Multiple sources of evidence for density dependence in the endangered Hawaiian stilt (Himantopus mexicanus knudseni)

Abstract

Hawaiian stilts (Himantopus mexicanus knudseni) are an endangered subspecies of the Black-necked stilt endemic to the Hawaiian Islands. Despite long-term study, the main drivers of Hawaiian stilt population dynamics are poorly understood. We tested for density dependence using two sources of evidence: a 30-year time series of annual estimated range-wide abundance, and two 15+ year time series of reproductive success. Using separate methods with independent data, sources allowed us to make up for the potentially positive bias of one approach with the more conservative nature of the second. We compared nonlinear density-dependent and density-independent population model fits to our time-series data, using both frequentist and Bayesian state-space approaches. Across both approaches, density-dependent models best fit observed population dynamics, with lower AICc and cross-validation statistics compared to density-independent models. Among density-dependent models, a conditional model in which density-independent dynamics occur below a population size threshold (~850–1,000 birds), and then density-dependent dynamics occur above that threshold, performed best across Bayesian and frequentist model comparisons, with the Ricker model ranked next or equivalently. Our analysis of reproduction data revealed a strong negative effect of local adult density on nest success (proportion of nests hatching at least one chick) at Kealia National Wildlife Refuge on Maui, where few alternative breeding habitats are available, but no such effect at another site where many nearby alternative wetlands are available. These congruent results across independent datasets and analytical approaches support the hypothesis that Hawaiian stilts exhibit density dependence across their range.

1 INTRODUCTION

The factors influencing the dynamics of animal populations are of long-standing and fundamental interest to theoretical and applied ecology. The existence, relative influence and functional form of density dependence—the influence of a population's size in finite space on its growth rate—have gained special attention as subjects of debate (e.g., Dennis & Taper, 1994; Turchin, 1990) and ongoing research effort (e.g., Riotte-Lambert, Benhamou, & Chamaillé-Jammes, 2015; Williams & Levine, 2018). Detecting and understanding density-dependent population dynamics is also important for applied research on threatened and managed populations, such as designing realistic demographic models and estimating extinction risk (e.g., Morris & Doak, 2002; Williams, Nichols, & Conroy, 2002). Density dependence is often a key concern in population viability analysis (Beissinger & Westphal, 1998; Henle, Sarre, & Wiegand, 2004; Zabel, Scheuerell, McClure, & Williams, 2006), making its study an important component of the conservation and management of endangered species (e.g., Ferrer et al., 2014; Ferrer & Donázar, 1996; Rose, Cowan, Winemiller, Myers, & Hilborn, 2001; Runge & Johnson, 2002; Singer, Harting, Symonds, & Coughenour, 1997).

Lebreton and Gimenez (2013) highlighted the difficulties of diagnosing and estimating the strength of density-dependence using time series of population data and regression of population density against individual measures of demographic rates. The latter method is thought to be conservatively biased, tending to underestimate density dependence, but working well in parallel with the former method, which can show a positive bias. Other problems with the former (population time-series based) method include the correlation and problematic identifiability of intrinsic growth rate and density dependence terms, for which the authors proposed explicit accounting for detection error and the use of external information to constrain modeled values of intrinsic growth rate r (Lebreton & Gimenez, 2013). These solutions can be simultaneously achieved using state-space population models fit in a Bayesian framework, in which demographic data are used to generate informative priors, bounding estimates of r (Delean, Brook, & Bradshaw, 2013; Lebreton & Gimenez, 2013). Such state-space models include an additional detection term that adds variance (observation error) according to a specified (typically normal) distribution to population counts at each time step. Additionally, Bayesian modeling approaches allow the explicit definition of prior distributions from which parameter values are sampled (Kéry & Schaub, 2012). The primary drawbacks of this powerful method of analysis are needing a long-time series, possibly 30 years or more (Lebreton & Gimenez, 2013), and sufficient demographic data to provide prior distributions for model parameters (particularly r) that are independent from population data used to fit the model.

Long-term survey data on bird populations offer a valuable opportunity for investigating density dependence using data-demanding approaches like state-space modeling. Particularly for threatened species of vertebrates, data on reproductive success and population size are sometimes collected in tandem across time to support management, which provides opportunities for investigating density dependence using multiple sources of evidence. These threatened taxa often also have narrow geographic ranges, allowing analyses to take place on a closed population. Time series of population data can be especially informative for species with narrow geographic ranges, such as island-endemic taxa, which have clear range boundaries, and make up a large proportion of the threatened avifauna (Blackburn, Cassey, Duncan, Evans, & Gaston, 2004; Johnson & Stattersfield, 1990; Milberg & Tyrberg, 1993).

The Hawaiian Islands are hotspots for avian endemism and extinction (Ziegler, 2002) and they support many threatened bird taxa that have been monitored for decades (Pratt, Atkinson, & Banko, 2009; Reed, Elphick, Zuur, & Ieno, 2011; Scott, Conant, & Van Riper, 2001). Most of Hawai'i's native forest birds are declining despite the protection of large amounts of forest upland, and some persist at such low numbers that density-dependence may have negligible impacts on current population dynamics (Gorresen, Camp, Reynolds, Woodworth, & Pratt, 2009; Reynolds & Snetsinger, 2001). Hawaiian waterbirds by contrast, may experience density-dependent dynamics in their limited habitats, which have management-relevant impacts on population growth rates (Reynolds, Breeden, & Klavitter, 2013; Walters & Reynolds, 2013). Additionally, remaining populations of Hawaiian forest birds often occupy dense or inaccessible forest habitat, adding detection error as a significant confounding factor for analysis (Camp, Reynolds, Gorresen, Pratt, & Woodworth, 2009; Freckleton, Watkinson, Green, & Sutherland, 2006; Marques, Thomas, Fancy, & Buckland, 2007; Reynolds & Snetsinger, 2001). By contrast, Hawai'i's endangered waterbirds occupy easily accessible, open habitats. In addition, Hawaiian stilts (Himantopus mexicanus knudseni) spend most of their time in the open, away from emergent vegetation. These factors combined make detection errors significantly lower stilts than for forest birds or other waterbirds in Hawai'i (Camp, Brinck, Paxton, & Leopold, 2014; Chang, 1990; Reed, Elphick, Zuur, et al., 2011). The goal of this article is to show evidence of density-dependent population dynamics in the Hawaiian stilt, an endangered waterbird endemic to the Hawaiian Islands, using long-term data sets on population size and reproduction.

There exists one intuitive hypothetical mechanism by which density-dependent feedback might occur in Hawaiian stilts. When population densities are high, the aggressive territorial behavior of adult stilts can lead to violent and occasionally fatal attacks on conspecific chicks and adults, sometimes with extensive chick fatalities, and damage to eggs and nests (A. Nadig USFWS pers. comm.; CVR and JMR pers. Obs.; Coleman, 1981; Cyanotech, 2006; Robinson, Reed, Skorupa, & Oring, 1999). Such impacts could cause nontrivial reductions in reproductive output and thus annual population growth rate. We test for density-dependence using both frequentist and Bayesian state-space analyses of population time series, as well as regression of a key demographic parameter against site-specific population density, and validate our findings using simulated population trajectories. Our use of complementary methods benefits from the advantages of each method while balancing conservative versus potentially positive bias in detecting density dependence.

2 MATERIAL AND METHODS

2.1 Study system

The Hawaiian stilt is a subspecies of the Black-necked stilt that is listed as endangered by the United States federal government and by the state of Hawai'i (Mitchell et al., 2005; USFWS, 2011). Stilts are found in coastal freshwater and saline wetlands throughout the main Hawaiian Islands (O'ahu, Kaua'i, Hawai'i, Moloka'i and Maui), and they showed population declines during the 20th century due to habitat loss, hunting and impacts from exotic invasive plants and predators (Coleman, 1981; Griffin, Shallenberger, & Fefer, 1989; Reed, DesRochers, Vanderwerf, & Scott, 2012; USFWS, 2011; Van Rees & Reed, 2014). With the establishment of national wildlife refuges and other protected areas that are actively managed for these species, Hawaiian stilt populations have shown a long-term increasing trend, with current statewide population estimates around 1,700 individuals (Hawai'i Biannual Waterbird Survey, unpubl. data; Reed, Elphick, Zuur, et al., 2011). Because there are no currently available methods to remove this species' threats, it is considered conservation reliant (Reed et al., 2012; Underwood, Silbernagle, Nishimoto, & Uyehara, 2013).

2.2 Count data

We used a time series of statewide stilt population counts from the Hawai'i Biannual Waterbird Survey (Hawai'i DOFAW, 2017), which consists of both summer (August) and winter (January) counts along consistent routes in all known major habitats for the subspecies (Camp et al., 2014; Engilis & Pratt, 1993). We restricted our analyses to statewide population counts, rather than conducting analyses on a by-island basis, because Hawaiian stilts disperse frequently between the main islands of Hawai'i, and they readily colonize newly restored or created habitats (Banko, 1988; Coleman, 1981; Engilis & Pratt, 1993; Reed, Silbernagle, Evans, Engilis, & Oring, 1998), suggesting that stilts in Hawai'i form one large population. As a result, by-island analyses would have inter-island dispersal as an additional source of variance in population size over time. Because access to the island of Ni'ihau was often sporadic for the Biannual Waterbird Surveys (the island is privately owned), and because movement of stilts between Kaua'i and Ni'ihau is understood to be frequent (citations above), we verified whether excluding data from these islands affected our results by repeating all analyses on a time series that excluded counts from these islands.

Although the Biannual Waterbird Survey is not an exhaustive census of the stilt population in Hawai'i, these counts consistently cover all major habitats in the state in the same 1–3 days to avoid double-counting individuals. Survey methods, routes and sites are consistent across years, implying that observation error is relatively constant over time. Although there has been turnover of observers over the course of the study period, we believe that detection error has remained relatively constant across time, given that methods have not changed and that all observers are federal or state biologists or volunteers with experience in wetland bird surveys. Inexperienced surveyors are trained by experienced surveyors, and typically accompanied by them (Engilis & Pratt, 1993; JMR & CVR personal experience). Chang (1990) and Camp et al. (2014) both reported that detection error was very low for Hawaiian stilts because of their noncryptic plumage, aggressive behavior, tendency to occupy open habitats and almost constant vocalization. Hawaiian stilts are also visually unlike any other waterbird in Hawai'i, making the potential for misidentification virtually null. Chang (1990) reports that, of the individuals present, on average 58% are detected within 5 min of a survey, with near-100% detection for this species within 60 min. Juvenile dispersal typically involves movement between large (surveyed) habitats on the island, making it unlikely that juveniles are any less likely to be detected than adults (JMR, pers. obs.). Camp et al. (2014) also argued that summer counts may be more reliable than winter counts, because the stilts are more aggressive toward humans near nesting areas during the breeding season, and because adult birds hold stable territories during that time, reducing movement between wetlands or islands over the course of a count. Accordingly, the Biannual Waterbird Survey (particularly in summer) is likely a high-quality estimate of the population size of Hawaiian stilts for the state. Summer versus winter waterbird counts might to differ due to three sources of variation that would be difficult to differentiate. These are: (a) imperfect (though high) detectability of stilts at a wetland during surveys, (b) summer counts consisting of hatch-year birds and winter counts consisting only of adults, thus adding the additional variation of environmental stochasticity in egg production and hatch-year survival to summer counts, and (c) incomplete closure of the surveyed population. With regards to the last, a small, varying, but unknown proportion of the statewide population is missed each year in surveys due to the inaccessibility of minor habitat sites (mainly private properties). Given these compounded differences, we elected to retain and use both datasets rather than analyze only one of them.

We also assimilated long-term data on the number of nests, nest success and local stilt abundance using data from annual reports from two Hawaiian stilt breeding sites, Kealia National Wildlife Refuge on Maui (hereafter Kealia; years 1997–2010), and Roland Pond at the Chevron Hawai'i Refinery on O'ahu (hereafter Roland pond; years 1992–2012). Data from both sites involved intensive daily or weekly monitoring of stilt nests and nest fates, as well as counts of abundance over the course of the breeding season.

2.3 Analyses of population time series

We analyzed a 30-year time series (1986–2015) of statewide population count data for evidence of density dependent dynamics using several approaches with complementary advantages and disadvantages. We conducted all statistical analyses in R (version 3.5.1, R Core Team, 2018). We repeated all of the following analyses for both winter and summer counts. As a method of validation, we repeated them on 30-year time series of simulated density-dependent and density-independent population data. These data were simulated using the Ricker and exponential population models, respectively, and included process error and parameter estimates based on Reed, Elphick, & Oring 1998 and the observed datasets (see Tables S1–S5, and code in Table S11). In each case, the starting population size was set to be similar to the starting population size in our observed summer time series.

As a general test for density-dependent population dynamics, we approximated density-dependent versus density-independent dynamics with process variation using linear models with ln-transformed population size at time t as a response variable, and ln-transformed population size at time t − 1 as an offset. This allowed the dependent variable to be a rate, by assigning a constant slope to the relationship between population sizes at t and t − 1. This simpler method provided the advantage of allowing easily interpretable R² and effect size values and a well-established basis for comparison of model performance (AIC_C; Akaike, 1973; Hurvich & Tsai, 1989). Prior to analyses, we visually checked our datasets for ln-normality. Our linear models were as follows:

(1)

(2)

Equation (1) assumes a constant growth rate with process variation, while Equation (2) includes an added coefficient of previous population size, implying a linear relationship between previous population size and growth rate in the current year. We assessed the relative explanatory power and fit of these models by comparing their R² and AIC values.

We also assessed more complex (nonlinear models) of population dynamics by fitting curves using both frequentist and Bayesian state-space approaches. Fitting and comparing nonlinear models offers the key advantage of examining the potential form of population dynamics among a suite of models with distinct behavior and assessing their relative explanatory power. We chose to implement these models first using a frequentist framework, again because of a well-established method of model comparison, but also implemented the models in a Bayesian framework, to take advantage of prior knowledge of the Hawaiian stilt's biology using informative priors for parameters, and to account for detection using a state-space framework.

For the frequentist approach, we used the nonlinear least squares approach (implemented using the nls function in R) to fit three commonly used density dependent and one density-independent population model to the time-series datasets. We used the exponential model as our density-independent model, and the Ricker, Gompertz and theta-logistic models as various forms of density-dependent population dynamics. We implemented these models on population size data in the natural logarithmic scale as follows:

(3)

(4)

(5)

(6)

Here, r is the intrinsic population growth rate (r_max) and K is carrying capacity; and theta ( θ) and alpha (α) are shape parameters for the theta-logistic and Gompertz equations, respectively. In addition to these models, we also used a conditional model of density dependence, which exhibited exponential growth below a given cutoff population density, and Ricker density-dependent growth after the cutoff. This allowed us to model circumstances in which population growth rate was not affected by population density until after a threshold population density. In other words, the conditional model avoided a potentially implausible dynamic in the Ricker and other density-dependent models, wherein density-dependent reductions in growth rate (albeit small ones) occur at any increase in density, even if density remains quite low overall. We chose this additional model because many population viability models (including Vortex; Lacy, Miller, & Traylor-Holzer, 2017) use ceiling models, in which density dependent dynamics typically occur at carrying capacity (e.g., McGowan, Catlin, Shaffer, Gratto-Trevor, & Aron, 2014) or at some proportion of K (Lacy et al., 2017; e.g., van Rees & Reed, 2018). We repeated a suite of models with cutoff points from 1 to 2,000 (covering nearly all possible population values given our observed datasets) to find the optimal population size for the transition to density dependence, comparing these models using AIC.

We implemented Bayesian analyses using Markov-Chain Monte Carlo sampling in JAGS (version 4.3.0; Plummer, 2003) using the packages Runjags (Denwood, 2016), rjags (Plummer, 2018) and jagsUI (Kellner, 2018) in R. We compared the same nonlinear models as mentioned in our frequentist approach (above section), but we additionally implemented both a state-space and non-state-space version of each model. In state space models, detection rate is included by adding an additional error term to the model, observation error, which is independent from the process variation that affects actual population size (Kéry & Schaub, 2012). Thus, each observed population count is assumed to be the product of the previous year's population size, the growth rate with process error and the observation error around the real population of that year. We treated the observation error term as a normally distributed value, whose mean we defined using informative priors (see below). Non-state-space models lacked this parameter and assumed that observed population counts were the same as actual population counts (the product only of the previous year's population and the growth rate with process error).

For our Bayesian analyses, we used informative priors for r_max, observation error, and process variation of population growth rate, theta, alpha and K, using current knowledge of the life history of the Hawaiian stilt. Following Delean et al. (2013), we calculated r_max using a demographic model and set our prior for r_max as a normal distribution centered around our calculated value. By using demographic rates calculated independently from field data with a large sample size (see Reed et al., 2015; Reed, Elphick, & Oring, 1998) to generate informative priors for r_max, we avoid potential bias in the estimation of that parameter given limitations in the length or total observed variation in a population time series (Fagan, Lynch, & Noon, 2010). We calculated this value using the deterministic r from a population viability model for the Hawaiian stilt in Vortex (version 10; Lacy, 1993, Lacy & Pollak, 2017). We modified this model by setting nest success, brood size and juvenile survival to 100% of their observed range of values to create an estimate of maximum possible growth rate for the subspecies. Details on brood size are given in Reed, Elphick, & Oring (1998), and it varies very little across individuals. Maximum brood size (clutch size) for Hawaiian stilts is 3, and this species in single-brooded (Reed, Elphick, & Oring, 1998). Brood size relative to population size has not been tracked, and would be difficult to estimate because chicks (along with their parents) move frequently after hatching, and because brood size can change rapidly due to predation and conspecific attacks. We used a graphical calculator to generate a standard deviation that would allow this prior distribution to encompass a range of feasible values around this mean. We used gamma distributions for the priors of both process variation and observation error, and set parameters for the former based on simulated variation in growth rate (Reed, Elphick, & Oring, 1998), and the latter based on estimates of ~90% detection rates for Hawaiian stilts (Camp et al., 2014; Chang, 1990). In both cases, we defined these functions such that values considerably higher than that dictated by prior knowledge were possible, to minimize bias in results. We chose minimally informative priors for K, theta and alpha, which were uniform distributions bounded by realistic values. In the case of K, we bounded the prior distribution at 1,000 and 3,000, based on long-term observed maximal population counts. We established theta and alpha priors (−3 to 3 and −5 to 5, respectively) visually using a graphing calculator and restricting our analyses to functional forms for which we could find feasible biological explanations.

We compared Bayesian state-space models using the cross-validation (C-V) method (Hooten & Hobbs, 2015). This method involves iteratively excluding (“dropping”) each point in the predictor dataset and predicting its value using a model trained on all other data points. A combined metric, the C-V metric is calculated that is the average deviation of predicted values from actual (excluded) values. This yields an integrated measure of the predictive power or accuracy of the model, which can be compared among models; it is currently accepted as the best method of comparing Bayesian statistical models (Hooten & Hobbs, 2015). We implemented C-V using the drop-k function in Runjags, which iteratively excludes each set of k points in a dataset and repeats model fitting, then attempts to predict the missing value. We used the following equation as our C-V metric, using a k value of 1 (see script in Table S7):

(7)

where T is the length of the total time series, i is the sample being temporarily left out of the model, is the model estimate of the missing datapoint, and is the actual missing value.

Finally, we examined the relationship between discrete population growth rate lambda (λ; N _t/N _{t − 1}) and population size (N _t, as an index of density) for our time series using a linear regression (assuming normal distribution) implemented with the lm() function in R. This allowed us to test for an effect of population density on realized growth rates across time. We verified the normality of data and model residuals visually and using Pearson's chi-square test for normality (implemented in using the package nortest in R; Gross & Ligges, 2015). R code used in all-time series analyses is available in Tables S7–S9, and JAGS code is available in Tables S12–S31.

2.4 Analyses of reproduction data

Because density-dependent population dynamics may be driven by reduced reproductive output under higher population density, we also looked for a negative relationship between a metric of reproductive success and population density. We added this approach to diagnosing density dependence to include a second, more conservative test to our analysis (Lebreton & Gimenez, 2013). We assessed the relationship between peak adult stilt abundance (a proxy for density) and nest success at two sites: Kealia National Wildlife Refuge, Maui and Roland pond, Chevron Refinery, O'ahu), using the linear regression methods outlined above. Kealia is actively managed for Hawaiian waterbirds, with regular predator trapping and control of exotic invasive plants, both of which can strongly impact reproductive success in Hawaiian stilts (Coleman, 1981; USFWS, 2011). The refuge supports high (on the order of 800–1,000 birds) numbers of Hawaiian stilts throughout the year and is the primary breeding habitat on Maui. Kealia is about 101 ha in area, although it is important to note that the habitat area, which varies due to hydrological conditions and vegetative cover, is much smaller and highly dynamic. Ponds at the Chevron refinery (~2.5 ha in total), by contrast, have had sporadic predator control and are not managed specifically for Hawaiian stilts, and have total population counts on the order of 100–150 birds. The Chevron refinery supports a relatively small proportion (<20%) of O'ahu's total breeding population of Hawaiian stilts in a typical year. We selected these sites both for the availability of long-term data on adult abundance and nest success and for their contrasting conditions, which may provide examples where hypothesized density-dependent dynamics do (Kealia NWR) and do not (Roland pond in the Chevron Refinery) occur.

Stilt abundance, nesting activity and nest success at both sites were monitored weekly throughout the breeding season (starting when courtship behaviors were first observed and continuing to September) and nests were considered successful if they hatched any chicks. Hawaiian stilts nest in bare mudflats and construct simple nests with dead vegetation, which are easily detectable during visual surveys (Chang, 1990; Coleman, 1981). Chicks are precocial, leaving the nest within a few days of hatching and moving around in their parents' territory which typically makes them easy to detect (Chang, 1990; Coleman, 1981). In addition, Hawaiian stilts are intensely territorial during breeding, leading to highly conserved space use that may further minimize detection error. Although nest success only captures part of the potential impacts of aggressive territorial behavior in Hawaiian stilts, we used it as a surrogate for the overall intensity of reproductive impacts from aggressive interactions, lacking information on early-stage chick survival between sites.

We performed linear regressions on the effect of annual maximum population size on mean annual nest success (as a percentage) using the same functions and tests of normality as with our regression of r against population size (above). We used only data for which the amount of wetland area surveyed and available for stilt breeding was known and consistent between years to ensure that population counts approximated density. R code for our analyses on reproduction data is available in Table S10.

2.5 Data management

All datasets used in this study are available in Tables S32–S37 and from the corresponding author upon request.

3 RESULTS

3.1 Analyses of population time series

Summer counts of Hawaiian stilts varied from 850 to 1,991 (mean = 1,382, SD = 427.9), and winter counts ranged from 663 to 2,125 (mean = 1,486, SD = 271.6148; Figure 1).

Figure 1

Time-series data of statewide Hawaiian stilt population counts from summer (August) and winter (January) surveys between 1986 and 2015

3.1.1 Frequentist analyses

For both our winter and summer time series, among linear-models of population dynamics (see Equations 1 and 2 in methods section), approximations of density-dependent dynamics with process variation (Equation 2) better explained variation in our observed data than did density-independent models (Equation 1; Table 1). Among nonlinear models of population dynamics, conditional models (in which density-dependent effects occurred after a critical population size) had the lowest AIC values for both summer and winter datasets, with the next best model having a ΔAIC >3 (Table 1) in our winter dataset and >6 in our summer dataset. AIC values for these conditional models were minimized at threshold values of 860–1,020 birds for our summer dataset and 910–930 for our winter dataset. This indicates that the statewide population size appears to change from density-independent for smaller values to density-dependent for larger values. For both datasets, the exponential (density independent) model had a ΔAIC of >5 from the best model, and was outperformed by the Ricker, density-dependent linear approximation and Conditional models. The 95% confidence intervals for parameter estimates for our top ranked models (Conditional, Ricker and linear density-dependent) in both datasets did not overlap zero, while all parameters for density-independent models contained zero.

Table 1. Comparison of model performance between linear-approximation and nonlinear models of density-dependent and density-independent population growth using 30-year time series of Hawaiian stilt range-wide abundance estimates for both Winter (nonbreeding) and Summer (breeding) seasons

Dataset	Model	ΔAICc	Parameter	95% CI
Winter	Conditional	0.0	r K	0.10–0.48* 960–1800*
Winter	Ricker	3.596	r K	0.04–0.66* 1,080–1850*
Winter	Density-dependent (linear)	3.594	Intercept DD term	0.04–0.66* −2.3E-5 to −4.5E-4
Winter	Theta-logistic	5.55	r theta (θ) K	−6.50 to 8.10 −32.6 to 39.6 −70,600 to 72,600
Winter	Exponential	5.97	r	−0.07 to 0.12
Winter	Density-independent (linear)	5.97	r	−0.07 to 0.12
Winter	Gompertz	13.34	r K	−1.9 to 2.2 −33,100 to 39,100
Summer	Conditional	0.0	r K	0.19–0.62* 1,300–1,700*
Summer	Density-dependent (linear)	6.354	Intercept DD term	0.07–0.73* −4.04E-5 to −0.005*
Summer	Ricker	6.346	r K	0.068–0.73* 1,300–1,760*
Summer	Theta-logistic	7.859	r theta (θ) K	−10.1 to 11.8 −43.2 to 50.2 −92,000 to 94,000
Summer	Exponential	9.294	r	−0.049 to 0.077
Summer	Density-independent (linear)	9.294	r	−0.049 to 0.077
Summer	Gompertz	13.374	r K	−1.280 to 1.65 −19,800 to 25,800

• Note: Asterisks (*) indicate model parameters for which the 95% confidence interval does not overlap 0. The conditional model is density-independent for population sizes below a given population threshold, and density dependent (Ricker model) above that threshold. r is instantaneous growth rate, K is carrying capacity, DD term is the density dependence term in a Ricker-approximating linear model and theta (θ) is the shape parameter in the theta-logistic growth equation.

Linear regression of discrete observed population growth rate (λ) against population size the previous year yielded statistically significant negative relationships for both winter and summer datasets (β = −.00033, p = .01 for Summer, β = −.00011, p = .02 for Winter). The R²s were .22 for our summer data and .18 for our winter data.

3.1.2 Bayesian analyses

Bayesian models run on our summer dataset had much lower C-V values than did those from our winter dataset, indicating a smaller average difference between model predictions and observed population sizes (higher predictive power; Table 2). For both datasets, the C-V values of state-space models were smaller than their non-state-space counterparts by 10–15, with the exception of the Gompertz models, where the non-state-space versions had substantially higher C-V values. For the summer dataset, the Ricker and conditional model showed higher performance than other models among both state-space and non-state-space models, although the exponential model also showed a good performance. For the winter dataset, the exponential and conditional model performed best among state-space models, while the conditional model was the best model by a margin of more than 10 units among non-state-space models.

Table 2. Comparison of model performances among nonlinear Bayesian state-space (SS) and non-state-space (NSS) models of population growth using the 30-year time series dataset of Hawaiian stilt range-wide abundances

Dataset	Model	C-V metric (no. stilts)
Winter	Conditional (SS)	220.50
	Gompertz (SS)	435.70
	Exponential (SS)	220.33
	Theta-logistic (SS)	238.62
	Ricker (SS)	227.93
	Conditional (NSS)	234.75
	Gompertz (NSS)	326.76
	Exponential (NSS)	254.63
	Theta-logistic (NSS)	249.38
	Ricker (NSS)	249.29
Summer	Conditional (SS)	106.74
	Gompertz (SS)	281.26
	Exponential (SS)	110.48
	Theta-logistic (SS)	144.08
	Ricker (SS)	106.35
	Conditional (NSS)	118.05
	Gompertz (NSS)	231.29
	Exponential (NSS)	122.22
	Theta-logistic (NSS)	119.84
	Ricker (NSS)	120.18

• Note: C-V is the cross-validation metric, calculated using leave-one-out cross validation, wherein each point in the time series dataset was iteratively excluded and the model re-fit, and the difference between the observed and estimated data point calculated. The C-V metric represents the average deviation (in stilt abundance) per data point between the observed and predicted abundance estimates for each model. Conditional model is described in Table 1. The best models among state-space (SS) and non-state-space (NSS) are shown in bold for each dataset.

3.1.3 Analyses on simulated population time series

Model comparison results for our simulated population time series using a density-dependent process matched those observed for our dataset in all analyses. Just as in our winter dataset, exponential state-space models performed nearly as well as the Ricker and Conditional models in our Bayesian analysis (see Table S3). Meanwhile, exponential models performed best based on the C-V metric for our density-independent dataset. Simulated datasets of length 50 and 100 years showed wider disparities among C-V metrics between exponential and density-dependent non-state-space population models, with the Ricker and conditional models performing best in density-dependent simulated datasets and the exponential and theta-logistic performing best for density-independent simulated datasets (Tables S4 and S5).

As a reminder, count for the Biannual Waterbird Surveys from island of Ni'ihau was sporadic. Because waterbirds can move seasonally between Kaua'i and Ni'ihau, it might bias total counts. Consequently, we wondered if our results would change qualitatively if we excluded counts from Kaua'i and Ni'ihau. They did not (see Table S6, for those analyses).

3.2 Analyses of reproduction data

Mean annual nest success (the proportion of nests which hatched at least one chick) at our two monitored sites were not significantly different from each other (t test adjusted for unequal variance, p = .13, df = 29.7), and ranged from 0.33 to 1 for Roland pond (mean = 0.64, SD = 0.22) and from 0.24 to 0.8 for Kealia (mean = 0.533, SD = 0.15). Maximum annual adult counts at Roland pond ranged from 44 to 175 (mean = 91.1, SD = 34.0), and from 386 to 1,079 at Kealia (mean = 71.3, SD = 192.08). We found a highly significant negative relationship between the maximum number of adult stilts and annual average nest success for Hawaiian stilts at Kealia (p < .04, β = −4.4 × 10^–4 R² = .32), but not at Roland Pond (p = .83, R² = .01; Figure 2). The coefficients estimated for both locations were of similar magnitude, but in the case of Roland pond, the p-value was not significant.

Figure 2

Scatterplots and regression lines of annual maximum counts of adult Hawaiian stilts and mean annual nest success at (a) Kealia National Wildlife Refuge on Maui and (b) Roland Pond at the Chevron Refinery, O'ahu. Linear regressions were highly significant for Kealia NWR but not for Roland Pond

4 DISCUSSION

We evaluated a 30-year time series of estimated range-wide abundance of Hawaiian stilts, plus two time series (16 and 21 years) of nest success and peak annual abundance from closely monitored breeding sites for evidence of density-dependent dynamics, and found evidence across a suite of datasets and methods of analysis. This work will inform the management and population modeling of an endangered island-endemic subspecies, and also demonstrates a new method of creating data-informed conditional models of density-dependent population growth; that is, identifying the population size (N < K) where population growth switches from density-independent growth as smaller population sizes and density-dependent growth at larger sizes.

Our suite of analyses on the population time series strongly supports the hypothesis that Hawaiian stilts exhibit density-dependent population dynamics at the range-wide level, above a statewide population threshold of between 860 and 1,020 individuals. Our model selection among both linear approximations of density dependent dynamics and nonlinear frequentist models consistently ranked exponential (density-independent) models lower (i.e., poorer fit) than density-dependent models. In the case of nonlinear frequentist models, simpler models like the Ricker and conditional model (density-independent population growth below a threshold population size, with density-dependent growth above that threshold) performed best, while more complex models like the Gompertz and theta-logistic models typically underperformed for at least one dataset. The poor performance of these two models may be due to difficulties in parameter estimation (for the theta-logistic; Clark, Brook, Delean, Akçakaya, & Bradshaw, 2010), or because these additional parameters contribute to model complexity that cannot be resolved without a longer time series. A strong negative relationship between annual discrete population growth rate and population size adds further credence to the notion that stilt populations exhibit density-dependent dynamics. Generally high C-V values for all models were likely due to large interannual variation in stilt counts, and the relatively short length of our time series (see below).

Multimodel inference among Bayesian models remains an under-explored and somewhat problematic topic (Hooten & Hobbs, 2015; Spiegelhalter, Best, Carlin, & van der Linde, 2014). For our Bayesian analyses of nonlinear population models in both seasons, the exponential models were included in the best performing models for our winter dataset (Table 2). In this case, the exponential model was not qualitatively different in performance when compared to the best-performing density-dependent models. Our population size time-series dataset spans 30 years, making it a reasonable length for ecological studies and at the higher end of the recommended minimum lengths for time-series analysis (Andow & Kiritani, 2016; Brook & Bradshaw, 2006; but see Lebreton & Gimenez, 2013; and Ziebarth, Abbott, & Ives, 2010). Given the difficulty of measuring density dependence in wild populations (Lebreton & Gimenez, 2013), a time series of this length is advantageous, but potentially not sufficient. Even though density-dependent dynamics have been detected in shorter time series (e.g., Andow & Kiritani, 2016), evidence of density dependence tends to be weaker in established, stable populations (Brouwer et al., 2009; Ferrer & Penteriani, 2008; Soutullo, Liminana, Urios, Surroca, & Gill, 2006). Consequently, it is plausible that our time-series datasets are insufficiently large to support analysis with complex state-space models, particularly because our population appears to have been near apparent carrying capacity for much of the observation period. Our validation experiment using simulated data supports this, given that we saw similar patterns in model comparison in a 30-year dataset simulated using a density-dependent process, while longer (50- and 100-year) datasets showed a more accurate selection of the true process model. The parallel patterns in model ranking between our observed data and those of the simulated 30-year dataset give greater credence to our detection of density-dependent dynamics in the observed datasets. Simulation studies using similar but more formal approaches have been used to good effect in other systems for testing for density dependence (e.g., Robinson, McGowan, & Devers, 2017).

The high detection rates of Hawaiian stilts, especially in summer, may give greater weight to the results of the analyses performed on our summer datasets, as well as non-state-space forms of our Bayesian nonlinear population models. Notably, density-dependent models performed much better than density-independent models among non-state-space versions, although these typically had higher C-V values (i.e., lower predictive power) than state-space models. Given a priori knowledge that detection error may be minor in Hawaiian stilts, and that the C-V metric does not penalize for model complexity, the higher performance of state-space models may be primarily due to the greater flexibility granted by adding an additional, potentially extraneous model parameter. If this line of reasoning is followed, then the support of density-dependent dynamics from our Bayesian nonlinear analysis of Hawaiian stilt population time series is greater and more consistent.

Finally, we detected a strong negative relationship between maximum annual adult density and mean annual nest success at Kealia National Wildlife Refuge on Maui, consistent with density-dependent impacts on reproductive success at higher densities of adult stilts. This supports our a priori hypothesis that aggressive territoriality in stilts may have negative reproductive impacts (Coleman, 1981). We found no such statistical support, however, for density-dependent dynamics at Roland pond in the Chevron refinery on O'ahu.

The different ecological and landscape contexts of these two sites may be responsible for the difference in observed population dynamics. Anecdotal reports of chick mortality at Kealia are widespread and impressive, with one biologist reporting more than 50 dead chicks encountered in a single day at the refuge (A. Nadig, U.S. Fish and Wildlife Service, pers. comm.). By contrast, reports of dead chicks are fewer at Roland pond (Eijzenga, 2006, 2007), and it is suspected that most of the adult breeding population at the site reflects strong site fidelity and not current habitat quality (A. Nadig, USFWS pers. comm.). The ratio of average annual peak abundance of adult stilts to average annual nest count for Kealia is nearly twice that of Roland pond (9.2 and 4.8 adults per nest, respectively). This may increase the probability of negative intraspecific interactions. Depredation by introduced mammalian predators is a dominant driver of nest success for Hawaiian stilts (Coleman, 1981; Underwood et al., 2013), so inconsistent predator management at Roland pond may have introduced variation in nest success that masked or prevented any density-dependent dynamics.

From a landscape perspective, Kealia is one of only two major nesting sites for Hawaiian stilts on Maui and is somewhat isolated from the other site. Roland pond, in contrast, is a relatively small wetland embedded in a network of many more, much larger stilt nesting habitats on O'ahu. The maximum annual count of adult stilts at Roland pond represents on average 18% of the total summer waterbird survey abundance of on O'ahu across years, while reported maxima at Kealia taken at various times of the year have, at times, exceeded the island-wide counts during the formal biannual waterbird counts on Maui. This suggests that large, temporary immigrations to that wetland occur, and may create conditions in which density-dependent factors are important. Habitat management at the wetland specifically to support higher numbers of stilts, combined with the lack of alternative habitats nearby, likely keep population densities relatively high. By contrast, breeding stilts at Roland pond have access to a suite of larger alternative nesting areas, all of which would be easily accessible along the island's flat southern coastal plain, not requiring any inland transit. The combination of lower habitat quality and the potential for easy alternative nesting sites might prevent the levels of crowding that lead to density-dependent dynamics in this taxon.

The results outlined above add to a growing body of work on bird species that exhibit density-dependent population dynamics, in parameters like population growth rate (e.g., Grear et al., 2009; Wilkin, Garant, Gosler, & Sheldon, 2006), reproduction (Bennetts, Fasola, Hafner, & Kayser, 2000; Nummi & Saari, 2003), and age at first reproduction (Cooper, Murphy, Redmond, & Dolan, 2009; Ferrer, Otalora, & García-Ruiz, 2004). In fact, based on a meta-analysis, Sibly and Hone (2002) suggested that density dependence is so pervasive that one should focus more on detecting the form of the relationship rather than its presence. To this end, we found strong support for a conditional density-independent—density-dependent model, which we think is more realistic than models that do not invoke density dependence until a population is at carrying capacity. In our study system, this might be driven by minimum densities (or maximum territory sizes) at which eggs and chicks are affected by aggressive adults from neighboring territories. Prior to this point, such impacts are trivial, but beyond this those impacts increase according to a density-dependent function like the Ricker. Such dynamics would imply that per capita reproductive output decreases before the actual carrying capacity of a habitat is reached, and that high population densities at some point may run counter to some management objectives.

Our investigation of multiple sources of evidence for density-dependent dynamics in Hawaiian stilts suggests that such dynamics do occur in the subspecies, particularly in sites managed for predators and habitat. Consequently, attempts at modeling Hawaiian stilt populations should include some form of density dependence, to account for the dynamics observed here. The only published population model of this species to date (Reed, Elphick, & Oring, 1998) included density dependence, but its form and strength were unclear. Our results also suggest that current population sizes are pushing the upper limit of the number of birds the islands can support under current conditions of habitat availability and management activities.

ACKNOWLEDGMENTS

Funding in support of this project was provided by the U.S. Fish and Wildlife Service. We are grateful to Nick Dorian, Elizabeth Crone, Matthew Kamm and Farshid Ahrestani for advice and guidance on the use of JAGS and Bayesian state-space models. We thank Joy Hiromasa Browning and Mike Nishimoto (USFWS) for sharing monitoring reports on stilt reproduction, and Adonia Henry (Scaup & Willet, LLC) for providing quality-controlled versions of biannual waterbird survey data. We also thank Arleone Dibben-Young, Martha Kawasaki and Mike Silbernagle for their insights on the behavior and natural history of Hawaiian stilts. We are additionally grateful to Conor McGowan, three anonymous reviewers and the handling editor of this manuscript for their helpful comments that improved upon an earlier draft of the article.