Estimating nest abundance while accounting for time-to-event processes and imperfect detection

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1 Ecology, 95(9), 2014, pp Ó 2014 by the Ecologcal Socety of Amerca Estmatng nest abundance whle accountng for tme-to-event processes and mperfect detecton GUILLAUME PE RON, 1,4,5 JOHANN WALKER, 2 JAY ROTELLA, 3 JAMES E. HINES, 4 AND JAMES D. NICHOLS 4 1 Colorado Cooperatve Fsh and Wldlfe Research Unt, Department of Fsh, Wldlfe, and Conservaton Bology, 1484 Campus Delvery, Colorado State Unversty, Fort Collns, Colorado USA 2 Ducks Unlmted, Inc., Great Plans Regonal Offce, 2525 Rver Road, Bsmarck, North Dakota USA 3 Department of Ecology, Montana State Unversty, Bozeman, Montana USA 4 Patuxent Wldlfe Research Center, U.S. Geologcal Survey, Beech Forest Road, Laurel, Maryland USA Abstract. Brds and ther populaton dynamcs are often used to understand and document anthropogenc effects on bodversty. Nest success s a crtcal component of the breedng output of brds n dfferent envronments; but to obtan the complete pcture of how brd populatons respond to perturbatons, we also need an estmate of nest abundance or densty. The problem s that raw counts generally underestmate actual nest numbers because detecton s mperfect and because some nests may fal or fledge before beng subjected to detecton efforts. Here we develop a state-space superpopulaton capture recapture approach n whch nference about detecton probablty s based on the age at frst detecton, as opposed to the sequence of re-detectons n standard capture recapture models. We apply the method to ducks n whch (1) the age of the nests and ther ntaton dates can be determned upon detecton and (2) the duraton of the dfferent stages of the breedng cycle s a pror known. We ft three model varants wth or wthout assumptons about the phenology of nest ntaton dates, and use smulatons to evaluate the performance of the approach n challengng stuatons. In an applcaton to Blue-wnged Teal Anas dscors breedng at study stes n North and South Dakota, USA, nestng stage (egg-layng or ncubaton) markedly nfluenced nest survval and detecton probabltes. Two ndvdual covarates, one bnary covarate (presence of grazng cattle at the nest ste), and one contnuous covarate (Robel ndex of vegetaton), had only weak effects. We estmated that 5 10% of the total number of nests were avalable for detecton but were mssed by feld crews. An addtonal 6 15% were never avalable for detecton. These percentages are expected to be larger n less ntense, more typcal samplng desgns. User-frendly software nestabund s provded to assst users n mplementng the method. Key words: Anas dscors; Blue-wnged Teal; capture recapture; habtat choce; nest success; North and South Dakota, USA; populaton densty; populaton dynamcs; Prare Pothole Regon. INTRODUCTION Brds and ther populaton dynamcs are often used to understand and document anthropogenc effects on bodversty, such as habtat fragmentaton (Newmark and Stanley 2011) and clmate change (Jguet et al. 2010), but also a pror bengn actvtes such as nature toursm (Kerbrou et al. 2009). Nest success s often used to measure the effect of perturbatons, and t has been the subject of a rch lterature (Mayfeld 1961, Rcklefs 1969, Pollock and Cornelus 1988, Martn 1995, Dnsmore et al. 2002, Rotella et al. 2004, Jones and Geupel 2007, Newmark and Stanley 2011). Nest success, however, s only one component of brd breedng productvty. Total nest abundance, or breedng densty, may also vary n tme and space. For example, Manuscrpt receved 20 September 2012; revsed 5 March 2014; accepted 7 March Correspondng Edtor: E. G. Cooch. 5 E-mal: peron_gullaume@yahoo.fr 2548 ndvduals may try to avod areas where ther reproductve success would be lower than average (Pe ron et al. 2010), or, n contrast, they may fall vctm to ecologcal traps when anthropogenc change alters habtat qualty but not the habtat cues that the brds use (Robertson and Hutto 2006). However, estmatng total nest abundance requres that two potental methodologcal ptfalls are addressed. Frst comes the problem of mperfect detecton, that s, observers n the feld generally cannot detect all of the nests that are avalable at the tme of surveys. Second, nests are subject to tme-to-event processes (Kalbflesch and Prentce 2002, Collett 2003): eggs eventually hatch and nestlngs fledge, and nests also rsk depredatons and nest abandonment. In other words, nests have a fnte lfe span durng whch they are avalable for detecton. As a result, the count of detected nests generally underestmates the total number of nests n an area. The same problem also exsts for other mmoble or sessle objects n nature, such as plants, benthc nvertebrates, dung ples, and terrtoral anmals.

2 September 2014 NEST DENSITY ESTIMATION 2549 Here we propose to use capture recapture models, and n partcular superpopulaton models (Crosbe and Manly 1985, Schwarz and Arnason 1996), to address those questons and correct raw counts of detected nests for mperfect detecton and fnte lfe span (Wllams et al. 2002). Frst we develop a general framework presentng the challenges assocated wth nference about nests and other mmoble or statonary objects n nature. Then we present a specfc case of the model talored for Anatdae brd speces. We analyze a data set from Blue-wnged Teal Anas dscors at study stes n the Prare regon of North and South Dakota, USA. The number of offsprng produced each season by a unt of habtat (or a populaton) s of crtcal mportance to assess habtat management efforts, to project populaton dynamcs, and to study habtat choce n an evolutonary context. Fnally, we revew methodologcal bottlenecks remanng after our work, as well as model features that could readly be ncluded n future model extensons. The analyses n ths artcle can be reproduced usng R-package nestabund, whch has a user-frendly nterface (avalable onlne). 6 GENERAL FRAMEWORK: TIME-TO-EVENT PROCESSES AND BIRD NESTS The dfferent stages of the nestng season (egg-layng, ncubaton, chck-rearng) are well known; and the bologcal process by whch a nest passes through these three stages may appear trval (Fg. 1). But, when combned wth a mortalty rsk (depredaton and nest abandonment), and wth mperfect detecton, ths process produces a large number of possble observatons. A nest may be detected durng any of the three stages and may fal at any pont of the cycle, whch s sometmes referred to as nterval censorng (Stanley and Newmark 2010). For each nest we defne D 1 as the date at nest ntaton (frst egg lad); D 2 as the date at clutch completon and start of ncubaton; D 3 as the hatchng date; and D 4 as the fledgng date. These dates are lnked by the nest-specfc stage duratons: L 1, the length of the egg-layng perod; L 2, the length of the ncubaton perod; and L 3, the length of the chck-rearng perod. A last varable of nterest s E, the ntal clutch sze at the start of ncubaton. Data collected from each nest are f, the date when the nest was frst detected; z, the date t was last observed before ether fledgng or falng; s, whch equals 1 f the nest succeeded and zero otherwse; and E, the clutch sze (f avalable; not recorded f the nest faled before clutch completon). In the followng, to smplfy the presentaton at ths stage, we assume that (1) the study area s vsted daly; (2) detected nests are montored untl ther fate s recorded; (3) there s a determnstc relatonshp between clutch sze and duraton of the egg-layng perod: L 1 ¼ ae, where a s 6 the number of eggs lad per day and s consdered known (a ¼ 1 n many speces ncludng Blue-wnged Teal); and (4) the duraton of the ncubaton perod s the same for all nests: L 2 s a constant. In addton to these, the model may requre assumptons about the parametrc form of the dstrbutons for the duraton of the chck-rearng perod L 3, nest ntaton date D 1, and clutch sze E. Model parameters and notaton are summarzed n Table 1. Let s then frst express the probablty of nest success (probablty that a nest survves to produce at least one fledglng) for a nest wth known values of the covarates D 1, L 1, L 2, L 3 : Y D2 1 Y D3 1 Y D4 1 Prðs ¼ 1 j D 1 ; L 1 ; L 2 ; L 3 Þ¼ u ð1þ u ð2þ u ð3þ : ¼D 1 ¼D 2 ¼D 3 ð1þ Second, let Pr(z j D 1, L 1, L 2, L 3,s ¼ 0) be the probablty that the same nest fals on date z: Prðz j D 1 ; L 1 ; L 2 ; L 3 ; s ¼ 0Þ ¼ 8 " Y z 1 >< >: " Y D2 1 ¼D1 u ð1þ " Y D2 1 ¼D1 #" Y D3 1 ¼D2 u ð1þ u ð2þ ¼D1 #" Y z 1 ¼D2 #" Y z 1 ¼D3 u ð1þ u ð2þ u ð3þ # 1 u ð1þ z ; f D 1, z, D 2 # 1 u ð2þ z ; f D 2 z, D 3 : # 1 u ð3þ z ; f D 3 z, D 4 Note that f falure occurs before the clutch s completed, then clutch sze E s not recorded but L 1 does not enter the formula so nformaton about E s not requred. Thrd, let Pr( f j D 1, L 1, L 2, L 3 ) be the probablty that the nest s detected on day f but not earler, condtonal on survval untl day f: Prðf j D 1; L 1; L 2; L 3Þ¼ 8 >< ½ YD2 1 >: ¼D1 1 p ð1þ ½ YD2 1 ¼D1 Š½ YD3 1 ¼D2 1 p ð1þ 1 p ð2þ ½ Yf 1 Š½ Š½ ¼D1 Yf 1 ¼D2 Yf 1 ¼D3 1 p ð1þ 1 p ð2þ 1 p ð3þ 0; f f, D 1 ð2þ Šp ð1þ f ; f D 1, f, D 2 Šp ð2þ f ; f D 2 f, D 3 Šp ð3þ f ; f D 3 f, D 4 Note that Eq. 3 devates from the standard way of modelng detecton hstory data n superpopulaton capture recapture models. The usual assumpton for moble organsms s that ntal and subsequent detec- : ð3þ

3 2550 GUILLAUME PE RON ET AL. Ecology, Vol. 95, No. 9 P 0 ¼ Z Z Z " Prðs ¼ 1 j D 1 ; ae; L 2 ; L 3 Þ E D 1 L 3 3 Prð not found j D 1 þ ae þ L 2 þ L 3 ; D 1 ; ae; L 2 ; L 3 Þ FIG. 1. Generalzed breedng cycle wth notaton used n ths artcle: D 1, D 2, D 3, and D 4 are, respectvely, the nest ntaton date, the date at the start of ncubaton, hatchng date, and fledgng date. They are separated by the egg-layng perod lastng L 1 days, the ncubaton perod lastng L 2 days, and the chck-rearng perod lastng L 3 days. tons are governed by the same probabltes, such that detecton hstory data collected after the ntal encounter are adequate for nference about detecton. However, followng ntal detectons, locatons of mmoble objects such as nests are known, so that subsequent detectons are near-certan and provde lttle or no nformaton about ntal detecton probablty. Inference about detecton probablty s here made condtonal on D 1, so that we know the number of tmes the nest escaped detecton. Eqs. 1 and 2 descrbe the state process of transton between stages and nest falure or success. Eq. 3 descrbes the observaton process. These processes are lnked to the data as follows for the kth detected nest: Prðf k ; z k ; s k ; E k Þ¼ PrðE k Þ Z D 1 Z L 3 ½½s k Prðs ¼ 1 j D 1 ; ae k ; L 2 ; L 3 Þ þð1 s k ÞPrðz k j D 1 ; ae k ; L 2 ; L 3 ; s ¼ 0ÞŠ 3 Prðf k j D 1 ; ae k ; L 2 ; L 3 ÞŠPrðL 3 ÞPrðD 1 Þ 3 dl 3 dd 1 ð4þ where Pr(E), Pr (L 3 ), and Pr(D 1 ) denote the dstrbuton among nests of clutch sze, chck rearng perod, and ntaton date, respectvely. Then, we compute the probablty of not fndng a nest that s n the superpopulaton, denoted P 0. Let s frst defne the probablty of not fndng a nest that s ntated on day D 1 and s avalable for detecton untl day z (z. D 1 ): Prð not found j z; D 1 ; L 1 ; L 2 ; L 3 Þ¼ 8 " Y z 1 # 1 p ð1þ ; f D 1, z, D 2 >< ¼D1 ¼D1 ¼D1 " Y D2 1 #" Y z 1 # 1 p ð1þ 1 p ð2þ ; f D 2 z, D 3 : " Y D2 1 #" Y D3 1 #" Y z 1 # >: 1 p ð1þ 1 p ð2þ 1 p ð3þ ; f D 3 z, D 4 ¼D2 Then the probablty of mssng a nest for whch we do not know the values of z, D 1, L 1, L 3 s: ¼D2 ¼D3 ð5þ þ X D4 1 z¼d 1þ1 Prðz j D 1 ; ae; L 2 ; L 3 ; s ¼ 0Þ 3 Prð}not found} j z; D 1 ; L 1 ; ae; L 3 Þ 3 PrðL 3 ÞPrðD 1 ÞPrðEÞdEdL 3 dd 1 : ð6þ Fnally, denotng H the vector contanng all model parameters, the lkelhood of the complete data set s obtaned by summng across all C observed nests and N C non-observed nests (Schwarz and Arnason 1996): LðH jff k ; z k ; s k ; E k g k¼1...c Þ } " Y C # N! Prð f k ; z k ; s k ; E k Þ ðn CÞ! PN C 0 : ð7þ k¼1 REALISTIC MODEL VARIANTS In ths secton we descrbe modfcatons brought to the prevously descrbed general model to make t relevant for real-lfe data sets. Further potental modfcatons and extensons are tackled n the dscusson. Frst, the study perod does not compulsorly encompass a representatve secton of the breedng season; thus, only part of the dstrbuton of D 1 s represented n the data. Ths s readly accommodated by restrctng the nference to the superpopulaton of nests that are avalable for detecton durng at least one ste vst (Schwarz and Arnason 1996). In ths case, nstead of modelng the dstrbuton for D 1, we estmate entry probabltes: these represent for each tme nterval the probablty that a nest of the superpopulaton s ntated durng ths nterval. Second, to reduce dsturbance to nestng brds, most stes are typcally vsted not on a daly bass, but rather at ntervals of several days. We used nterval survval rates rather than daly survval. Last, the homogenety assumpton (the as- TABLE 1. Model parameters. Notaton Bologcal meanng N Superpopulaton sze. In the most general models ths ncludes all nests ntated n the study area. In restrcted applcatons (staggered-entry model) ths represents nests that are avalable for detecton durng at least one samplng occason. It ncludes renestng attempts after the falure of ntal attempts and does not dscrmnate between those and ntal attempts. u ðsþ Daly nest survval for day n state s. p ðsþ Detecton probablty for day n state s. d Expected ntaton date. r Standard devaton of ntaton date. #

4 September 2014 NEST DENSITY ESTIMATION 2551 sumpton that detected nests are representatve of undetected nests) mght not be verfed. In ths study, ths ssue s partly addressed through the use of ndvdual covarates,.e., nest-specfc varables that are beleved to mpact on nest fate (survval and/or detecton). For example, nest success probablty may vary wth the local predator assemblage and ntaton date (Calero-Restra et al. 2013). In superpopulaton models, ths nvolves summng over all possble nestspecfc covarate values n order to estmate P 0, the probablty of not fndng a nest wth unknown covarates. The use of covarates also allows one to control for potental covaraton between nest success and nest detecton probablty (e.g., va varaton n nest concealment). A last set of modfcatons was brought to talor the model to Anatdae, and Blue-wnged Teal, n partcular. Anatdae consttute an deal group of speces for applyng our method for the frst tme because several features allow smplfcaton of the model. Frst, by usng the candlng method, we know when the detected nests were ntated (D 1 ). Although the candlng method can be mprecse, for example f the eggs are staned, we assume here as a frst approxmaton, and followng most prevous works on duck breedng success, that there s no error. We can then compute exactly how many tmes a nest went undetected (Eq. 3). Second, ducks have precocal young that leave the nest wthn hours of hatchng, so we can fx L 3 ¼ 0,.e., nest success s measured by hatchng success. Thrd, ncubaton length L 2 s known and does not need to be estmated (L 2 ¼ 24 days n Blue-wnged Teal). Fnally, when computng P 0 (Eq. 6) we dd not consder all possble values of E, but rather fxed t to the modal value (E ¼ 10 eggs), so that the length of the egg-layng perod was fxed to L 1 ¼ 10 d. APPLICATION CASE Feld procedures and data descrpton Our applcaton case s based on data collected n 2012; we collected nest observatons on four plots owned by Ducks Unlmted n the Prare Pothole Regon of North and South Dakota. Plot centrods were located at W, N; W, N; W, N; and W, N, and plot areas were 56, 59, 90, and 88 ha, respectvely. Land cover of the plots was domnated by herbaceous perennal vegetaton. Two of the plots were grazed by cattle durng part of the nestng season. We searched the study plots for nests at approxmately weekly ntervals (range 6 9 d) usng all-terran vehcles and chan-drags (Klett et al. 1986; see Plate 1). When we found a nest, we recorded date, speces, and nest age. Nest age was determned by candlng 2 3 eggs from each nest to assess ther developmental stage (Weller 1956, Ackerman and Eagles-Smth 2010). To descrbe varaton n nest concealment, we recorded Robel s ndex of vegetaton at the nest ste (Robel et al. 1970). We marked each nest wth a unquely numbered cm fberglass rod placed 5 m north of the nest ste, and we recorded the geographc coordnates of the nest wth a handheld GPS recever. We revsted prevously dscovered nests at ntervals of 2 7 days untl the eggs hatched or the nest faled. In total, 401 duck nests were detected (44, 61, 153, and 143 n each of the four stes). Of these, 207 (31, 35, 71, 70) were Blue-wnged Teal nests. Teal nests, on average, were ntated 8.5 d before they were detected (SD ¼ 4.5 d), and overall 6.2 d after the frst day of the study (SD ¼ 11.9 d). The average Robel ndex of vegetaton at each nest was 1.73 (SD ¼ 0.63); 90 nests had cattle grazng on the feld they were located n (stes 2 and 4 only). In ths study, we focused on the followng effects actng on nest success and detecton probablty: ste, Robel ndex of vegetaton, presence of grazng cattle, and nestng stage. Female ducks spend less tme attendng ther nest durng the egg-layng perod; thus, a nest mght be less lkely to both be depredated and detected by humans durng the egg-layng perod than durng the ncubaton perod. Specfctes of the present data set Our data are exceptonal n some ways compared to other nest-searchng data. Frst, the study was desgned to encompass most of the breedng season at each of the four stes, so that we could safely assume that almost the whole range of nest ntaton dates was sampled and only the tals of the dstrbutons were lackng. Second, we knew the fate of all of the detected nests (no rght-censorng). Thrd, the feld effort (number of ste vsts, detecton probablty durng each vst, and nterval between them) was ntensve enough that we were confdent that P 0 was small. We also observed that the dstrbuton of ntaton dates among detected nests was sgnfcantly platykurtc (wth broad shoulders and few very early and late nests; Anscombe-Glynn kurtoss test: kurtoss ¼ 2.40, P ¼ 0.02; Fg. 4). Ths could be expected due to the ftness benefts of nestng early and the fact that hens generally renest after falure. There was no skewness (d Agostno test: skew ¼ 0.35; P ¼ 0.17). These elements suggested that the rased cosne dstrbuton, wth a kurtoss value of ;2.41 (close to the 2.40 observed value), was more adapted to ths specfc data set than the Gaussan dstrbuton. Ths dstrbuton has two parameters akn tomeanandstandarddevatonthatwencludednthe parameter set. Data analyss As prevously ntroduced, we ft three varants of superpopulaton models that dffered n the way the temporal dstrbuton of nest ntaton dates was modeled. Frst, we ft a standard superpopulaton model n whch a set of entry probablty parameters represented the temporal dstrbuton. In ths model, we estmated the number of nests avalable for detecton

5 2552 GUILLAUME PE RON ET AL. Ecology, Vol. 95, No. 9 FIG. 2. Smulaton results from 148 data sets. (a) Varaton n percentage bas n superpopulaton sze estmate wth nterval duraton between two vsts to the study stes, when the study starts on three dfferent dates. Average age on day 1 represents the dfference n days between peak ntaton date and start of the study (negatve f the peak date s after the start of the study). (b) Varaton n percentage bas n superpopulaton sze estmate wth nest detecton probablty, for three levels of nest success. Bas s the percentage dfference between true and estmated superpopulaton sze. The standard devaton of ntaton date was always 12 days. Each pont corresponds to an average across 6 9 scenaros dependng on the occurrence of numercal nstabltes (see Results: Smulaton study). durng at least one ste vst. These nests were thus ntated wthn a specfc tme frame determned by the frst and last vsts to the stes, and they survved at least untl the next ste vst. Second, we ft the model n whch nest ntaton date was normally dstrbuted. For ths model we used the Gauss-Hermte quadrature to compute P 0 (Golub and Welsch 1969, Gmenez and Choquet 2010). Thrd, we ft the model n whch nest ntaton date followed the rased cosne dstrbuton. For ths we used a Remann sum approxmaton to compute P 0 ;.e., we assgned a P 0 value to each possble 24-h nterval and then summed over 24-h ntervals. In the latter two models, the estmated superpopulaton sze ncludes all nests, even those that were ntated after the study perod or that ded before beng subjected to the detecton process. Ths was made possble by the strong assumpton about temporal dstrbuton. Note that we dd not ft a ste-specfc dstrbuton, but used the same dstrbuton for all four stes. We dd not use the Akake nformaton crteron to decde upon a preferred model varant, because the estmated quanttes were actually not the same, and because of the techncal dfferences n the way the lkelhood was computed (dfferent approxmatons). To ft models wth the effect of the bnary ndvdual covarate presence of cattle, we ntroduced a ste-specfc parameter proporton of nest stes that have grazng cattle. To ft models wth the effect of the contnuous covarate Robel ndex, we assumed that Robel ndex was normally dstrbuted among nests and used the Gauss- Hermte quadrature to compute P 0. Statstcal sgnfcance of the bologcal effects (ste, nestng stage, Robel ndex, and presence of cattle) was determned usng the 95% confdence nterval (95% CI) obtaned by nversng the Hessan of the log-lkelhood functon at ts mnmum. If the 95% CI ncluded zero (or overlapped n the case of ste effects), we consdered the effect as nonsgnfcant. We also ran a parametrc bootstrap (250 replcatons) to compute 95% CI on transformed estmates and derved quanttes. Smpler models requred a few mnutes to converge, but the full model wth both covarates and staggered entry could requre up to several hours on some desktop computers. The varant wth Gaussan dstrbuton was the fastest to run. We somewhat reduced computaton tme by frst fttng a closed-populaton capture recapture model and then usng ts estmate as the startng value n the optmzaton routne for the superpopulaton model. We provde user-frendly software nestabund as an R- package to reproduce our analyses (see footnote 6). SIMULATIONS We generated and analyzed data for 108 scenaros, mostly wth the am of llustratng problematc stuatons pertanng to our new developments, not to superpopulaton capture recapture models n general. Across scenaros, nest success (percentage hatchng) could be 25%, 50%, or 90% (correspondng to daly nest survval of 3.2, 3.9, 5.8 on the logt scale); detecton probablty could be 0.2, 0.5, or 0.8; mean ntaton date could be 10 days before the start of the study, at the start of the study, or 20 days after t; the nterval between successve samplng occasons could be 5, 10, or 15 days. Note that an nterval duraton of 15 days meant that nests had at most two chances of beng detected before fledgng/falng. In all scenaros, L 1 and L 2 were fxed to the values for Blue-wnged Teal. Standard devaton of ntaton date was always 20 days and the dstrbuton was Gaussan. The study ended on day 100, largely after the last nest fledged. We smulated a total superpopulaton of 1000 nests. We analyzed these data usng the Gaussan model only and computed the bas (percentage dfference between true and estmated superpopulaton sze) for the varous scenaros. We ft

6 September 2014 NEST DENSITY ESTIMATION 2553 Estmates for the model ncludng all consdered effects actng on survval and detecton probabltes of Blue-wnged Teal Anas dscors nests n North and South Dakota (USA) study stes, wth 95% confdence ntervals (lower 2.5th and upper 97.5th quantles). TABLE 2. Parameter Estmate 2.5th quantle 97.5th quantle Survval Ste Ste Ste Ste Egg-layng vs. ncubaton Cattle Robel Detecton Ste Ste Ste Ste Incubaton vs. egg-layng Cattle Robel Notes: These estmates correspond to the parameters from logt-lnear regressons of survval and detecton probabltes aganst ste dentty (ste 1 to 4), nestng stage (egg-layng or ncubaton), presence of cattle near the nest, and Robel ndex of vegetaton densty near the nest. Sgnfcant parameters (as per the 95% confdence ntervals) are ndcated n bold. lnear models n order to dentfy the study desgn varables that led to largest bas. RESULTS Smulaton study Frst, we observed numercal nstabltes (P 0 estmated near 1.0 and populaton sze estmate approachng nfnty) for all scenaros wth both low detecton probablty (0.2) and long ntervals between ste vsts (10 or 15 d); these scenaros were excluded from further analyss and we recommend not usng the method wth such data. Negatve bas on superpopulaton sze estmate ncreased wth nterval duraton ( 1.70% % bas/d, estmate 6 SE; ANOVA P, 0.001; Fg. 2a) and wth tme dfference between the start of the study and peak nest ntaton date ( 0.78% 6 SE 0.13% bas/d; P, 0.001; Fg. 2a). Low values of survval probablty and of detecton probablty also led to larger bases (P, 0.001; Fg. 2b). We thus observed as expected that (1) scenaros wth long ntervals between nest searches led to underestmatng superpopulaton sze because detected nests were a nonrepresentatve sample and (2) when nest success or detecton probablty were low, ths bas was amplfed. In favorable scenaros (nest success 50% or above, detecton probablty 50% or above, nterval duraton of 5 days, and study startng at or before peak ntaton date), the bas was always below 7% n absolute value and the estmate was unbased, on average, over eght smulatons. In summary, our smulatons hghlghted the mportance of (1) tmng the start of study at or before peak nest ntaton date; and (2) reducng the nterval between vsts (at least when the tmng of the frst vst s correct). Smulatons also revealed that (3) efforts to ncrease detecton probablty wthn each samplng occason are not as rewardng as ncreasng the number of samplng occasons wthn a gven tme frame. If the study starts after peak ntaton date or nterval length exceeds onethrd of the nestng perod, we recommend swtchng to the standard staggered-entry formulaton of the model. Dakotas Blue-wnged Teal example The statstcally sgnfcant effects were: ste effects on survval probablty, cattle effect on survval probablty, and stage effect on both survval and detecton probabltes (Table 2). The statstcally nonsgnfcant effect of the Robel ndex of vegetaton was n the drecton of lower detecton and lower survval probabltes for stes wth more vegetaton. In the followng, we present parameter estmates from the model wth only sgnfcant effects and wth the rased cosne dstrbuton (results from the Gaussan model were qualtatvely smlar). Daly nest survval was almost 1.0 durng the egg-layng stage (Fg. 3). Indeed, no nest was recorded as havng faled before the expected start of ts ncubaton perod, and only seven nests were recorded as havng faled before the vst mmedately followng the start of ncubaton. Daly nest survval was greater when cattle were present (effect on the logt scale, 1.00; 95% CI ¼ ; Fg. 3). However, ths effect was largely compensated for by a negatve effect of beng n stes 2 and 4, where the majorty of nests had cattle around (Fg. 3, Table 3). Nests located where cattle were absent n stes 2 and 4 had an almost zero probablty of succeedng (95% CI ¼ %). A major dfference between models wth contnuous phenology functons and the staggered entry ( standard ) model s that pont estmates of nest success were hgher n the latter (Table 3), because the standard approach s based on nests that survved untl at least the next samplng occason, whch s a bas toward hgh survval. Detecton probablty

7 2554 GUILLAUME PE RON ET AL. Ecology, Vol. 95, No. 9 FIG. 3. Daly nest survval probablty (mean wth 95% CI) for Blue-wnged Teal nests n North and South Dakota study stes. Square shadng ndcates the stage: black, survval durng egg-layng; gray, survval durng ncubaton. For stes 2 and 4 only, sold symbols represent nests n felds wth cattle; open symbols are nests n felds wthout cattle. S1, S2, S3, and S4 desgnate the four stes. ncreased from 0.40 (95% CI ¼ ) durng the egg-layng stage to 0.73 durng the ncubaton stage (95% CI ¼ ). In the standard approach, the probablty of not detectng a nest was lowest n ste 1 (Table 3, last columns). Ths dfference was not due to detecton probablty, whch dd not vary across stes; t was due to survval probablty, whch was hghest n ste 1, so that more nests survved long enough to be subjected to the detecton process. Indeed, the estmated probablty of nest success was above 60% n ste 1 but around 20% n the three other stes (Table 3). However, there also was a dfference n both the tmng of vsts and the ste-specfc phenology, whch meant that a greater proporton of nests faled before the start of the study n ste 1 (Fg. 4). Breedng densty (nest abundance per hectare) vared from 0.61 nests/ha (95% CI ¼ nests/ha) n ste 1 to 1.01 (95% CI ¼ nests/ ha) n ste 3 (Table 3). Ste 1 appeared to be the most productve ste n terms of successful nests per hectare, but was the least densely occuped of the stes (Table 3). DISCUSSION Our method provdes a synthetc framework to estmate the abundance of mmoble objects that can be aged upon detecton, when detecton probablty s below 1.0 and objects are submtted to mortalty rsks n addton to tme-to-event processes. Such objects nclude nests, as n our developed example, but also benthc organsms (usng growth marks to age the anmals), plants (usng lfe stages), dung ples (usng a classfca- Estmate of superpopulaton sze, probablty of not detectng a nest, hatchng success, and productvty for the four study stes and the three model varants, for Blue-wnged Teal nests n North and South Dakota study stes. TABLE 3. Model wth Gaussan dstrbuton Model wth rased cosne dstrbuton Standard model wth entry probabltes Parameter, by ste Estmate 95% CI Estmate 95% CI Estmate 95% CI Superpopulaton sze Ste Ste Ste Ste Probablty of not detectng a nest Ste Ste Ste Ste Hatchng success (%) Ste Ste Ste Ste Productvty (successful nests/ha) Ste Ste Ste Ste Notes: Models only ncluded sgnfcant bologcal effects. The superpopulaton s not defned n the same way for the frst two models compared to the thrd model; hence, the dfference n pont estmates (see Realstc model varants). Gven are pont estmates and 95% confdence nterval boundares from a parametrc bootstrap.

8 September 2014 NEST DENSITY ESTIMATION 2555 FIG. 4. Densty dstrbuton of the age of Blue-wnged Teal nests at the start of the study at stes n North and South Dakota. Negatve values ndcate nests that were ntated after the start of the study. For all panels, the shaded hstogram s the dstrbuton among detected nests not accountng for the detecton process. The unshaded, black-outlned hstogram bars show estmated daly entry rates from the standard superpopulaton model accountng for the detecton process; the rghtmost bar spreads over the full potental age range for nests that are present at the frst vst to a ste (34 days). For the bottom panel (all stes), the sold lne s the estmated dstrbuton usng the rased cosne model and accountng for the detecton process; the dashed lne s the estmated dstrbuton usng the Gaussan model (neglectng excess kurtoss) and accountng for the detecton process. ton of decay state), and even terrtoral vertebrates (usng growth marks n horns, teeth, shells, and so forth). The method uses data prevously ntended at estmatng only the success of the tme-to-event processes, not abundance. In brd populaton studes, nest success, however, s only one component of reproductve output, so the ablty to use these same data to estmate total offsprng producton for a populaton or focal area s an mportant step forward. The key to nference about abundance s nformaton about detecton probablty. We emphasze that the probablty structure for detecton (Eq. 3), ndependently of the structure representng nest ntaton dates, represents an nnovaton because t provdes a means for nference about nest abundance. Informaton about varaton n reproductve output s useful for (1) gudng management of brd habtat, (2) montorng speces response to clmate and land use changes, and (3) studyng the relatonshp between habtat choce, as measured by nest abundance, and habtat qualty, as measured by breedng success. Our results can be drectly used to compare the relatve values of dfferent study stes,.e., to measure spatally explct ftness. In our example analyss, we show a twofold varaton n breedng productvty (successful nests per hectare) across stes. Nest success was the most mport drver of productvty n ths system, but wthout correctng for nest abundance, the between-ste varaton would have been largely overestmated. Ste 1 was both the most successful and the least densely occuped of the four study stes, whch also suggests that teal may be unable to predct ste-specfc breedng success when returnng from sprng mgraton, or they may be unable to adapt habtat choce accordng to antcpated ste-specfc breedng success. Regardng the effect of nest-ste vegetaton densty on breedng success, our results are statstcally nonsgnfcant, but suggest that nest stes wth denser vegetaton are less successful, as prevously found n steppe-breedng brds (e.g., Wth 1994), possbly because mammalan nest predators use denser vegetaton patches for concealment from raptors. When should one or the other model varant be used? Our applcaton case was, when compared to our smulated scenaros, n the range where estmaton s relable va models wth contnuous phenology functons

9 2556 PLATE 1. GUILLAUME PE RON ET AL. Ecology, Vol. 95, No. 9 Duck nest found usng the chan-drag method at a Dakota (USA) study ste. Photo credt: Ducks Unlmted. (Gaussan or rased cosne); most mportantly, peak nest ntaton occurred about a week after the start of the study and nterval length was about a week; hgh detecton probablty also helped to reduce P0 below 25%, of whch only 5 10% were nests actually mssed by field crews (the rest beng never avalable for detecton). As we have hghlghted, for sparser data, the contnuous dstrbuton models are not well adapted and the staggered-entry approach should be preferred (denoted TIME n the software nestabund). The quantty to be estmated s then the number of nests avalable durng at least one samplng occason. There are a number of post hoc correctons avalable to account for nests that are ntated and fal between successve samplng occasons (Schwarz et al. 1993), but no correcton s possble for nests that are ntated after the study or long before t starts. Further model extensons An advantage of our model for nference about nest success at the scale of the sampled populaton s the formal ncorporaton of both success and detecton parameters that can be modeled as functons of habtat covarates, nestng stage, and layng date. If both detecton probablty and nest success vary accordng to any of these covarates, then nests wth hgher detecton probabltes could be overrepresented n the sample of detected nests. In most standard approaches to nference about nest success, there s no ablty to adjust the estmate of nest success by these dfferent probabltes of nest ncluson n the sample, whereas under our approach ths can be done. However, even after accountng for varaton assocated wth dentfied covarates, t s always possble that substantal heterogenety remans n ether nest success or detecton probablty. If ths heterogenety s characterzed by a covarance between probabltes of detecton and nest success (e.g., hgher probabltes of success for more concealed nests), then msleadng nferences could stll result. A possble model extenson would be to ncorporate ether finte mxture (e.g., Norrs and Pollock 1996) or random effects approaches (e.g., Natarajan and McCulloch 1999) to deal wth (possbly dependent) heterogenety n probabltes of nest detecton and success. Other potental mprovements that we foresee nclude ste specficty n the dstrbuton of nest ntaton dates. Indeed, n our teal example we suspect that the breedng season started earler at ste 1, whch led to overestmaton of P0 and N for ths ste when fittng a model where all stes had the same phenology. Second, the ablty to jontly analyze data from multple speces appears to be useful n the context of duck studes n whch multple speces co-occur. Thrd, between-nest varaton n L3 and E may be ntroduced usng a mxture-model approach wth proporton vectors descrbng the probablty that a nest from the superpopulaton takes some gven L3 and E values. Fourth, rght-censorng (the fact that the field season may end before some nests have ether succeeded or faled) could readly be accommodated by censorng Eq. 1 at the approprate tme. Last, our model was nspred by duck studes, but can readly be adapted for brd speces n whch age determnaton of the unhatched eggs s dfficult, but

10 September 2014 NEST DENSITY ESTIMATION 2557 the speces have altrcal young that stay n the nest after hatchng (passernes, Columbdae, and so on). In these speces, t s often possble to determne the age of the nestlngs (e.g., Hanson and Kossack 1963). Then, nformaton about egg age s replaced by nformaton about nestlng age; nest ntaton date s replaced by clutch completon date ; and the quantty estmated becomes the number of nests that reach the ncubaton stage rather than the number of nests that are ntated. In other words, nference about detecton probablty can be based on nests that eventually hatch at least one egg. Implementaton of these extensons would be especally useful n ncreasng the applcablty of our approach to the majorty of brd speces for whch unhatched eggs are not easly aged. ACKNOWLEDGMENTS Ths work was supported by the U.S. Fsh and Wldlfe Servce. We thank three revewers for ther helpful comments. LITERATURE CITED Ackerman, J. T., and C. A. 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