Chaotic attractors and physical measures for some density dependent Leslie population models

Ilie Ugarcovici, Howard Weiss

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7 Scopus citations

Abstract

Following ecologists' discoveries, mathematicians have begun studying extensions of the ubiquitous age structured Leslie population model that allow some survival probabilities and/or fertility rates to depend on population densities. These nonlinear extensions commonly exhibit very complicated dynamics: through computer studies, some authors have discovered robust Hénon-like strange attractors in several families. Population biologists and demographers frequently wish to average a function over many generations and conclude that the average is independent of the initial population distribution. This type of 'ergodicity' seems to be a fundamental tenet in population biology. In this paper we develop the first rigorous ergodic theoretic framework for density dependent Leslie population models. We study two generation models with Ricker and Hassell (recruitment type) fertility terms. We prove that for some parameter regions these models admit a chaotic (ergodic) attractor which supports a unique physical probability measure. This physical measure, having full Lebesgue measure basin, satisfies in the strongest possible sense the population biologist's requirement for ergodicity in their population models. We use the celebrated work of Wang and Young 2001 Commun. Math. Phys. 218 1-97, and our results are the first applications of their method to biology, ecology or demography.

Original languageEnglish (US)
Pages (from-to)2897-2906
Number of pages10
JournalNonlinearity
Volume20
Issue number12
DOIs
StatePublished - Dec 1 2007

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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