Pre-images of invariant sets of a discrete-time two-species competition model

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Abstract

In this paper, we explore the structure of pre-images of invariant sets of a discrete-time two-species competition model with singularity at the origin. We first show that this competition model is persistent with respect to the total population of two species, i.e. all initial conditions in ℝ 2/{(0,0)} are attracted to a compact set which is bounded away from the origin. Then we study the properties of pre-images of this system and give the explicit structure of all pre-images of invariant sets for this system under certain parameter range. These results are analogous to the one-dimensional discrete system. Our study is the first step to explore the structure of the basins of attractions of interior attractors of a general discrete-time two-species model (e.g. the locally asymptotically stable interior period-2 orbit). Finally, we discuss how our results give useful insights on the future study for coexistence of the species and list some open problems related to our system.

Original languageEnglish (US)
Pages (from-to)1709-1733
Number of pages25
JournalJournal of Difference Equations and Applications
Volume18
Issue number10
DOIs
StatePublished - Oct 1 2012

Keywords

  • basins of attraction
  • competition models
  • critical curves
  • invariant sets
  • persistent
  • rank-k pre-images

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

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