Demographic responses of a site-faithful and territorial predator to its fluctuating prey: long-tailed skuas and arctic lemmings

Authors


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 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 non-stationary 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.

Introduction

Classic ecological theory demonstrates that variability in vital rates is inherently detrimental to population 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, Haridas & Lee 2006) have shown that some positive effects of environmental variability are possible 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 (Jonzén et al. 2010; van de Pol 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 autocorrelation in environmental variables. Nonlinearities can indeed transform the colour of the environmental noise (Laakso, Kaitala & Ranta 2001a, 2001b, 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 non-stationary nature of its lemming prey population dynamics. The model is parametrized 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 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. Ims, Henden & Killengreen 2008; Kausrud et al. 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 non-cyclic dynamics (Steen, Yoccoz & Ims 1990; Angerbjörn, Tannerfeldt & Lundberg 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 & Ylönen 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 non-stationary 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 environmental 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 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 unobserved) for population persistence and shows that 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’ (Katzner et al. 2011; Penteriani, Ferrer & Delgado 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.

Materials and 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 (Wiley & Lee 1998; Sittler, Aebischer & Gilg 2011), 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 seem synchronous to that of Traill island, though the amplitude of fluctuations is smaller (Fig. 1).

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).

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/non-cyclic alternance, and it produces cycles with very skewed distributions, as often observed in lemmings. In contrast, the second model, called loglinear 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 log-normal 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

display math(eqn 1)

with K the threshold density marking the onset of density dependence, math formula a maximal population growth rate, and γ the abruptness of density dependence. This model has the desirable property that when math formula is large, γ almost only affects periodicity and math formula 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 (math formula) in its centered form

display math(eqn 2)

where math formula.

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 math formula. Productivity (breeding output) math formula depends on the density of lemmings math formula in year t. The fraction of breeders that survive each year is math formula, and we assume for simplicity that once they acquire a territory, breeders do not lose it (consistent with the fact that they come back each year irrespective of whether there are enough lemmings to reproduce).

The yearly production then enters the juvenile stage (see Fig. 2; Table 1). We use a stage-based framework for simplicity, with math formula the number of juveniles. We assume that the annual survival probability of juveniles is a constant math formula, 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 math formula).

Table 1. Table of parameters for the skua and lemmings models. DD: Density dependence. MS: Maynard Smith model. AR(2): Loglinear second-order autoregressive model
ModelParameter nameSymbolReference valueUnit
Long-tailed skuaAdult survival (breeder and floater) math formula 0·93 math formula
Adult stage duration math formula math formula year
Juvenile survival math formula 0·75 math formula
Average duration of the juvenile stage1/ϕ4year
Number of available territories math formula 25NA
Threshold density of the productivity math formula 6±2 math formula
Abruptness parameter of the productivity η 3N
Asymptotic productivity math formula 1·75 math formula
Lemming MSMax growth rate math formula 10 math formula
Threshold density K 2·35 math formula
DD abruptness γ 6NA
Lemming AR(2)Mean log density m 1·5NA
Direct DD math formula –1·76NA
Delayed DD math formula –0·58NA

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, Kokko & Pietiäinen (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 math formula as the total number of available territories.

This model is akin to a ‘musical chair’ or ‘lottery’ contest: there are math formula ‘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, math formula.

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

display math(eqn 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):

display math(eqn 4)

Other functional forms are possible, provided the function is sigmoid. The importance of a sigmoid shape comes from two facts: (i) we know empirically the curve accelerates at low densities from the data (Fig. 2b), and (ii) it has to decelerate at large densities, because the maximum number of eggs is two.

We chose an asymptote math formula because the maximum number of eggs layed is 2, and there is always some nest predation (Meltofte & Høye 2007). math formula 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 data set (data from both sites), there is a large margin for error in the parametrization, given the data scarcity around the inflection point. Accordingly we consider two additional values for the threshold, i.e. math formula (Fig. 2b).

Figure 2.

(a) Life cycle graph of the skua model. B: Breeders (have a territory); F: Floaters (do not have, and wait for a territory); J: Juveniles (cannot reproduce yet). See Table 1 for parameters interpretations and values. (b) Productivity function π(N) and comparison to empirical data from both sites.

While the total number of territories, math formula, is approximately 20 in the extended Zackenberg area based on maximum observed data, we chose 25 to get a conservative estimate of the maximum number of breeder territories in the population.

The adult annual survival probability math formula is directly related to the average duration of the adult stage math formula which is itself closely related to longevity. For instance, math formula corresponds to an average adult life span of 10 years and math formula 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 math formula 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, math formula because the number of floaters (math formula) is large when compared to the number of free territories math formula. In that case, the habitat is saturated, and the matrix multiplication for the breeder compartment of the model yields math formula (Appendix S1).

Hence the number of breeders is fixed to math formula at all times (provided floater abundance is large), and is dependent only on their survival rate. Combining this result with (eqn 3), we can then calculate the number of juveniles as

display math(eqn 5)

where math formula is the productivity, hereby obtaining the number of juveniles as a simple recurrence equation. Unfortunately, because the sequence math formula 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 (eqn 5), we obtain

display math(eqn 6)

math formula increases with math formula, and decreases with ϕ (less juveniles if they mature faster). math formula 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)

display math(eqn 7)

where math formula is the average duration of the adult stage. The factor math formula is the number of juveniles recruiting annually into the floater compartment per breeder. Let us call this the ‘effective adult production’, say math formula. Then we arrive at a simple expression for the floater-to-breeder ratio

display math(eqn 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 (math formula) and production of juveniles has already stopped, we have math formula, where math formula is the loss of floaters to the breeder compartment (demonstration in Appendix S2). math formula decreases with math formula for the observed values, and math formula is also the common ratio of this arithmetico-geometric sequence. Thus, the larger adult survival math formula, the slower the decline in floater numbers.

We performed a sensitivity analysis of the model to the duration of the lemming low-density period (from 0 to 30 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 math formula (see Appendix S2) where math formula is the initial floater-to-breeder ratio and math formula 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 math formula, which means that T increases but decelerates with math formula. In situations where the initial floater-to-breeder ratio is small (e.g. math formula) breeder numbers can decline fast (0–2 years), while in situations where math formula is close to 1 or more, it will take between 5 and 10 years for reasonable values of math formula. See section “'Effect of periods without lemmings on skua population persistence'” for more results on how T is influenced by math formula.

Figure 3.

(a) Effect of a period of lemming scarcity on skua population dynamics. Lemmings are in individuals per ha, and bird numbers are abundances (territories/breeding pairs for breeders). The lemming trough (a 20-year long period with no cyclic peaks and generally low population density) is simulated using math formula instead of math formula in the MS model (Table 1). Other parameters are: math formula, the rest of skua parameters as in Table 1. (b) Graph showing skua abundance (all 3 compartments) after 150 years math formula vs. lemming trough duration (math formula) for various adult survival rates math formula (0.9,0.93,0.95). More long-lived phenotypes are less likely to suffer from lemming lows (b). Other parameters are as indicated in Table 1.

The effect of variability in lemming densities depends on the mean

Let us assume that math formula can be represented as a simple random sequence, characterized by a mean and variance (we show in the following sections that temporal autocorrelation does not matter). Starting from the juvenile (eqn 5), we can write the following expectation (assuming the process is ergodic, i.e. averaging over time and realizations yields the same result)

display math(eqn 9)

Defining math formula the long run average of juvenile abundance, we obtain

display math(eqn 10)

where math formula is math formula (the expected productivity). So we recover the same expression as in the constant-productivity case (section “'Number of floaters, assuming habitat saturation'”), except here math formula, and by extension math formula which is linearly related to math formula, include an expected instead of constant productivity. How math formula depends on lemming interannual variability can be seen with a second-order Taylor development (Appendix S1, using primes for derivatives), which eventually leads to:

display math(eqn 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 math formula. Thus, depending on the value of average lemming abundance math formula

  1. A large negative effect of increased lemming variability (CV) is expected at high math formula (math formula), because math formula and this is multiplied by math formula which is large.
  2. A small positive effect of increased lemming variability is expected at low math formula (math formula), because math formula but this is multiplied by math formula 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 math formula 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 loglinear 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 math formula (positive effect) or math formula (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 math formula 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 math formula increases cycle amplitude but not variability in a statistical sense. Interestingly, the effect of increasing math formula 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 rodent variability

As shown above, a key quantity in the model is the temporal average of skua productivity math formula, that converges in the long-term limit (T→∞) to the expectation of π(N) with respect to all possible N values, denoted Eπ(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 math formula and large negative effects at large math formula. 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 (Ef(X)) = ∫f(x)ψ(x)dx is valid. Formally, this is even true for any ergodic stochastic process, which is a reasonable assumption for lemming densities math formula in the cyclic regime, and expected productivity is therefore obtained with the formula

display math(eqn 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 math formula, but greater analytical insight can be obtained if we replace the sigmoid productivity function by a step function, i.e. π(N) = 0 for math formula and math formula above math formula. The equation then becomes

display math(eqn 13)

where math formula is the probability that N is above the reproductive threshold 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.

Figure 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 (coefficient 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 data sets, 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 even when the 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 math formula is large with respect to average lemming density (such as on Greenland), it becomes appropriate to think of a binary sequence of skua reproductive events, math formula. We show in the preceding section (Fig. 5 and Appendix S4) that such common simplification (Andersson 1980; Brommer, Kokko & Pietiäinen 2000) yields qualitatively similar results to a sigmoid function for reproduction.

Let us consider an expected skua productivity math formula, 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 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 math formula 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 (eqn 7), we obtain the expression

display math(eqn 14)

The derivation is provided in Appendix S1. We see that math formula nonlineary depends on math formula. While we have poor ‘guesstimates’ for math formula, we can be relatively confident in all other parameters. Andersson (1976) suggests math formula is in the range [0.7;0.85]. This assumes however no emigration from the population; in contrast, if math formula 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 math formula.

Figure 6 shows the relationship to juvenile survival math formula. 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 (math formula), 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 (Ims, Henden & Killengreen 2008; Kausrud et al. 2008; Gilg, Sittler & Hanski 2009). The phenomenon might be generated by climate change, although other interpretations are possible, and it cannot be excluded that some populations are erupting elsewhere. Indeed rodent dynamics have been non-stationary over long time-scales, alternating between cyclic and non-cyclic periods (Angerbjörn, Tannerfeldt & Lundberg 2001; Henden, Ims & Yoccoz 2009).

Given that relatively high rodent densities are necessary for breeding in long-tailed skuas, that breeding output is strongly saturating (maximum two eggs), and that they are strongly site faithful (Andersson 1976; Andersson 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; Andersson 1981). Therefore, a demographic model including survival processes and population structure is needed to understand the consequences of non-stationary lemming dynamics for long-tailed skua populations. Progressive simplifications of the full version of the model allowed us to isolate its essential components, 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 variability 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. However, the temporal autocorrelation in the lemming values would matter if juvenile survival was dependent on N. In this (hypothetical) case the cohort produced at time t−1, because math formula was high, depends on math formula to survive during year t (as frequently found for owls, Brommer, Kokko & Pietiäinen 2000). But for skuas, where juveniles depend on marine food immediately after they fledge (i.e. survival is independent of math formula), 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.

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, Haridas & Lee 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 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. Such nonlinear averaging effects stem from Jensen's inequality (Jensen 1906), and are well-observed in various areas of ecology (e.g. McNamara & Houston 1992; Boyce, Haridas & Lee 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 (Figs 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.

Figure 4.

(a) Effect of variability in lemming densities on the average floater (upper panel) and juvenile (lower panel) skua densities, computed with the loglinear AR(2) model with constant log-mean (average over 20 000 time steps after equilibrium has been reached). This amounts to assume that the true mean lemming density is variable though the median is constant (medianmath formula, with math formula). We consider two treatments, either math formula (blue plain line) or math formula (green dashed line). We chose to focus on the median/log-mean to facilitate the comparison with Henden et al. (2008), but the discrepancy between mean and median is however small. (b) Location of mean values of lemming density on the productivity curve (clearly within the convex part of the productivity curve – it would be the same for medians, see Figs 5 and Appendix S4). These mean values below 4 lemmings/ha suggest that the effect of lemming variability on average skua productivity in NE Greenland is positive for both populations and both after/before 2000 (see Fig. 1).

Figure 5.

Effect of mean lemming density (m) and variability (coefficient of variation, CV) on the expected productivity (for math formula). The upper panels use the threshold function for π(N) while the 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 of lemming values is asymmetric. Indeed, for large CV the log-normal distribution is skewed to the left. This implies that for a constant mean, increasing CV pushes more and more values to low lemming densities. Hence, even though at low lemming mean, increasing CV first has a positive effect, when CV is already large more variability is not helpful provided the mean lemming density m stays constant. Note that here math formula and math formula where μ and σ are the mean and s.d. of the associated normal distribution. The median of the log-normal is math formula. The three symbols are empirical data points on mean and CV in Traill island (filled circle: pre-2000, empty circle: post-2000) and Zackenberg (blue).

Figure 6.

(a) Relationship between the critical period between peaks and juvenile survival (the critical period maintains positive floater numbers), and (b) the time to breeder decline in absence of lemming peaks, when floaters are initially present, for various adult longevity/survival values (other parameters in Table 1). We consider math formula = 10 years (filled line), math formula = 20 (dashed line), and math formula = 30 (dotted line). In (a), we see the period between peaks has an accelerating relationship to juvenile survival. Two values of math formula are marked by bars, math formula, 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 math formula greatly change the critical cycle period. In contrast, math formula marks the survival value for which even a normal lemming cycle of 5 years will not allow the persistence of a population, and below this value small changes in math formula generate small changes to the critical period. Note math formula only changes the maximum value of math formula, i.e. adult longevity does not change the shape of the curve. In (b), we show the time between the stoping of juvenile production and the decline of the breeder numbers, as a function of the initial floater-to-breeder ratio (note panel (a) assumed virtually no floaters).

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 to 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 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 reproduction 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 survival) 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 & Rudolf 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 & Rudolf 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 food density (Brommer, Kokko & Pietiäinen 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).

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. 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 population 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, populations would probably not survive for much longer in NE Greenland, as >10 years without lemming 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 BioBasis 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.

Data Accessibility

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

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