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

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.

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Notes

  1. 1.

    For this reason we use the name gamma-models for (1.1).

References

  1. 1.

    Allwright, D.J.: Hypergraphic functions and bifurcations in recurrence relations. SIAM J. Appl. Math. 34, 687–691 (1978)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bellows, T.S.: The descriptive properties of some models for density dependence. J. Anim. Ecol. 50, 139–156 (1981)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Coppel, W.A.: The solution of equations by iteration. Proc. Camb. Philos. Soc. 51, 41–43 (1955)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Cull, P.: Population models: stability in one dimension. Bull. Math. Biol. 69, 989–1017 (2007)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Kot, M.: Elements of Mathematical Ecology. Cambridge University Press, New York (2001)

    Google Scholar 

  12. 12.

    Levin, S.A., May, R.M.: A note on difference delay equations. Theor. Popul. Biol. 9, 178–187 (1976)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Liz, E.: A new flexible discrete-time model for stable populations. Discrete Contin. Dyn. Syst. B 23, 2487–2498 (2018)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Liz, E., Röst, G.: Global dynamics in a commodity market model. J. Math. Anal. Appl. 398, 707–714 (2013)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control system. Science 197, 287–289 (1977)

    Article  Google 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)

    MathSciNet  Article  Google Scholar 

  20. 20.

    May, R.M., Oster, G.F.: Bifurcations and dynamic complexity in simple ecological models. Am. Nat. 110, 573–599 (1976)

    Article  Google Scholar 

  21. 21.

    Maynard Smith, J., Slatkin, M.: The stability of predator–prey systems. Ecology 54, 384–391 (1973)

    Article  Google 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)

    Google 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)

    Google 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)

    Article  Google Scholar 

  26. 26.

    Singer, D.: Stable orbits and bifurcation of maps of the interval. SIAM J. Appl. Math. 35, 260–267 (1978)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Solow, R.M.: A contribution to the theory of economic growth. Q. J. Econ. 70, 65–94 (1956)

    Article  Google Scholar 

  28. 28.

    Zheng, J., Kruse, G.H.: Stock–recruitment relationships for three major Alaskan crab stocks. Fish. Res. 65, 103–121 (2003)

    Article  Google Scholar 

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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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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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Keywords

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

Mathematics Subject Classification

  • Primary 39A10
  • 39A30
  • Secondary 34K20