# Bringing It All Together: Multi-species Integrated Population Modelling of a Breeding Community

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## Abstract

Integrated population models (IPMs) combine data on different aspects of demography with time-series of population abundance. IPMs are becoming increasingly popular in the study of wildlife populations, but their application has largely been restricted to the analysis of single species. However, species exist within communities: sympatric species are exposed to the same abiotic environment, which may generate synchrony in the fluctuations of their demographic parameters over time. Given that in many environments conditions are changing rapidly, assessing whether species show similar demographic and population responses is fundamental to quantifying interspecific differences in environmental sensitivity and highlighting ecological interactions at risk of disruption. In this paper, we combine statistical approaches to study populations, integrating data along two different dimensions: across species (using a recently proposed framework to quantify multi-species synchrony in demography) and within each species (using IPMs with demographic and abundance data). We analyse data from three seabird species breeding at a nationally important long-term monitoring site. We combine demographic datasets with island-wide population counts to construct the first multi-species Integrated Population Model to consider synchrony. Our extension of the IPM concept allows the simultaneous estimation of demographic parameters, adult abundance and multi-species synchrony in survival and productivity, within a robust statistical framework. The approach is readily applicable to other taxa and habitats.

Supplementary materials accompanying this paper appear on-line.

## Keywords

Bayesian inference Long-term monitoring Mark-resight-recovery State-space model Survival Synchrony## 1 Introduction

Understanding population dynamics and trends is critical for the management of species, be they threatened, invasive or harvested. A long tradition exists of monitoring wildlife population abundance and the demographic rates that drive its fluctuation, with statistical approaches developed to help scientists and managers understand the environmental drivers of change in these parameters (Williams et al. 2002). Of special relevance for our study are long-term monitoring programs, which can act as essential ecosystem sentinels for environmental change and provide the time-scale required for an improved understanding of the relationships between demography, population and environment at a multi-decadal scale, particularly for long-lived species (Wooller et al. 1992).

Long-term wildlife studies are often intensive and generate a wealth of data. When population counts and demography-related data are collected, these data can be analysed together in a single Integrated Population Model (IPM; Besbeas et al. 2002). By constructing a single likelihood function from all the data on abundance and demography, the IPM estimates are a compromise across the various sources of information, often achieving improved parameter estimation or allowing estimation of parameters that could not be obtained from the datasets in isolation (e.g. productivity: Besbeas et al. 2002, immigration: Abadi et al. 2010). Various types of IPMs have been proposed to date, corresponding to a variety of sources of data, but they have traditionally dealt with a single species.

Species do not occur in isolation but exist within communities and ecosystems. Expanding the IPM framework by incorporating multiple species allows new questions of ecological interest to be addressed, by modelling the association or direct interaction between species. The only example of multi-species IPM to date (Péron and Koons 2012) models competition between two species; predator–prey interactions could be studied with a similar structure. A related ecological question is synchrony. Sympatric species are exposed to the same abiotic environment, and this common exposure may generate synchrony in some aspect of these species’ response to their common environment, including population trends and the temporal variation of demographic parameters. The study of synchrony (and asynchrony) is relevant to understanding community structure and its response to environmental changes, and can provide clues to guide further research (McCarthy 2011). Traditional approaches to modelling multi-species synchrony usually involve pairwise species comparisons. An alternative approach, based on a truly multi-species view, was recently proposed to partition the between-year variance in a demographic parameter into a ‘synchronous’ component (common to all species within a set), and species-specific ‘asynchronous’ components, as well as to estimate the proportion of each component accounted for by environmental covariates (Lahoz-Monfort et al. 2013, 2014; Swallow et al. 2016). In this paper, we combine data integration in two different conceptual dimensions: across demography for each species (IPM) and across species to estimate multi-species synchrony, thus extending the traditional concept of single-species IPM (ssIPM) in the first multi-species IPM (msIPM) defined to estimate synchrony.

Our data come from the Isle of May, Scotland (\(56^{\circ }11^{\prime }\)N, \(2^{\circ }34^{\prime }\)W), one of the four ‘Key Site’ seabird colonies in UK’s Seabird Monitoring Programme, where detailed monitoring of abundance, breeding success and adult survival of various seabird species is carried out. We construct a multi-species IPM for three alcid species: the Atlantic puffin *Fratercula arctica*, the common murre (or guillemot) *Uria aalge* and the razorbill *Alca torda*; hereafter puffin, murre and razorbill, respectively. We start by describing the structure of independent ssIPMs for each species. These are then modelled jointly, estimating population abundance, demographic parameters including survival and productivity, and multi-species synchrony in these parameters, in a robust way. The datasets used in this study can be obtained from the first author by request.

## 2 Single-Species Integrated Population Models (ssIPM)

Methodologies used to collect field data are described elsewhere (Harris and Wanless 1988, 1989, 2011; Harris et al. 2015): mark-resight data of individuals marked as breeding adults of unknown age, total counts of chicks leaving the colony (referred to as fledged even though murre and razorbill chicks are flightless when they leave) from a number of monitored nests, and colony-wide counts of breeding pairs, conducted annually for murres and razorbills and less frequently for puffins. For murres, datasets are also available on the proportions of breeding pairs that skipped breeding in different years and mark-resight-recovery data from individuals banded as chicks, which contributes valuable information regarding immature survival and pre-recruitment emigration. We use data from 1984 to 2009 and denote the \(T=26\) years of data by \(t=1,{\ldots },T\). We use parameter subscripts (or superscripts in likelihood functions) to identify species (razorbill: *R*; puffin: *P*; murre: *M*) and *S* when describing general model structures to refer to any species within a set. The following sections describe the specific datasets and single-species IPMs (ssIPMs) for the three species, with a detailed account of parameters involved and their relationship through the IPM.

### 2.1 Breeding Success Data

*t*. We represent the data using vectors \({\varvec{C_S}} =\left\{ {{{C_S}} \left( t\right) :t=1,\ldots , T} \right\} \) and \({\varvec{E_S}} =\left\{ {{{E_S}} \left( t \right) :t=1,\ldots , T} \right\} \), and the full dataset as \({\varvec{P_S}} =\left\{ {{\varvec{C_S, E_S}}} \right\} \). Letting \({\varvec{\rho _S}} =\left\{ {\rho _S \left( t \right) :t=1,\ldots , T} \right\} \) be the set of year-specific productivity parameters, the likelihood corresponding to the binomial model for a breeding success (‘BS’) dataset is

### 2.2 Non-breeding Data (Murres)

*t*, out of a number of monitored individuals \(\xi _{mM} \left( t\right) \) which ranged between 155 and 389 (mean = 310 murres/year). Given this dataset \({\varvec{\xi _M}} =\left\{ {\xi _{bM} \left( t\right) , \xi _{mM} \left( t \right) :t=1,\ldots , T} \right\} \), the non-breeding process can be modelled with a binomial distribution, \(\xi _{bM} \left( t\right) \sim \hbox {bin}\left( {\xi _{mM} \left( t\right) ,B\left( t \right) }\right) \), where \(B\left( t\right) \) is the probability of a pair breeding in year

*t*. Letting \({\varvec{B}}=\left\{ {B\left( t\right) : t=1,\ldots , T} \right\} \), the likelihood for this ‘non-breeding’ model is

### 2.3 Mark-Resight Data: Adult Survival

Between 1984 and 2009, 163 breeding razorbills, 578 breeding puffins and 837 breeding murres were individually colour-banded and remained individually identifiable throughout their lives. Searches were made for these birds in subsequent years. The resulting adult Mark-Resight dataset MR(A), \({\varvec{m_S}} \) for each species *S*, is modelled using the open-population Cormack–Jolly–Seber (CJS) model (reviewed e.g. in McCrea and Morgan 2015), which estimates year-dependent survival and resight probabilities. We assumed no adult emigration (estimated parameters are thus true survival) and fully year-dependent survival probabilities, \({\varvec{s_{aS}}} =\left\{ {s_{aS} \left( t\right) :t=1,\ldots , T-1} \right\} \). Based on a previous analysis (Lahoz-Monfort et al. 2011), we use year-specific resight probability \({\varvec{p}_{\varvec{S}}^*} =\left\{ {p_S^*\left( t\right) :t=1,\ldots ,T-1} \right\} \) and account for whether an individual was resighted the season before (1-year ‘trap dependence’, with constant \(a_S\)). Full details of the MR model and its multinomial likelihood \(L_{\mathrm{MR}\left( \mathrm{A}\right) }^S \left( {{\varvec{m_S}} |{\varvec{s_{aS}}}, {\varvec{p}_{\varvec{S}}^*}, a_S}\right) \) are given in McCrea and Morgan (2015).

### 2.4 Mark-Resight-Recovery Data: Juvenile Survival (Murres)

A total of 6569 murre chicks were banded between 1984 and 2009 (annual totals: 113–325; mean: 253). Large-scale banding and resighting of puffin and razorbill chicks were not possible due to logistical constraints. Each murre chick was given a unique colour-band on one leg (with an individual code) and a numbered hard metal band on the other. Two areas were used: a 400-m length of cliff (‘area A’) and a nearby skerry (‘area B’) of lesser visibility (where banding of 1356 chicks occurred only until 1997). Full details about field methods are given in Harris et al. (2007). From 1985 to 2010, regular searches were made during the breeding season for banded murres that had returned to the Isle of May. This resulted in 11,388 individual resightings (excluding initial capture but otherwise including birds seen more than once in a breeding season) which translated into 4738 detections in the mark-resight history (raw resightings include birds seen more than once in a season). In addition, 248 banded murres were reported dead elsewhere which allowed us to estimate true survival and fidelity separately, as opposed to apparent survival (their combined effect) in MR studies (Burnham 1993).

- (i)
\(\phi _{a,t} \left( r\right) :\) probability that a bird in state \(r=\left\{ {0,1} \right\} \) aged

*a*at year*t*survives until age \(a+1\). We assume same survival for any state: \(\phi _{a,t} \left( 1\right) =\phi _{a,t} \left( 0 \right) =\phi _{a,t} \); - (ii)
\(\psi _{a,t} \left( {r,s}\right) :\) probability that a bird in state \(r=\left\{ {0,1} \right\} \) aged

*a*in year*t*, moves to state \(s=\left\{ {0,1} \right\} \) by age \(a+1\), given that it is alive at this age. Fidelity is \(\psi _{a,t} \left( {1,1}\right) =F_{a,t} \) and permanent emigration is \(\psi _{a,t} \left( {1,0}\right) =1-F_{a,t} \). Also, \(\psi _{a,t} \left( {0,1}\right) =0, \psi _{a,t} \left( {0,0}\right) =1\); - (iii)
\(p_{a,t} \left( r\right) :\) probability that a bird alive in state \(r=\left\{ {0,1} \right\} \) aged

*a*at year*t*is resighted at this age. As birds that emigrate permanently cannot be resighted, \(p_{a,t} \left( 0\right) =0\). We denote resightings at the Isle of May as \(p_{a,t} \left( 1\right) =p_{a,t} \); - (iv)
\(\lambda _{a,t} \left( r\right) :\) ‘reporting’ probability, i.e. probability that a bird in state \(r=\left\{ {0,1} \right\} \) aged

*a*at year*t*that dies before age \(a+1\) is recovered dead and its numbered metal band reported (before age \(a+1)\). We assume \(\lambda _{a,t} \left( 1\right) =\lambda _{a,t} \left( 0\right) =\lambda _{a,t}\).

*a*at year

*t*, to state \(s=\left\{ {0,1} \right\} \) at age \(b+1\) and is unobserved between these ages:

*a*at year

*t*, remains unobserved until it is resighted at age \(b+1\) in state \(s=\left\{ {0,1} \right\} \):

*b*and \(b+1\), given that it was last observed alive in state \(r=\left\{ {0,1} \right\} \) aged

*a*at time

*t*:

*a*at year

*t*is not seen again alive or dead during the rest of the study:

*a*in year

*t*and next seen alive in state \(s=\left\{ {0,1} \right\} \) aged \(b+1\); (ii) \(d_{a,b,t} \left( r\right) \): number of birds recovered dead at age

*b*that were last observed alive in state \(r=\left\{ {0,1}\right\} \) aged

*a*in year

*t*; and (iii) \(v_{a,t} \left( r\right) \): number of birds seen alive (including initial release) for the last time in state \(r=\left\{ {0,1} \right\} \) aged

*a*in year

*t*, and not recovered dead at a later encounter occasion.

*y*the standardised years (from 1 to \(T-1)\).

Some colour bands on immatures became worn and dropped off, so colour-band loss and recruitment into an area of low visibility are in principle confounded with emigration as individuals become unobservable alive but the stainless steel numbered bands may still be reported once the bird dies. These two processes can be separated from ‘true’ fidelity with the help of an IPM, as they impact very differently on population counts (Reynolds et al. 2009). We define the probability \(\psi \) that an adult (marked as chick) retains a readable band and recruits (or continues breeding) at a visible location. Assuming \(F_a =1\) and that \(\psi \) only applies to birds that have started breeding (therefore adults), we can model the ‘retention of colour bands and recruitment to a visible location’ using the ‘fidelity’ parameter \(\psi _{a,t} \left( {1,1}\right) =F_{a>6,t} =\psi \) for \(a>6\).

*a*in year

*t*in state 1, multinomial cell probabilities and corresponding observed cell numbers are \(\big \{O_{a,a,t} ({1,1}),\ldots , O_{a,A,t} ({1,1}),D_{a,a,t} (1),\ldots , D_{a,A,t} (1),\chi _{a,t} (1) \big \}\) and \(\left\{ {n_{a,a,t} \left( {1,1}\right) ,\ldots , n_{a,A,t} \left( {1,1}\right) ,d_{a,a,t} \left( 1\right) ,\ldots , d_{a,A,t} \left( 1\right) ,v_{a,t} \left( 1\right) } \right\} .\)

### 2.5 Breeding Population Counts: Population Model

In an IPM, population counts are modelled using a state-space population model (Buckland et al. 2004), which consists of two linked models. For each species S, the system process model describes the true population abundance \(N_{xS} \left( {t+1}\right) \) for the different age classes *x* at year \(t+1\) as a function of the previous year’s abundance. The structure of the population model for each species will have a degree of complexity (and realism) that depends on the datasets available and the ecology of the species. We specifically keep track of female abundance, which is sufficient to model the number of breeding pairs as our species are monogamous (Gaston and Jones 1998). A number \(N_{aS} \left( t\right) \) of adult breeding females in year *t* will produce a single egg. Each egg has a probability \(\rho _S \left( t\right) \) (overall productivity in year *t*) of hatching and the chick surviving until fledging, and a factor 0.5 takes into account that on average half of the chicks will be females (balanced sex ratio at fledging). Only a fraction of these fledglings will survive their first winter. The number of ‘age 1’ females at time \(t+1\) can be modelled as a binomial distribution: \(N_{1S} \left( {t+1}\right) \sim \hbox {bin}\left( {N_{aS} \left( t \right) ,\rho _S \left( t\right) s_{1S} \left( t\right) /2}\right) \), with \(s_{1S} \left( t\right) \) the survival probability over the first year of life. The number of immature females of increasing age can be modelled in the same way using binomial distributions with corresponding age-specific survival.

We model recruitment using the median value of age at first breeding, denoted \(d_S \) for species *S*. We use \(d_R =5\) (median from Skokholm Island in Wales, \(n = 20\); Lloyd and Perrins 1977), \(d_P =7\) (median from the Isle of May, \(n = 108\); Harris and Wanless 2011); and \(d_M =6\) (median from the Isle of May, \(n = 42\); Harris et al. 1994). Pre-breeders \(N_{d-1,S} \left( t\right) \) represent the number of females in the year before first breeding. A non-negligible fraction of puffins and murres, and we assume razorbills, hatched at the Isle of May permanently emigrate and recruit to other colonies (Harris et al. 1996; Harris and Wanless 2011). We also assume that survival over the winter immediately before recruiting is equal to that of adult birds \(s_{aS} \), hence the new recruits \(R_S \left( t\right) \) to the female adult population in year *t* will be \(R_S \left( t\right) \sim \hbox {bin}\left( {N_{d-1,S} \left( {t-1}\right) , F_S s_{aS} \left( {t-1}\right) }\right) \), where \(F_S \) is pre-breeding fidelity.

*s*. Razorbill and puffin new recruits can thus be modelled as \(R_S \left( t\right) \sim \hbox {bin}\left( {N_{aS} \left( {t-d_S}\right) , \rho _S \left( {t-d_S}\right) \phi _{cS} s_{aS} \left( {t-1}\right) /2}\right) .\) From the adult population at time \(t-1\), individuals will survive to year

*t*with probability \(s_{aS} \): \(S_S \left( t\right) \sim \hbox {bin}\left( {N_{aS} \left( {t-1}\right) , s_{aS} \left( {t-1}\right) }\right) \). The total number of breeding females at year

*t*will be the sum of surviving adults and new female recruits: \(N_{aS} \left( t\right) =S_S \left( t \right) +R_S \left( t\right) \). Established breeding adults of the three species virtually never move to other colonies (Gaston and Jones 1998) so we assume no emigration (\(F_{aS} =1\)). A small pre-breeder immigration into the Isle of May population (Lloyd 1974; Halley and Harris 1993; Harris and Wanless 2011) occurs but our models assume no immigration due to lack of data to estimate it. Letting \(\rho _S \left( {t-d_S}\right) \frac{1}{2}\phi _{cS} =\tau _S \left( {t-d_S}\right) \), the ‘likelihood’ of the system process model can be written as

### 2.6 Breeding Population Counts: Observation Model

Finally, the likelihood of the state-space population model for each species \(S\,(L_{\mathrm{POP}}^{S})\) is the product of the likelihood of the observation model \((L_{\mathrm{OBS}}^{S})\) and the system process model \((L_{N}^{S})\). This represents the complete-data likelihood, which includes the unobserved data (true population abundances). This expression is not easily evaluated (e.g. in frequentist inference); we circumvent this limitation using Bayesian inference, as explained in Sect. 4.

### 2.7 Joint Likelihood: ssIPMs

List of parameters involved in the msIPM, specifying in which model component for which species they appear.

Parameters | \(\hbox {BS}_{R}\) | \(\hbox {MR(A)}_{R}\) | \(\hbox {POP}_{R}\) | \(\hbox {BS}_{P}\) | \(\hbox {MR(A)}_{P}\) | \(\hbox {POP}_{P}\) | \(\hbox {BS}_{M}\) | \(\hbox {NB}_{M}\) | \(\hbox {MR(A)}_{M}\) | \(\hbox {MRR(C)}_{ M}\) | \(\hbox {POP}_{M}\) |
---|---|---|---|---|---|---|---|---|---|---|---|

\({\varvec{\delta _\rho }}, \sigma _{\delta \rho }^2 \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | |||||

\({\varvec{\delta _\phi }}, \sigma _{\delta \phi }^2 \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | ||||

[S] \(\beta _{\rho R}, {\varvec{\varepsilon _{\rho R}}}, \sigma _{\varepsilon \rho R}^2 \) | \(\checkmark \) | \(\checkmark \) | |||||||||

[S] \(\beta _{\phi R}, {\varvec{\varepsilon _{\phi R}}}, \sigma _{\varepsilon \phi R}^2 \) | \(\checkmark \) | \(\checkmark \) | |||||||||

\({\varvec{\rho _R}} \) | \(\checkmark \) | \(\checkmark \) | |||||||||

\({\varvec{s_{aR}}}, {\varvec{p}}_{\varvec{R}}^*, a_R, {\varvec{p_R}}, \overline{{\varvec{p}}}_{\varvec{R}} \) | \(\checkmark \) | \(\checkmark \) | |||||||||

\(\phi _{cR}, \sigma _{xR}^2, R_R, S_R, N_R \) | \(\checkmark \) | ||||||||||

[S] \(\beta _{\rho P}, {\varvec{\varepsilon _{\rho P}}}, \sigma _{\varepsilon \rho P}^2 \) | \(\checkmark \) | \(\checkmark \) | |||||||||

[S] \(\beta _{\phi P}, {\varvec{\varepsilon _{\phi P}}}, \sigma _{\varepsilon \phi P}^2 \) | \(\checkmark \) | \(\checkmark \) | |||||||||

\({\varvec{\rho _P}} \) | \(\checkmark \) | \(\checkmark \) | |||||||||

\({\varvec{s_{aP}}}, {\varvec{p}}_{\varvec{P}}^*, a_P, {\varvec{p_P}}, \overline{{\varvec{p}}}_{\varvec{P}}\) | \(\checkmark \) | \(\checkmark \) | |||||||||

\(\phi _{cP}, \sigma _{xP}^2, R_P, S_P, N_P \) | \(\checkmark \) | ||||||||||

[S] \(\beta _{\rho M}, {\varvec{\varepsilon _{\rho M}}}, {{\sigma }}_{{{\varepsilon \rho M}}}^2 \) | \(\checkmark \) | \(\checkmark \) | |||||||||

[S] \(\beta _{\phi M}, {\varvec{\varepsilon _{\phi M}}}, {{\sigma }}_{{{\varepsilon \phi M}}}^2 \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | ||||||||

\({\varvec{\rho _M}} \) | \(\checkmark \) | \(\checkmark \) | |||||||||

\({\varvec{B}}\) | \(\checkmark \) | \(\checkmark \) | |||||||||

\({\varvec{s_{aM}}}, {\varvec{p}}_{\varvec{M}}^*, a_M, {\varvec{p_M}}, \overline{{\varvec{p}}}_{\varvec{M}} \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | ||||||||

\({\varvec{s_1}}, s_2, s_{35}, F_5, F_6, \psi , {\varvec{r}},\alpha _0, \alpha _1 \) | \(\checkmark \) | \(\checkmark \) | |||||||||

\({\varvec{p_2^A, p_3^A, p_{45}^A, p_a^A, p_2^B, p_3^B, p_{45}^B, p_a^B}}\) | \(\checkmark \) | \(\checkmark \) | |||||||||

\({\phi }_{cM}, {\sigma }_{xM}^2, R_M, S_M, N_M\) | \(\checkmark \) |

## 3 Multi-Species Integrated Population Model (msIPM)

*S*can be derived from the random effects variances as \(I_{\phi S} =\frac{\hat{\sigma }_{\delta \phi }^2}{\hat{\sigma }_{\delta \phi }^2 +\hat{\sigma }_{\varepsilon \phi S}^2}\) and \(I_{\rho S} =\frac{\hat{\sigma }_{\delta \rho }^2}{\hat{\sigma }_{\delta \rho }^2 +\hat{\sigma }_{\varepsilon \rho S}^2}\). They represent the synchrony of species S with the rest of the species: the amount of between-year variance for species

*S*that is common to all the others (common random terms \(\delta (t))\). High values for a species indicate that most of its year-to-year variation is synchronous to the set. ‘Community-level’ synchrony parameters (common random terms and their variances) are shared across species, rendering the model multi-species. Table 1 lists the estimated parameters for the three species, and where they appear in the likelihood.

## 4 Bayesian Analysis

## 5 Results

### 5.1 Multi-species IPM

We obtain one million MCMC samples (thinned to 1/40th to reduce memory requirements) after a burn-in of one million samples (5.9 days on a 3.4 GHz processor). Marginal posteriors are obtained using MCMC for 1031 model parameters and summarized by the median and symmetric 95% Credible Intervals. Of these, 390 are derived deterministically from others.

Figure 1 shows the estimated true adult female population abundance for the period 1984–2009 for the three species. Puffins show the greatest change, strongly increasing from 11,390 in 1984 to 70,540 in 2006, followed by an unprecedented population crash (37%) to 44,710 pairs two years later. Despite population counts only being available for five years after 1990, the IPM is able to fit well the initial steady increase and estimates the population peak as taking place in 2006. Murre abundance shows a similar pattern but with less variation: steady increase for most of the period (from \(\sim \)13,000 to 18,450 in 2004) followed by a substantial decline (19%). Estimates fit the counts reasonably well, save for some discrepancy in 1991–1993 and 2002; despite its more flexible structure, some assumptions are still made (e.g. constant immature survival over their first winter). Razorbill numbers are substantially smaller but follow a similar pattern of steady increase (from \(\sim \)1500 to 3045 in 2006) followed by a drop (22%) in 2008; another population drop is apparent in 2003. Estimates fit the general pattern of counts well, but with greater annual variation: model structure may be slightly rigid (e.g. constant combined immature survival \(\phi _{cR} \) imposed by lack of data on immature survival); attempts to fit more flexible models (e.g. year-specific \(\phi _{cR} \left( t \right) )\) gave very imprecise estimates of \(\phi _{cR} \). The model captures the general population trend, in agreement with the demographic variation.

### 5.2 Multi-species: Synchrony and Shrinkage of the Estimates

Estimates (median and 95% CI in brackets) of the msIPM constant parameters for puffin, murre and razorbill.

Puffin | Murre | Razorbill | |
---|---|---|---|

\(\hat{\sigma }_{xs}\) | 5551 (2501, 11,920) | 1503 (1055, 2284) | 358 (237, 550) |

Trap-dep. \(\hat{a}_s \) | 1.928 (1.633, 2.227) | 3.240 (2.840, 3.618) | 1.826 (1.234, 2.428) |

Intercept \(\hat{\beta }_{\phi s} \) | 2.436 (2.119, 2.823) | 2.789 (2.548, 3.056) | 2.319 (1.976, 2.721) |

\(\hat{\sigma }_{\varepsilon \phi s} \) | 0.625 (0.391, 0.970) | 0.261 (0.019, 0.597) | 0.519 (0.101, 0.961) |

\(\hat{\sigma }_{\delta \phi } \) | 0.493 (0.272, 0.748) | ||

\(\hat{I}_{\phi s} \) | 0.383 (0.108, 0.697) | 0.787 (0.270, 0.999) | 0.477 (0.122, 0.968) |

Intercept \(\hat{\beta }_{\rho s} \) | 0.890 (0.627, 1.159) | 1.002 (0.751, 1.250) | 0.689 (0.518, 0.870) |

\(\hat{\sigma }_{\varepsilon \rho s} \) | 0.498 (0.300, 0.762) | 0.491 (0.344, 0.702) | 0.109 (0.006, 0.337) |

\(\hat{\sigma }_{\delta \rho } \) | 0.357 (0.237, 0.540) | ||

\(\hat{I}_{\rho s} \) | 0.340 (0.127, 0.690) | 0.344 (0.138, 0.651) | 0.913 (0.545, 1.000) |

\(\hat{\alpha }_0 \) | NA | −3.084 (−3.225, −2.949) | NA |

\(\hat{\alpha }_1 \) | NA | −0.687 (−0.837, −0.540) | NA |

\(\hat{\phi }_c \) | 0.761 (0.621, 0.905) | NA | 0.501 (0.402, 0.614) |

## 6 Discussion

Depending on data availability and model structure, demographic integration can sometimes permit estimation of parameters which cannot be estimated from independent analyses of the individual datasets involved (Besbeas et al. 2002). Our IPMs allow the estimation of combined juvenile survival for razorbills and puffins, two species without direct data about the fate of individual juvenile birds (a common situation for long-lived, pelagic species). IPMs also improve the estimation of abundance, compared to naïve counts with observation error. Despite the apparent complexity of the msIPM (641 non-derived parameters), the amount of data is correspondingly very large (e.g. 17,303 resightings for the three species combined), as it is the result of combining eight datasets, an important investment in field effort over this period of time.

IPMs can also separate true population abundance and observation error. Using counts without allowing for observation error may dilute the relationship of abundance to other processes of interest (e.g. density-dependence, Freckleton et al. 2006). IPMs also allow the estimation of abundance in years when counts are not available. In our study, this enabled us to identify when the puffin population crashed on the Isle of May. While population models can be populated with independent estimates of demographic parameters, their joint estimation within an IPM ensures that demography ‘agrees’ with an (imperfect) observation of the variation of population abundance (counts). Associated with species integration in the msIPM (and synchrony estimation) is some degree of shrinkage towards the community mean, which in our case was only slight for adult survival and even smaller for productivity; this is also associated with increases in precision. As expected, the effect is stronger for the species that contributes the least data.

IPMs commonly have many parameters. Methods and guidelines for checking potential identifiability issues and overall goodness-of-fit of IPMs are still under development, but exploring each separate sub-model may provide informative detail about the origin of a potential lack of fit in the overall model (Besbeas and Morgan 2014). In our case, trap dependence in resight probabilities was already introduced in the model following an assessment of the MR components (Lahoz-Monfort et al. 2011) and the more complex murre chick MRR model was scrutinized in a previous analysis (Lahoz-Monfort et al. 2014). Model components based on independent binomial trials for each year, whose mean is estimated from a single data point per year, have perfect fit. Finally, population estimates are in line with population counts, so that at least a systematic lack of fit appears unlikely. Formal parameter redundancy methods for IPMs have only been developed recently (Cole and McCrea 2016). Model selection was also conducted locally for model components when needed (e.g. MRR).

Independence between census and demographic datasets is a key assumption for forming the IPM joint likelihood by multiplying different likelihood components (Besbeas et al. 2009). Our datasets do not strictly meet this assumption, as some are obtained from the same colony areas and therefore include information from different life history aspects of the same individuals, also counted in the census. In practice, population counts can be considered independent because the island-wide counts are much larger than the sample of monitored birds. The impact of lack of independence in IPMs has yet to be thoroughly studied but it is likely to depend strongly on the nature of the dependency and degree of overlap in the number of shared individuals. The mixed results reported in the literature (c.f. simulation results of Besbeas et al. 2009; Abadi et al. 2010) may be caused by one of the datasets contributing most of the information for the parameter under study. Extending the framework to encompass several species may create new forms of dataset dependence across species.

Multi-species IPMs could also be extended to incorporate spatial aspects. For example, our multi-species synchrony IPM could be expanded to include other breeding colonies in the Northeast Atlantic with the same alcid community (or more generally, several populations of a set of species). Such a *multi-species multi-population IPM* would combine our idea with that of multi-population IPMs (Cave et al. 2010) and could allow the estimation of multi-population synchrony (Schaub et al. 2015).

Integrated population modelling makes explicit the relationship between changes in demographic rates and their impact on population fluctuations, and may bring insights into drastic population changes. Our extension of the IPM concept to encompass sympatric populations of several species allows at the same time the estimation of multi-species synchrony in a robust framework, and opens the door to further methodological developments.

## Notes

### Acknowledgements

J.J.L.-M. was funded by EPSRC/NERC Grant EP/1000917/1 and by the Centre for Ecology & Hydrology. We thank the many people who helped with data collection on the Isle of May, particularly Mark Newell. Part of the fieldwork was funded by the Joint Nature Conservation Committee’s integrated Seabird Monitoring Programme. Scottish Natural Heritage allowed us to work on the Isle of May. Ring recoveries come from the BTO Ringing Scheme, funded by a partnership of the British Trust for Ornithology, the Joint Nature Conservation Committee (on behalf of Natural England, Scottish Natural Heritage and the Countryside Council for Wales and also on behalf of the Council for Nature Conservation and the Countryside in Northern Ireland), The National Parks and Wildlife Service (Ireland) and the ringers themselves. We thank the associate editor and two anonymous referees for comments on an earlier version of the paper.

## Supplementary material

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