2.2 Matrix Model Projections
- For each species, an age-structured matrix model (Caswell, 2001) simulates the population’s progress through time in terms of abundance and age distribution, based on species-specific demographic rates and count estimates. The model assumes individuals to be grouped into discrete year age-classes, and all members of an age-class are considered equal with respect to their demographic vital rates (i.e., survival, growth and reproduction). The model dynamics involves predicting the population numbers at age in the next year given its previous year’s numbers and vital rates.
- The generic population model can be written in compact form as:
- where
is the population vector with elements 
denoting the number of individuals at each age-class 
at year 
, 
is the numbers at age-class in the following year, and 
represents the 
projection matrix (also known as the Leslie matrix). The projection matrix defines the expected contribution of individuals in each age-class in a given year to each age-class in the subsequent year. - Models used in this analysis were built under the following assumptions, for all considered species:
- Models represent an annual post-breeding census over a period of
year steps. Therefore, the model annual cycle comprises a census immediately after fledging on the first day of the biological year, with the first age-class (
) containing newly hatched birds, followed by a 12-month period of survival. Then, on the first day of the subsequent year, surviving animals increment in age, adult age-classes reproduce and resultant chicks fledge, and the next census is carried out. - Reproduction is considered to be confined to adult birds, with age of first breeding being species-specific.
- Population size is density independent, and therefore projections will either increase to infinity or decrease to extinction, given sufficient time, which is biological unrealistic.
- Density dependent regulation in seabirds is through two mechanisms, either compensatory or depensatory. Adopting a density independent model will over- and underestimate potential impacts if compensatory or depensatory processes are occurring, respectively (Horswill and Robinson 2015). Currently there are insufficient data to include density dependence into the population modelling framework since its effect within populations is a complex interaction between resource availability, colony size and other local factors, such as predation (Horswill and Robinson 2015).
- Population is considered a closed system, i.e., age distributions are not affected by migration exchanges between neighbouring colonies.
- The final age-class
is an aggregated age group, representing years-old birds and older. This implies the absence of senescence, i.e., the survival and reproductive performances of the oldest individuals remain constant over time. The value of , and hence the size of the projection matrix, of each species is determined by either the age of first breeding or the oldest adult age-class for which survival data was available (the largest of the two values).
- Based on the above assumptions, the expanded version of the generic population model used in this analysis can be expressed as:
where 
denotes the annual productivity rate of age-class , expressed as the annual average number of fledged young per breeding pair; and 
represents the annual survival transition rate of animals of age-class 
, i.e. the average proportion of birds in age-class that will survive the whole year and transition to age-class 
. Elements in the top row of the projection matrix (i.e. half of the productivity rate multiplied by the survival rate) reflect the annual fecundity rate per capita of each adult age-class. The population age ratios for each species from Forth Islands SPA is shown in Table 2.4.
- Stochasticity can be simply added by Monte-Carlo simulation if input parameters consist of estimates and associated measures of uncertainty. Variability can be provided by repeated simulations of population trajectories, where parameter values fluctuate as random draws from appropriate distributions.
Table 2.4: Population age ratios for species at the Forth Islands SPA and modelled using PVA
Species | 0-1 | 1-2 | 2-3 | 3-4 | 4-5 | 5-6 | Adult |
Gannet | 0.184 | 0.096 | 0.074 | 0.061 | 0.049 | - | 0.536 |
Guillemot | 0.17 | 0.092 | 0.074 | 0.06 | 0.058 | 0.052 | 0.494 |
Herring gull | 0.186 | 0.138 | 0.118 | 0.1 | 0.08 | - | 0.378 |
Kittiwake | 0.184 | 0.104 | 0.093 | 0.079 | - | - | 0.54 |
Lesser black-backed gull | 0.199 | 0.1 | 0.089 | 0.079 | 0.067 | - | 0.466 |
Puffin | 0.145 | 0.128 | 0.115 | 0.099 | 0.072 | - | 0.442 |
Razorbill | 0.148 | 0.109 | 0.089 | 0.08 | 0.066 | - | 0.508 |
2.3 Model Parameterisation
- Input demographic parameters use SPA-specific estimates when available. In cases where local estimates were unavailable, preference was given to broader scale estimates based on combined independent studies collated in Horswill and Robinson (2015), as advised in NatureScot’s Scoping advice (7th December 2021). In the absence of local estimates, combined regional and national level estimates are believed to generate parameter values that express more accurately the underlying degree of uncertainty in model simulations. Parameters for the models are given in sections 2.7, along with sources.
- The colony counts for each of the SPAs were provided from JNCC as two validated datasheets of all colony count data for the UK and Ireland within the Seabird Monitoring Programme (SMP) database for 1998 – 2019 and 1998-2021 to HiDef on 10th January 2022. For the species of interest here, the database summarised counts by subsites and whole SPAs; “counts” are recorded as individuals or Apparently Occupied Nests (AON) or Apparently Occupied Sites (AOS). For guillemot and razorbill, counts of individuals were converted to estimates of the population size by multiplying by the correction factor 1.34. Where counts were provided as AON/AOS, these were treated as equivalent to pairs and therefore doubled to arrive at a number of individuals and then multiplied by the correction factor of 1.34, which is an Isle of May specific correction factor but which has been applied in previous applications to guillemot and razorbill. The colony population sizes used in the models, and the year of the counts from which they are derived, are given in Table 2.6. Ideally, counts should be concurrent across breeding colonies of interest. However, for many SPAs, counts are divided by subsite and not all subsites are censused every year. Entire counts for SPAs comprising multiple subsites are often only achieved over a period of years.
2.4 Simulation Parameterisation
- Models were run from starting year to 50 years post-impact (assumed to be 2027), simulated 5,000 times to obtain indicative population trends and estimates of uncertainty surrounding those trends. Outputs from the models were extracted at both 35-year and 50-year post impact time points to reflect the lifetime of the Project (35 years) and beyond. Models were run for each species/SPA combination separately taking the associated adult population size estimate as a starting condition (Table 2.6).
- The starting year for simulations of each species/SPA combination corresponds to the most recent year of their population size estimate and is assumed to be without error in the simulations. Specific mortality scenarios are presented, with differential impacts for adults and immature age-classes for species with immature-specific additional deaths. These specific mortality scenarios are adjusted for sabbatical birds (i.e., fraction of adults skipping breeding in a given year) a priori.
- Additionally, a range of absolute additional adult mortalities per annum, from 0 to a species-specific maximum value by incremental steps (section 2.8), were used as impact scenarios. Although impacts are only reported with respect to the adult numbers, impacts within the simulations were also applied proportionally to immature age-classes classes (based upon the stable age distribution from eigen-decomposition of the Leslie matrix. No density dependence or demographic stochasticity (as defined by the nepva tool) was assumed, whereas environmental stochasticity (as per the nepva) was specified via beta distributions on survival and productivity rates. Species-specific maximum brood sizes were assumed.
- Impacted vs unimpacted comparisons were based on a matched runs approach, whereby stochasticity is applied to the population before impacts are applied, i.e., survival and productivity rates simulated at each time step are the same for the unimpacted and impacted populations, before additional impact mortalities are deducted from simulated survivals for the impacted populations. Productivity rates were assumed to be unaffected by wind farm effects.
2.5 Regional Projections
- Calculations were made on a regional basis for seven species. Regions were defined as consisting of the SPAs within Table 2.5 i.e within breeding season foraging range (mean max distance +-1sd; Woodward et al. 2019).
- For each of these SPAs, the specific mortality scenarios used within each of the individual species PVAs were assumed. Regional estimates are in essence a sum of projected population sizes, at each timepoint, for each of the constituent SPAs.
- In detail 5000 simulated population projections were run for each species, SPA and impact scenario. These were summed over SPAs for each projection year, within each species and impact scenario. This provides 5000 regional population simulations for each species and impact scenario. The summary statistics and counterfactuals are calculated thereafter in the standard fashion.
- For SPAs not deemed to be impacted under a regional scenario, but who are included in the regional population definition (not all scenarios are deemed to impact the same set of SPAs), are included in the regional population through their baseline/unimpacted population traces. All individual SPA projections are treated as independent.
Table 2.5: Special Protection Areas (SPAs) included in the regional assessment
Species | Site |
|---|---|
Gannet | Forth Islands SPA |
Flamborough and Filey Coast | |
North Rona and Sula Sgeir | |
Fair Isle | |
Noss | |
Hermaness, Saxa Vord and Valla Field | |
Guillemot | Forth Islands SPA |
St Abb's Head to Fast Castle SPA | |
Fowlsheugh SPA | |
Farne Islands SPA | |
Buchan Ness to Collieston Coast | |
Troup, Pennan and Lion’s Heads | |
East Caithness cliffs | |
Kittiwake | Forth Islands SPA |
Fowlsheugh SPA | |
St Abb's Head to Fast Castle SPA | |
Farne Islands SPA | |
Buchan Ness to Collieston Coast | |
Troup, Pennan & Lions Head | |
East Caithness cliffs | |
North Caithness cliffs | |
Coquet island | |
Flamborough and Filey Coast | |
Puffin | Forth Islands SPA |
Farne Islands | |
North Caithness cliffs | |
Razorbill | Forth Islands SPA |
St Abb's Head to Fast Castle SPA | |
Fowlsheugh SPA | |
Troup, Pennan and Lion’s Heads | |
Farne Islands | |
East Caithness cliffs | |
Flamborough and Filey Coast | |
Herring gull | Forth Islands SPA |
Fowlsheugh SPA | |
Farne Islands SPA | |
St Abb’s Head to Fast Castle SPA | |
Lesser-Black Back gull | Forth Islands SPA |
Farne Islands SPA | |
Coquet Island |
2.6 Starting Population Sizes
Table 2.6: Population counts for SPAs in terms of numbers of breeding adults
SPA | Population Size (Breeding Adults) | Year(s) of Counts | |
|---|---|---|---|
Gannet | Forth Islands | 150,518 | 2014 |
Flamborough and Filey Coast | 26,784 | 2017 | |
North Rona and Sula Sgeir | 22,460 | 2013 | |
Fair Isle | 9,942 | 2021 | |
Noss | 27,530 | 2019 | |
Hermaness, Saxa Vord and Valla Field | 51,160 | 2014 | |
Guillemot | Forth Islands | 34,580 | 2018 + 2021 |
Farne Islands | 85,816 | 2019 | |
St Abb's Head to Fast Castle | 61,408 | 2016 - 2018 | |
Fowlsheugh | 91,358 | 2018 | |
Buchan Ness to Collieston Coast | 39,553 | 2019 | |
Troup, Pennan and Lion's Heads | 31,893 | 2017 | |
Herring gull | Forth Islands | 11,868 | 2019 - 2021 |
Farne Islands | 1,496 | 2019 | |
St Abb's Head to Fast Castle | 612 | 2016 - 2020 | |
Fowlsheugh | 1,414 | 2018 | |
Kittiwake | Forth Islands | 9,034 | 2021 |
Farne Islands | 8,804 | 2019 | |
Flamborough and Filey Coast | 91,008 | 2017 | |
Coquet Island | 932 | 2021 | |
St Abb's Head to Fast Castle | 10,904 | 2020 | |
Fowlsheugh | 26,542 | 2018 | |
Buchan Ness to Collieston Coast | 22,590 | 2019 | |
Troup, Pennan and Lion's Heads | 21,232 | 2017 | |
East Caithness Cliffs | 48,920 | 2015 | |
North Caithness Cliffs | 7,712 | 2015/2016 | |
West Westray | 5,486 | 2017 | |
Lesser-Black Back gull | Forth Islands | 4,006 | 2018 - 2021 |
Farne Islands | 1,362 | 2019 | |
Coquet Island | 40 | 2019 | |
Puffin | Forth Islands | 87,240 | 2017 - 2021 |
Farne Islands | 87,504 | 2019 | |
North Caithness Cliffs | 3,034 | 2019 | |
Razorbill | Forth Islands | 7,878 | 2015/2016 |
Farne Islands | 572 | 2017 | |
Flamborough and Filey Coast | 37,476 | 2015-2018 | |
St Abb's Head to Fast Castle | 3,928 | 2017 - 2021 | |
Fowlsheugh | 17,817 | 2016 - 2018 | |
Troup, Pennan and Lion's Heads | 6,054 | 2018 | |
East Caithness Cliffs | 40,117 | 2019 |
*‘These data were extracted from the Seabird Monitoring Programme Database (https://app.bto.org/seabirds). Data have been provided to the SMP by the generous contributions of nature conservation and research organisations, and many volunteers throughout Britain and Ireland.’ There is no requirement for individuals or organisations to acknowledge the SMP when they use data they have collected themselves.