p = 1 – (1 – r)^N_i
where N_i is the number of individuals in site i.
We typically have several sites (i.e. pentads) with different number of individuals. The distribution of individuals across M sites may be given a probabilistic distribution such as a Poisson(lambda) (see Royle-Nichols for more details). The reporting rate of the species of interest (number of detections at site i/total number of visits to site i) may be modelled as a binomial process such that:
L(d_i) = Sum_k binomial(d_i | J_i, p_k) * Poisson(k | lambda) for k = 0, 1, 2,..K
Where d_i is the number of detections at site i, k is the (potential) number of individuals at site i, J_i is the total number of visits to site i, p_k is the probability of detecting at least one individual at site i, provided that there are k individuals present and lambda is the mean number of individuals across sites.
The likelihood of the detection history in all sites is then
L(d) = Pi_M L(d_i)
Building on this basic model we define a temporal extension, in which the mean number of individuals across site changes every year with a constant mean rate psi. We then model the reporting rate of the species in year t as we just explained:
L(d_it) = Sum_k binomial(d_it | J_it, p_k) * Poisson(k | lambda_t) for k = 0, 1, 2,..K
Note that the index t marks the year the data corresponds to and that lambda may now change between years. We then set a model for lambda such that
Lambda_t+1 = psi * lambda_t
Where psi corresponds to the mean rate of change of the population.
We further incorporate all we know about the life history of the Black Harrier to model psi.
**
To understand the effect of added mortality produced by wind farms on Black Harrier populations, we first defined a model for the population dynamics of the species, then we simulated population trajectories under different scenarios. Since wildlife populations are complex systems and sources of information are often quite limited \citep{Heppel200}, simulations need to incorporate, not only different levels of wind farm mortality, but also inter-annual variation and uncertainty in life history parameters. We used a Bayesian approach whereby we defined prior distributions that capture the information about life history parameters available to us (see \citealt{Allen_2018}). In absence of annual population counts, but with a rough estimate of population decline in the last two decades (see below), we ran prior predictive simulations to understand how life history parameters relate to population growth rate for the Black Harrier, and well as possible relationships between life history parameters. With this information at hand we can constrain simulations to parameter values that are consistent with what we know about Black Harrier populations. 
We began by defining a deterministic, age-structured model that helped us study basic properties of the system \cite{Caswell2001}. Then, we progressively incorporated different sources of uncertainty and variability to make a more realistic model \cite{Fieberg_2001,Newman2014}
To structure the life history of the Black Harrier, we defined three age classes fledglings  - individuals that have left the nest up to 1 year old , sub-adults for individuals that are  1-2 years old, and adults - for individuals 2 years and older. Adult individuals are the only birds capable of breeding. In addition to an age structure, we defined a survival rate, which is the probability of survival in a given year, and a fertility, which is the number of offspring per adult bird per year (note that fertility usually refers to offspring per female, but we have redefined this to simplify notation later).
We used the following conventions: i) population census are conducted post-breeding,  ii) years span the period from one breeding event to the next, iii) birds breed on their birthday once they reach the maturity, iv) at any time there is equal number of males and females, and v) we assume a closed, spatially invariant population. Each year harriers go through three phases: survive, age and reproduce, in this order. This means that a fledgling that survives the first year, becomes a sub-adult, and a surviving sub-adult, an adult. It also means that an sub-adult individual that survives, becomes and adult and is immediately capable of breeding. This is important to ensure that sub-adult individuals in a given year will be able to reproduce in the following year as adults (see figure \ref{316349} and  \citealt{Kendall_2019}).