-term persistence is whether there

are floaters around or not. From equation 7, we obtain the expression

p =lAϕsJ0.5πm1− sJ + ϕsJ

(14)

The derivation is provided in Appendix S1. We see that p nonlineary depends on sJ . While we have

poor “guesstimates” for sJ , we can be relatively confident in all other parameters. Andersson (1976)

suggests sJ is in the range [0.7; 0.85]. This assumes however no emigration from the population; in

contrast, if sJ represents apparent survival, we could have much lower estimates. This is quite plausible

because juveniles are likely to disperse from source populations to other areas. Therefore, it seems

relevant to investigate how the critical period depends on sJ .

Figure 6 shows the relationship to juvenile survival sJ , and that for our assumed parameter values,

skua populations can withstand long periods (e.g. >10 years) without lemmings. It also suggests that

any measurement of low apparent survival (sJ < 0.5), provoking local extinction in 10 years without

lemmings, points to a potential metapopulation structure for long-tailed skuas.

Discussion

In this paper, we analyse a detailed, empirically-based model of long-tailed skua population dynamics,

based on skua demographic data and lemming counts from NE Greenland. The main motivation

for this study was the ongoing changes in lemming dynamics (collapsing cycles, i.e. a main food

shortage from the predator viewpoint) in some of the best studied populations, in both Norway and

Greenland (Kausrud et al., 2008; Ims, Henden & Killengreen, 2008; Gilg, Sittler & Hanski, 2009). The

phenomenon might be generated by climate change, although other interpretations of this phenomenon

are possible, and it cannot be excluded that some populations are erupting elsewhere. Indeed rodent

dynamics have been nonstationary over long timescales, alternating between cyclic and non-cyclic

periods (Angerbjorn et al., 2001; Henden, Ims & Yoccoz, 2009).

Given that relatively high rodent densities are necessary for breeding in long-tailed skuas, that



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breeding output is strongly saturating (maximum two eggs), and that they are strongly site faithful

(Andersson, 1976, 1981), it seems at first surprising that skua populations manage to persist through

long periods of lemming scarcity. In this context, repeated years with failed breeding can appear

worrying from a conservation perspective. However, such breeding philopatry has been shown to be

adaptive for birds that live long, such as long-tailed skuas (Andersson, 1980, 1981). Therefore, a

demographic model including survival processes and population structure is needed to understand the

consequences of nonstationary lemming dynamics for long-tailed skua populations. We constructed

such a model using long-term empirical data. Progressive simplifications of the full version of the

model allowed us to isolate the essential components of the model, and verify simulation results by

analytical approximations.

We found that the ability of skua populations to persist during a lemming shortage depended on the

number of floaters prior to the shortage. For surviving a shortage of 10 years, the floater-to-breeder

ratio should be around one according to the model. In turn, floater numbers before the shortage

depend on the average productivity of breeders during a normal lemming cycle, itself depending on

the probability distribution of lemming densities but not on its temporal autocorrelation.

In section “Coefficient of variation is more relevant than cycle amplitude”, we show that the only

components of the lemming model that really matter to the skua population are the mean and vari-

ability of lemming density N (in the stationary case, without prolonged lemming troughs). The cyclic

nature of the sequence is actually of no importance - one can permutate all the values - because skua

productivity π(N) is the only lemming-dependent quantity in the model. The temporal autocorre-

lation in the lemming values would instead matter if juvenile survival was dependent on N . In this

(hypothetical) case, the cohort produced at time t−1 because Nt−1 was high depends on Nt to survive

during year t (as frequently found for owls, Brommer et al., 2000). But for skuas, where juveniles

depend on marine food immediately after they fledge (i.e. survival is independent of Nt), temporal

autocorrelation does not matter. Similarly, van de Pol et al. (2011) found also no important effects of

temporal autocorrelation in some weather variables. It seems however that in their case other causes

are involved, such as opposite effects of temporal autocorrelation on various demographic components.



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THEORETICAL IMPLICATIONS OF THE SKUA-LEMMING INTERACTION

Positive effects of lemming variability mediated by skua territoriality and longevity

Interannual variability in lemming abundances is found to positively affect skua populations in NE

Greenland. This contrasts with the classical perception that environmental variation negatively affects

demography, but is in line with current theory (Drake, 2005; Boyce et al., 2006; Barraquand & Yoccoz,

2013).

Our model reveals that floaters are likely to be very numerous in healthy skua populations (almost

as numerous as breeders), and this is a direct consequence of the strong territorial system and longevity

of long-tailed skuas. Large number of floaters can buffer population changes in lemming-poor periods.

Floater densities, in turn, are affected by lemming fluctuations (within the normal lemming cycle)

through the temporal average of breeder productivity.

Assuming a period of stationary lemming dynamics, variability in the lemming cycle, as opposed

to a nearly constant lemming density, is beneficial to skua populations relying on a low average food

supply (i.e. below the inflection point of their sigmoid productivity) and detrimental to populations

relying on a high average lemming density. This is because of the nonlinearity of the productivity

function, that has a sigmoid shape for skua populations. Such non-linear averaging effects stem from

Jensen’s inequality (Jensen, 1906), and are well-observed in various areas of ecology (e.g. McNamara

& Houston, 1992; Boyce et al., 2006). For the populations studied in NE Greenland, the temporal

averages of lemming density suggest that more variability in lemming densities would actually be

beneficial (Fig. 4b, 5). It is likely, however, that increases in CV, for a constant average, cannot

benefit productivity when CV is already very large. The distribution of rodent densities is indeed

quite skewed towards low values, and such skewness increases with increased variability.

Evolutionary implications of positive effects of lemming variability

The possibility of positive effects of lemming variability suggests there might be selection for more

and more convex reproduction norms and very variable reproductive output, which has been termed

“demographic lability” (Koons et al., 2009), in contrast to the demographic buffering of life-history

traits (Stearns & Kawecki, 1994; Pfister, 1998; Gaillard & Yoccoz, 2003). Demographic buffering, or

selection for a less variable demographic trait, is expected to happen on traits that contribute largely

on population growth (e.g. adult survival/longevity in long-lived animals), though Koons et al. (2009)

suggest demographic lability is possible as well. This was shown with a density-independent matrix



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model including traits as sigmoidal functions of an environmental variable. Actually, we performed

in another study detailed analyses of such density-independent models varying at the same time re-

production and survival rates, using sigmoid functions convex at low densities (Barraquand & Yoccoz,

2013). The results suggest that demographic lability is more likely to happen in the reproductive rate

if survival is high and varies little - i.e. is demographically “buffered”. Longevity (i.e. high adult sur-

vival) and territoriality (generating density-dependence in recruitment) are instrumental in facilitating

positive effects of environmental variability, and therefore selection for reproduction rates accelerating

at low average prey densities.

The classic literature on life-history evolution in stochastic environments (e.g. Wilbur & Rudolph,

2006) further suggests that the relationship between a strongly prey-driven fertility and longevity has a

somewhat chicken-or-egg nature. Iteroparity, and longevity with it, can evolve in response to stochastic

fertility (Wilbur & Rudolph, 2006). Thus, either reaction norms are pronounced and convex because

longevity is high – or longevity is high because reaction norms are convex and amplify environmental

variability. In the case of the long-tailed skua and related pelagic birds, the phylogenetic signal for high

longevity (see discussion in Andersson, 1976) suggests that convex reaction norms are the adaptation

and high longevity the evolutionary constraint. This is further corroborated by the fact that although

skuas can eat other prey - and could therefore have a more constant reproductive output – they

have specialized on fluctuating rodents. In conclusion, we have here a demographic lability of the

reproduction rate which is likely favoured by the demographic buffering of the survival rate.

In the case of the long-tailed skua, there is a decoupling (i.e. lack of temporal covariance) between

survival of juveniles and reproduction probability. Because of this decoupling, it is clear that it pays

to have a convex reaction norm for low mean prey density. Note however, that some foxes and owls

have juvenile survival dependent on (future) food density (Brommer et al., 2000; Meijer et al., 2013).

It is less clear how such convex reaction norm in reproductive success could be advantageous in those

cases, because high investments in reproduction in good rodent years might be offset by poor survival

the next year. Actually our last results (Barraquand & Yoccoz, 2013) suggest that the results of

Henden et al. (2008), which focuses on such species, are largely due to changes in the mean rather

than variability (the median, or log-mean, was kept constant in Henden et al., 2008).



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THE FUTURE OF GREENLANDIC LONG-TAILED SKUA POPULATIONS

On the importance of floaters

The lemming cycle has been ‘down’ for more than a decade (last peak year 1998) on Traill island and, to

a lesser extent, in the Zackenberg valley (Schmidt et al., 2012). However, this does not imply necessarily

that long-tailed skua populations are endangered. We show that skua populations are typically able

to withstand 10 years of lemming scarcity - or maybe more - if adult and juvenile survival rates are

as high as we currently think they are. However, whether or not long-tailed skua will persist in NE

Greenland depends largely on the floater-to-breeder ratio when there was a lemming cycle (<2000, the

cycle was probably present back to the 1950s, Schmidt et al., 2008), which is unknown.

So far, it has been difficult to assess whether there are indeed many floaters in studied long-

tailed skua populations, because capture-recapture data are too scarce. In general, the ecology and

conservation literatures recognize more and more that bird populations can include a large number of

floaters, and that floaters can have a great demographic impact (Penteriani, Ferrer & Delgado, 2011).

In skuas, the non-territorial fraction of the population might be either non-reproducing at-sea, failing

to reproduce on land in suboptimal areas, or even searching new places. Although adult skuas seem

site-tenacious (Andersson, 1981), it is unclear what juveniles do. In Great Horned Owls, where similar

models have been formulated (Rohner, 1996), models predicted a floater-to-breeder ratio slightly below

but close to one. A floater-to-breeder ratio about 4 has even been recently suggested in populations of

imperial eagles relying on genetic analyses (Katzner et al., 2011). The expected floater-to-breeder ratio

can be investigated thanks to theoretical models (Kokko & Sutherland, 1998; Pen & Weissing, 2000),

but such models better lend themselves to qualitative rather than quantitative conclusions (see however

Hunt, 1998, and for a more data-rich example, van de Pol et al., 2007). We think therefore the most

pressing need to understand how populations of long-tailed skua (and similar bird species) function is

to estimate the sizes of all population compartments, and also whether and how local populations are

connected.

Open or closed populations?

An important question, suggested by the possibly high floater-to-breeder ratios in the model, is: do

floaters emigrate when too numerous? Additionally, are skua always as site-tenacious as the seminal

paper of Andersson (1976) suggests? Are juveniles philopatric? The answer to these questions will

determine the pattern of connectivity between skua populations, and the importance of local popula-



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tion persistence to circumpolar persistence. If adult long-tailed skuas are site-tenacious and juveniles

philopatric, then the extinction risk of local populations (e.g. at Traill island or Zackenberg) has an

important impact on large-scale persistence. In this case, the currently observed lemming-poor periods

could be survived only if floaters were initially as numerous as breeders. But in that scenario, popu-

lations would probably not survive for much longer in NE Greenland, as >10 years without lemmings

peaks have already gone by (Schmidt et al., 2012).

However, even if adults are site-tenacious, local skua populations could be connected thanks to

juvenile dispersal among Arctic regions (e.g. populations of Greenland between themselves or with

Canada; Fennoscandia with Siberia). In this case, what matters is the circumpolar persistence, i.e.

the balance of local colonization and extinction events. In that scenario, a collapse of some skua

populations in Greenland would not matter much in terms of conservation, in case lemming cycles

are maintained in other places in the Arctic. We know from telemetry data that long-tailed skuas

can migrate very long distances in a short period of time (e.g. wintering as far south as South Africa,

Sittler, Aebischer & Gilg, 2011; Gilg et al., 2013), which means they have largely the ability to disperse.

However, whether local populations are actually connected at a circumpolar scale is currently unknown.

More empirical studies using colour-ringing, telemetry, or genetics, are therefore needed to measure

survival and dispersal rates, especially for juveniles, in order to better understand the demography of

such long-lived birds.

Acknowledgements

The research presented here owes much to two long-term monitoring programs: Zackenberg BioBa-

sis program (http://www.zackenberg.dk/monitoring/biobasis/), funded by the Danish Environmental

Protection Agency, and that of the GREA (Groupe de Recherche en Ecologie Arctique, http://grearctique.free.fr/)

at Karupelv valley, Traill island. FB was funded by the Biodiversa ECOCYCLES program. OG was

supported by the French Polar Institute (IPEV; “Interactions” program 1036). We thank X. Lambin,

T. Cornulier, and A. Millon for comments on a previous version of the manuscript. We also thank

two anonymous reviewers and the associate editor for constructive suggestions on the presentation of

results and their evolutionary implications.



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Data Accessibility

The data and computer codes are available on Dryad at doi:10.5061/dryad.8041k

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in time. The American Naturalist, 168, 398–411.

Wiley, R.H. & Lee, D.S. (1998) Long-tailed Jaeger (Stercorarius longicaudus). The Birds of North

America (eds. A. Poole & F. Gill), 365, p. 24. Philadelphia, PA.



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Tables

Table 1: Table of parameters for the skua and lemmings models. DD: Density-dependence. MS:Maynard Smith model. AR(2): Log-linear second-order autoregressive model.

Model Parameter name Symbol Reference value UnitLong-tailed skua Adult survival (breeder and floater) sA 0.93 year−1

Adult stage duration lA 1/(1− sA) = 14.3 yearJuvenile survival sJ 0.75 year−1

Average duration of the juvenile stage 1/ϕ 4 yearNumber of available territories KB 25 NA

Threshold density of the productivity Nthresh 6± 2 indivs.ha−1

Abruptness parameter of the productivity η 3 NAsymptotic productivity πm 1.75 year−1

Lemming MS Max growth rate RN 10 year−1

Threshold density K 2.35 indivs.ha−1

DD abruptness γ 6 NALemming AR(2) Mean log density m 1.5 NA

Direct DD Ω1 -1.76 NADelayed DD Ω2 -0.58 NA



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Figures

1990 1995 2000 2005 2010

1

2

3

4

5

6

7

8

9

10

11

Lemming density N (individuals per ha)

Time (Year)

Traill island

Zackenberg

Figure 1: Collared lemming dynamics at Karupelv Valley, Traill Island (Gilg, Hanski & Sittler, 2003;Gilg, Sittler & Hanski, 2009) and Zackenberg area (Schmidt et al., 2012).



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(1-R(Bt))s

A

(1-φ)sJ s

A

F

BJ

π(Nt)

R(Bt)s

Aφs

J

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Nthresh

Nthresh

Nthresh

Traill data

(a)

(b)

=6

=4

=8


Average number of youngs produced

Zackenberg data

Figure 2: (a) Life-cycle graph of the skua model. B: Breeders (have a territory); F: Floaters (donot have, and wait for a territory); J: Juveniles (cannot reproduce yet). See Table 1 for parametersinterpretations and values. (b) Productivity function π(N) and comparison to empirical data fromboth sites.



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20 40 60 80 1000

5

10

15

Lem

min

gs

20 40 60 80 1000

10

20

Juve

nile

s

20 40 60 80 1000

5

10

Flo

ater

s

20 40 60 80 100

15

20

25

Bre

eder

s

Time (years)

0 10 20 300

2

4

6

8

Juve

nile

s

s

A = 0.9

sA = 0.93

sA = 0.95

0 10 20 300

2

4

6

8

Flo

ater

s

0 10 20 300

5

10

15

Bre

eder

s

Length of lemming trough (years)

(a) (b)

Figure 3: (a) Effect of a period of lemming scarcity on skua population dynamics. Lemmings arein numbers per ha, and bird numbers are abundances (territories/breeding pairs for breeders). Thelemming trough (a 20-year long period with no cyclic peaks and generally low population density) issimulated using RN = 0.5 instead of RN = 10 in the MS model (Table 1). Other parameters are:KN = 2.35, γ = 6, sA = 0.95, the rest of skua parameters as in Table 1. (b) Graph showing skuaabundance (all 3 compartments) after 150 years Nend vs. lemming trough duration (llow) for variousadult survival rates sA (0.9,0.93,0.95). There is a threshold duration of the ‘bad period’, after whichthere are no breeders left in the population, but more long-lived phenotypes are less likely to sufferfrom lemming lows (Fig. 3b). Other parameters are as indicated in Table 1.



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(a)

(b)

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Average number of youngs produced

Nthresh

=4

Nthresh

=6

Nmean

Traill <2000

Nmean

Traill >=2000

Nmean

Zackenberg

0.35 0.4 0.45 0.5 0.550

5

10

15

Floaters

0.35 0.4 0.45 0.5 0.5559

60

61

62

0.35 0.4 0.45 0.5 0.55

5

10

Juveniles

AR(2) standard deviation in log−scale σ

0.35 0.4 0.45 0.5 0.5522

22.5

23

Figure 4: (a) Effect of variability in lemming densities on the average floater (upper panel) andjuvenile (lower panel) skua densities, computed with the log-linear AR(2) model with constant log-mean (average over 20 000 timesteps after equilibrium has been reached).This amounts to assumethat the true mean lemming density is variable though the median is constant (median = eE(xt), withxt = ln(Nt)). We consider two treatments, either E(xt) < Nthresh (blue plain line) or E(xt) > Nthresh

(green dashed line). We chose to focus on the median/log-mean to facilitate the comparison withHenden et al. (2008), but the discrepancy between mean and median is however small. (b) Locationof mean values of lemming density on the productivity curve (clearly within the convex part of theproductivity curve - it would be the same for medians, see Figs. 5 and 6). These mean values below4 lemmings/ha suggest that the effect of lemming variability on average skua productivity in NEGreenland is positive for both populations and b after/before 2000 (see Fig. 1).



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Mean

CV

Productivity (threshold model)

2 4 6

0.2

0.4

0.6

0.8

1

1.2

1.4

0.5

1

1.5

0 0.5 1 1.50

0.5

1

1.5

2

Coefficient of variation

Exp

ecte

d pr

oduc

tivity

∝ 1

− c

df

m<N

thresh

m=Nthresh

m>Nthresh

Mean

CV

Productivity (sigmoid model)

2 4 6

0.2

0.4

0.6

0.8

1

1.2

1.4

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.50

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1

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1.4

Coefficient of variation

Exp

ecte

d pr

oduc

tivity

val

ue (

sigm

oid)

m<N

thresh

m=Nthresh

m>Nthresh

Figure 5: Effect of mean lemming density (m) and variability (coefficient of variation, CV) on theexpected productivity (for Nthresh = 5). The upper panels use the threshold function for π(N) whilethe lower panels use a sigmoid. This makes the effect of variability smoother in the lower panels,though qualitatively very similar. The effect of variability is asymmetric because the distribution oflemming values is asymmetric. Indeed, for large CV the lognormal distribution is skewed to the left.This implies that for a constant mean, increasing CV pushes more and more values to low lemmingdensities . Hence, even though at low lemming mean, increasing CV first has a positive effect, when CVis already large more variability is not helpful provided the mean lemming density m stays constant.Note that herem = eµ+σ2/2 and CV =

√eσ2 − 1 where µ and σ are the mean and s.d. of the associated

normal distribution. The median of the lognormal is eµ. The three symbols are empirical data pointson mean and CV in Traill island (filled circle: pre-2000, empty circle: post-2000) and Zackenberg(blue).



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0

5

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0 0.2 0.4 0.6 0.8 1

Critica

l be

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-pea

ks p

erio

d p~

Juvenile Survival sJ

0

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0 0.2 0.4 0.6 0.8 1

Critica

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-pea

ks p

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d p~


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Critica

l be

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d p~


0

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0 0.5 1 1.5 2

Tim

e t

o b

ree

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Initial floater-to-breeder ratio ρ0

0

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0 0.5 1 1.5 2

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e


0

5

10

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20

0 0.5 1 1.5 2

Tim

e t

o b

ree

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e


(a) (b)

Figure 6: (a) Relationship between the critical period between peaks and juvenile survival (the criticalperiod maintains positive floater numbers), and (b) the time to breeder decline in absence of lemmingpeaks, when floaters are initially present, for various adult longevity/survival values (other parametersin Table 1). We consider lA=10 years (filled line), lA=20 (dashed line), and lA=30 (dotted line). In (a),we see the period between peaks has an accelerating relationship to juvenile survival. Two values of sJare marked by bars, sJ = 0.75, which is the value assumed by the models and taken from Andersson(1976) worse-scenario guesstimates. In that part of the curve, small changes in sJ greatly change thecritical cycle period. In contrast, sJ = 0.5 marks the survival value for which even a normal lemmingcycle of 5 years will not allow the persistence of a population, and below this value small changes insJ generate small changes to the critical period. Note lA only changes the maximum value of p, i.e.adult longevity does not change the shape of the curve. In (b), we show the time between the juvenileproduction stops and the breeder numbers start declining, as a function of the initial floater-to-breederratio (note panel (a) assumed virtually no floaters).



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