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A New Formula to Get Sharp Global Stability Criteria for One-Dimensional Discrete-Time Models

  • Eduardo Liz
  • Sebastián Buedo-Fernández
Article
  • 12 Downloads

Abstract

We present a new formula that makes it possible to get sharp global stability results for one-dimensional discrete-time models in an easy way. In particular, it allows to show that the local asymptotic stability of a positive equilibrium implies its global asymptotic stability for a new family of difference equations that finds many applications in population dynamics, economic models, and also in physiological processes governed by delay differential equations. The main ingredients to prove our results are the Schwarzian derivative and some dominance arguments.

Keywords

Global stability Discrete-time model Mackey–Glass equation Gamma-model Schwarzian derivative 

Mathematics Subject Classification

Primary 39A10 39A30 Secondary 34K20 

Notes

Acknowledgements

The authors sincerely thank Víctor Jiménez López (Universidad de Murcia, Spain) and Ábel Garab (Alpen-Adria-Universität Klagenfurt, Austria) for useful discussions, encouraging comments and relevant remarks, and an anonymous reviewer for his/her helpful comments. Eduardo Liz acknowledges the support of the research Grant MTM2017-85054-C2-1-P (AEI/FEDER, UE). The research of Sebastián Buedo-Fernández has been partially supported by Ministerio de Educación, Cultura y Deporte of Spain (Grant No. FPU16/04416), Consellería de Cultura, Educación e Ordenación Universitaria, Xunta de Galicia (Grant Nos. GRC2015/004 and R2016/022), and Agencia Estatal de Investigación of Spain (Grant MTM2016-75140-P, cofunded by European Community fund FEDER).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada IICampus Marcosende, Universidad de VigoVigoSpain
  2. 2.Departamento de Estatística, Análise Matemática e OptimizaciónUniversidade de Santiago de Compostela, Facultade de Matemáticas, Campus VidaSantiago de CompostelaSpain

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