### Study area and species

Field work was conducted from May through July of 2004–2007 on a 10 km^{2} alpine study site (Pika Camp) in the Ruby Range Mountains of the southern Yukon Territory, Canada. In 2008, we also returned to the site to census marked individuals to allow 4 years of survival data. Mean daily temperature at the site averaged 6.5°C during the breeding period (May through July) and −13.3°C during the winter months (November through March).

Rock and white-tailed ptarmigan are precocial, ground-nesting birds within the genus *Lagopus* (family Phasianidae). Ptarmigan are socially monogamous during breeding although polygyny occurs regularly. Males of these two species remain with the female on the territory until approximately mid-incubation after which they leave and form small flocks with other males and failed female breeders. Females will re-nest following clutch failure but only produce one brood per year, which stay with the female until late August to late September [13, 15, 16, 27]. The two species are sympatric in the study region but show some segregation in their breeding habitat and display intra- and inter-specific territoriality. Rock ptarmigan typically select lower alpine meadows while white-tailed ptarmigan select steeper, rocky slopes at higher elevations [34]. Densities of rock and white-tailed ptarmigan respectively ranged from about 4–5 and 2–3 pairs/km^{2}. For additional detail on habitat and other study site characteristics see Wilson and Martin [13, 34].

### Field methods

We used ground nets and noose poles to capture males and females during the period after territory establishment and prior to nesting (~ 1 May – 20 May). All individuals were color-marked with an aluminum band on one leg and a numbered plastic color band on the other. Females were fitted with a 4 or 7 g radio-transmitter (Holohil Inc., Carp, Ontario) to facilitate nest finding. Individuals were aged as second-year (SY, 10–11 months) or after-second year (ASY, 22+ months of age) based on the pigmentation on the outer primaries and primary coverts following Weeden and Watson [35]. The total number of breeding females monitored for reproduction by age class were as follows: rock ptarmigan SY - 11, rock ptarmigan ASY - 65, white-tailed ptarmigan SY - 23, white-tailed ptarmigan ASY – 28 (note that some individuals were monitored as both SY and ASY birds as well as multiple years as ASY birds). Date of first egg was estimated by observing nests during laying, back-dating from hatch or floating an egg during incubation following the method of Westerskov [36]. Incubation was assumed to have begun with the laying of the penultimate egg [37, 38]. Clutch size for first and second attempts was determined as the maximum number of eggs laid per nesting attempt (i.e. completed clutch), defined as those where egg number was constant over two consecutive nest checks. In the lining of each nest we also placed a small Ibutton temperature logger (Maxim Products, Dallas, TX), which keeps a continuous record of nest temperature and allowed us to determine the precise time of nest failure or hatch. During incubation, nests were checked visually every 3–5 days to determine if they were still active and females were only flushed off the nest when we needed to switch the Ibutton every 10–12 days. We monitored nests more frequently as the expected hatch date approached to ensure an accurate measurement of the number of chicks hatched and when they left the nest. Broods were re-located every 3 to 7 days to re-count the chicks to estimate juvenile survival. If a nest failed, the female was located every few days to determine whether and when she initiated a re-nest attempt.

Radio-collars were removed from two-thirds of the females after breeding. For the other one-third, we left the collars on through the winter and this allowed us to conduct surveys of the immediate study area and adjacent regions (approximately 100 km^{2}) in the following spring to estimate the extent of breeding dispersal. With these surveys, we found no evidence that females of either species that had bred at the study site in the previous year had moved elsewhere within this 100 km^{2} range. We cannot rule out the possibility that they had moved beyond this range but overall this suggests that our estimates of apparent survival are likely close to true survival and not biased by permanent emigration from the study area. The use of animals in this research adheres to the ethical standards of Canada as approved by the University of British Columbia Animal Care Committee (permit no. A05-0450).

### Calculation of demographic rates

Many demographic rates used here were previously estimated in Wilson and Martin [13, 20], but we summarize those analyses used in our population model. Because of the ASY-biased age structure for rock ptarmigan, we had a low sample size of SY females to estimate the mean and variance in the number of hatched young they produced. Data for white-tailed ptarmigan indicated that SY females hatched about 20% fewer young than ASY females and this difference is similar to average estimates for both species elsewhere [16, 39, 40]. Therefore, we estimated the mean and variance in the number of hatched young for ASY female rock ptarmigan and assumed this rate would be 20% lower for SY females with equal variance. Among ptarmigan generally, the probability of re-nesting is age dependent with 2 and 3 year-old females showing a greater propensity to re-nest than 1-yr old females [41, 42]. We calculated the observed re-nest probability for all individuals combined and adjusted by age assuming SY females had a 20% lower re-nest propensity than ASY females. We previously found no influence of female age on daily nest survival [15]. Therefore we assumed constant nest survival with age for both species and estimated rates separately for first and second nest attempts.

We used program MARK for analyses of adult survival [43]. Adult survival was previously calculated in Wilson and Martin [13] but we expanded on previous estimates to obtain separate survival rates for SY and ASY females of each species. Juvenile survival was directly estimated during the chick stages (June through August) and both species had similar rates with a mean of 0.52-0.55 over this period [15, 20]. To determine annual juvenile survival, we combined these values with an estimate from September through April from the literature. Survival of juvenile willow and white-tailed ptarmigan from September to the following spring averages about 0.45-0.55 [44]. There are no detailed studies of juvenile survival in rock ptarmigan. To parameterize the population model, we assumed a rate of 0.5 from independence through April, which when combined with our field data prior to independence yield an average annual rate of about 0.27. However, because of uncertainty in this estimate, we ran most model projections with mean values of juvenile survival ranging between 0.22 and 0.32.

### Population model

Population growth rates were calculated using female-based matrix models [

6,

7,

10] calculated in Matlab Vers 7.1 [

45]. Variation in the size and age-structure of a population from time

*t* to time

*t* + 1 can be computed from:

${\mathbf{n}}_{\text{t}+1}=\mathbf{A}{\mathbf{n}}_{\text{t}}$

(1)

where n is a vector describing the age, stage or size-structure of the population and A is a population projection matrix. We used a two-age pre-breeding model with second-year (SY) and after-second year (ASY) females as the two classes:$\mathbf{A}=\left[\begin{array}{cc}\hfill H{Y}_{\mathit{SY}}*{S}_{\mathit{JUV}}\hfill & \hfill H{Y}_{\mathit{ASY}}*{S}_{\mathit{JUV}}\hfill \\ \hfill {S}_{\mathit{SY}}\hfill & \hfill {S}_{\mathit{ASY}}\hfill \end{array}\right]$where HY_{SY} and HY_{ASY} are the number of female young hatched annually by SY and ASY females, S_{JUV} is the survival rate of juveniles from hatch to the following breeding season, and S_{SY} and S_{ASY} are the survival rates of adult SY and ASY females, respectively.

From the above model we calculated the population growth rate λ

_{1} (the dominant eigenvalue) and from λ

_{1}, the subdominant eigenvalue (λ

_{2}), reproductive value (v, left eigenvector) and the stable-age distribution (w, right eigenvector). We also determined the average number of female offspring produced per female over her lifetime (net reproductive rate, Ro) as:

$\text{Ro\hspace{0.17em}}=\sum _{x=0}^{n}{\text{s}}_{\text{x}}\phantom{\rule{0.12em}{0ex}}{\text{\hspace{0.17em}f}}_{\text{x}}$where s

_{x} is the probability of survival to age x and f

_{x} is the fecundity of females at age x. After determining the net reproductive rate for both species, we calculated the generation time as:

$\text{T}=\text{ln}\left(\text{Ro}\right)/\text{\hspace{0.17em}ln}\left({\lambda}_{1}\right)$

(2)

Matrix models are also useful for calculating elasticity [

6,

7,

10], the proportional change in λ

_{1} in response to a proportional change in a demographic rate r

_{i}.

${\text{Er}}_{\text{i}}=\partial {\lambda}_{1}/{\lambda}_{1}/\partial {\text{r}}_{\text{i}}/{\text{r}}_{\text{i}}$

(3)

Although elasticity measures provide an estimate of how influential a particular demographic rate is, it is important to note that the variability in a rate must also be considered to predict which rates ultimately have the greater influence on population growth. To provide a more informative analysis of elasticity, we decomposed the fecundity terms (HY

_{SY} and HY

_{ASY}) into the primary rates that influence the number of young hatched; clutch size, nest success and the probability of re-nesting after failure. This model had the following structure:

$\mathbf{A}=\left[\begin{array}{cc}\hfill {F}_{\mathit{SY}}\hfill & \hfill {F}_{\mathit{ASY}}\hfill \\ \hfill {S}_{\mathit{SY}}\hfill & \hfill {S}_{\mathit{ASY}}\hfill \end{array}\right]$For rock ptarmigan, F1 and F2 are equal to:

${\text{F}}_{\text{SY}}=\left[{\text{C}1}_{\text{SY}}\phantom{\rule{0.12em}{0ex}}\text{ns}1+\left(1-\text{ns}1\right)\phantom{\rule{0.12em}{0ex}}{\text{r}1}_{\text{SY}}\phantom{\rule{0.12em}{0ex}}\text{C}2\phantom{\rule{0.12em}{0ex}}\text{ns}2\right]\phantom{\rule{0.37em}{0ex}}{\text{\hspace{0.17em}S}}_{\text{JUV}}\phantom{\rule{0.37em}{0ex}}0.5$

(4)

${\text{F}}_{\text{ASY}}=\left[{\text{C}1}_{\text{ASY}}\phantom{\rule{0.12em}{0ex}}\text{ns}1+\left(1-\text{ns}1\right)\phantom{\rule{0.12em}{0ex}}{\text{r}1}_{\text{ASY}}\phantom{\rule{0.12em}{0ex}}\text{C}2\phantom{\rule{0.12em}{0ex}}\text{ns}2\right]\phantom{\rule{0.37em}{0ex}}{\text{\hspace{0.17em}S}}_{\text{JUV}}\phantom{\rule{0.5em}{0ex}}0.5$

(5)

where C1

_{SY} and C1

_{ASY} are the size of the first clutch for SY and ASY females respectively, C2 is the size of the 2nd clutch (equal for both age groups), ns1 and ns2 is nest success for the first and second attempt respectively, r1

_{SY} and r1

_{ASY} are the re-nest probabilities for SY and ASY females, and S

_{JUV} is juvenile survival. Fifty percent of the young were assumed to be female and therefore estimates were multiplied by 0.5. Equations for white-tailed ptarmigan are the same except ASY females were assumed to have a probability of re-nesting a second time [

13]:

${\text{F}}_{\text{SY}}=\left[{\text{C}1}_{\text{SY}}\phantom{\rule{0.12em}{0ex}}\text{ns}1+\left(1-\text{ns}1\right)\phantom{\rule{0.12em}{0ex}}{\text{r}1}_{\text{SY}}\phantom{\rule{0.12em}{0ex}}\text{C}2\phantom{\rule{0.12em}{0ex}}\text{ns}2\right]\phantom{\rule{0.37em}{0ex}}{\text{\hspace{0.17em}S}}_{\text{JUV}}\phantom{\rule{0.37em}{0ex}}0.5$

(6)

${\text{F}}_{\text{ASY}}=\left[{\text{C}1}_{\text{ASY}}\phantom{\rule{0.12em}{0ex}}\text{ns}1+\left(1-\text{ns}1\right)\phantom{\rule{0.12em}{0ex}}{\text{r}1}_{\text{ASY}}\phantom{\rule{0.12em}{0ex}}\text{C}2\phantom{\rule{0.12em}{0ex}}\text{ns}2+\left(1-\text{ns}1\right)\left(1-\text{ns}2\right)\phantom{\rule{0.12em}{0ex}}{\text{\hspace{0.17em}r}1}_{\text{ASY}}\phantom{\rule{0.12em}{0ex}}{\text{\hspace{0.17em}r}2}_{\text{ASY}}\phantom{\rule{0.12em}{0ex}}\text{C}2\phantom{\rule{0.12em}{0ex}}\text{ns}2\right]\phantom{\rule{0.37em}{0ex}}{\text{\hspace{0.17em}S}}_{\text{JUV}}\phantom{\rule{0.37em}{0ex}}0.5$

(7)

where r1_{ASY} and r2_{ASY} are the ASY re-nest probabilities for 2nd and 3rd attempts. If a 3rd attempt was initiated, we assumed clutch size and nest success was the same as for the second attempt. For both species, survival of SY and ASY females were represented by S1 and S2 as for the previous matrix.

The above estimates were calculated from a deterministic model without variation. To better examine the potential for uncertainty we also examined growth rates using a stochastic population model [46]. We introduced environmental stochasticity to the model by allowing demographic rates to be drawn at random from a specified distribution and simulated 1000 population trajectories each for 25 years following the approach of Morris and Doak [10]. The number of hatched young were randomly drawn from a stretched beta distribution with mean and variance equal to the observed values across the four years. The maximum and minimum values for this distribution were assigned based on likely upper and lower limits for the two species [37, 38]. Annual survival of second year and after-second year females were assumed equal and drawn from a beta distribution with mean and variance approximated from field data. For earlier analyses we separated process and sampling variance in survival using the approaches outlined in [10]. However, the resulting estimates of process variance were low relative to sampling variance and led to a range of annual survival rates that appeared to be too narrow given knowledge of species biology. Therefore, we chose to use total variance in our models instead even though it is a conservative approach that includes process and sampling variance e.g. [41, 47]. Annual juvenile survival was also drawn from a beta distribution but simulations were run with a mean rate of 0.22, 0.27 and 0.32 to incorporate the uncertainty described earlier.

To include the potential effects of covariation in the vital rates on population performance [46, 48], we allowed rates to be correlated within years. Because only 3 and 4 years of data were available for survival and fecundity respectively, it was not possible to examine correlations among demographic rates and therefore, we assigned correlations based on likely values given the species biology. Within-year correlations between SY and ASY fecundity and, between SY and ASY survival are likely strong and we assigned a correlation coefficient (r) of each = 0.9. Annual survival rates of juveniles and adults (combined SY and ASY) may also be correlated although the relationship could vary. A negative relationship might be observed if there are strong density-dependent effects of adults on juveniles, while a positive relationship may be more likely if environmental conditions affect all age groups equally. To represent this range of possibilities, we ran the model with juvenile-adult survival correlations = −0.5, 0 and 0.5, and evaluated how each affected population growth. We assumed there were no within-year correlations between the reproductive and survival components. Between-year correlation might occur when factors such as climatic conditions, predator abundance, disease or food supply are temporally autocorrelated [10, 49]. However, because of the difficulty in identifying these effects with short term data, we assumed there were no between-year correlations in this analysis. Density dependence is also a key process in population dynamics but is difficult to incorporate in matrix projection models because many years of data are required to identify the functional relationship [6, 10]. Rather than assume what this relationship might be, we chose not to include density dependence here.