Review of methods used to calculate scale of artificialnesting structures proposed as a compensation measure for Kittiwake mortality at offshore wind farms

Rhoades, J., Johnston, D.T., Humphreys, E.M. & Boersch-Supan, P.H.
BTO RESEARCH REPORT 788, British Trust for Ornithology 2025

Appendix 3: Computational details of the recommended approach for calculating compensation scale

We recommend that calculations of the size of the compensation population are based on a structured population model to ensure that selected combinations of demographic parameters are constrained to those that yield a net addition of recruits into the metapopulation, while at the same time falling into realistically achievable parameter ranges (cf. Appendix 2, Figure B). This mirrors approaches taken in other wind energy related contexts (e.g. Fielding & Haworth, 2010; Millsap et al., 2022). We note that the current evidence base for what constitutes a realistically achievable demographic parameter value is limited, given the scarcity of observational data from birds breeding on offshore structures in the Southern North Sea. Adaptive management actions and/or updated recommendations about suitable demographic parameter ranges may be required as such information becomes available. Here we use a population model that employs an age-structure following Horswill et al. (2022), which can be considered a simplified version of the age structure underpinning the Hornsea 3 approach. In addition, we explicitly model dispersal between the ANS and the metapopulation. The objective of the calculation is to find the minimum size of the breeding population required to produce a number of annual recruits into the breeding class that equals or exceeds the estimated collision mortality. We implemented the population model in the software package R (R Core Team, 2024) and calculated asymptotic properties using functions from the popbio package (Stubben & Milligan, 2007), as per Appendix 2. However, our results should be insensitive to software choices – within the limits of numerical precision - as population models are built around the same underlying mathematics. It is also important to note that population models are approximations of the biological dynamics and – as outlined in the main text - there is usually insufficient information on parameter values to justify anything other than relatively simple projection matrices. It is therefore more important to provide an unambiguous model description and parameter definitions to provide the transparency needed for others to repeat the models in same or different software, while allowing further tests of the model’s sensitivity to the particular parameter values adopted in the model.

The calculation proceeds in the following steps:

Step 1:

Select appropriate values for survival $\varphi_i$, productivity $F$, and recruitment probability $b$ (i.e. proportion of birds that begin breeding at a particular age class). Here we use the same survival and productivity rates as in the Hornsea 3 approach, and a recruitment probability of 25% for 3 year old birds and 75% for non-breeders in subsequent age classes (following Horswill et al., 2022).

Step 2:

Estimate maximum sustainable net dispersal rate $\delta$ which allows the asymptotic population growth rate $\lambda$ to remain equal or larger than 1. We used the eigen.analysis function in popbio for this purpose, as per Appendix 2.

Step 3:

Determine the stable age structure for the projection matrix using the maximized value for the net dispersal rate $\delta$. We used the eigen.analysis function in popbio for this purpose. The stable age distribution is a vector of proportions which we designate $p={p_1,p_2,p3,p_4}$ where $p_i$ designates the proportion of individuals in stage $i$ of the population.

Step 4:

The compensation population, i.e. the required breeding population size $P$ (in breeding pairs) to compensate a predetermined annual mortality $M$ (specified as individuals across both sexes) - effectively an annual harvest - is then calculated as $$P=\frac{M}{2(p_2 \varphi_a b \delta + p_3 \varphi_a (1-b) \delta)}p_4$$ Where the factor 2 accounts for the fact that the model considers female offspring only, $p_2\varphi_ab\delta$ describes the surviving individuals recruiting from $N_2$ into the dispersing breeder class, $p_3\varphi_a(1-b)\delta$ describes the surviving individuals recruiting from $N_3$ into the dispersing breeder class, and $p_4$ describes the proportion of breeders in the ANS population.