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


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.

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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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Liz, E., Buedo-Fernández, S. A New Formula to Get Sharp Global Stability Criteria for One-Dimensional Discrete-Time Models. Qual. Theory Dyn. Syst. 18, 813–824 (2019). https://doi.org/10.1007/s12346-018-00314-4

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

Mathematics Subject Classification

  • Primary 39A10
  • 39A30
  • Secondary 34K20