Persistence and global stability for a class of discrete time structured population models

Hal L. Smith; Horst R. Thieme

doi:10.3934/dcds.2013.33.4627

Article Contents

2013, Volume 33, Issue 10: 4627-4646. Doi: 10.3934/dcds.2013.33.4627 open access

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Persistence and global stability for a class of discrete time structured population models

Hal L. Smith^1, and
Horst R. Thieme^2,

1.
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804
2.
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287

Received: August 31, 2012

Revised: December 31, 2012

Published: September 2013

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Abstract

We obtain sharp conditions distinguishing extinction from persistence and provide sufficient conditions for global stability of a positive fixed point for a class of discrete time dynamical systems on the positive cone of an ordered Banach space generated by a map which is, roughly speaking, a nonlinear, rank one perturbation of a linear contraction. Such maps were considered by Rebarber, Tenhumberg, and Towney (Theor. Pop. Biol. 81, 2012) as abstractions of a restricted class of density dependent integral population projection models modeling plant population dynamics. Significant improvements of their results are provided.

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Mathematics Subject Classification: Primary: 37N25, 92D25; Secondary: 37B25.

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