Ecology

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After the successful protection of Himalayan areas on the border of China and Nepal, we propose that the two nations should create the world’s highest international peace park by combining the Qomolangma and Sagarmatha national parks. This would align with United Nations Sustainable Development Goal 17, to achieve sustainable development through international cooperation (see go.nature.com/3ixmini).
Competing Interests
The authors declare no competing interests. More

Climate change drove evolution, biogeography, and demographyPhylogenetic results (Fig. 1 and Supplementary Fig. 2) confirm previous findings, recovering Aptenodytes (king and emperor penguins) as the sister clade to all other crown penguins, with brush-tailed (Pygoscelis) penguins in turn sister to two clades uniting the banded (Spheniscus) and little (Eudyptula) penguins and the yellow-eyed (Megadyptes) and crested (Eudyptes) penguins6,7,9. Biogeographical reconstructions (Fig. 1, Supplementary Figs. 3–4 and Supplementary Data 1) support a Zealandian origin for penguins6,7. Stem penguins radiated extensively in Zealandia before dispersing to South America and Antarctica multiple times, following the eastward-flowing direction of the Antarctic Circumpolar Current (ACC) (Fig. 1). Crown penguins most likely arose from descendant lineages in South America, before dispersing back to Zealandia at least three times. Interestingly, at least two such dispersals occurred before the inferred onset of the ACC system, suggesting that early stem penguins were not dependent on currents to disperse over long distances. A second pulse of speciation coincides with the onset of the ACC, though understanding whether this pattern is real or an artifact of fossil sampling requires more collecting from early Eocene localities. We infer an age of ~14 Ma for the origin of crown penguins, which is more recent than the ~24 Ma age recovered in genomic analyses, not including fossil taxa7 (Supplementary Fig. 2b) and coincides with the onset of global cooling during the middle Miocene climate transition4,10 (Supplementary Fig. 3a). This young age suggests that expansion of Antarctic ice sheets and the onset of dispersal vectors such as the Benguela Current11 during the middle to late Miocene facilitated crown penguin dispersal and speciation, as hinted at by fossil evidence12.Incongruences between species trees and gene trees were identified, e.g., alternate topologies occurred at high frequencies ( >10%) for several internal branches (Fig. 1c; Supplementary Fig. 5). These patterns indicate that gene tree discordance may be caused by incomplete lineage sorting (ILS) or introgression events. By quantifying ILS and introgression via branch lengths from over 10,000 gene trees, we found that the rapid speciation within crown penguins was accompanied by >5% ILS content within the ancestors of Spheniscus, Eudyptula, Eudyptes, and several subgroups within Eudyptes (Fig. 2a). Our dated tree provides a temporal framework for this rapid radiation: the four extant Spheniscus taxa are all inferred to have split from one another within the last ~3 Ma, and likewise the nine extant Eudyptes taxa likely split from one another in that same time (Fig. 1b). Many closely related penguin species/lineages are known to hybridize in the wild (see supplementary methods). Consistent with this, multiple analyses suggest that introgression also contributes to species tree—gene tree incongruence (Supplementary Figs. 6–9 and Supplementary Data 2; also see Supplementary Methods for further details). This could explain the most notable conflict in previous phylogenetic results, which showed inconsistency over whether Aptenodytes alone7 or Aptenodytes and Pygoscelis together4,5 represent the sister clade to all other extant penguins. Introgression was detected between the ancestor of Aptenodytes and the ancestor of other extant penguins, and is inferred to have occurred when the range of these ancestors overlapped in South America (Fig. 2a and Supplementary Data 2). Introgression ( >9%) was also detected between Eudyptula novaehollandiae and Eudyptula minor, and several introgression events were especially pervasive in Eudyptes (Fig. 2a and Supplementary Fig. 6).Fig. 2: Incomplete lineage sorting, introgression events, and demographic history among penguins.a Model of incomplete lineage sorting (ILS) and introgression events estimated from QuIBL and hybrid pairwise sequentially Markovian coalescent (hPSMC) results. hPSMC was only run for 20 species pairs (see b). Numbers on branches represent the proportion (%) of ILS (orange branches) or introgression (blue lines, blue dashed lines, and blue dotted lines) detected by QuIBL. Proportions 50 km; see Supplementary Data 117), while taxa that decreased towards the end of the LGP (e.g., S. humboldti, S. demersus, M. a. antipodes and likely M. a. richdalei) tend to be residential, and forage inshore; see Supplementary Data 1. Taxa that disperse farther may have overcome local impacts of global climate cooling during the LGP (e.g., changes in sea-ice extent, prey abundance and terrestrial glaciation, however see18) largely by relocating to lower latitudes (e.g.,14), whereas locally-restricted taxa may have been more prone to sudden population collapses.Penguins have the slowest evolutionary rates among birdsThe integrated evolutionary speed hypothesis (IESH) proposes that temperature, water availability, population size, and spatial heterogeneity influence evolutionary rate19. Life history traits also impact the evolutionary rate, but such relationships remain incompletely understood in birds20. Penguins are long-lived, large-bodied, and produce few offspring, thus providing an ideal case study in how life history may impact evolutionary rate. We tested the IESH using three proxies for evolutionary rate: substitution rate, P and K2P distances between lineages and their ancestors (Supplementary Fig. 12 and Supplementary Data 3). We found that penguins and their sister group (Procellariiformes) had the lowest evolutionary rates of the 17 avian orders sampled by21 (Fig. 3a, Supplementary Fig. 13, and Supplementary Data 3). Because other aquatic orders also show slow rates (e.g., the aquatic Anseriformes show a significantly slower rate than their terrestrial sister group Galliformes), we hypothesize that the rate in penguins represents the culmination of a gradual slowdown associated with increasingly aquatic ecology. Intriguingly, we detected a trend toward decreasing rate over the first ~10 Ma of crown penguin evolution, followed by a marked uptick ~2 Ma, which suggests the onset of glacial-interglacial cycles contributed to a recent increase in evolutionary rates in penguins (Fig. 3b).Fig. 3: Evolutionary rates in birds.a Evolutionary rate in avian orders based on a ~19 Mbp alignment of highly conserved genome regions. Sphenisciformes and Procellariiformes have the lowest evolutionary rate among modern bird orders (One-sided Wilcoxon Rank sum test, P values  0.1)). Numbers at the tips represent the sample size in each group. Numbers at nodes represent the divergence times (Ma) between each order and its sister taxon and red dots within the boxplots indicate average values. We did not attempt to estimate the evolutionary rates for orders containing less than three sampled species (gray font; Musophagiformes, Mesitornithiformes, and Struthioniformes). Boxplots show the median with hinges at the 25th and 75th percentile and whiskers extending 1.5 times the interquartile range. Some bird images were downloaded from phylopic.org and were licensed under the Creative Commons (CC0) 1.0 Universal Public Domain Dedication. b Evolutionary rates inferred for extant penguin lineages at internal nodes from the maximum clade credibility tree, calculated using a 500 Mbp genome alignment. Gray shadows represent the 95% credible intervals. c–e Correlations between c, body mass and generation time (P value  More

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LED trapsWe used a commercially available portable light trap (Eco-chu trap, Konan Shisetsu Kanri, Okinawa, Japan) to modify the light source. A prototype trap equipped with 12 UV-LED bulbs was developed to catch the green chafer Anomala albopilosa (Hope)35, but it was not sufficiently attractive to stink bugs. Light sources with different numbers of LEDs, from 12 to 84, were used. Either or both bullet-type UV-LED bulbs (NS395L-ERLO; 395 nm, 20 mA, Nitride Semiconductors, Tokushima, Japan) and green LED bulbs (NEPG510S; 525 nm, 20 mA, Nichia, Tokushima, Japan) were used. LEDs were arranged vertically on a stainless-steel cylinder (4.8 cm in diameter, 20 cm in height). Light sources with 12 LEDs were arranged in six rows around the circumference. Each row was arranged as two LEDs at 7.8 cm intervals. Adjacent LEDs were arranged in a left-handed spiral (depression angle of 53°, at approximately 2.5 cm intervals). Light sources with 21 LEDs were arranged in eight rows around the circumference. Each row was arranged as two or three LEDs at 7.2 cm intervals. Adjacent LEDs were arranged in a left-handed spiral (depression angle of 63°, at approximately 2.0 cm intervals). Light sources with 42 LEDs were arranged in eight rows around the circumference. Each row was arranged as five or six LEDs at 3.6 cm intervals. Adjacent LEDs were arranged in a left-handed spiral (elevation angle 63°, at approximately 2.0 cm intervals). Light sources with 84 LEDs were arranged in eight rows around the circumference. Each row was arranged as 10 or 11 LEDs at 1.8 cm intervals. Adjacent LEDs were arranged in a left-handed spiral (elevation angle 63°, at approximately 2.0 cm interval). When both UV and green LEDs were used, both LEDs were arranged alternately in a row (Fig. 4). The cylinder with the LEDs was covered with a transparent acrylic cylinder (9.8 cm in diameter, 20 cm in height).Figure 4Photograph of combined UV and green LED trap used in the experiments.Full size imageThe light source was mounted on a funnel (31 cm in diameter, 24 cm in height), and the lower part of the light source was approximately 100 cm above the ground. A cylindrical chamber (23 cm in diameter, 20 cm in height) was placed under the funnel so that insects that were attracted to the light fell into the funnel and were trapped. The legs of the trap were anchored to the ground using steel stakes. A dimethyl-dichloro-vinyl-phosphate (DDVP) plate containing 10.7 g dichlorvos (Bapona, Earth Chemical, Tokyo, Japan) was placed inside the chamber to kill the insects. The lights were turned on at 18:00 and turned off at 6:00 the next day. The power for the lights was supplied by rechargeable car batteries (N-40B19R/SB; DC 12 V, 28 Ah, Panasonic, Osaka, Japan) or domestic electricity power supplies (AC100V).Emission spectra of combined UV and green lightThe spectral intensity of combined UV and green light was measured using a high-speed spectrometer (HSU-100S, Asahi Spectra, Tokyo, Japan) in a dark room. An attached sensor fiber was placed 50 cm in front of the light source. The measurement was performed five times, the light source was rotated for each measurement to minimize the angle effect, and the average was used as a representative value. The UV- and green-LED emission spectra showed single peaks at wavelengths of 400 and 526 nm, respectively (Fig. 5). Calculated light intensities of UV (350–450 nm) and green (451–600 nm) regions were 2.12 × 1017 and 2.03 × 1017 photons m−2 s−1, respectively; that is, the light intensities of UV- and green-LEDs were almost equal.Figure 5Emission spectra of light source with UV- and green-LEDs. The light source was composed of alternating 42 UV-LEDs and 42 green-LEDs. The intensity of light was measured using a high-speed spectrometer (HSU-100S). An attached sensor fiber was placed 50 cm in front of the light source.Full size imageField evaluation of attractiveness to light sourcesField experiments were conducted at three locations in Japan: Central Region Agricultural Research Center (CARC), Hokuriku Research Station (37° 07′ 00″ N, 138° 16′ 23″ E) in Niigata; Yamaguchi Prefectural Agriculture & Forestry General Technology Center (YPATC) (34° 09′ 37″ N, 131° 29′ 47″ E) in Yamaguchi; and Okinawa Prefectural Agricultural Research Center (OPARC) (26° 06′ 18″ N, 127° 40′ 53″ E) in Okinawa. The distribution of Nezara spp. varies among the regions in Japan. Only N. antennata is distributed in Niigata, and only N. viridula is distributed in Okinawa. Both N. antennata and N. viridula were found in Yamaguchi.Experiment 1: Attractiveness of UV light at different intensitiesField experiments to evaluate the attractiveness of UV light at different intensities were conducted from August 2 to 29, 2017, around a soybean field at the CARC in Niigata and from July 12 to September 9, 2019, in grassland at the OPARC in Okinawa. Light traps with different numbers of UV-LEDs (12, 21, 42, and 84) were used as light sources. Each of the four LED traps was spaced more than 30 m apart and placed randomly around the soybean field or grassland. Due to time constraints, the numbers of N. viridula and N. antennata captured in traps were counted every 3–4 days at Niigata (total eight replicates) and every 7 days in Okinawa (total eight replicates). The traps were randomly repositioned every week to minimize the effect of trap location. The raw capture data for each trap are listed in Supplementary Table S1.Experiment 2: Attractiveness of green light at different intensitiesField experiment to evaluate the attractiveness of green light at different intensities was conducted from July 5 to August 5, 2019, around a soybean field at the YPATC in Yamaguchi. Light traps with different numbers of green LEDs (12, 21, 42, and 84) were used as light sources. Light trap with 84 UV-LEDs was used as the positive control. Each of the five LED traps was spaced more than 30 m apart and placed randomly around the soybean field. The numbers of Nezara bugs captured in traps were counted every 3–4 days (total nine replicates). The traps were randomly repositioned every week. The raw capture data for each trap are listed in Supplementary Table S2.Experiment 3: Attractiveness of combined-UV and green lightField experiments to evaluate the attractiveness of combinations of UV- and green-LEDs were conducted from June 13 to September 4, 2017, in the grassland at the OPARC in Okinawa, and from July 15 to September 1, 2017, around a soybean field at the YPATC in Yamaguchi. Light traps with 84 UV-LEDs, 84 green-LEDs, and a combination of 42 UV-LEDs and 42 green-LEDs were used as light sources. Each of the three LED traps was spaced more than 30 m apart and placed randomly around the soybean field or grassland. Although insects other than Nezara bugs (mainly coleopteran species) were captured in the light traps, for soybean pests, the funnel-type light traps are intended for monitoring large coleopteran and heteropteran insects ( > 1 cm). Therefore, we targeted and counted insects that meet these conditions. Statistical analysis was performed on species with a total capture number of more than 20 individuals in the three traps. The species were as follows: in addition to Nezara bugs, heteropteran bugs, Piezodorus hybneri (Gmelin), Glaucias subpunctatus (Walker), Halyomorpha halys (Stål), and Plautia stali Scott, as well as coleopteran beetles, Anomala albopilosa (Hope), A. cuprea Hope, A. rufocuprea Motschulsky, and Holotrichia parallela Motschulsky. The insects captured in traps were counted for each species every 7 days at Okinawa (total 12 replicates) and every 3–4 days at Yamaguchi (total 14 replicates). The traps were randomly repositioned every week. The raw capture data for each trap are listed in Supplementary Table S3.Data analysisIn Experiment 1, the effect of UV light intensities for trap catches were analyzed using a nonparametric one-tailed Shirley–Williams test under an assumption that higher light intensity attracts larger amounts of insects. In Experiment 2, the effect of green light intensities for trap catches were analyzed using the Shirley–Williams test. Subsequently, the attractiveness of each green light was compared to that of UV light using Wilcoxon matched pairs signed-rank test. In Experiment 3, the effect of light sources for trap catches was analyzed using the Friedman test, followed by the Wilcoxon signed-rank test, with Bonferroni correction for multiple comparisons. Statistical analyses were performed using R version 4.2.0 (R Core Team, 2022). More

All figures were produced using the R package ggplot2 v3.3.534.eBird and covariate dataeBird data are structured as follows. Birders submit observations as species checklists with counts of each species they identify. They report associated metadata, such as location, date and time, duration of the observation period, number of observers, and sampling protocol25,26,31. The birder indicates whether their checklist is “complete”; complete checklists yield inferred zeroes for all species not reported on a checklist.We retrieved the eBird Basic Dataset containing all eBird observations and sampling metadata. We extracted all complete checklists that occurred within the U.S. state of California between April 1 and June 30, 2019. Four survey-level covariates were retrieved from eBird checklist metadata as detection covariates: number of observers, checklist duration, date of year, and time of day; any checklist that failed to report one or more of these variables was dropped. Corresponding to best practices for use of eBird data, we filtered the data for quality according to the following criteria: we discarded checklists other than those following the “Stationary” survey protocol (observations made at a single spatial location) with duration shorter than 4 hours and at most 10 observers in the group31,35.We selected twenty circular regions of high sampling intensity with 10 km radii across California (Supplemental Fig. 1). These spanned the state’s many habitats including coastal, agricultural, wetland, and mountain areas, and contained active birding areas such as parks and human population centers. In each subregion, we selected 10 species with the highest reporting rate (proportion of checklists including that species) and 10 representing an intermediate reporting rate. An additional 10 species were selected that were detected in many regions to enable cross-region comparisons, yielding 407 species-subregion (SSR) datasets (with overlaps between the two species selection protocols; see Supplemental Section 2 for the full algorithm). Across 20 subregions, we accepted 6094 eBird checklists for analysis, each with an associated count (potentially zero) for each species. Observations were aggregated to sampling sites defined by a 50 m spatial grid. The 50 m grid was chosen to conservatively identify related surveys and was not motivated by biological processes, nor does it represent the sampling area of each survey. In this context, the concept of “closure” in the latent state is already suspect due to the fact that eBird checklist sampling areas are inconsistent. Data were processed in R using the ‘auk’ package36,37.An elevation surface for the state of California was retrieved from WorldClim at (8.3 times 10^{-3}) decimal degrees resolution using the R package raster38,39. This commonly used covariate was included as a baseline spatial covariate to enable comparison of estimation properties across sites, but its biological relevance to abundance is not crucial to our analysis31. Land cover data were retrieved from the LandFire GIS database’s Existing Vegetation Type layer40. For each unique survey location, a 500 m buffer was calculated around the reported location, and the percent of the buffer which was water, tree cover, agriculture or other vegetation (shrub or grassland) was calculated. We used the following five site-level covariates: elevation, and percent of the landscape within a 500 m buffer of the site that was water, trees, agricultural land, or other vegetation. We included six checklist-level covariates: duration, number of observers, time of day, time of day squared, Julian date, and Julian date squared. Covariates were dropped in datasets where only a single unique value was observed for that covariate.Model implementation and selectionWe considered four variants of the N-mixture model and two variants of the GLMM comprising a total of 6 distinct models, defined by the distributions used in the model or sub-model.The GLMM for count data that we considered is defined as$$begin{array}{*{20}l} {y_{{ij}} sim D(mu _{{ij}} ,[theta ])} hfill \ {log (mu _{{ij}} ) = beta _{0} + {mathbf{x}}_{{ij}}^{T} user2{beta } + alpha _{i} } hfill \ {alpha _{i} sim {mathcal{N}}(0,sigma _{alpha } )} hfill \ end{array}$$where (y_{ij}) is th jth observation at site i, D is a probability distribution (which may contain an extra parameter (theta) to account for overdispersion), (mu _{ij}) represents the mean expected count and is a logit-linear combination of observed site- and observation-level covariates (x_{ij}), (beta) are coefficients representing the effect of those covariates, (beta _0) is a log-scale intercept corresponding to the expected log count at the mean site (i.e. with all centered covariates set to 0), and (alpha _i) is the random effect of site i following a normal distribution. Due to the right skew of (exp (y_{ij})), by log-normal distribution theory the log of the expected count at the mean site is (beta _0 + 0.5 sigma _{alpha }^2). We considered two forms of this model, where D was either a Poisson or a negative binomial distribution, in the latter case with the extra parameter (theta).The N-mixture model is defined as$$begin{array}{*{20}l} {y_{{ij}} sim D_{w} (N_{i} ,p_{{ij}} ,[theta _{w} ])} hfill \ {N_{i} sim D_{b} (lambda _{i} ,[theta _{b} ])} hfill \ {{text{logit}}(p_{{ij}} ) = {text{}}{text{logit}}(p_{0} ) + {mathbf{x}}_{{ij(w)}} {mathbf{beta }}_{w} } hfill \ {log (lambda _{i} ) = log (lambda _{0} ) + {mathbf{x}}_{{i(b)}} {mathbf{beta }}_{b} } hfill \ {p_{0} = e^{{frac{{phi _{1} + phi _{2} }}{2}}} } hfill \ {lambda _{0} = e^{{frac{{phi _{1} – phi _{2} }}{2}}} } hfill \ end{array}$$where (D_b) and (D_w) are probability distributions representing between- and within-site variation, respectively; (N_i) is a site-level latent variable normally representing the “true” abundance at site i; (p_{ij}) is the detection probability of each individual on the jth observation event at site i; (lambda _i) is the mean abundance at site i; and (x_{(w)}) and (x_{(b)}) are covariate vectors for detection and abundance, respectively, with corresponding coefficients (beta _w) and (beta _b). For reasons described below, we reparameterize the intercept parameters of the N-mixture submodels, (log (lambda _0)) and (text{ logit }(p_0)), in terms of two orthogonal parameters (phi _1 = log (lambda _0 p_0)) and (phi _2=log (p_0 / lambda _0)). Now (phi _1) and (phi _2) represent the expected log count and the contrast between detection and abundance, respectively, at the mean site. This parameterization allows us to investigate stability of parameter estimation. The log-scale expected count of the N-mixture model is (phi _1 = log (lambda _0 p_0)), analogous to (beta _0 + 0.5 sigma _{alpha }^2) in the GLMM (see Supplemental Section 6). Each submodel distribution D could include or not include an overdispersion parameter ((theta _w) and (theta _b)), yielding four possible N-mixture model variants: binomial-Poisson (B-P), binomial-negative binomial (B-NB), beta-binomial-Poisson (BB-P), and beta-binomial-negative binomial (BB-NB)8,11.We chose to fit models with maximum likelihood estimation (MLE) for computational feasibility and because key diagnostic tools, such as AIC and methods for checking goodness-of-fit and autocorrelation, were best suited to MLE estimation15. We fit N-mixture models with the nimble and nimbleEcology R packages starting with a conservatively large choice of K, the truncation value of the infinite sum in the N-mixture likelihood calculation33,41 (see Supplemental Section 4 for a discussion of maximum likelihood estimation with NIMBLE). We fit GLMMs with the R package glmmTMB42. We applied forward AIC selection to choose the best covariates for each model with each dataset (illustrated in Fig. S1). One spatial covariate (elevation) and two checklist metadata covariates (duration and number of observers) were treated as a priori important and were included in all models. In the N-mixture model, checklist-specific sampling metadata were only allowed in the detection submodel, while land cover covariates and the interactions between them were allowed in both the detection and abundance submodels. Interactions were dropped in datasets when interaction values showed a correlation of > 0.8 with one of their first-order terms. In N-mixture models, additions to both submodels were considered simultaneously during forward AIC selection.For comparisons between models, we selected a heuristic threshold of (Delta text {AIC} > 2) to say that one model is supported over another30.Fit, estimation, and computationGoodness-of-fitWe used the Kolmogorov-Smirnov (KS) test, a p-value based metric, to evaluate goodness-of-fit on each selected model. For GLMMs, residuals were obtained using the DHARMa R package’s ‘simulateResiduals’ and the KS test was applied using the ‘testUniformity’ function43. For N-mixture models, we considered the site-sum randomized quantile (SSRQ) residuals described by Knape et al.15, computing these for each N-mixture model and running a KS test against the normal CDF. We assumed that covariate effects did not vary by space within subregions and chose not to use spatially explicit models31,44. To test this assumption, we applied Moran’s I test to the SSRQ or DHARMa-generated residuals for each site or observation.Parameter estimationWe compared two abundance parameters of interest across models: coefficients for elevation and log expected count at a standard site (in the GLMM, (beta _0 + 0.5 sigma _alpha ^2); in the N-mixture model, (log (lambda _0 p_0))). We examined absolute differences in point estimates and the log-scale ratios between their standard errors.Stability of estimated parametersAttempting to decompose the expected value of observed data into within- and between-site components can lead to ridged likelihood surfaces with difficult-to-estimate optima. Kéry found that instability of model estimates with increasing K occurred when there was a likelihood tradeoff between detection and abundance, resulting in a tendency in abundance toward positive infinity restrained only by K10. Dennis et al. showed that N-mixture models could in fact yield estimates of absolute abundance at infinity18. We interpreted this as a case of a boundary parameter estimate rather than non-identifiability and explored it by reparametrizing as follows. We estimated the intercepts for detection and abundance with two orthogonal parameters (rotated in log space) (phi _1 = log (lambda _0 p_0)) and (phi _2 = log (p_0 / lambda _0)), where (lambda _0) and (p_0) are real-scale abundance and detection probability at the mean site. We hypothesized that in unstable cases, (phi _1), log expected count, is well-informed by the data, but (phi _2), the contrast between abundance and detection, is not well-informed, corresponding to a likelihood ridge as (phi _2 rightarrow -infty) due to detection probability approaching 0 and abundance approaching infinity. This reparameterization isolates the likelihood ridge to one parameter direction, similar to a boundary estimate as (exp (phi _2) rightarrow 0). Boundary estimates occur in many models and are distinct from non-identifiability in that they result from particular datasets. Confidence regions extending from a boundary estimate may include reasonable parameters, reflecting that there is information in the data. We defined a practical lower bound for (phi _2). When (phi _2) was estimated very near that bound, we conditioned on that boundary for (phi _2) when estimating confidence regions for other parameters.In the N-mixture case, diagnosing a boundary estimate for (phi _2) is made more difficult by the need to increase K for large negative (phi _2) to calculate the likelihood accurately. We used an approach like that of Dennis et al.18 to numerically diagnose unstable cases. For each N-mixture variant in each SSR, the final model was refitted twice, using values of K 2000 and 4000 greater than the initial choice. Estimates were considered unstable if the absolute value of the difference in AIC between these two large-K refits was above a tolerance of 0.1. We monitored whether MLE estimates of (phi _1) and (phi _2) also varied with increasing K.Evaluating the fast N-mixture calculationWe extended previous work by Meehan et al. to drastically improve the efficiency of N-mixture models using negative binomial or beta-binomial distributions in submodels45 (see Supplemental Section 3).We ran benchmarks of this likelihood calculation for a single site against the traditional algorithm, which involves iterating over values of N to compute a truncated infinite sum. We calculated the N-mixture likelihood at 5,000 sites and compared the computation time between the two methods for all four N-mixture model variations. We ran benchmarks along gradients of (text {length}(y_i)) (number of replicate observations at the simulated site) and K (the upper bound of the truncated infinite sum) for each variant. 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