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.
Breeding Success Data
Breeding success data (Reed et al. 2015) consist of a series of yearly counts of chicks \(C_S \left( t\right) \) that fledge from a number of monitored marked adult pairs \(E_S \left( t\right) \) that make a breeding attempt. As all three species lay a single egg, data can be modelled as a binomial variable \(C_S \left( t\right) \sim \hbox {bin}\left( {E_S \left( t\right) ,\rho _S \left( t \right) }\right) \), where \(\rho _S \left( t\right) \) is the productivity of species S, in year 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
$$\begin{aligned} L_{\mathrm{BS}}^S \left( {{\varvec{P_S}} |{\varvec{\rho _S}}}\right) =\mathop {\prod }\limits _{t=1}^T \left( {{ \begin{array}{l} {E_S \left( t\right) } \\ {C_S \left( t\right) } \\ \end{array}}}\right) \rho _S \left( t\right) ^{C_S \left( t\right) }\left\{ {1-\rho _S \left( t \right) } \right\} ^{E_S \left( t\right) -C_S \left( t\right) }. \end{aligned}$$
The annual number of monitored pairs ranged from 73 to 194 pairs (mean \(=\) 135) for razorbills, 32 to 196 pairs (mean \(=\) 159; only 1984 and 1985 had fewer than 100 burrows monitored) for puffins, and 656 to 1014 (mean \(=\) 828) for murres.
Non-breeding Data (Murres)
Every year, a small proportion of murre pairs (typically below 10%) do not lay an egg. Non-breeding has been monitored at the Isle of May by counting the number of murres \(\xi _{bM} \left( t\right) \) that do not skip breeding at any particular year 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
$$\begin{aligned} L_{\mathrm{NB}}^M \left( {{\varvec{\xi _M}} |{\varvec{B}}}\right) =\mathop {\prod }\limits _{t=1}^T \left( {{ \begin{array}{l} {\xi _{mM} \left( t\right) } \\ {\xi _{bM} \left( t\right) } \\ \end{array}}}\right) B\left( t\right) ^{\xi _{bM} \left( t\right) }\left\{ {1-B\left( t \right) } \right\} ^{\xi _{mM} \left( t\right) -\xi _{bM} \left( t \right) }. \end{aligned}$$
A small number of razorbill and puffin pairs also do not breed in a given season, but data are not available to model these processes.
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).
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).
We construct the likelihood corresponding to the chick mark-resight-recovery data ‘MRR(C)’ with a generic age- and year-dependence structure, based on a computationally efficient multi-state approach with sufficient statistic matrices (McCrea 2012). We define two mutually exclusive states: State \(r = 1\) (‘Isle of May’; recruited to this population); State \(r = 0\) (‘Emigrated’; recruited into another breeding colony). Birds in State 0 (unobserved) do not contribute to population abundance at the Isle of May, although bands can be recovered from dead birds in this state. For resightings of murres aged \(a=1,\ldots ,A\) during years \(t=1,\ldots ,T\) (no recoveries after \(t=T\)), we can define the model parameters (dropping species subscript M for brevity):
-
(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: \(\phi _{a,t} \left( 1\right) =\phi _{a,t} \left( 0 \right) =\phi _{a,t} \);
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(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, \(\psi _{a,t} \left( {0,1}\right) =0, \psi _{a,t} \left( {0,0}\right) =1\);
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(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 \(p_{a,t} \left( 1\right) =p_{a,t} \);
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(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 \(\lambda _{a,t} \left( 1\right) =\lambda _{a,t} \left( 0\right) =\lambda _{a,t}\).
Based on McCrea (2012), we define the following probabilities, for our particular case:
(i) \(Q_{a,b,t} \left( {r,s}\right) :\) probability that a bird migrates from state \(r=\left\{ {0,1} \right\} \) aged a at year t, to state \(s=\left\{ {0,1} \right\} \) at age \(b+1\) and is unobserved between these ages:
$$\begin{aligned} Q_{a,b,t} \left( {1,0}\right)= & {} \left\{ {{ \begin{array}{ll} {\phi _{a,t} \left( {1-F_{a,t}}\right) ,} &{}\quad {a=b} \\ {\phi _{a,t} \left\{ {\left( {1-F_{a,t}}\right) Q_{a+1,b,t+1} \left( {0,0}\right) +F_{a,t} \left( {1-p_{a+1,t+1}}\right) Q_{a+1,b,t+1} \left( {1,0}\right) } \right\} ,} &{} \quad {a<b} \\ \end{array}}} \right. \\ Q_{a,b,t} \left( {1,1}\right)= & {} \left\{ {{ \begin{array}{ll} {\phi _{a,t} F_{a,t},} &{} \quad {a=b} \\ {\phi _{a,t} F_{a,t} \left( {1-p_{a+1,t+1}}\right) Q_{a+1,b,t+1} \left( {1,1}\right) ,} &{} \quad {a<b} \\ \end{array}}} \right. \\ Q_{a,b,t} \left( {0,0}\right)= & {} \left\{ {{ \begin{array}{ll} {\phi _{a,t},} &{}\quad {a=b} \\ {\phi _{a,t} Q_{a+1,b,t+1} \left( {0,0}\right) ,} &{}\quad {a<b} \\ \end{array}}} \right. \\ Q_{a,b,t} \left( {0,1}\right)= & {} 0. \end{aligned}$$
(ii) \(O_{a,b,t} \left( {r,s}\right) :\) probability that a bird in state \(r=\left\{ {0,1} \right\} \) aged a at year t, remains unobserved until it is resighted at age \(b+1\) in state \(s=\left\{ {0,1} \right\} \):
$$\begin{aligned} O_{a,b,t} \left( {1,1}\right)= & {} Q_{a,b,t} \left( {1,1}\right) p_{b+1,t+b-a+1}\\ O_{a,b,t} \left( {1,0}\right)= & {} O_{a,b,t} \left( {0,1}\right) =O_{a,b,t} \left( {0,0}\right) =0. \end{aligned}$$
(iii) \(D_{a,b,t} \left( r\right) :\) probability that a bird is recovered dead between ages b and \(b+1\), given that it was last observed alive in state \(r=\left\{ {0,1} \right\} \) aged a at time t:
$$\begin{aligned} D_{a,b,t} \left( 1\right)= & {} \left\{ {{ \begin{array}{ll} {\left( {1-\phi _{a,t}}\right) \lambda _{a,t},} &{}\quad {a=b} \\ {\left( {1-\phi _{b,t+b-a}}\right) \lambda _{b,t+b-a} \left\{ {Q_{a,b-1,t} \left( {1,0}\right) +\left( {1-p_{b,t+b-a}}\right) Q_{a,b-1,t} \left( {1,1}\right) } \right\} ,} &{} \quad {a<b} \\ \end{array}}} \right. \\ D_{a,b,t} \left( 0\right)= & {} 0. \end{aligned}$$
(iv) \(\chi _{a,t} \left( r\right) :\) probability that a bird alive in state \(r=\left\{ {0,1} \right\} \) at age a at year t is not seen again alive or dead during the rest of the study:
$$\begin{aligned} \chi _{a,t} \left( 0\right)= & {} \left\{ {{ \begin{array}{ll} {1,} &{} {t=T} \\ {\left( {1-\lambda _{a,t}}\right) \left( {1-\phi _{a,t}}\right) +\phi _{a,t} \chi _{a+1,t+1} \left( 0\right) ,} &{} {t<T} \\ \end{array}}} \right. \\ \chi _{a,t} \left( 1\right)= & {} \left\{ {{ \begin{array}{ll} {1,} &{} {t=T} \\ {\left( {1-\lambda _{a,t}}\right) \left( {1-\phi _{a,t}}\right) +\phi _{a,t} \left\{ {\left( {1-F_{a,t}}\right) \chi _{a+1,t+1} \left( 0 \right) +F_{a,t} \left( {1-p_{a+1,t+1}}\right) \chi _{a+1,t+1} \left( 1 \right) } \right\} ,} &{} {t<T} \\ \end{array}}} \right. . \end{aligned}$$
The MRR dataset can be summarized using a set of sufficient statistics: (i) \(n_{a,b,t} \left( {r,s}\right) \): number of birds observed in state \(r=\left\{ {0,1} \right\} \) at age 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.
Given that no murres are resighted in state 0, only the following terms are nonzero: \(n_{a,b,t} \left( {1,1}\right) , d_{a,b,t} \left( 1\right) \) and \(v_{a,t} \left( 1\right) \). The full age- and year-dependent likelihood of the MRR(C) dataset, taking into account restrictions in the relationships of the indices, is:
$$\begin{aligned} L\left( {{\varvec{n,d,v}}|{\varvec{\phi , \psi , p,\lambda }}}\right)= & {} \mathop {\prod }\limits _{a=1}^{A-1} \mathop {\prod }\limits _{b=a}^{A-1} \mathop {\prod }\limits _{t=a}^{\left( {T-1+a-b}\right) } \left\{ {O_{a,b,t} \left( {1,1}\right) ^{n_{a,b,t} \left( {1,1}\right) }\times D_{a,b,t} \left( 1\right) ^{d_{a,b,t} \left( 1\right) }} \right\} \\&\times \mathop {\prod }\limits _{a=1}^A \mathop {\prod }\limits _{t=a}^{T-1} \chi _{a,t} \left( 1 \right) ^{v_{a,t} \left( 1\right) }, \end{aligned}$$
which requires calculating the terms \(\chi _{a,t} \left( 0\right) , Q_{a,b,t} \left( {0,0}\right) , Q_{a,b,t} \left( {1,0}\right) \) and \(Q_{a,b,t} \left( {1,1}\right) \). Based on our previous analysis of this MRR dataset (Lahoz-Monfort et al. 2014), we simplify the above fully age- and time-dependent likelihood using the following age and year model structure (adult: defined as age \(a>5\) years): (i) year-specific first-year survival parameters \(s_1 \left( t\right) \) and adult survival \(s_a \left( t\right) \); constant for \(2\mathrm{nd}\) and \(3\mathrm{rd}\)-to-\(5\mathrm{th}\) years of life (parameters \(s_2 \) and \(s_{35} =s_3 =s_4 =s_5 )\); (ii) year-specific resight probabilities, for three age classes (\(p_2 \left( t\right) , p_3 \left( t\right) \) and \(p_{45} \left( t\right) \) for 2, 3 and ‘4-to-5’ year olds) and adults \(p_a \left( t\right) \), estimated independently for each banding area (A or B, indicated by superscript). We fix \(p_1 =0\) as young murres do not return to their natal colony in their first year; (iii) we estimate fidelity in the two years before recruitment (\(F_5 \) and \(F_6 )\) but let \({F}_{1}={F}_{2}={F}_{3}={F}_{4}=1\) for younger birds (uncommon recruitment at that early age) and \(F_a =1\) for adults; (iv) a general trend of decreasing reporting probabilities has been noticed in several species in the UK (Robinson et al. 2009), so we fit a linear trend with time in \(\lambda _{a,t}\) (on the logit scale) common to all ages: \(\alpha _0 +\alpha _1 y\), with 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 \(a>6\).
For banding area A, let: \({\varvec{p_2^A}} =\left\{ {p_2^A \left( t \right) :t=3,\ldots , T} \right\} , {\varvec{p_3^A}} =\left\{ {p_3^A \left( t \right) :t=4,\ldots , T} \right\} , {\varvec{p_{45}^A}} =\left\{ {p_{45}^A \left( t\right) :t=5,\ldots , T} \right\} , {\varvec{p_a^A}} =\left\{ {p_a^A \left( t\right) :t=7,\ldots , T} \right\} \); for area B: \({\varvec{p_2^B}} =\left\{ {p_2^B \left( t\right) :t=3,\ldots , 17} \right\} , {\varvec{p_3^B}} =\left\{ {p_3^B \left( t\right) :t=4,\ldots , 18} \right\} , {\varvec{p_{45}^B}} =\left\{ {p_{45}^B (t):t=5,\ldots , 20} \right\} , {\varvec{p_a^B}} =\left\{ {p_a^B \left( t\right) :t=7,\ldots , T} \right\} \); and the complete parameter sets as \({\varvec{p_M^C}} =\big \{\varvec{p_2^A ,p_3^A, p_{45}^A}, \varvec{p_a^A, p_2^B, p_3^B, p_{45}^B, p_a^B} \big \}, {\varvec{F_M}} =\left\{ {F_5, F_6} \right\} \), and immature survival \({\varvec{s_{iM}}} =\left\{ {{\varvec{s_1}}, s_2, s_{35}} \right\} \). We treat areas A and B as two distinct datasets, \(\left\{ {{\varvec{n_A, d_A, v_A}}} \right\} \) and \(\left\{ {{\varvec{n_B, d_B, v_B}}} \right\} \), and construct the likelihoods for both areas, \(L_A^M \) and \(L_B^M \). The overall likelihood of the complete chick MRR(C) dataset can be constructed by multiplying both:
$$\begin{aligned}&L_{\mathrm{MRR}\left( \mathrm{C}\right) }^M \left( {{\varvec{n_M, d_M, v_M}} |{\varvec{s_{iM}, s_{aM}, p_M^C}}, \alpha _0, \alpha _1, {\varvec{F_M}}, \psi }\right) \\&\quad =L_A^M \left( {{\varvec{n_A, d_A, v_A}} |{\varvec{s_{iM}, s_{aM}, p^{A}}},\alpha _0, \alpha _1, {\varvec{F_M}}, \psi }\right) \\&\qquad \times L_B^M \left( {{\varvec{n_B, d_B, v_B}} |{\varvec{s_{iM}, s_{aM}, p^{B}}},\alpha _0, \alpha _1, {\varvec{F_M}}, \psi }\right) . \\ \end{aligned}$$
It is easier to handle this likelihood by realizing that it is product-multinomial (McCrea 2012). For releases aged 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 \(\left\{ {n_{a,a,t} \left( {1,1}\right) ,\ldots , n_{a,A,t} \left( {1,1}\right) ,d_{a,a,t} \left( 1\right) ,\ldots , d_{a,A,t} \left( 1\right) ,v_{a,t} \left( 1\right) } \right\} .\)
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.
In practice, we do not have enough data on immature razorbills and puffins to separate pre-breeder emigration from mortality, or to estimate age-dependent survival probabilities, so for these species we use a ‘combined survival’ parameter \(\phi _{cS} \) which combines survival since fledging to the year before recruitment, and fidelity. We use letter \(\phi \) following a common naming convention (White and Burnham 1999) to denote ‘apparent survival’ (where permanent emigration and mortality are confounded) instead of ‘true survival’ 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
$$\begin{aligned}&L_N^S \left( {{\varvec{R_S}}, {\varvec{S_S}} |\phi _{cS}, {\varvec{s_{aS}}}, {\varvec{\rho _S}}}\right) \\&\quad =\mathop {\prod }\limits _{t=d+1}^T \left[ {\left( {{ \begin{array}{l} {N_{aS} \left( {t-d_S}\right) } \\ {R_S \left( t\right) } \\ \end{array}}}\right) \left\{ {\tau _S \left( {t-d_S}\right) s_{aS} \left( {t-1}\right) } \right\} ^{R_S \left( t\right) }} \right. \\&\qquad \times \left\{ {1-\tau _S \left( {t-d_S}\right) s_{aS} \left( {t-1}\right) } \right\} ^{N_{aS} \left( {t-d_S}\right) -R_S \left( t\right) } \\&\qquad \left. \times \left( {{ \begin{array}{l} {N_{aS} \left( {t-1}\right) } \\ {S_S \left( t\right) } \\ \end{array}}}\right) \left\{ {s_{aS} \left( {t-1}\right) } \right\} ^{S_S \left( t\right) }\left\{ {1-s_{aS} \left( {t-1}\right) } \right\} ^{N_{aS} \left( {t-1}\right) -S_S \left( t\right) } \right] . \end{aligned}$$
\(L_N^S \) is not a true likelihood strictly speaking (it does not involve the observed data) but rather a description of the unobserved underlying population changes.
For murres, we have direct information regarding immature survival (MRR(C) dataset) so we incorporate immature survival and fidelity parameters defined in Sect. 2.4 into the population model. New recruits \({\varvec{R_M}} =\left\{ {R_M \left( t\right) :t=7,\ldots , T} \right\} \), surviving adult females \({\varvec{S_M}} =\left\{ {S_M \left( t \right) :t=7,\ldots , T} \right\} \) and adult breeding females \({\varvec{N_{aM}}} =\left\{ {N_{aM} \left( t\right) : t=7,\ldots , T} \right\} \) can be modelled as
$$\begin{aligned}&R_M \left( t\right) \sim \hbox {bin}\left( {N_{aM} \left( {t-6}\right) , B\left( {t-6}\right) \rho _M \left( {t-6}\right) \frac{1}{2}s_1 \left( {t-6}\right) s_2 s_{35}^3 F_5 F_6 s_{aM} \left( {t-1}\right) }\right) ,\\&S_M \left( t\right) \sim \hbox {bin}\left( {N_{aM} \left( {t-1}\right) , s_{aM} \left( {t-1}\right) }\right) ,\\&N_{aM} \left( t\right) =R_M \left( t\right) +S_M \left( t\right) . \end{aligned}$$
Letting \(B\left( {t-6}\right) \rho _M \left( {t-6}\right) \frac{1}{2}s_1 \left( {t-6}\right) s_2 s_{35}^3 F_5 F_6 =\tau _M \left( {t-6}\right) \), the system process model is
$$\begin{aligned}&L_N^M \left( {{\varvec{R_M, S_M}} |{\varvec{s_M, F_M, s_{aM}, \rho _M}}}\right) \\&\quad =\mathop {\prod }\limits _{t=7}^T \left[ {\left( {{ \begin{array}{l} {N_{aM} \left( {t-6}\right) } \\ {R_M \left( t\right) } \\ \end{array}}}\right) \left\{ {\tau _M \left( {t-6}\right) s_{aM} \left( {t-1}\right) } \right\} ^{R_M \left( t\right) }} \right. \\&\qquad \times \left\{ {1-\tau _M \left( {t-6}\right) s_{aM} \left( {t-1}\right) } \right\} ^{N_{aM} \left( {t-6}\right) -R_M \left( t\right) } \\&\qquad \left. {\times \left( {{ \begin{array}{l} {N_{aM} \left( {t-1}\right) } \\ {S_M \left( t\right) } \\ \end{array}}}\right) \left\{ {s_{aM} \left( {t-1}\right) } \right\} ^{S_M \left( t\right) }\left\{ {1-s_{aM} \left( {t-1}\right) } \right\} ^{N_{aM} \left( {t-1}\right) -S_M \left( t\right) }} \right] . \end{aligned}$$
Breeding Population Counts: Observation Model
An observation model relates an imperfect observation of abundance (counts) to the true state of the system: the true abundance of breeding females \(N_{aS} \left( t\right) \). Island-wide population counts \(x_S \left( t\right) \) of adult breeding pairs (and hence females) have been conducted annually for murres and razorbills and less frequently for puffins. We model these counts with a normally distributed observation error \(x_S \left( t\right) \sim \hbox {N}\left( {N_{aS} \left( t\right) ,\sigma _{xS}^2}\right) \), for \(t=d_s +1,{\ldots },T\). This assumes that false negatives are approximately as likely as false positives. Counts for the first \(d_S \) years cannot be related to \(N_{aS}\) abundance modelled as a function of parameters and immature abundance since there is no direct source of information about the abundance of younger age classes for the first \(d_S \) years. We use these to initialize the population model for that period by setting informative normal priors for the adult population, assuming same variance as for observation error: \(N_{aS} \left( t\right) \sim \hbox {N}\left( {x_S \left( t\right) , \sigma _{xS}^2}\right) \), for \(t=1,{\ldots },d_s\). Letting \({\varvec{x_S}} =\left\{ {{{x_S}} \left( t\right) :t=d_S +1,\ldots , T} \right\} \), the observation process likelihood is
$$\begin{aligned} L_{\mathrm{OBS}}^S \left( {{\varvec{x_S}} |{\varvec{N}}_{aS}, \sigma _{xS}^2}\right) =\mathop {\prod }\limits _{t=d+1}^T \left[ {\frac{1}{\sigma _{xS} \sqrt{2\pi }}\cdot \exp \left( {-\frac{\left\{ {x_S \left( t\right) -N_{aS} \left( t \right) } \right\} ^{2}}{2\sigma _{xS}^2}}\right) } \right] . \end{aligned}$$
Puffin counts are only available for 7 non-consecutive years: 1984 and 1989 are used as initialization priors (with missing years interpolated linearly) and the model is fitted to counts from 1992, 1998, 2003, 2008 and 2009, but is able to estimate adult population for all years.
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.
Joint Likelihood: ssIPMs
Assuming independence between the different datasets involved, the joint likelihood of each single-species IPM can be found by multiplying the likelihoods of the different components:
$$\begin{aligned}&L_{\mathrm{IPM}}^S \left( {{\varvec{x_S, m_S, P_S}} |{\varvec{R_S, S_S}}, \phi _{cS}, {\varvec{s_{aS}}}, {\varvec{p}}_{\varvec{S}}^*, a_S, {\varvec{\rho _S}}}\right) \\&\quad =L_{\mathrm{POP}}^S \left( {{\varvec{x_S}} |{\varvec{R}}_S, {\varvec{S_S}}, \phi _{cS}, {\varvec{s_{aS}}}, {\varvec{\rho _S}}, \sigma _{xS}^2}\right) \times L_{\mathrm{MR}\left( \mathrm{A}\right) }^S \left( {{\varvec{m_S}} |{\varvec{s_{aS}}}, {\varvec{p}}_{\varvec{S}}^*, a_S}\right) \times L_{\mathrm{BS}}^S \left( {{\varvec{P_S}} |{\varvec{\rho _S}}}\right) . \end{aligned}$$
For murres, this joint likelihood contains also the components related to the extra datasets:
$$\begin{aligned}&L_{\mathrm{IPM}}^M \left( {{\varvec{x_M, m_M, n_M, d_M, v_M, P_M, \xi _M}} |{\varvec{N_M, s_{aM}, \rho _M}}, \sigma _{xM}^2, {\varvec{p}}_{\varvec{M}}^*, a_{\mathrm{M}}, {\varvec{s_M, p_M^C}}, \alpha _0, \alpha _1, {\varvec{F_M}}, \psi , {\varvec{B_M}}}\right) \\&\quad =L_{\mathrm{POP}}^M \left( {{\varvec{x_M}} |{\varvec{R_M, S_M, s_M, F_M, s_{aM}, \rho _M}}, \sigma _{xM}^2}\right) \times L_{\mathrm{MR}\left( \mathrm{A}\right) }^M \left( {{\varvec{m_M}} |{\varvec{s_{aM}}}, {\varvec{p}}_{\varvec{M}}^*, a_M}\right) \\&\qquad \times L_{\mathrm{MRR}\left( \mathrm{C}\right) }^M \left( {{\varvec{n_M, d_M, v_M}} |{\varvec{s_{iM}, s_{aM}, p_M^C}}, \alpha _0, \alpha _1, {\varvec{F_M}}, \psi }\right) \times L_{\mathrm{BS}}^M \left( {{\varvec{P_M}} |{\varvec{\rho _M}}}\right) \times L_{\mathrm{NB}}^M \left( {{\varvec{\xi _M}} |{\varvec{B_M}}}\right) . \end{aligned}$$
We use different adult resight probabilities for murres marked as chicks (\({\varvec{p_M^C}})\) than for those marked as adults (\({\varvec{p_M}}\), derived from \({\varvec{p}}_{\varvec{M}}^*\) and \({\varvec{a_M}})\), as high resight probabilities are expected for the latter (more resight effort and highly likely to return to the same breeding spot where banded). Adult survival \({\varvec{s_{aM}}}\) is a common parameter for the adult MR likelihood \(L_{MR\left( A\right) }^M\) and the chick MRR likelihood \(L_{\mathrm{MRR}\left( \mathrm{C}\right) }^M\). Table 1 summarizes the shared parameters.
Table 1 List of parameters involved in the msIPM, specifying in which model component for which species they appear.