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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: frederic.barraquand@uit.no
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-
ing
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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
References
Andersson, M. (1976) Population ecology of the long-tailed skua (Stercorarius longicaudus Vieill.).
Journal of Animal Ecology, 45, 537–559.
Andersson, M. (1980) Nomadism and site tenacity as alternative reproductive tactics in birds. Journal
of Animal Ecology, 49, 175–184.
Andersson, M. (1981) Reproductive tactics of the Long-tailed Skua Stercorarius longicaudus . Oikos,
37, 287–294.
Andersson, M. & Erlinge, S. (1977) Influence of predation on rodent populations. Oikos, 29, 591–597.
Angerbjorn, A., Tannerfeldt, M. & Lundberg, H. (2001) Geographical and temporal patterns of lem-
ming population dynamics in Fennoscandia. Ecography, 24, 298–308.
Barraquand, F. & Yoccoz, N. G. (2013)When can environmental variability benefit population growth?
Counterintuitive effects of nonlinearities in vital rates. Theoretical Population Biology, 89, 1–11.
Boyce, M.S., Haridas, C.V. & Lee, C.T. (2006) Demography in an increasingly variable world. Trends
in Ecology & Evolution, 21, 141–148.
Brommer, J., Kokko, H. & Pietiainen, H. (2000) Reproductive effort and reproductive values in periodic
environments. The American Naturalist, 155, 454–472.
Brommer, J., Pietiainen, H., Ahola, K., Karell, P., Karstinen, T. & Kolunen, H. (2010) The return
of the vole cycle in southern Finland refutes the generality of the loss of cycles through ‘climatic
forcing’. Global Change Biology, 16, 577–586.
Cornulier, T., Yoccoz, N. G., Bretagnolle, V., Brommer, J. E., Butet, A., Ecke, F., Elston, D. A.,
Framstad, E., Henttonen, H., Hornfeldt, B., Huitu, O., Imholt, C., Ims, R. A., Jacob, J., Jedrzejew-
ska, B., Millon, A., Petty, S. J., Pietiainen, H., Tkadlec, E., Zub, K. & Lambin, X. Europe-wide
dampening of population cycles in keystone herbivores. (2013) Science, 340, 63–66.
This article is protected by copyright. All rights reserved.
This article is protected by copyright. All rights reserved.
Acc
epte
d A
rtic
le
Drake, J. M. (2005) Population effects of increased climate variation. Proceedings of the Royal Society
B: Biological Sciences 272, 1823-1827.
Gaillard, J. M. & Yoccoz, N. G. Temporal variation in survival of mammals: a case of environmental
canalization? Ecology, 84, 3294–3306.
Garcia-Carreras, B. & Reuman, D. C. (2011) An empirical link between the spectral colour of climate
and the spectral colour of field populations in the context of climate change. Journal of Animal
Ecology, 80, 1042–1048.
Getz, W.M. (1996) A hypothesis regarding the abruptness of density dependence and the growth rate
of populations. Ecology, 77, 2014–2026.
Gilg, O., Hanski, I. & Sittler, B. (2003) Cyclic dynamics in a simple vertebrate predator-prey commu-
nity. Science, 301, 866–868.
Gilg, O., Sittler, B., Sabard, B., Hurstel, A., Sane, R., Delattre, P. & Hanski, I. (2006) Functional and
numerical responses of four lemming predators in high arctic Greenland. Oikos, 113, 193–216.
Gilg, O., Sittler, B. & Hanski, I. (2009) Climate change and cyclic predator–prey population dynamics
in the high Arctic. Global Change Biology, 15, 2634–2652.
Gilg, O., Moe, B., Hanssen, S.A., Schmidt, N. M., Sittler, B., Hansen, J., Reneerkens, J., Sabard,
B., Chastel, O., Moreau, J., Phillips, R. A., Oudman, T., Biersma, E. M., Fenstad, A. A., Lang,
J. & Bollache, L. (2013) Trans-equatorial migration routes, staging sites and wintering areas of a
high-Arctic avian predator: the long-tailed skua (Stercorarius longicaudus). PloS one, 8, e64614.
Grenfell, B.T., Price, O.F., Albon, S.D. & Clutton-Brock, T.H. (1992) Overcompensation and popu-
lation cycles in an ungulate. Nature, 355, 823–826.
Hanski, I., Hansson, L. & Henttonen, H. (1991) Specialist predators, generalist predators, and the
microtine rodent cycle. Journal of Animal Ecology, 60, 353–367.
Hanski, I. & Henttonen, H. (1996) Predation on competing rodent species: a simple explanation of
complex patterns. Journal of Animal Ecology, 65, 220–232.
Henden, J.-A., Bardsen, B.J., Yoccoz, N.G. & Ims, R.A. (2008) Impacts of differential prey dynamics
on the potential recovery of endangered arctic fox populations. Journal of Applied Ecology, 45,
1086–1093.
This article is protected by copyright. All rights reserved.
This article is protected by copyright. All rights reserved.
Acc
epte
d A
rtic
le
Henden, J.-A., Ims, R.A. & Yoccoz, N.G. (2009) Nonstationary spatio-temporal small rodent dynamics:
evidence from long-term Norwegian fox bounty data. Journal of Animal Ecology, 78, 636–645.
Hunt, W. (1998) Raptor floaters at Moffat’s equilibrium. Oikos, 82, 191–197.
Ims, R.A., Henden, J.-A. & Killengreen, S.T. (2008) Collapsing population cycles. Trends in Ecology
& Evolution, 23, 79–86.
Jensen, J. (1906) Sur les fonctions convexes et les inegalites entre les valeurs moyennes. Acta Mathe-
matica, 30, 175–193.
Jonzen, N., Pople, T., Knape, J. & Skold, M. (2010) Stochastic demography and population dynamics
in the red kangaroo Macropus rufus. Journal of Animal Ecology, 79, 109–116.
Katzner, T.E., Ivy, J.A., Bragin, E.A., Milner-Gulland, E. & DeWoody, J.A. (2011) Conservation
implications of inaccurate estimation of cryptic population size. Animal Conservation, 14, 328–332.
Kausrud, K.L., Mysterud, A., Steen, H., Vik, J.O., Østbye, E., Cazelles, B., Framstad, E., Eikeset,
A.M., Mysterud, I., Solhøy, T. & Stenseth, N.C. (2008) Linking climate change to lemming cycles.
Nature, 456, 93–97.
Kokko, H. & Sutherland, W. (1998) Optimal floating and queuing strategies: consequences for density
dependence and habitat loss. The American Naturalist, 152, 354–366.
Koons, D. N., Pavard, S., Baudisch, A., & Metcalf, J. E. (2009) Is life-history buffering or lability
adaptive in stochastic environments? Oikos, 118, 972–980.
Laakso, J., Kaitala, V & Ranta, E. (2001) How does environmental variation translate into biological
processes? Oikos, 92, 119–122.
Laakso, J., Kaitala, V & Ranta, E. (2001) Non-linear biological responses to disturbance: consequences
on population dynamics. Ecological Modelling, 162, 247–258.
Lewontin, R. & Cohen, D. (1969) On population growth in a randomly varying environment. Proceed-
ings of the National Academy of Sciences, 62, 1056–1060.
Maynard Smith, J. (1974) Models in ecology. Cambridge University Press, Cambridge, United Kind-
gom.
This article is protected by copyright. All rights reserved.
This article is protected by copyright. All rights reserved.
Acc
epte
d A
rtic
le
Maynard Smith, J. & Slatkin, M. (1973) The stability of predator-prey systems. Ecology, 54, 384–391.
McNamara, J.M. & Houston, A.I. (1992) Risk-sensitive foraging: a review of the theory. Bulletin of
Mathematical Biology, 54, 355–378.
Meijer, T., Elmhagen, B., Eide, N. & Angerbjorn, A. (2013) Life history traits in a cyclic ecosystem:
a field experiment on the arctic fox Oecologia, in press
Meltofte, H. & Høye, T.T. (2007) Reproductive response to fluctuating lemming density and climate
of the Long-tailed Skua Stercorarius longicaudus at Zackenberg, Northeast Greenland, 1996-2006.
Dansk Ornithologisk Forenings Tidsskrift, 101, 109–119.
Pen, I. & Weissing, F. (2000) Optimal floating and queuing strategies: the logic of territory choice.
The American Naturalist, 155, 512–526.
Penteriani, V., Ferrer, M. & Delgado, M.M. (2011) Floater strategies and dynamics in birds, and their
importance in conservation biology: towards an understanding of nonbreeders in avian populations.
Animal Conservation, 14, 233–241.
Pfister, C. A. (1998) Patterns of variance in stage-structured populations: evolutionary predictions
and ecological implications. Proceedings of the National Academy of Sciences, 95, 213–218.
Rohner, C. (1996) The numerical response of great horned owls to the snowshoe hare cycle: conse-
quences of non-territorial floaters’ on demography. Journal of Animal Ecology, 65, 359–370.
Royama, T. (1992) Analytical population dynamics. Chapman and Hall, London, UK.
Schmidt, N.M., Berg, T.B., Forchhammer, M.C., Hendrichsen, D.K., Kyhn, L.A., Meltofte, H. & Høye,
T.T. (2008) Vertebrate predator–prey tnteractions in a seasonal environment. Advances in Ecological
Research, 40, 345–370.
Schmidt, N.M., Ims, R.A., Høye, T.T., Gilg, O., Hansen, L.H., Hansen, J., Lund, M., Fuglei, E.,
Forchhammer, M.C. & Sittler, B. (2012) Response of an arctic predator guild to collapsing lemming
cycles. Proceedings of the Royal Society B: Biological Sciences, 279, 4417–4422.
Sittler, B., Aebischer, A. & Gilg, O. (2011) Post-breeding migration of four Long-tailed Skuas (Ster-
corarius longicaudus) from North and East Greenland to West Africa. Journal of Ornithology, 152,
375–381.
This article is protected by copyright. All rights reserved.
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Acc
epte
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rtic
le
Stearns, S. C. & Kawecki, T. J. (1994) Fitness sensitivity and the canalization of life-history traits.
Evolution, 48, 1438–1450.
Steen, H., Yoccoz, N.G. & Ims, R.A. (1990) Predators and small rodent cycles: An analysis of a 79-year
time series of small rodent population fluctuations. Oikos, 59, 115–120.
Stenseth, N.C. (1999) Population cycles in voles and lemmings: density dependence and phase depen-
dence in a stochastic world. Oikos, 87, 427–461.
Sundell, J. & Ylonen, H. (2008) Specialist predator in a multi-species prey community: boreal voles
and weasels. Integrative Zoology, 3, 51–63.
Tuljapurkar, S. & Haridas, C. V. (2006) Temporal autocorrelation and stochastic population growth
Ecology Letters, 9, 327–337.
Ugland, K.I. & Stenseth, N.C. (1985) On the evolution of reproductive rates in populations with
equilibrium and cyclic densities. Mathematical Biosciences, 74, 59–87.
van de Pol, M., Pen, I., Heg, D. & Weissing, F. J. (2007) Variation in habitat choice and delayed re-
production: adaptive queuing strategies or individual quality differences? The American Naturalist,
170, 530–541.
van de Pol, M., Vindenes, Y., Sæther, B.-E., Engen, S., Ens, B. J., Oosterbeek, K. & Tinbergen, J. M.
(2010) Effects of climate change and variability on population dynamics in a long-lived shorebird.
Ecology, 91, 1192–1204.
van de Pol, M., Vindenes, Y., Sæther, B.-E., Engen, S., Ens, B. J., Oosterbeek, K. & Tinbergen,
J. M. (2011) Poor environmental tracking can make extinction risk insensitive to the colour of
environmental noise. Proceedings of the Royal Society B: Biological Sciences, 278, 3713–3722.
Wilbur, H. M. & Rudolf, V. H. (2006) Life-history evolution in uncertain environments: bet hedging
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
Lemming density N (individuals per ha)
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
Lemming density N (individuals per ha)
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
0.2
0.4
0.6
0.8
1
1.2
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
10
15
20
0 0.2 0.4 0.6 0.8 1
Critica
l be
twe
en
-pea
ks p
erio
d p~
Juvenile Survival sJ
0
5
10
15
20
0 0.2 0.4 0.6 0.8 1
Critica
l be
twe
en
-pea
ks p
erio
d p~
Juvenile Survival sJ
0
5
10
15
20
0 0.2 0.4 0.6 0.8 1
Critica
l be
twe
en
-pea
ks p
erio
d p~
Juvenile Survival sJ
0
5
10
15
20
0 0.5 1 1.5 2
Tim
e t
o b
ree
der
de
clin
e
Initial floater-to-breeder ratio ρ0
0
5
10
15
20
0 0.5 1 1.5 2
Tim
e t
o b
ree
der
de
clin
e
Initial floater-to-breeder ratio ρ0
0
5
10
15
20
0 0.5 1 1.5 2
Tim
e t
o b
ree
der
de
clin
e
Initial floater-to-breeder ratio ρ0
(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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