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

  • Eduardo LizEmail author
  • Sebastián Buedo-Fernández


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.


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

Mathematics Subject Classification

Primary 39A10 39A30 Secondary 34K20 



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).


  1. 1.
    Allwright, D.J.: Hypergraphic functions and bifurcations in recurrence relations. SIAM J. Appl. Math. 34, 687–691 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bellows, T.S.: The descriptive properties of some models for density dependence. J. Anim. Ecol. 50, 139–156 (1981)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Buedo-Fernández, S., Liz, E.: On the stability properties of a delay differential neoclassical model of economic growth. Electron J. Qual. Theory Differ. Equ. 43, 1–14 (2018)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Coppel, W.A.: The solution of equations by iteration. Proc. Camb. Philos. Soc. 51, 41–43 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cull, P.: Population models: stability in one dimension. Bull. Math. Biol. 69, 989–1017 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Day, R.H.: Irregular growth cycles. Am. Econ. Rev. 72, 406–414 (1982)Google Scholar
  7. 7.
    El-Morshedy, H.A., Jiménez López, V.: Global attractors for difference equations dominated by one-dimensional maps. J. Differ. Equ. Appl. 14, 391–410 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gopalsamy, K., Trofimchuk, S.I., Bantsur, N.R.: A note on global attractivity in models of hematopoiesis. Ukr. Math. J. 50, 3–12 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ivanov, A.F., Sharkovsky, A.N.: Oscillations in singularly perturbed delay equations. Dyn. Rep. Expos. Dyn. Syst. (N.S.) 1, 164–224 (1992)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Jiménez López, V., Parreño, E.: L.A.S and negative Schwarzian derivative do not imply G.A.S. in Clark’s equation. J. Dyn. Differ. Equ. 28, 339–374 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kot, M.: Elements of Mathematical Ecology. Cambridge University Press, New York (2001)CrossRefzbMATHGoogle Scholar
  12. 12.
    Levin, S.A., May, R.M.: A note on difference delay equations. Theor. Popul. Biol. 9, 178–187 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liz, E.: Local stability implies global stability in some one-dimensional discrete single-species models. Discrete Contin. Dyn. Syst. Ser. B 7, 191–199 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Liz, E.: A global picture of the gamma-Ricker map: a flexible discrete-time model with factors of positive and negative density dependence. Bull. Math. Biol. 80, 417–434 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Liz, E.: A new flexible discrete-time model for stable populations. Discrete Contin. Dyn. Syst. B 23, 2487–2498 (2018)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Liz, E., Pinto, M., Robledo, G., Trofimchuk, S., Tkachenko, V.: Wright type delay differential equations with negative Schwarzian. Discrete Contin. Dyn. Syst. 9, 309–321 (2003)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Liz, E., Röst, G.: Global dynamics in a commodity market model. J. Math. Anal. Appl. 398, 707–714 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control system. Science 197, 287–289 (1977)CrossRefzbMATHGoogle Scholar
  19. 19.
    Mallet-Paret, J., Nussbaum, R.D.: A differential-delay equation arising in optics and physiology. SIAM J. Math. Anal. 20, 249–292 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    May, R.M., Oster, G.F.: Bifurcations and dynamic complexity in simple ecological models. Am. Nat. 110, 573–599 (1976)CrossRefGoogle Scholar
  21. 21.
    Maynard Smith, J., Slatkin, M.: The stability of predator–prey systems. Ecology 54, 384–391 (1973)CrossRefGoogle Scholar
  22. 22.
    Quinn, T.J., Deriso, R.B.: Quantitative Fish Dynamics. Oxford University Press, New York (1999)Google Scholar
  23. 23.
    Sedaghat, H.: Nonlinear Difference Equations: Theory with Applications to Social Science Models, Mathematical Modelling: Theory and Applications, vol. 15. Kluwer Academic Publishers, Dordrecht (2003)zbMATHGoogle Scholar
  24. 24.
    Sharkovsky, A.N., Kolyada, S.F., Sivak, A.G., Fedorenko, V.V.: Dynamics of One-Dimensional Maps, Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht (1997)CrossRefzbMATHGoogle Scholar
  25. 25.
    Shepherd, J.G.: A versatile new stock-recruitment relationship for fisheries, and the construction of sustainable yield resources. J. Conserv. Int. Explor. Mer. 40, 67–75 (1982)CrossRefGoogle Scholar
  26. 26.
    Singer, D.: Stable orbits and bifurcation of maps of the interval. SIAM J. Appl. Math. 35, 260–267 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Solow, R.M.: A contribution to the theory of economic growth. Q. J. Econ. 70, 65–94 (1956)CrossRefGoogle Scholar
  28. 28.
    Zheng, J., Kruse, G.H.: Stock–recruitment relationships for three major Alaskan crab stocks. Fish. Res. 65, 103–121 (2003)CrossRefGoogle Scholar

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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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