Persistence versus extinction for a class of discrete-time structured population models

Wen Jin, Hal Smith, Horst Thieme

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


We provide sharp conditions distinguishing persistence and extinction for a class of discrete-time dynamical systems on the positive cone of an ordered Banach space generated by a map which is the sum of a positive linear contraction A and a nonlinear perturbation G that is compact and differentiable at zero in the direction of the cone. Such maps arise as year-to-year projections of population age, stage, or size-structure distributions in population biology where typically A has to do with survival and individual development and G captures the effects of reproduction. The threshold distinguishing persistence and extinction is the principal eigenvalue of (Formula presented.) provided by the Krein-Rutman Theorem, and persistence is described in terms of associated eigenfunctionals. Our results significantly extend earlier persistence results of the last two authors which required more restrictive conditions on G. They are illustrated by application of the results to a plant model with a seed bank.

Original languageEnglish (US)
Pages (from-to)821-850
Number of pages30
JournalJournal Of Mathematical Biology
Issue number4
StatePublished - Mar 1 2016


  • Basic reproduction number
  • Basic turnover number
  • Eigenfunctional
  • Krein-Rutman theorem
  • Net reproductive number
  • Persistence threshold
  • Plant population
  • Seed bank
  • Stability

ASJC Scopus subject areas

  • Modeling and Simulation
  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics


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