Parameterizing the growth-decline boundary for uncertain population projection models

Authors

Joan Lubben, University of Nebraska-LincolnFollow
Derek Boeckner, University of Nebraska - Lincoln
Richard Rebarber, University of Nebraska - LincolnFollow
Stuart Townley, University of Exeter, UK
Brigitte Tenhumberg, University of Nebraska - LincolnFollow

Document Type

Article

Date of this Version

3-2009

Comments

Published in Theoretical Population Biology 75:2–3 (March-May 2009), pp. 85–97; doi: 10.1016/j.tpb.2008.11.004 Copyright © 2008 Elsevier Inc. Used by permission. http://0-www.sciencedirect.com/science/journal/00405809

Abstract

We consider discrete time linear population models of the form n(t + 1) = An(t) where A is a population projection matrix or integral projection operator, and represents a structured population at time t. It is well known that the asymptotic growth or decay rate of n(t) is determined by the leading eigenvalue of A. In practice, population models have substantial parameter uncertainty, and it might be difficult to quantify the effect of this uncertainty on the leading eigenvalue. For a large class of matrices and integral operators A, we give sufficient conditions for an eigenvalue to be the leading eigenvalue. By preselecting the leading eigenvalue to be equal to 1, this allows us to easily identify, which combination of parameters, within the confines of their uncertainty, lead to asymptotic growth, and which lead to asymptotic decay. We then apply these results to the analysis of uncertainty in both a matrix model and an integral model for a population of thistles. We show these results can be generalized to any preselected leading eigenvalue.