Describe the model for the population dynamics.
We use a stochastic, age-structured population matrix model to represent the dynamics of the Black Harrier populations  \cite{Caswell2001,Fieberg_2001,Newman2014} . We use the following conventions: i) population census are conducted post-breeding,  ii) years span the period from one breeding event to the next, ii) birds breed on their birthday once they reach the maturity. We define four age classes: chick - for individuals that are zero years old, juvenile - for individuals that are one year old, immature - for individuals two years old, and adults - for individuals older than two years.
The first task of a bird in a year is to survive, which they do with a rate given by \(\phi_a\) . The subscript \(a\in\left\{1,2\right\}\) indicates whether the bird is still younger than one year (1) or older (2). Black Harriers have lower survival rates on their first year as pointed by WHO? Thus, \(\phi_a\) is a random variable with distribution \(beta\left(\alpha_a, \beta_a\right)\) and the distribution of \(\phi_1\)has \(mean_{\phi_1} = 0.5\) and the distribution of \(\phi_2\) has \(mean_{\phi_2} = 0.75\) different age classes in the population.
Only adult birds reproduce, and their fecundity (number of chicks per nest) is given by \(\rho\), a \(multinomial\) random variable that may take on the values \(\rho = \{0,1,2,3,4\}\), with probabilities \(P(\rho) = (0.25,0.25,0.2,0.2, 0.1)\).