Population Ecology

, Volume 52, Issue 2, pp 279–287

Nonlinear responses of wolverine populations to declining winter snowpack

Authors

    • Wildlife Biology ProgramUniversity of Montana
  • Eric Post
    • Department of BiologyPennsylvania State University
Original Article

DOI: 10.1007/s10144-009-0189-6

Cite this article as:
Brodie, J.F. & Post, E. Popul Ecol (2010) 52: 279. doi:10.1007/s10144-009-0189-6

Abstract

Understanding the population-level impacts of climate change is critical for effectively managing ecosystems. Predators are important components of many systems because they provide top−down control of community structure. Ecological theory suggests that these species could be particularly susceptible to climate change because they generally occur at low densities and have resource-limited populations. Yet, our understanding of climate-change impacts on predators is hindered by the difficulty in assessing complex, nonlinear dynamics over the large spatial scales necessary to depict a species’ general response to abiotic forcing. Here we use fur-return data to characterize population dynamics of a snow-adapted carnivore, the wolverine, across most of its North American range. Using novel modeling techniques, we simultaneously measured the impact of winter snowpack on wolverine dynamics across critical thresholds in snowpack depth and two domains of population growth. Winter snowpack declined from 1970 to 2004 in nearly the entire region studied, concordant with increases in Northern Hemisphere temperature anomalies. Fur returns have declined in many areas; our models show that snowpack has strong, nonlinear effects on wolverine population dynamics. Importantly, wolverine harvests dropped the fastest in areas where snowpack declined most rapidly and also where snowpack had the greatest effect on population dynamics. Moreover, declining snow cover appears to drive trends in wolverine population synchrony, with important implications for overall persistence. These results illustrate the vulnerability and complex responses of predator populations to climate change. We also suggest that declining snowpack may be an important and hitherto little-analyzed mechanism through which climate change alters high-latitude ecosystems.

Keywords

Climate changeGlobal warmingGulo guloHarvestPopulation dynamicsTime-series analysis

Introduction

A number of studies have assessed the impact of climate change on the dynamics of vertebrate populations (e.g., Post and Stenseth 1998; Stenseth et al. 1999). For example, changes in temperature and precipitation may alter distribution (Peterson et al. 2002; McKenney et al. 2007) and abundance (McLaughlin et al. 2002) of various organisms. We also have an increasing realization of how anthropogenically driven changes in large-scale climate drivers, such as the North Atlantic Oscillation, affect the demography of vertebrates across the Northern Hemisphere (Stenseth et al. 1999; Post and Forchhammer 2002; Saether et al. 2008).

Large predators are strong interactors in many communities through their influence on herbivore dynamics (Hebblewhite et al. 2002; Wilmers et al. 2007) and community diversity (Estes and Palmisan 1974). Yet, ecological theory suggests that predators could be especially susceptible to climate change for several reasons. First, impacts of changing climate will likely be of greatest concern for species that have higher “background” levels of extinction risk. Carnivores are generally thought to be extinction prone because their naturally low population densities (Carbone and Gittleman 2002) are associated with increased extinction risk (Purvis et al. 2000). Second, predators in three-trophic-level communities (including nearly all terrestrial food webs involving large vertebrates) are thought to be resource-limited (Oksanen et al. 1981; Oksanen and Oksanen 2000). Thus, predators may respond more strongly to bottom−up abiotic forcing than do herbivores whose populations are buffered from the bottom−up effects of climate change by top−down effects of predation (Wilmers and Post 2006; Wilmers et al. 2007).

Determining how populations respond to climate change can be very difficult due to the complexity of population dynamics that are simultaneously driven by density dependence (Stenseth et al. 1998, 1999) and nonlinear responses to abiotic conditions (Ellis and Post 2004; Tyler et al. 2008). For example, most studies assessing the ecology of climate change examine the impacts of temperature and precipitation either alone or in additive combination (e.g., Peterson et al. 2002; Araujo et al. 2005; McKenney et al. 2007). Understanding the complex, nonlinear, interactions between these two factors may prove equally critical for predicting organisms’ responses to climate change (Tyler et al. 2008). Nonlinearity may arise from condition-dependent pseudochaotic dynamics (May 1976), trophic interactions (Royama 1992), or through the actions of multiple population equilibria (Bjørnstad and Grenfell 2001). Indeed, nonlinear models often explain more of the variance in population time series than do linear functions (Ellis and Post 2004). A commonly used class of nonlinear models in time series analysis are self-exciting threshold autoregressive (SETAR) models. These are piecewise linear functions in which model terms and coefficients can differ above and below a critical threshold (Tong 1990). Nearly all previous use of SETAR models in ecology has focused on one-dimensional thresholds, such as critical values of density (Stenseth et al. 1998; Ellis and Post 2004) or population growth (Framstad et al. 1997; Post et al. 2002). Yet, populations in nature may display phase dependence in more than one dimension concurrently (Tyler et al. 2008). Thus, when assessing the impact of climate change on populations, it is critical to incorporate appropriately complex estimations of nonlinearity in dynamical response.

The difficulty in assessing species responses to climate change is further exacerbated by the necessity of considering large spatial scales. Recent work suggests that only analyses across large regions or over much of a species’ distribution can adequately depict the general population’s response to climate change (Post et al. 2009). Large-scale analyses are important not only to elucidate spatial variation in population response to climate (Post et al. 2009) but also to effectively account for population synchrony (Post and Forchhammer 2002). Synchronous dynamics tend to reduce population persistence (Heino et al. 1997; Palmqvist and Lundberg 1998). Climatic drivers can synchronize widely separated populations (Post and Forchhammer 2002), and climate change can increase interpopulation synchrony (Post and Forchhammer 2004).

For our study, we used recently developed modeling techniques that analyze population processes across two dimensions of phase dependence concurrently (cf. Tyler et al. 2008). We assessed the population response of a snow-adapted carnivore, the wolverine (Gulo gulo), to snowpack dynamics across two phases of population growth and two phases of snowpack depth. We focused on wolverines because they have morphological adaptations that make them a particularly snow-adapted species. For example, wolverines have several physiological adaptations to snow, including a stocky build with heat-retentive pelage (Aubrey et al. 2007) and a low foot loading (Telfer and Kelsall 1984). Snowpack may be important for wolverine foraging and movement efficiency over the course of the winter (Lofroth et al. 2007) and may be particularly critical in early spring when the animals den and nurse neonates (Magoun and Copeland 1998; Lofroth et al. 2007). Thus, if changes in snowpack are important for natural communities, they should manifest especially strongly in this snow-dependent species. We considered snowpack and population dynamic processes across the six provinces and territories of western and northern Canada. Specifically, our objectives were to test whether: (1) annual snowpack levels have detectable influence on wolverine population dynamics, (2) temporal trends in snowpack correspond to trends in wolverine harvest levels, and (3) spatial synchrony in snowpack dynamics drives synchrony in wolverine population dynamics, as population synchrony is a known determinant of extinction risk (Heino et al. 1997; Palmqvist and Lundberg 1998).

Materials and methods

We used time series of wolverine fur returns from Statistics Canada for the six provinces from 1970 to 2005. Equivalent data sets for lynx have been extensively used by ecologists for nearly seven decades (Elton and Nicholson 1942; Moran 1949; Stenseth et al. 1998, 1999) and have been instrumental in the development of many quantitative population analysis techniques (Tjostheim and Auestad 1994; Royama 2005). In contrast, the fur-return data for other species have received scant attention. The use of harvest data to assess population dynamics has been criticized on the grounds that capture rates are not good proxies for abundance (Winterhalder 1980). Nevertheless, despite an imperfect correlation between annual harvest and true population abundance, variation in harvest rates has been shown to effectively capture the interannual dynamics in abundance in a variety of Arctic and sub-Arctic species (e.g., Cattadori et al. 2003; Post and Forchhammer 2004).

On its Web site, Statistics Canada (http://www.statcan.gc.ca/), states that harvest “data originate from royalty payments, export taxes and other administrative sources. Telephone follow-up is conducted to ensure complete response”. Furthermore, they state that “the data accuracy of this census survey is high as the response rate is normally 100%, although occasionally reports are provided late” and that “quality is viewed as excellent as the data are closely scrutinized by each individual province and corrected before being provided to Statistics Canada where the data are validated by comparing it to previous periods and other provinces”. We subjected each time series to autoregression analysis using the global model (see Forchhammer et al. 1998; Post and Stenseth 1999; Stenseth et al. 1999; Post 2005):
$$ \hat{X}_{t} = \beta_{0} + (1 + \beta_{1} )X_{t - 1} + \sum\limits_{i = 2}^{3} {(\beta_{t - i} )X_{t - i} } + \sum\limits_{j = 0}^{2} {(\beta_{t - j} )C_{t - j} } $$
(1)
where Xt is the natural log of harvest rate (here used as a proxy of abundance) in year t, Ct is the climate measure (e.g., annual snowpack) in year t, i and j are time lags for the effects of density and climate, respectively, β0 is the regression intercept or the intrinsic population growth rate without the influence of density or climate (Forchhammer and Asferg 2000), (1 + β1) is the autoregression coefficient, and βh (for h = 1–6) are covariate coefficients. We first tested for the most parsimonious AR dimension (up to 3), then determined whether the addition of climate (snowpack) terms at 0, 1, and 2 year laged improved model fit, assessed using corrected Akaike Information Criteria (AICc) and compared with Akaike model weights. Once we found the most parsimonious log-linear population model, we assessed whether model fit was improved (i.e., AICc score lowered) with nonlinear SETAR models (Stenseth et al. 1998; Ellis and Post 2004). As in previous studies (Framstad et al. 1997; Post et al. 2002), we used the threshold of population growth R = 0. Thus, climate may act differently on populations when they are in expanding versus declining phases (Ellis and Post 2004). Concurrently, we searched for critical thresholds in snowpack depth and examined wolverine population responses to snowpack across both sides of these thresholds. We included “year” in all models to remove temporal trends in the data. We also calculated the temporal trend in snowpack in each province (δ) as the slope of the regression of the natural log of snowpack over time. Finally, we calculated discrete population growth rate for each population as λ = er, where r is the slope of the regression of the natural log of abundance over time.

We obtained weather-station data from Environment Canada. These stations, in addition to measuring monthly temperature and precipitation, also record data on snowpack depth. On its Web site, Environment Canada (http://www.climate.weatheroffice.ec.gc.ca/climateData/canada_e.html) states that: “The vast majority of observational data is accurate but the database contains some incorrect values, which show up from time to time. Environment Canada continues to review quality control procedures, both as current data is observed and incorporated into the database, and retrospectively for historical data”. We also used metrics of Northern Hemisphere temperature anomalies from the National Climatic Data Center, part of the US National Oceanic and Atmospheric Administration (NOAA). We assessed temporal trends in snowfall, winter temperature, and early mean snowpack depth, all averaged from December through March, for the six western Canadian provinces and territories (hereafter province): Alberta, British Columbia, Manitoba, Northwest Territories (including Nunavut), Saskatchewan, and Yukon Territory. For each province, we took the mean winter snowfall, temperature, and snowpack depth averaged across all the weather stations in the province. Some stations failed to report some or all weather data in certain months. To reduce the variance in the climate time series due to individual stations coming on- and off-line, we filled in the missing data with expected values obtained from linear regression of the climate data at that site. As snowpack may be particularly important in the spring, when the animals den and nurse young, we assessed the influence of both winter and early spring snowpack on wolverine population dynamics. Our data do not include many of the intricacies of snowpack physical properties, such as density, chemical characteristics, and structural heterogeneity. Yet, snowpack depth alone has critical impacts on mammalian behavior and demography (DelGiudice et al. 2002; Morrison et al. 2003; Garroway and Broders 2007) and is used here as an index of the overall snowpack conditions confronting animals in winter. Finally, we assessed temporal synchrony in population and snowpack dynamics. We calculated measures of synchrony using correlations between time series of wolverine harvest numbers and snowpack depth for all pairwise combinations of provinces (cf. Post and Forchhammer 2004).

Results

Annual snowpack depths declined across all six western provinces (slope = −0.21 to −0.46, P ≤ 0.03; Fig. 1) except the Yukon (P = 0.08; Table 1). Mean snowpack depths were significantly negatively correlated with Northern Hemisphere temperature anomalies (averaged from December to March) in all provinces (Pearson’s R = −0.41 to −0.59, P ≤ 0.01) except the Northwest Territories (R = −0.31, P = 0.06).
https://static-content.springer.com/image/art%3A10.1007%2Fs10144-009-0189-6/MediaObjects/10144_2009_189_Fig1_HTML.gif
Fig. 1

Mean spring (March 31) snowpack depth from 1968 to 2004 in the six western Canadian provinces and territories; Northwest Territories (NWT) includes Nunavut. Linear trend lines shown in provinces and territories where temporal changes are significant at α = 0.05; regression statistics given in Table 1

Table 1

Number of weather stations used for analysis in each of the six western Canadian provinces and territories from 1968 to 2004; Northwest Territories (NWT) includes Nunavut

Province/territory

Number of stations

Snowfall (cm)

Temperature (°C)

Snowpack (cm)

β

R2

P

β

R2

P

β

R2

P

Alberta

7

−1.08

0.19

0.01

0.09

0.13

0.03

−0.35

0.13

0.03

British Colombia

17

−1.42

0.16

0.02

0.05

0.15

0.02

−0.21

0.17

0.01

Manitoba

6

0.02

0.00

0.93

0.07

0.12

0.04

−0.46

0.17

0.01

NWT

13

0.27

0.09

0.08

0.05

0.09

0.07

−0.28

0.27

<0.01

Saskatchewan

8

−0.34

0.04

0.22

0.07

0.09

0.07

−0.40

0.21

0.01

Yukon

5

−0.35

0.03

0.28

0.01

0.00

0.84

0.24

0.08

0.08

Slope (β), R2 and P values are from linear regressions of annual winter snowfall and daily maximum temperature, both averaged from December to March, as well as spring (March 31) snowpack depth

Population dynamics (Fig. 2) in all provinces except Saskatchewan and Manitoba were substantially better fit by SETAR models (Tables 2, 3). Population dynamics in British Columbia strongly supported growth-phase SETAR models, where snowpack terms in the former differed between expanding and declining phases. Snowpack-phase SETAR models were well supported in Alberta and the Yukon Territory, where snowpack in the current year and at a 2-year lag importantly influence wolverine population dynamics above critical thresholds of snowpack depth. Snowpack terms were not important below the snowpack depth thresholds in either province.
https://static-content.springer.com/image/art%3A10.1007%2Fs10144-009-0189-6/MediaObjects/10144_2009_189_Fig2_HTML.gif
Fig. 2

Reported number of wolverines harvested annually in each of the six western Canadian provinces and territories from 1970 to 2004; Northwest Territories (NWT) includes Nunavut

Table 2

Most parsimonious log-linear, growth-phase, self-exciting threshold autoregressive (SETAR), and snowpack-phase SETAR population models for wolverines in each province; Northwest Territories (NWT) includes Nunavut

Province/territory

Log-linear model

Growth-phase SETAR model

Snowpack threshold (cm)

Snowpack-phase SETAR model

ωLin

ωGrowth

ωSnow

Alberta

\( \hat{X}_{t} = C_{t - 1} \)

R < 0: \( \hat{X}_{t} = {\text{year}} \)

R ≥ 0: \( \hat{X}_{t} = {\text{year}} \)

τ = 11

Below τ: \( \hat{X}_{t} = {\text{year}}\)

Above τ: \( \hat{X}_{t} = C_{t} + C_{t - 2} \)

0.01

0.04

0.94

BC

\( \hat{X}_{t} = X_{t - 1} + C_{t - 1} \)

R < 0: \( \hat{X}_{t} = X_{t - 1} + C_{t} + C_{t - 1} \)

R ≥ 0: \( \hat{X}_{t} = X_{t - 1} + C_{t - 2} \)

τ = 12

Below τ: \( \hat{X}_{t} = X_{t - 1} \)

Above τ: \( \hat{X}_{t} = X_{t - 1} + C_{t} + C_{t - 2} \)

0.00

0.99

0.01

Manitoba

\( \hat{X}_{t} = X_{t - 1} \)

R < 0: \( \hat{X}_{t} = X_{t - 1} \)

R ≥ 0: \( \hat{X}_{t} = X_{t - 1} + C_{t} + C_{t - 2} \)

τ = 19

Below τ: \( \hat{X}_{t} = X_{t - 1} \)

Above τ: \( \hat{X}_{t} = X_{t - 1} \)

0.70

0.20

0.10

NWT

\( \hat{X}_{t} = {\text{year}} \)

R < 0: \( \hat{X}_{t} = {\text{year}} \)

R ≥ 0: \( \hat{X}_{t} = {\text{year}} \)

τ = 31

Below τ: \( \hat{X}_{t} = C_{t - 2} \)

Above τ: \( \hat{X}_{t} = {\text{year}} \)

0.01

0.99

0.00

Saskatchewan

\( \hat{X}_{t} = X_{t - 1} + C_{t} \)

R < 0: \( \hat{X}_{t} = X_{t - 1} \)

R ≥ 0: \( \hat{X}_{t} = X_{t - 1} \)

τ = 18

Below τ: \( \hat{X}_{t} = X_{t - 1} \)

Above τ: \( \hat{X}_{t} = X_{t - 1} \)

0.84

0.13

0.02

Yukon

\( \hat{X}_{t} = C_{t - 2} \)

R < 0: \( \hat{X}_{t} = {\text{year}} \)

R ≥ 0: \( \hat{X}_{t} = {\text{year}} \)

τ = 29

Below τ: \( \hat{X}_{t} = {\text{year}} \)

Above τ: \( \hat{X}_{t} = C_{t} + C_{t - 2} \)

0.00

0.01

0.99

Climate terms (C) represented by spring (March 31) mean snowpack depth. SETAR models include critical thresholds (τ) of population growth. R = 0 (growth-phase) and the most parsimonious snowpack depth (snowpack phase), such that different parameters can affect population dynamics in expanding versus declining phases or in winters with low versus high mean snowpack. Year was also included in all models to remove temporal trends in the data. Akaike Information Criteria (AICc) weight (ω) shown for each model in each province; these may not sum to 1 in some cases due to rounding error

Table 3

Parameter coefficients for the most parsimonious population model for each province

Province/territory

Threshold

Parameter coefficients

Intercept

Year

Xt−1

Ct

Ct−1

Ct−2

Alberta

Below τ

138.722

−0.068

 

−0.003

 

0.023

Above τ

18.693

−0.008

    

British Colombia

R < 0

76.224

−0.036

0.004

0.012

0.004

−0.031

R ≥ 0

83.243

−0.039

0.076

   

Manitoba

None

22.944

−0.010

0.475

   

Northwest Territories

R < 0

−36.926

0.021

    

R ≥ 0

−28.101

0.016

    

Saskatchewan

None

50.675

−0.024

0.234

−0.003

  

Yukon

Below τ

6.305

−0.001

 

0.018

 

−0.010

Above τ

48.893

−0.022

    

See Table 2 for model-selection results for each province. Thresholds include τ (snowpack depth) and R (population growth)

Across provinces, wolverine population growth rate was positively related to temporal trends in winter snowpack (Fig. 3a). Thus, in provinces where winter snowpack levels are declining the fastest, wolverine populations tend to be declining most rapidly. Spring snowpack also appears to influence wolverine population dynamics. Population growth rate was negatively related to the snowpack coefficients averaged across the three population models for each province (Fig. 3b); where snowpack has strong positive effects on population dynamics, wolverine populations tend to be declining. The only province with a positive wolverine population growth rate is the Northwest Territories where snowpack levels, though declining, are still much higher and less variable than in other provinces (Fig. 1) and where snowpack negatively influences wolverine population dynamics.
https://static-content.springer.com/image/art%3A10.1007%2Fs10144-009-0189-6/MediaObjects/10144_2009_189_Fig3_HTML.gif
Fig. 3

a Temporal trend in winter (December–March average) snowpack (δ) versus wolverine population growth rate (λ); linear trend-line shown. b Impact of spring (March 31) snowpack on wolverine population dynamics, measured as the snow coefficient in the autoregressive time-series models (averaged between log-linear and self-exciting threshold autoregressive models) versus wolverine population growth rate (λ); linear trend-line shown

We detected a strong positive relationship between synchrony in wolverine population dynamics and synchrony in snowpack dynamics for pairs of provinces (linear regression: slope = 1.23, R2 = 0.78, P < 0.01; Fig. 4a). However, when we removed the temporal trends from the snowpack and population data and reanalyzed synchrony with the detrended data, the relationship became nonsignificant (slope = 0.17, R2 = 0.06, P = 0.37; Fig. 4b).
https://static-content.springer.com/image/art%3A10.1007%2Fs10144-009-0189-6/MediaObjects/10144_2009_189_Fig4_HTML.gif
Fig. 4

Pearson’s correlation coefficients between pairwise combinations of provinces in their snowpack dynamics and wolverine population dynamics: a raw data, and b data with the temporal trends removed

Discussion

Spring snowpack has been declining across western and northern Canada (in five out of six provinces) over the last 30 years, likely with important repercussions for species and ecosystems in the region. The results presented here reveal patterns consistent with the hypothesis that declining snowpack is negatively affecting the population dynamics of a snow-dependent carnivore, the wolverine. Wolverine populations are declining most quickly in provinces where snowpack levels are declining the fastest. Wolverine populations in which dynamics are positively affected by snowpack depth are declining faster than those that are negatively affected or not affected by snowpack. Moreover, we detected synchrony between wolverine populations and snowpack dynamics across the study range. Trends in declining snow appear to be driving trends in wolverine synchrony (Fig. 4a), even though annual fluctuations in snow cover (i.e., the detrended data in Fig. 4b) do not appear to be driving annual fluctuations in wolverine synchrony. Synchrony in climatic conditions has been shown to affect synchrony in population dynamics in Canada lynx (Stenseth et al. 1999); the same pattern could occur in wolverines if snowpack influences dispersal of individuals among populations (Schwartz et al. 2009). Synchrony among and between populations and climate are important to consider because synchrony (Cattadori et al. 2003) can increase overall extinction risk (Heino et al. 1997; Palmqvist and Lundberg 1998). Together, these results could have very important implications for wolverine viability and persistence across most of the species’ North American range in the face of widespread declines in winter snowpack.

Why would declining snowpack negatively affect wolverines? If reduced snowpack limits dispersal (Schwartz et al. 2009), individuals could potentially be precluded from successfully establishing new home ranges. Declining snowpack could also reduce reproductive success (Magoun and Copeland 1998) or alter the availability or procurement efficiency of food in winter and early spring (Persson 2005; Lofroth et al. 2007). Lower snowpack levels may reduce the density of ungulate carcasses through increased ungulate survival (Wilmers and Post 2006), and lower per-capita hunting success of wolves (Mech et al. 2001). Landa et al. (1997) suggest that wolverine reproductive success may depend on small-mammal abundance, which may, in turn, be affected by snowpack via positive relationships between snowpack depth and small-mammal overwinter survival (Korslund and Steen 2006). By potentially limiting wolverine dispersal (Schwartz et al. 2009), reduced snowpack could threaten population viability as well as genetic structure.

In addition to simply suggesting impacts of climate change on a threatened mammalian species, these analyses may help managers alleviate said impacts. We show here that wolverine populations may be affected by climate change via demographic impacts of declining snowpack. Yet, wolverine populations are also affected by human harvest (Golden et al. 2007; Lofroth and Ott 2007). Squires et al. (2007) showed that harvest of wolverines in Montana was the single most important driver of adult mortality and that adult mortality is the vital rate that drives population dynamics. Whereas there is little that managers on the ground can do to improve snow conditions, they can easily adjust trapping seasons or quotas. Harvest-induced mortality of wolverines is thought to be additive to natural sources of mortality (Krebs et al. 2004), implying that adjusting harvest is an effective management tool for regulating wolverine populations. We suggest that, for wolverine populations negatively affected by declining snowpack, harvest levels could be reduced as a way to offset the impacts of changing abiotic conditions and preclude irreversible declines in abundance.

Some of the biggest changes that global warming may wreak on extratropical terrestrial ecosystems are alterations to snowpack levels (Jones et al. 2001). Snowpack depth, density, and physical properties are extremely important for structuring natural communities in temperate and arctic systems (Jones et al. 2001). However, these snowpack characteristics are determined by very intricate interactions between temperature and precipitation (Pomeroy and Brun 2001). The specific timing of changes in temperature and precipitation can have profound impacts on snowpack (Tews et al. 2007). Simple annual or monthly time series of temperature and precipitation, such as those used in many climate-change studies (e.g., Araujo et al. 2005; McKenney et al. 2007), may be inadequate to adequately depict snowpacks or assess climate-change impacts on high-latitude species, particularly those that are snow dependent (Pomeroy and Brun 2001).

Acknowledgments

This project was supported by a David H. Smith Conservation Research Fellowship to JB. We are grateful to Michael Schwartz, Mark Hebblewhite, and Justina Ray for discussion and constructive comments on previous versions of the manuscript.

Copyright information

© The Society of Population Ecology and Springer 2009