Analytical expressions for the eigenvalues, demographic quantities, and extinction criteria arising from a three‐stage wildlife population matrix

Abstract

AbstractWe used the symbolic solution for the roots of a cubic polynomial to derive expressions for the eigenvalues of a three‐stage population projection matrix. As well, we obtained expressions for eigenvectors, moduli, damping ratios, sensitivities, and elasticities. The equations reveal the existence of “superparameters;” natural groupings of vital rates that drive population dynamics. We show that growth rates can be calculated using (at most) three superparameters in place of as many as nine original vital rates, potentially simplifying data collection. Necessary and sufficient conditions for extinction can be summarized equivalently by three superparameter inequalities. For a common life history (Noon & Biles, 1990, J Wildl Manag, 54, 18–27) four vital rate parameters are reduced to two superparameters. The results are applicable to population viability and recovery analysis and harvest planning.Two recommendations for resource managers Superparameters can be estimated to determine population growth rates. Superparameters can be used to conduct rapid assessments of extinction risk.