Sensitivity analysis

To understand what stages in the life history of the harriers have the greatest impact on the dynamics of the population, we analyse the elasticities associated with the transition matrix \(\bar{\mathbf{A}}\) \citep{Kroon00,Caswell2001}. The elasticities matrix \(\mathbf{E}\) has the same dimensions as \(\bar{\mathbf{A}}\) and entries \(e_{i,j}\), such that
\(\begin{equation} e_{i,j} = \frac{a_{i,j}}{\lambda}\frac{v_i w_j}{\langle\mathbf{v},\mathbf{w}\rangle} \end{equation} \)
where \(a_{i,j}\)is the entry of row \(i\) and column \(j\) in \(\bar{\mathbf{A}}\), and \(\mathbf{v}\) and \(\mathbf{w}\) are the left and right eigenvectors of \(\bar{\mathbf{A}}\), respectively. The entries in the elasticity matrix represent the proportional change in \(\zeta\) as a response to a small proportional change in the entries of \(\bar{\mathbf{A}}\). Thus, elasticities may be interpreted as proportional partial derivatives with respect to changes in different life history stages of the harriers.