On the gamma-logistic map and applications to a delayed neoclassical model of economic growth

  • Sebastián Buedo-FernándezEmail author
Original Paper


In this work, we study the stability properties of a delay differential neoclassical model of economic growth, based on the original model proposed by Solow (Q J Econ 70:65–94, 1956). We consider a logistic-type production function, which comes from combining a Cobb–Douglas function and a linear pollution effect caused by increasing concentrations of capital. The difference between the production function and the classical logistic map comes from the presence of a parameter \(\gamma \in (0,1)\) in the exponent of one factor. We call this new function the gamma-logistic map. Our main purpose is to obtain sharp global stability conditions for the positive equilibrium of the model and to study how the stability properties of such equilibrium depend on the relevant model parameters. This study is developed by using some properties of the gamma-logistic map and some well-known results connecting stability in delay differential equations and discrete dynamical systems. Finally, we also compare the obtained results with the ones written in related articles.


Delay differential equation Neoclassical growth model Global stability Gamma-logistic map 

Mathematics Subject Classification

34K20 91B62 



The author thanks Prof. Eduardo Liz for all his ideas, work and suggestions throughout the discussion of the model and the improvement of the document. Moreover, the author also acknowledges all the valuable comments coming from the referee process, which led to clearer explanations and a better motivation of the model. This research has been partially supported by Ministerio de Educación, Cultura y Deporte of Spain (Grant No. FPU16/04416), Consellería de Cultura, Educación e Ordenación Universitaria da Xunta de Galicia (Grant Nos. ED481A-2017/030, GRC2015/004 and R2016/022) and Agencia Estatal de Investigación of Spain (Grant No. MTM2016-75140-P, cofunded by European Community fund FEDER).

Compliance with Ethical Standards

Conflict of interest

The author declares that there is no conflict of interest.


  1. 1.
    Avilés, L.: Cooperation and non-linear dynamics: an ecological perspective on the evolution of sociality. Evol. Ecol. Res. 1, 459–477 (1999)Google Scholar
  2. 2.
    Barro, R.J., Sala-i-Martin, X.: Economic Growth, 2nd edn. MIT Press, Cambridge (2004)zbMATHGoogle Scholar
  3. 3.
    Buedo-Fernández, S., Liz, E.: On the stability properties of a delay differential neoclassical model of economic growth. Electron. J. Qual. Theo. Differ. Equ. 2018, 1–14 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Day, R.H.: Irregular growth cycles. Am. Econ. Rev. 72, 406–414 (1982)Google Scholar
  5. 5.
    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
  6. 6.
    Eskola, H.T., Parvinen, K.: On the mechanistic underpinning of discrete-time population models with Allee effect. Theor. Popul. Biol. 72, 41–51 (2007)CrossRefzbMATHGoogle Scholar
  7. 7.
    Győri, I., Trofimchuk, S.: Global attractivity in \(x^{\prime }(t) = -\delta x(t) + pf(x(t-\tau ))\). Dyn. Syst. Appl. 8, 197–210 (1999)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ivanov, A.F., Liz, E., Trofimchuk, S.: Global stability of a class of scalar nonlinear delay differential equations. Differ. Equ. Dyn. Syst. 11, 33–54 (2003)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ivanov, A.F., Sharkovsky, A.N.: Oscillations in singularly perturbed delay equations. Dyn. Report. (N.S.) 1, 164–224 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Liz, E.: Delayed logistic population models revisited. Publ. Mat. 309–331 (2014).
  11. 11.
    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
  12. 12.
    Liz, E.: A new flexible discrete-time model for stable populations. Discrete Contin. Dyn. Syst. B 23, 2487–2498 (2018)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Liz, E., Röst, G.: Dichotomy results for delay differential equations with negative Schwarzian derivative. Nonlinear Anal. Real World Appl. 11, 1422–1430 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Liz, E., Ruiz-Herrera, A.: Attractivity, multistability, and bifurcation in delayed Hopfield’s model with non-monotonic feedback. J. Differ. Equ. 255, 4244–4266 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Matsumoto, A., Szidarovszky, F.: Delay differential neoclassical growth model. J. Econ. Behav. Organ. 78, 272–289 (2011)CrossRefGoogle Scholar
  16. 16.
    Matsumoto, A., Szidarovszky, F.: Asymptotic behavior of a delay differential neoclassical growth model. Sustainability 5, 440–455 (2013)CrossRefGoogle Scholar
  17. 17.
    Maynard Smith, J.: Mathematical Ideas in Biology. Cambridge University Press, London (1968)CrossRefGoogle Scholar
  18. 18.
    Sedaghat, H.: The impossibility of unstable, globally attracting fixed points for continuous linear mappings of the line. Am. Math. Month. 104, 356–358 (1997)CrossRefzbMATHGoogle Scholar
  19. 19.
    Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Texts in Applied Mathematics, vol. 57. Springer, New York (2011)CrossRefGoogle Scholar
  20. 20.
    Smith, H.L., Thieme, H.R.: Dynamical Systems and Population Persistence. Graduate Studies in Mathematics, vol. 118. American Mathematical Society, Providence (2011)Google Scholar
  21. 21.
    Solow, R.M.: A contribution to the theory of economic growth. Q. J. Econ. 70, 65–94 (1956)CrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Departamento de Estatística, Análise Matemática e Optimización, Facultade de MatemáticasUniversidade de Santiago de CompostelaSantiago de CompostelaSpain

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