Parameterizing the growth-decline boundary for uncertain population projection models

Joan Lubben, Derek Boeckner, Richard Rebarber, Stuart Townley, Brigitte Tenhumberg

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


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 n (t) 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.

Original languageEnglish (US)
Pages (from-to)85-97
Number of pages13
JournalTheoretical Population Biology
Issue number2-3
StatePublished - Mar 2009


  • Asymptotic growth rate
  • Integral projection model
  • Population projection matrix
  • Robustness

ASJC Scopus subject areas

  • Ecology, Evolution, Behavior and Systematics


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