The population parameters \(\rho\) and \(\phi\) appear with a subscript \(t\) in addition to the subscript associated with the different age classes, to represent the fact that they are random parameters that may change in different years. The transition matrix \(\mathbf{A}_t\) imposes population dynamics represented in figure \ref{316349}. The rate at which the population changes from time \(t\) to time \(t+1\) is given by the main eigenvalue of the matrix \(\mathbf{A}_t\), which we denote by \(\lambda\) \cite{Caswell2001}. Because the population parameters are random, the population will change at different rates in different years. Thus, we define the expected population rate of change \(\text{E}(\lambda) = \tilde{\lambda}\) as the rate of change of the population when survival and fecundity are set to their expected values \(\text{E}(\phi_a)\) and \(\text{E}(\rho)\). The transition matrix resulting from plugging expected values of survival and fecundity in matrix \(\mathbf{A}_t\) is denoted by \(\tilde{\mathbf{A}}_t\).
The variation in population rate of change across different years is estimated by HOW?
In addition, to understand what stages of the life cycle of the harriers have the greatest impact on the dynamics of the population, we analyse the elasticities associated with the transition matrix \(\tilde{\mathbf{A}}_t\) \cite{Kroon00,Caswell2001}. The elasticities matrix \(\mathbf{E}\) have the same dimensions as \(\tilde{\mathbf{A}}_t\) and entries \(e_{i,j}\), such that