Testing hypotheses on distribution shifts and changes in

phenology of imperfectly detectable species

Thierry Chambert1,2*,WilliamL. Kendall3, JamesE. Hines2, JamesD. Nichols2, Paolo Pedrini4,

J. HardinWaddle5, GiacomoTavecchia6, SusanC.Walls7 and Simone Tenan4

1Department of EcosystemScience andManagement, Pennsylvania State University, University Park, PA 16802, USA;2PatuxentWildlife ResearchCenter, U.S. Geological Survey, Laurel, MD 20708, USA; 3ColoradoCooperative Fish andWildlife

ResearchUnit, U.S. Geological Survey, Colorado State University, Fort Collins, CO80523, USA; 4Vertebrate Zoology Section,

MUSE –Museo delle Scienze, Corso del Lavoro e della Scienza 3, Trento 38122, Italy; 5NationalWetlandsResearchCenter,

U.S. Geological Survey, Lafayette, LA 70506, USA; 6Population EcologyGroup, Institut Mediterrani d’Estudis Avanc�ats(IMEDEA), CSIC-UIB,MiquelMarqu�es 21, Esporles, Mallorca 07190, Spain; and 7Southeast Ecological ScienceCenter, U.S.

Geological Survey, Gainesville, FL 32653, USA

Summary

1. With ongoing climate change, many species are expected to shift their spatial and temporal distributions. To

document changes in species distribution and phenology, detection/non-detection data have proven very useful.

Occupancy models provide a robust way to analyse such data, but inference is usually focused on species spatial

distribution, not phenology.

2. Wepresent amulti-season extension of the staggered-entry occupancymodel of Kendall et al. (2013,Ecology,

94, 610), which permits inference about the within-season patterns of species arrival and departure at sampling

sites. The new model presented here allows investigation of species phenology and spatial distribution across

years, as well as site extinction/colonization dynamics.

3. We illustrate the model with two data sets on European migratory passerines and one data set on North

American treefrogs. We show how to derive several additional phenological parameters, such as annual mean

arrival and departure dates, from estimated arrival and departure probabilities.

4. Given the extent of detection/non-detection data that are available, we believe that this modelling approach

will prove very useful to further understand and predict species responses to climate change.

Key-words: closure assumption, detection, occupancy modelling, species distribution models, spe-

cies phenology, staggered-entrymodel

Introduction

Understanding and predicting how climate change is affecting

the spatial and temporal (i.e. phenology) distributions of spe-

cies are important tasks of ecological research (Parmesan

2006; McMahon et al. 2011). To document shifts in species

range and phenology, detection/non-detection data, which

usually require less effort than other animal survey methods

(Dickinson, Zuckerberg & Bonter 2010), can be particularly

useful, as they allow covering large spatial and temporal

scales (Parmesan & Yohe 2003). These survey methods are

also relevant for examining range expansions of invasive spe-

cies (Bled, Royle & Cam 2011) and the spread of pathogens

and infectious disease in the wild (e.g. Chestnut et al. 2014).

Although still sometimes ignored, it is well known that detec-

tion/non-detection data can virtually never be equated to

presence/absence information (Bailey, MacKenzie & Nichols

2014), as the detection of a species is usually imperfect (Mac-

Kenzie et al. 2006). If not accounted for, this source of error

can induce important biases when estimating the spatial dis-

tribution of species (K�ery, Gardner & Monnerat 2010; Rota

et al. 2011), an issue often disregarded in the ecological litera-

ture on climate change effects (but see Tingley et al. 2012).

The issue of imperfect detection is especially relevant to stud-

ies on climate change, which is likely to affect not only species

distribution and phenology, but also detection probability

itself (Møller 2011).

To deal with sources of observational error in detection/

non-detection data, static and dynamic occupancy models

have been developed (MacKenzie et al. 2002, 2003). As these

models allow estimation of the proportion of area occupied

by a species or occupancy probability, as well as occupancy

dynamics over multiple years (i.e. site colonization and

extinction probabilities), they provide a very flexible and use-

ful tool to investigate shifts in distribution for species that are

not detected perfectly (K�ery, Gardner & Monnerat 2010). To

disentangle occupancy probability from the probability of

detection given species presence, occupancy models rely on*Correspondence author. E-mail: thierry.chambert@gmail.com

© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society

Methods in Ecology and Evolution 2015 doi: 10.1111/2041-210X.12362

repeated survey visits and the assumption that the occupancy

status of the species at a site does not change between visits

(‘closure assumption’; MacKenzie et al. 2006; Kendall et al.

2013). Because of the closure assumption, occupancy models

have traditionally not been used to investigate trends in spe-

cies phenology. However, Roth, Strebel & Amrhein (2014)

recognized the usefulness of detection/non-detection data to

assess phenological trends and developed a new model within

the static occupancy framework that permits direct estima-

tion of the mean dates of arrival and departure of a species

from given sites (e.g. breeding sites). They used an over-dis-

persed Poisson process to model dates of arrival and depar-

ture within the single sampling season considered. Although

we are convinced that their approach will be useful in many

cases, the fact that the model assumes a specific distributional

form (Poisson or other) for arrival and departure dates could

be a limitation for some study systems. An alternative

approach to investigate species phenology, in addition to spe-

cies spatial distribution, with detection/non-detection data is

provided by the recent model development of Kendall et al.

(2013), which relaxes the closure assumption and allows esti-

mation of conditional probabilities of arrival and departure

of the species to/from sampled sites. The two main limits of

the model presented by Kendall et al. (2013), referred to as

the ‘staggered-entry model’, are as follows: (i) it is a single-

season model that does not allow estimation of inter-annual

dynamics in occupancy and (ii) it does not directly provide

means to estimate other parameters of direct interest to

address questions about phenology, such as residency times,

mean dates of arrival and departure or within-season proba-

bility of presence.

Our purpose is to fill these gaps by (i) extending the stag-

gered-entrymodel ofKendall et al. (2013) to includemulti-sea-

son occupancy dynamics and (ii) showing how estimates and

standard errors of derived parameters of interest can be

obtained. Here, our emphasis is on using this approach to

investigate inter-annual variation, especially trends, in both

occupancy (spatial distribution) and within-season residence

patterns (phenology). We illustrate applications of this new

model with two data sets on Palearctic migratory passerines,

for which we investigate hypotheses about inter-annual shifts

(trends) in distribution along an altitudinal gradient and in the

phenology of migration. We also apply the method to a data

set on a North American treefrog species at sites that were

monitored year-round by automated acoustic recording to

illustrate the flexibility of our approach to different study sys-

tems (Appendix S1, Supporting information).

Model description

The data are modelled in the same fashion as any multi-year

occupancy study accounting for imperfect detection (MacKen-

zie et al. 2003). Each year t = 1, . . . T, a fixed number of

i = 1,. . .R sites is visited j = 1,. . .Jt times in an attempt to

detect the presence of the focal species. This sampling situation

corresponds to the robust design of Pollock (1982), where

years represent primary occasions and visits represent second-

ary occasions. We note that the temporal scale of primary

occasions can be different than year-to-year. Also, missing

observations within this sampling design are allowed. For any

site i within the study area, we define the data yi = {xit,j}, forany given year t and survey occasion j, where xit,j takes value 1

if the species was detected and 0 otherwise. The data-generat-

ing process can be modelled as the results of three processes.

For a detection (xit,j = 1) to occur, (i) the site i must be occu-

pied by the species in year t (where by ‘occupied’, we mean that

at least one individual of the focal species is present at the site

during at least 1 sampling occasion of year t), (ii) the species

must be ‘present’ (i.e. available for detection) on that site on

occasion j, and (iii) the species must be detected by an

observer.

We now describe each of the three levels of this model in

more detail. First, at the annual level, we consider that any site

is either occupied or unoccupied by the species of interest;

inter-annual transitions from unoccupied (occupied) to occu-

pied (unoccupied) are modelled using colonization (extinction)

parameters (MacKenzie et al. 2003). We define the following

parameters: wi1 is the probability that a site (i) is occupied the

first year of the study, eit is the probability that a site occupiedin year t becomes extinct in year t + 1, and cit is the probabilitythat a site unoccupied in year t is colonized, and thus becomes

occupied, in year t + 1. Assuming no false detections, if a site is

unoccupied by the focal species in year t, no detection can

occur that year. At the within-year level, instead of assuming

closure, we consider the possibility that, at a site occupied in

year t, a species might not be present and available for detec-

tion during the entire sampling season. We consider that the

species can enter and depart the site only once each season

(Kendall et al. 2013). We thus define three possible states

describing the availability (for detection) status of the species

on a site: (i) has not yet arrived (unavailable); (ii) has arrived

and not yet departed (available); and (iii) has departed

(unavailable). To remain consistent with the literature (e.g.

Kendall et al. 2013), we use the terms ‘arrive’ and ‘depart’ to

describe unidirectional changes in the availability status of the

species within primary occasions (years). These changes might

not always correspond to physical ‘arrival’ and ‘departure’ of a

species on a site. For instance, availability for detection might

be determined by the species starting to emit breeding calls (see

Appendix S1). Finally, we model the detection process, defin-

ing pit,j the probability of detecting the species on site i, in year

t, at survey occasion j, given that the species is available for

detection (i.e. the site is occupied in year t and the species is

present on occasion j).

The arrival/departure process, which describes the pheno-

logical dynamics of the species within each year, is modelled by

adding two new parameters. First, eit,j is the probability of arri-

val of the species at site i between occasions j and j + 1, given

that the site is occupied in year t and the species has not yet

arrived by occasion j. Note that conditioning on occupancy at

some time during year t requires eit,J�1 = 1, that is if the species

has not arrived by period J � 1, it must do so between J � 1

and J. The probabilities of arrival between j and j + 1 can also

be expressed as conditional only on occupancy and not on

© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society, Methods in Ecology and Evolution

2 T. Chambert et al.

availability status at j. These unconditional arrival probabili-

ties (bit,j) are derived from eit,j as:

bit; j ¼ eit; jYj�1

m¼0

ð1� eit;mÞ:

Because of the conditioning on arrival at some time during

year t,

XJ�1

j¼0

bit; j ¼ 1:

Next, we define dit,j the probability of departure of the spe-

cies from site i before occasion j + 1, which is also conditional

on the site being occupied in year t and the species being avail-

able for detection on occasion j. As mentioned earlier, we

assume only one entry and a maximum of one departure from

the site during any given year t. Re-entries and multiple depar-

tures of the focal species within a season are not permitted,

although random movements in and out of sampled sites that

might occur between the initial entry and the final departure

are absorbed into the detection parameter (Kendall, Nichols

& Hines 1997). Transition probabilities between the three

availability states described above can be conveniently

described with a transitionmatrix (Dit,j:), where rows represent

the three states (1,2,3) at occasion j (top to bottom) and col-

umns represent states at occasion j + 1 (left to right). In any

given year t, at any site i, we have:

Dit; j ¼ð1� eit; jÞ eit; j 0

0 ð1� dit; jÞ dit; j0 0 1

24

35:

We now illustrate the probability structure for two example

detection histories, in the case of a study consisting of 2 years

and three sampling occasions per year: h1 = (011 101) and

h2 = (000 110), for sites 1 and 2, respectively. The probability

associated with the first example detection history is:

Prð011101Þ ¼wi½fei1;0ð1�pi1;1Þð1�di1;1Þþ ei1;1ð1� ei1;0Þgpi1;2ð1�di1;2Þpi1;3�� ð1� ei1Þ½ei2;0pi2;1ð1�di2;1Þð1�pi2;2Þð1�di2;2Þpi2;3�:

The probability associated with the second example detec-

tion history is:

Pr ð000110Þ ¼ ½ð1�wiÞci1 þwið1� p�i1Þð1� ei1Þ�� ½ei2;0pi2;1ð1� di2;1Þpi2;2fð1� di2;2Þð1� pi2;3Þþ di2;2g�:

In this equation, the derived parameter p�it corresponds tothe probability of being detected at least once in year t, across

all secondary occasions, given that the site was occupied that

year (Kendall et al. 2013). For any year t, this parameter is

defined as:

p�it¼feit;0pit;1gþfeit;0ð1�pit;1Þð1�dit;1Þpit;2þeit;1ð1�eit;0Þpit;2gþfeit;0ð1�pit;1Þð1�dit;1Þð1�pit;2Þð1�dit;2Þpit;3þeit;1ð1�eit;0Þð1�pit;2Þð1�dit;2Þpit;3þeit;2

ð1�eit;1Þð1�eit;0Þpit;3g:

For modelling purposes, encounter history probability

structures can be expressed concisely with matrix notation.

First, we describe the probabilities of within-season detec-

tion histories, which correspond to the likelihood developed

by Kendall et al. (2013) for the single-season staggered-

entry model. The within-season process determining species

availability or non-availability is described by the probabil-

ity matrix Dit,j defined above. We also define two diagonal

matrices for the probabilities of detection (Pit,j) and non-

detection (Qit,j), given the current availability state of a

site:

Pit; j ¼0 0 00 pit; j 00 0 0

24

35;

and

Qit; j ¼1 0 00 ð1� pit; jÞ 00 0 1

24

35:

We define a vector n0 = [1 0 0] that describes the availability

state of any site just before the field season (‘not arrived yet’).

From these probability matrices, we define the column vector

ph,it for the probability associated with within-season detection

history h for site i in year t, conditional on true site occupancy

state (occupied or unoccupied, respectively). The first row of

ph,it denotes the probability of within-season detection history

h in year t when the site is occupied (z = 1), while the second

row corresponds to this probability for an unoccupied site

(z = 0). When there is at least one detection in year t, this col-

umn vector takes the form:

ph;it ¼ n0QJtj¼1

Dit; j�1ðPit; jÞxit; jðQit; jÞð1�xit; jÞ !

13

0

264

375;

andwhen therewas no detection at all in year t, it becomes:

ph;it ¼ n0QJtj¼1

Dit; j�1Qit; j

!13

1

264

375:

Next, we specify the inter-season dynamics of site occu-

pancy, following the likelihood notation of the multi-season

occupancy model of MacKenzie et al. (2003). We define a row

vectorΦ0 for the initial probability of site occupancy:

U0 ¼ w1 1� w1½ �:We also define a matrix Φt of annual transition probabili-

ties, from year t to t + 1, between the two occupancy states:

Ut ¼ 1� et etct 1� ct

� �:

The probability of an observed detection history hi can then

be written as:

PrðhiÞ ¼ U0

�YT�1

t¼1

Diagðph;itÞUt

�ph;iT;

© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society, Methods in Ecology and Evolution

Open dynamic occupancy model 3

Parameters can be estimated with a likelihood or Bayesian

approach. Here, we implemented this model in program PRES-

ENCE, freely available online at http://www.mbr-pwrc.usgs.gov/

software/presence.shtml. A detailed example is provided in

Appendix S2.

We now show how to derive additional parameters of inter-

est to investigate annual patterns of variation in species phe-

nology and occupancy probability. First, we define the within-

season probability of presence ait,j as the probability that the

species is available at a given survey occasion within season t

(i.e. has arrived at site i prior to occasion j and has not yet

departed), given the site is occupied that season.We emphasize

that this parameter has a different interpretation than the ‘local

occupancy’ parameter of Hines et al. (2010), which is defined

for spatial replicate surveys. The parameter ait,j is obtained as:

ait; j ¼ n0Yjg¼1

Dit;g�1

" #m;

where m = [0 1 0]T. For instance, the ait,j values for the first

three survey occasions of any year twould be:

ait;1 ¼ eit;0;

ait;2 ¼ ð1� eit;0Þeit;1 þ eit;0ð1� dit;1Þ;

ait;3 ¼ ð1� eit;0Þð1� eit;1Þeit;2 þ ð1� eit;0Þeit;1ð1� dit;2Þþ eit;0ð1� dit;1Þð1� dit;2Þ:

Next, we show how to derive themean arrival and departure

occasions, which can then easily be translated into Julian and

calendar dates. We note that the derivation of these average

dates is better suited to studies performing daily sampling. If

sampling occasions are groups of days, as in our passerine data

examples, average dates can still be obtained using the median

sampling dates for each defined occasion, but this necessarily

adds some imprecision. The mean arrival occasion (ait) can be

obtained from unconditional entry probabilities as:

ait ¼XJtj¼1

bit; j�1 � j:

The formulation to derive annual mean departure occasions

(git; mean of last sampling occasion of presence) is a bit more

complex, as departure times depend not only on occasion-spe-

cific departure probabilities (dit,j), but also on the probability

of being available (ait,j) at that occasion j:

git ¼XJtj¼1

j � ait; j � dIit; j ¼XJtj¼1

j � n0Yjg¼1

Dit;g�1

" #m � dIit; j;

where I is an indicator function with value 1 for j < Jt and 0 for

j = Jt, which ensures the constraint that dit,J = 1 for the last

survey occasion. For instance, for three survey occasions in

year t, if we develop that expressionwewould have:

git ¼ 1� bit;0dit;1 þ 2� ½bit;0ð1� dit;1Þdit;2 þ bit;1dit;2�þ 3� ½bit;0ð1� dit;1Þð1� dit;2Þdit;3 þ bit;1ð1� dit;2Þdit;3þ bit;2dit;3�;

where dit,3 = 1, which is a mathematical constraint of the

model, as described. This model constraint simply acknowl-

edges that departure cannot be modelled beyond the last sur-

vey occasion; if the species has not departed by then, one

assumes that it must depart, with probability 1�0, any time

afterwards. Here, we use the term ‘departed’ to denote that the

species becomes totally unavailable for detection for the

remainder of the year; it does not necessarily mean physical

departure from the site. For instance, it could correspond to

frogs ceasing to emit calls when the breeding season is over,

despite frogs still being physically present at the site (see

Appendix S1). To obtain accurate estimates of mean departure

times, it is important that the sampling period extends beyond

(or at least close to) the last departure. Right truncation of the

data relative to the period of presence each year can induce

substantial uncertainty as to when the last departure happens,

and the weighted average of the last departure time could be

negatively biased. Similarly, bit,0 represents the probability thatthe species entered any time before the first occasion; the same

issue arises for the timing of the beginning of the sampling per-

iod and the interpretation of the mean arrival date. In the pres-

ence of left truncation, the weighted average probability of

entry by the first occasion might positively bias the estimated

mean arrival dates. Ideally, the dates to start and end the sam-

pling should be chosen to encompass the first entries and last

departures, such that bit,0 = 0 and dit,J�1 = 0. Left and right

truncations are important issues to consider in any phenology

study.

Finally, for any year t > 1, estimated annual probabilities of

site occupancy can be derived from initial occupancy, extinc-

tion and colonization probabilities as follows:

wit ¼ wit�1ð1� eit�1Þ þ ð1� wit�1Þcit�1

Standard errors for all these derived parameters can readily

be obtained with the delta method (Seber 1982), using vari-

ance–covariancematrices provided by program PRESENCE.

Example analyses

METHODS

We applied the model to presence/absence data on Eurasian siskin

(Carduelis spinus) and Common redstart (Phoenicurus phoenicurus).

These two passerines are short- and long-distance migrants, respec-

tively, betweenEurope andAfrica. The presence/absence datawere col-

lected during post-nuptial migration, from 1997 to 2010, at 31 and 32

southern Alpine sites for the siskin and the redstart, respectively. Sam-

pling sites were potential migratory stopovers distributed along an alti-

tudinal gradient, ranging from 190 to 1774 m a.s.l. (mean = 885 m;

SD = 612�2 m), and were located on alpine paths (12 sites), mountain

slopes (9) or at the bottom of alpine valleys (11). This studywas initially

implemented to characterize species-specific habitat use in the alpine

landscape during the migratory season. Migratory dynamics are domi-

nated by transient individuals on alpine passes, whereas valley bottoms

are important stopover sites, and mountain slopes are used by a mix-

ture of transient and stopping-over birds (Pedrini, Tenan & Spina

2012). On each sampling site, birds were trapped using mist nets that

© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society, Methods in Ecology and Evolution

4 T. Chambert et al.

were operated according to a standardized protocol over the years

(Pedrini et al. 2008). Each survey occasion j was defined as a trapping

period of five consecutive days. Variation in trapping effort was accom-

modated in themodel by including a covariate for detection probability

representing the number of days of trapping activity, ranging from one

to five, within each 5-day period. The sampling season, that is starting

and ending 5-day periods, was kept constant among years for each spe-

cies (23–27 September to 28 October–1 November for siskin; 14–18

August to 13–17 October for redstart). With pronounced climate

warming in the Alpine region (Auer et al. 2007) and, thus, milder

autumnal conditions, we expect a retraction or a shift of the distribu-

tion of the two passerine species towards higher elevation. In terms of

phenology, we expect a change in arrival dates and in the peak of abun-

dance of the two study species. The direction of the phenological

change would, however, be difficult to predict due to the heterogeneity

in species response observed during autumn migration (Lehikoinen,

Sparks & Zalakevicius 2004). Species could be expected to delay their

autumn departure, due tomilder conditions, or advance it in synchrony

with early spring events. To address these hypotheses, we used a model

selection approach to investigate variation across year (yr) and eleva-

tion (elev) for occupancy w, colonization (c) and extinction (e) proba-bilities, as well as variation across years and surveys for arrival (e),

departure (d) and detection (p) probabilities. We note that in the con-

text of this migratory-bird study, the term ‘occupancy’ is defined as the

use of a site as a migratory stopover at any point during our sampling

season. Similarly, colonization and extinction probabilities correspond

to changes, from 1 year to the next, in usage of a site as a migratory

stopover. For instance, a site ‘colonization’ means that the species was

not using the site as a migratory stopover in year t but used it in year

t + 1. Because of data limitations, and to avoid over-parameterization,

we considered trend relationships (linear and quadratic, on the logit

scale), rather than full time variation (i.e. a fixed effect parameter

defined for each sampling occasion), for effects of years and survey

occasions. The effort covariate was also included for detection proba-

bility.

RESULTS: S ISKIN ANALYSIS

The siskin data supported the open occupancymodel much better than

the closed model (DAIC = 99�71, see Table S1), where we emphasize

that the terms ‘open’ and ‘closed’ refer to within-season occupancy and

presence. We found substantial uncertainty in the best covariate struc-

ture of the open model, as seven models had an AIC weight (w) larger

than 0�05 (Table 1), a criterion we use here as a threshold for model

support. Despite this model uncertainty, there was clear evidence that

elevation was an important covariate for both colonization (summed

w = 0�87) and extinction (summed w � 1�00) probabilities. We also

found substantial support for a positive linear logistic annual trend in

colonization probability (summed w = 0�88) and only marginal sup-

port for a positive trend in extinction probability (summed w = 0�12).Using model-averaged derived estimates of annual probabilities of

occupancy, we found that changes in occupancy patterns were different

among sites at different altitudes. For low-elevation sites (<500 m),

occupancy showed a decline in the early years of the study followed by

an increase in more recent years, while at high-elevation sites, occu-

pancy was very stable across the 1997–2010 study period (Fig. 1a).

Observed patterns from the raw data (‘na€ıve occupancy’; Fig. 1a) do

not show any trend over time, but, in accordance with model predic-

tions, the proportion of sites that appeared to be occupied is lower at

low-elevation sites. We also note that these na€ıve estimates of occu-

pancy are biased low, a consequence of not accounting for imperfect

detection. Regarding phenology, there was no support for an annual

trend in survey-specific probabilities of arrival and departure (Table

S1). In all supportedmodels, departure probability was constant across

years and survey occasions and low (d = 0�007, SE = 0�01), and proba-bility of arrival was high for the first occasion (efirst = 0�56, SE = 0�13)and constant and lower for all subsequent occasions (esubseq = 0�31,SE = 0�08). The pattern of within-season presence (Fig. 1c) shows a

decelerated increasing trend across the sampling season, with a peak of

presence on the last survey (i.e. around November 1). Na€ıve patterns,

derived from the raw data, for occasion-specific presence and arrival

rates were also calculated and are shown on Fig. 1c. The trends over

occasions are similar to model predictions, but we also remark (i) that

na€ıve presence rates are strongly biased low and (ii) na€ıve arrival rates

are strongly biased low for the first occasion and then slightly biased

high for later occasions. This latter point emphasizes the need to correct

for imperfect detection to avoid biasing mean arrival times towards

later dates. For detection probability, we found strong support for an

additive effect of survey effort, year and survey occasions. Detection

showed an increasing trend both within and across years (Fig. 2a). The

within-year upward trend is probably due to an increase in siskin abun-

Table 1. Summary of the model selection results of the Eurasian siskin analysis. The seven most supported models (i.e. model weight ≥0�05), usedfor model averaging, are shown. The full model list is available in Supplement (Table S1). The covariate structure for each parameter is shown.

Model parameters are as follows: probability of initial occupancy (w), probability of colonization (c), probability of extinction (e), conditional proba-bilities of arrival (e) and departure (d) and detection probability (p). The covariates are as follows: elevation (‘elev’), year (‘yr’), within-season sam-

pling occasion (‘j’) and sampling effort (‘effort’), which is measured in number of days (1–5) that were surveyed during the five consecutive days

representing a sampling occasion. For entry probability, we also considered an alternative parameterization (‘First’): the probability of entry at first

occasion is distinct from subsequent entry probabilities, which are assumed equal. Linear logistic trends are indicated by ‘Lin()’, additive effects are

indicated by a ‘+’, interaction effects are indicated by a ‘*’, and ‘(.)’ indicates that the parameter is held constant

Model parameterization

AIC DAIC AICweight Num. Par.w c e e d p

(.) Lin(yr + elev) Lin(elev) (First) (.) Lin(yr + j) + effort 655�42 0�00 0�24 16

(.) Lin(yr + elev) Lin(yr + elev) (First) (.) Lin(yr + j) + effort 656�84 1�42 0�12 17

(.) Lin(yr) Lin(elev) (First) (.) Lin(yr + j) + effort 656�90 1�48 0�11 15

(.) Lin(yr*elev) Lin(elev) (First) (.) Lin(yr + j) + effort 657�01 1�59 0�11 17

(.) Lin(elev) Lin(elev) (First) (.) Lin(yr + j) + effort 657�08 1�66 0�10 15

elev Lin(yr + elev) Lin(elev) (First) (.) Lin(yr + j) + effort 657�12 1�70 0�10 17

(.) Lin(yr*elev) Lin(yr + elev) (First) (.) Lin(yr + j) + effort 658�40 2�98 0�05 18

© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society, Methods in Ecology and Evolution

Open dynamic occupancy model 5

dance as the migratory season advances, indicating that surveys

stopped before the end of the migratory season (i.e. right truncation of

sampling period). The increase of detection across years could be either

due to an observer artefact or linked to a shift of abundance towards

an earlier time in the season.

RESULTS: REDSTART ANALYSIS

For the redstart data, the open occupancy model also received

much more support than the closed model (DAIC = 30�68, see

Table S2). Two models received virtually all support from the data

(w = 0�86 and w = 0�12). The effect of elevation was strongly sup-

ported for initial occupancy, colonization and extinction probabili-

ties (w � 1�00). There was no support for an annual trend in

colonization probability and only marginal support for a negative

trend in extinction probability (w = 0�12, second best-supported

model; see Table S2). In contrast with the inter-annual pattern

observed for siskins, occupancy probability tended to decrease over

years for low-elevation sites and increase for high-elevation sites

(Fig. 1b). This pattern is in accordance with our predictions of a

shift in occupancy towards higher altitude. When looking at na€ıve

patterns, obtained directly from raw data (Fig. 1b), both high- and

low-elevation sites seem to have equivalent occupancy rates, both

biased low. Here again, this highlights the importance to correct for

imperfect detection (overall bias) and account for sampling effort,

as it is very likely that differences in occupancy among elevation

classes are confounded by various capture efforts at different alti-

tudes. Regarding phenological trends, we found support for an

annually constant decrease in conditional probabilities of arrival

throughout the migratory season. The supported decreasing trend

was linear on the logit scale, dropping from 0�68 (SE = 0�07) for

the first sampling occasion to 0�02 (SE = 0�007) for the penultimate

sampling occasion. There was also support for a quadratic trend

across sampling occasions in probabilities of departure (Table S2),

which increased from an annually averaged value of d < 0�001(SE < 0�001) at the first occasion to d = 0�41 (SE = 0�09) at the last

occasion. There was also support for an increasing trend in occa-

sion-specific departure rates over years, probably indicating a shift

towards earlier departures. The within-season probability of species

presence showed a strong concave quadratic pattern (Fig. 1d), with

a peak of presence in the middle of the sampling period (occasion

6), around September 8–September 12. Similarly to the siskin analy-

sis, na€ıve trends observed from the raw data were similar to model

predictions, but heavily biased. Na€ıve presence rates are biased low

1998 2002 2006 2010

0.0

0.2

0.4

0.6

0.8

Siskin

Year

Pr (

site

occ

upan

cy)

● ● ● ● ● ● ● ● ● ● ● ● ● ●

200 m410 m ●

850 m1700 m

1 2 3 4 5 6 7 8

0.0

0.2

0.4

0.6

0.8

1.0

Survey occasion

Pro

babi

lity

Pr (presence)

●

●●

●● ● ●

●

Pr (arrival)●

●

●●

●● ●

●

1998 2002 2006 2010

0.5

0.7

0.9

Redstart

Year

Pr (

site

occ

upan

cy)

● ● ● ● ● ● ● ● ● ● ● ● ● ●

200 m410 m ●

850 m1700 m

2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

Survey occasion

Pr (

pres

ence

)

Pr (presence)●

●

●● ● ● ● ● ● ● ● ● ●

Pr (arrival)●

●

●

●

●

●

● ● ● ● ●● ●

(a) (b)

(c) (d)

Fig. 1. Patterns of site occupancy probability (top row) across years and within-season probabilities of presence and arrival, conditional on occu-

pancy (bottom row), across survey occasions for Eurasian siskins (left column) and Common redstarts (right column). For site occupancy patterns

(top panels, a and b), model predictions are shown as open points, with lines, for four different elevations: 200, 410, 850 and 1750 m. Black points

represent observed annual ratios of the number sites with at least one detection to number of sites with no detection, for low (<500 m; black squares)

and high (≥500 m; black triangles) elevation sites. The peak of presence is shown by a dashed vertical line. In the two bottom panels (c and d), open

points represent model predictions for presence (triangles) and arrival (circles) probabilities. The probabilities of arrival are unconditional on the

availability status at the previous occasion. Black points are observed proportions derived from the raw data: (i) black triangles are the proportion of

detections to non-detections for each survey, for sites having at least one detection a given season (i.e. equivalent of a na€ıve estimate of probability of

presence); (ii) black circles represent the proportion of ‘first detections’ that occurred at any survey occasion (i.e. equivalent of a na€ıve estimate of

unconditional probability of arrival). All model predictions shownwere derived frommodel-averaged estimates (see text and Table 1). Note that SE

bars are omitted for better readability.

© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society, Methods in Ecology and Evolution

6 T. Chambert et al.

over all occasions, while na€ıve arrival rates are biased low for the

first occasion and slightly biased high for later occasions. The most

supported pattern for detection probability was similar to that of si-

skins, with an additive effect of survey effort, year and survey occa-

sion and an increasing trend both within and across years (Fig. 2b).

These detection patterns could indicate increasing redstart abun-

dance as the migratory season advances and a shift towards more

abundant eruptive events across years.

Discussion

We presented an extension of the staggered-entry model of

Kendall et al. (2013) that allows modelling the occupancy

dynamics of a species over several ‘seasons’, in addition to the

within-season dynamics of arrivals and departures.We showed

how to derive several phenological parameters of interest, such

as mean arrival and departure times (dates) and within-season

probabilities of presence at each occasion. An alternative mod-

elling approach to estimate arrival and departure dates, in

addition to occupancy probability, was developed by Roth,

Strebel & Amrhein (2014). An attractive feature of the latter

approach is that arrival and departure dates are directly esti-

mated from themodel, along with standard errors, without the

need to derive these parameters. However, this model requires

making potentially restrictive assumptions on the distribu-

tional form of arrival and departure dates, while ours does not.

Roth, Strebel&Amrhein (2014) used an overdispersed Poisson

process that allows for normal among-site variability in mean

arrival dates, but other distributions, such as a negative bino-

mial, could also be considered. Assuming such distributions

might be appropriate for species that display gradual patterns

of arrival and/or departure through the season, but we cannot

expect this to be valid for every species. Many animals display

more erratic or eruptive patterns, in which case we believe our

more flexible approach will prove more useful and appropri-

ate. For instance, some of the data sets analysed here provide

good examples of multiphasic abrupt arrival patterns: the

Eurasian siskin is known to be eruptive in the Southern Alps,

with movements regulated by seed productivity of coniferous

forests in the breeding grounds; green treefrogs display several

peaks of arrivals throughout the breeding season (see Appen-

dix S1). We also note that our approach can still handle cases

of smooth and gradual arrivals and departures.

ASSESSING SHIFTS IN SPECIES DISTRIBUTION AND

PHENOLOGY

Because it simply requires detection/non-detection data, the

model presented here provides an easy way to investigate

shifts in species spatial distribution and phenology. There

are several ways one can characterize phenological events,

such as emergence, breeding, migration, wintering or hiber-

nation. Previous studies often used na€ıve measures such as

first appearance dates (e.g. Sparks & Mason 2004), but also

measures based on percentile dates of first appearance (e.g.

Stefanescu, Pe~nuelas & Filella 2003). Biases due to sample

size sensitivity have been shown to occur for the former type

of measure but not the latter (Van Strien et al. 2008). How-

ever, when using raw observations, biases can also arise as a

result of imperfect detection, especially if species detectability

displays temporal variation (Moussus, Julliard & Jiguet

2010). For example, with imperfect detection, we often

expect species arrivals before the first date of detection and

species departures following the final detection (Pledger et al.

2009). The model presented here accounts for imperfect

detection and thus provides unbiased estimators of occasion-

specific arrival and departure probabilities, although poten-

tial biases in estimated phenological dates might still occur

from inappropriate sampling designs (see recommendations

in the next section). Here, the definition of arrival (depar-

ture) probability is based on times of arrival (departure) of

the first (last) individual at any site. Such definitions based

on first (last) individual arrival (departure) are often used in

phenology studies (Moussus, Julliard & Jiguet 2010), but

one might sometimes be more interested in characterizing

phenology more quantitatively (Jonz�en et al. 2006), using

peaks in species density or abundance. We showed how to

derive estimates of within-season probabilities of species

presence, but this parameter is still defined based on first and

last individuals that have arrived and departed, respectively,

from the different sample sites. We can, however, often

expect a strong correlation in the seasonal trend of this

parameter and that of species abundance. Alternatively, one

could use detection probability as a proxy for variation in

species abundance along the season, as detection is generally

closely linked to species abundance (Royle & Nichols 2003).

1 2 3 4 5 6 7 8

0.4

0.5

0.6

0.7

0.8

0.9

1.0 Siskin

Survey

Pr (

dete

ctio

n)

●

●

●

●

●

●

●

● 1997

●

●

●

●

●

●

●

● 2001

●

●

●

●

●

●

●● 2005

●

●

●

●●

●●

● 2009

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●

1 3 5 7 9 11 13

0.4

0.5

0.6

0.7

0.8

0.9

1.0 Redstart

Survey

Pr (

dete

ctio

n)

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●●

●●

●●

●●

●●

●

●●

●●

●●

●●

●●

● ● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●●

●●

●●

●●

●●

●

●●

●●

●●

●●

●●

● ● ●

1997

2001

20052009

(a) (b)

Fig. 2. Intra-annual patterns of detection

probability across sampling surveys for the sis-

kin (a) and the redstart (b) data sets. Four

years are shown to illustrate the increasing

trend. For the siskin data, there were 8 annual

surveys spanning from September 23 to

November 1. For the redstart data, 13 surveys

were performed each year between August 14

andOctober 17.

© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society, Methods in Ecology and Evolution

Open dynamic occupancy model 7

Our modelling approach could also be used to investigate

interactions and relationships between spatial distribution and

phenology. Although the relatively small sample sizes of our

passerine data sets did not allow us to attempt such models,

one could otherwise have considered investigating effects of

elevation on arrival and departure probabilities. The same

could be done with any type of site-specific covariate, such as

habitat characteristics. From such models, one could then eas-

ily compute site-specific ait,j, ait and git and produce topo-

graphic maps illustrating how the phenology of the species is

changing across a spatial gradient of interest.

We believe that the existence of flexible modelling

approaches such as those presented here has great potential

for phenological studies. In particular, we would hope that

investigations of phenology will move from descriptions of

‘change’ to studies that focus on discrimination of competing

hypotheses about phenology. Such hypotheses can be embed-

ded in the models described here, but associated predictions

require thought and explicit specification. For example, the

general hypothesis that breeding seasons are advanced (occur

earlier) in northern temperate areas can lead to the prediction

of earlier mean arrival date, mean departure date and mean

date of local presence, all of which can be investigated using

the derived statistics that we describe. However, such predic-

tions might also extend to model parameters such as probabil-

ities of entry and departure. Earlier breeding seasons could

translate into higher entry probabilities for earlier dates and

smaller conditional entry probabilities for later dates. Such

predictions for mechanistic model parameters are possible

within our framework, but require thought about the specific

processes that lead to phenological change.

SAMPLING CONSIDERATIONS

To avoid biases when estimating the phenology parameters

presented here, a few important sampling considerations

deserve emphasis. We note that these considerations are only

relevant for inferences focused on phenology; if the staggered-

entrymodel is simply used to deal with violations of the closure

assumptions, the issues discussed below can easily be over-

looked. In terms of sample sizes, as for any study seeking statis-

tical inference, it is desirable to increase the number of

surveyed sites, years and within-season survey occasions. This

is especially true for methods based on logistic regression types

of models (Stockwell & Peterson 2002). Naturally, financial

and logistical constraints are often limiting, but the increasing

availability of passive systems of detection (e.g. camera traps,

acoustic recorders, satellite images) and citizen science pro-

grams should help provide larger sample sizes, both spatially

and temporally (O’Connell, Nichols & Karanth 2010; New-

man et al. 2011). The model described here assumes that spe-

cies might enter and depart a site only once each season, with

continuous ‘presence’ at the site in between. For situations in

which multiple arrivals and departures are likely to occur, we

recommend the use of a robust design (replicate sampling)

within each season (e.g. two observers or two consecutive days

sampling each week) in order to use the standard multi-season

model ofMacKenzie et al. (2003) to test for complete openness

(Rota et al. 2009). Further, it is important that the number

and distribution of within-season surveys be adjusted to the

temporal scale at which one is trying to characterize the species

phenology. Typically, if mean dates of arrival/departure are

wanted, surveys should be performed daily; if a weekly tempo-

ral resolution is sufficient, weekly surveys will be appropriate.

For instance, in our two passerine case examples, sampling

occasions are defined as 5-day groups. The resolution of esti-

mated mean dates of arrival and departure is necessarily the

same, although it is always possible to use median dates as an

approximation.

Another very important study design consideration con-

cerns the time frame over which surveys are implemented.

For many phenological investigations, the goal is to minimize

biases due to left and right truncation of the sampling period

relative to the phenological season of interest. This issue is rel-

evant to any study concerned with a species phenology,

regardless of the modelling approach used. The idea is

straightforward: if one wants to know when a species arrives

at (departs from) some areas, one should start (stop) sampling

before (after) the first arrival (last departure) on any of the

sites. When arrival (departure) events occur before (after)

sampling started (ended), estimated dates of arrival (depar-

ture) will inevitably be positively (negatively) biased. Ideally,

one wants the entry probability for the first sampling occasion

to be zero and departure probability for some period prior to

the last sampling occasion to be 1, as it is the best indicator

that the species has not yet arrived at any site before the first

sampling occasion and has departed from all sites by the last

survey. We thus recommend using a priori knowledge or a

‘best informed guess’ to include at least one, preferably two,

survey(s) at each end of the sampling period, even when no

detections are expected. Continuous year-round monitoring,

as in our treefrog analysis (Appendix S1), can be useful for

species with protracted focal periods. When little is known, a

priori, about a species’ phenology, year-round monitoring can

also be used in a pilot study to determine the most appropri-

ate sampling period. In our siskin example, we suspect both

left and right truncation to occur some years, suggesting the

sampling season could be extended on both ends. However,

in this case, the presence of locally wintering siskins late in the

migratory season was another potential source of bias that

was considered when the sampling scheme was designed (Ped-

rini et al. 2008). For studies focused on phenology of migra-

tion, the existence of such transient, or even resident,

individuals of the focal species, the presence of which overlaps

with migrant individuals of interest, constitutes another

potential source of bias by inducing false-positive detections

(Sutherland, Elston & Lambin 2013). The issue is that the spe-

cies can be detected outside the period of presence of

migrants, which will negatively (positively) bias their dates of

arrival (departure). In the Common redstart example, some

individuals are local breeders at some of the sampled sites

and, therefore, there is a risk of detecting the species before

the first migrants have actually arrived; there are no locally

wintering individuals, so there is no such risk of bias at the

© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society, Methods in Ecology and Evolution

8 T. Chambert et al.

end of the migratory season. There is no simple and general

solution to such a study-specific issue; one option would sim-

ply be to avoid sampling sites where such sources of false

detections are suspected to exist. An alternative would be to

extend the model to deal with false-positive detections, as has

already been done for closed occupancy models (Royle &

Link 2006; Miller et al. 2011, 2013). Such extensions certainly

represent an exciting research area for model developments in

the near future.

Finally, we remind the reader that, besides their interest for

phenological studies, staggered-entry occupancy models can

be viewed more generally as a means of drawing inferences

when the closure assumption is violated. Reasons for sites

being open to changes in occupancy include those associated

with species phenology (breeding, migration, etc.), but also

more general metapopulation processes such as local coloniza-

tion and/or extinction. In such instances, the availability of the

staggered-entry open occupancy model for multiple seasons

should prove very useful, and some of the design consider-

ations noted above (e.g. daily surveys; survey dates that

bracket the expected season of occupancy) will not be impor-

tant in thesemore general cases.

Acknowledgements

Funding for this research was primarily provided by the U.S. Geological

Survey – Amphibian Research and Monitoring Initiative (ARMI). This is

ARMI contribution number 505. J. Barichivich, M. Brown and J. Hefner

assisted in the collection and examination of the frog calls. The passerine

data sets analysed came from the ‘Progetto Alpi’ monitoring program. The

research reported here is partially funded by the Autonomous Province of

Trento. We are grateful to all the ringers involved in the project, the Ital-

ian Ringing Centre (ISPRA) represented by Fernando Spina, and Giuseppe

Bogliani (University of Pavia). This is contribution number 1 of the ‘Prog-

etto Alpi’ monitoring initiative. Any use of trade, firm or product names is

for descriptive purposes only and does not imply endorsement by the U.S.

Government.

Data accessibility

Data files are all provided in Supporting Information.

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Received 19December 2014; accepted 23 February 2015

Handling Editor:DavidHodgson

Supporting Information

Additional Supporting Information may be found in the online version

of this article.

Appendix S1.Analysis ofGreenTreefrogCall data.

Appendix S2.Model implementation in programPRESENCE.

Data S1.Additional Tables S1 and S2.

Data S2.Datasets of analyses presented in the paper.

© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society, Methods in Ecology and Evolution

10 T. Chambert et al.