title: Persistence versus extinction for a class of discrete-time structured population models.
abstract: 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 (II−A)(−1)G'(0) 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.
authors.affiliation: School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA.
authors.forename: Wen
authors.initials: W
authors.lastname: Jin
authors.affiliation: School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA.
authors.forename: Hal L
authors.initials: HL
authors.lastname: Smith
authors.affiliation: School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA. thieme@math.asu.edu.
authors.forename: Horst R
authors.initials: HR
authors.lastname: Thieme
mesh_terms: D000818:Animals; D053476:Extinction, Biological; D005260:Female; D008297:Male; D055641:Mathematical Concepts; D008954:Models, Biological; D063245:Plant Development; D011157:Population Dynamics; D012098:Reproduction; D013997:Time Factors
keywords: Basic reproduction number; Basic turnover number; Eigenfunctional; Krein-Rutman theorem; Net reproductive number; Persistence threshold; Plant population; Seed bank; Stability