Open AccessAnalytical expressions for the eigenvalues, demographic quantities, and extinction criteria arising from a three‐stage wildlife population matrix
Author(s) -
Hanley Brenda,
Dennis Brian
Publication year - 2019
Publication title -
natural resource modeling
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.28
H-Index - 32
eISSN - 1939-7445
pISSN - 0890-8575
DOI - 10.1111/nrm.12207
Subject(s) - extinction (optical mineralogy) , eigenvalues and eigenvectors , population , noon , mathematics , matrix (chemical analysis) , wildlife , vital rates , stage (stratigraphy) , statistics , population growth , econometrics , ecology , biology , demography , physics , atmospheric sciences , paleontology , quantum mechanics , sociology , materials science , composite material
We 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 managersSuperparameters can be estimated to determine population growth rates. Superparameters can be used to conduct rapid assessments of extinction risk.