A note on global stability of three-dimensional Ricker models

Research output: Contribution to journalArticleScientificpeer-review

Abstract

In the recent paper [E. C. Balreira, S. Elaydi, and R. Luis, J. Differ. Equ. Appl. 23 (2017), pp. 2037-2071], Balreira, Elaydi and Luis established a good criterion for competitive mappings to have a globally asymptotically stable interior fixed point by a geometric approach. This criterion can be applied to three dimensional Kolmogorov competitive mappings on a monotone region with a carrying simplex whose planar fixed points are saddles but globally asymptotically stable on their positive coordinate planes. For three dimensional Ricker models, they found mild conditions on parameters such that the criterion can be applied to. Observing that Balreira, Elaydi and Luis' discussion is still valid for the monotone region with piecewise smooth boundary, we prove in this note that the interior fixed point of three dimensional Kolmogorov competitive mappings is globally asymptotically stable if they admit a carrying simplex and three planar fixed points which are saddles but globally asymptotically stable on their positive coordinate planes. This result is much easier to apply in the application.
Original languageEnglish
JournalJournal of Difference Equations and Applications
Volume25
Issue number1
Pages (from-to)142-150
Number of pages9
ISSN1023-6198
DOIs
Publication statusPublished - 2 Jan 2019
MoE publication typeA1 Journal article-refereed

Fields of Science

  • 111 Mathematics
  • Global stability
  • Ricker model
  • carrying simplex
  • global dynamics
  • phase portrait
  • 37Cxx
  • CARRYING SIMPLEX
  • DYNAMICS
  • CLASSIFICATION
  • BOUNDARY
  • Global stability
  • Ricker model
  • carrying simplex
  • global dynamics
  • phase portrait
  • 37Cxx
  • CARRYING SIMPLEX
  • DYNAMICS
  • CLASSIFICATION
  • BOUNDARY

Cite this

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title = "A note on global stability of three-dimensional Ricker models",
abstract = "In the recent paper [E. C. Balreira, S. Elaydi, and R. Luis, J. Differ. Equ. Appl. 23 (2017), pp. 2037-2071], Balreira, Elaydi and Luis established a good criterion for competitive mappings to have a globally asymptotically stable interior fixed point by a geometric approach. This criterion can be applied to three dimensional Kolmogorov competitive mappings on a monotone region with a carrying simplex whose planar fixed points are saddles but globally asymptotically stable on their positive coordinate planes. For three dimensional Ricker models, they found mild conditions on parameters such that the criterion can be applied to. Observing that Balreira, Elaydi and Luis' discussion is still valid for the monotone region with piecewise smooth boundary, we prove in this note that the interior fixed point of three dimensional Kolmogorov competitive mappings is globally asymptotically stable if they admit a carrying simplex and three planar fixed points which are saddles but globally asymptotically stable on their positive coordinate planes. This result is much easier to apply in the application.",
keywords = "111 Mathematics, Global stability, Ricker model, carrying simplex, global dynamics, phase portrait, 37Cxx, CARRYING SIMPLEX, DYNAMICS, CLASSIFICATION, BOUNDARY, Global stability, Ricker model, carrying simplex, global dynamics, phase portrait, 37Cxx, CARRYING SIMPLEX, DYNAMICS, CLASSIFICATION, BOUNDARY",
author = "Mats Gyllenberg and Jifa Jiang and Lei Niu",
year = "2019",
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doi = "10.1080/10236198.2019.1566459",
language = "English",
volume = "25",
pages = "142--150",
journal = "Journal of Difference Equations and Applications",
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A note on global stability of three-dimensional Ricker models. / Gyllenberg, Mats; Jiang, Jifa; Niu, Lei.

In: Journal of Difference Equations and Applications, Vol. 25, No. 1, 02.01.2019, p. 142-150.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

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N2 - In the recent paper [E. C. Balreira, S. Elaydi, and R. Luis, J. Differ. Equ. Appl. 23 (2017), pp. 2037-2071], Balreira, Elaydi and Luis established a good criterion for competitive mappings to have a globally asymptotically stable interior fixed point by a geometric approach. This criterion can be applied to three dimensional Kolmogorov competitive mappings on a monotone region with a carrying simplex whose planar fixed points are saddles but globally asymptotically stable on their positive coordinate planes. For three dimensional Ricker models, they found mild conditions on parameters such that the criterion can be applied to. Observing that Balreira, Elaydi and Luis' discussion is still valid for the monotone region with piecewise smooth boundary, we prove in this note that the interior fixed point of three dimensional Kolmogorov competitive mappings is globally asymptotically stable if they admit a carrying simplex and three planar fixed points which are saddles but globally asymptotically stable on their positive coordinate planes. This result is much easier to apply in the application.

AB - In the recent paper [E. C. Balreira, S. Elaydi, and R. Luis, J. Differ. Equ. Appl. 23 (2017), pp. 2037-2071], Balreira, Elaydi and Luis established a good criterion for competitive mappings to have a globally asymptotically stable interior fixed point by a geometric approach. This criterion can be applied to three dimensional Kolmogorov competitive mappings on a monotone region with a carrying simplex whose planar fixed points are saddles but globally asymptotically stable on their positive coordinate planes. For three dimensional Ricker models, they found mild conditions on parameters such that the criterion can be applied to. Observing that Balreira, Elaydi and Luis' discussion is still valid for the monotone region with piecewise smooth boundary, we prove in this note that the interior fixed point of three dimensional Kolmogorov competitive mappings is globally asymptotically stable if they admit a carrying simplex and three planar fixed points which are saddles but globally asymptotically stable on their positive coordinate planes. This result is much easier to apply in the application.

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