Skip to main content
SpringerLink
Log in

Logistic equation with continuously distributed lag and application in economics

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, we consider nonlinear dynamics with continuously distributed lags. A generalization of the logistic equation, its solution and economic models of logistic growth are proposed by taking into account continuously distributed lags. The logistic integro-differential equations are considered for exponential and gamma distributions of delay time. The integro-differential equations of the proposed model of logistic growth with distributed lag are represented by differential equations with derivatives of integer orders. The solution of the logistic integro-differential equations with exponentially distributed lag is obtained. Characteristic properties of nonlinear dynamics with continuously distributed lags are described. The main difference between dynamics with lag from standard dynamics without delay lies in the existence of a cutoff threshold of growth. We propose the principle of growth clipping by distributed lag, which states that the distributed lag can lead to the emergence of the cutoff threshold, below which growth is replaced by decline. For economy, this means that for production growth, the starting production should exceed a certain minimum (critical) value of production.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Verhulst, P.F.: Mathematical researches into the law of population growth increase. Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres de Bruxelles. 18, 1–42 (1845). (in French)

    Google Scholar 

  2. Kwasnicki, W.: Logistic growth of the global economy and competitiveness of nations. Technol. Forecast. Soc. Change 80(1), 50–76 (2013)

    Article  Google Scholar 

  3. Girdzijauskas, S., Streimikiene, D., Mialik, A.: Economic growth, capitalism and unknown economic paradoxes. Sustainability 4, 2818–2837 (2012)

    Article  Google Scholar 

  4. Tarasova, V.V., Tarasov, V.E.: Logistic map with memory from economic model. Chaos Solitons Fractals 95, 84–91 (2017). https://doi.org/10.1016/j.chaos.2016.12.012. (arXiv:1712.09092)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fick, E., Fick, M., Hausmann, G.: Logistic equation with memory. Phys. Rev. A 44(4), 2469–2473 (1991)

    Article  MathSciNet  Google Scholar 

  6. El-Sayed, A.M.A., El-Mesiry, A.E.M., El-Saka, H.A.A.: On the fractional-order logistic equation. Appl. Math. Lett. 20(7), 817–823 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. West, B.J.: Exact solution to fractional logistic equation. Phys. A Stat. Mech. Appl. 429, 103–108 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Area, I., Losada, J., Nieto, J.J.: A note on the fractional logistic equation. Phys. A Stat. Mech. Appl. 444(C), 182–187 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Tarasov, V.E., Tarasova, V.V.: Accelerator and multiplier for macroeconomic processes with memory. IRA Int. J. Manag. Soc. Sci. 9(3), 86–125 (2017). https://doi.org/10.21013/jmss.v9.v3.p1

    Google Scholar 

  10. Tarasova, V.V., Tarasov, V.E.: Concept of dynamic memory in economics. Commun. Nonlinear Sci. Numer. Simul. 55, 127–145 (2018). https://doi.org/10.1016/j.cnsns.2017.06.032. (arXiv:1712.09088)

    Article  MathSciNet  Google Scholar 

  11. Tarasov, V.E., Tarasova, V.V.: Criterion of existence of power-law memory for economic processes. Entropy 20(6), (2018) Article ID 414. 24 p. https://doi.org/10.3390/e20060414

  12. Allen, R.G.D.: Mathematical Economics, 2nd edn. Macmillan, London (1960). (First Edition 1956) 812 p. ISBN: 978-1-349-81547-0. https://doi.org/10.1007/978-1-349-81547-0

  13. Allen, R.G.D.: Mathematical Economics. Andesite Press, p. 568 (2015). ISBN: 978-1297569906

  14. Allen, R.G.D.: Macro-economic Theory, A Mathematical Treatment, p. 420. Macmillan, London (1968)

    Google Scholar 

  15. Phillips, A.W.: Stabilisation policy in a closed economy. Econ. J. 64(254), 290–323 (1954). https://doi.org/10.2307/2226835

    Article  Google Scholar 

  16. Phillips, A.W.H.: Collected Works in Contemporary Perspective. In: Leeson, R. (ed.). Cambridge University Press, Cambridge, p. 515 (2000). ISBN: 9780521571357

  17. Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 73–85 (2015). https://doi.org/10.12785/pfda/010201

    Google Scholar 

  18. Caputo, M., Fabrizio, M.: Applications of new time and spatial fractional derivatives with exponential kernels. Prog. Fract. Differ. Appl. 2(1), 1–11 (2016). https://doi.org/10.18576/pfda/020101

    Article  Google Scholar 

  19. Ortigueira, M.D., Tenreiro, Machado J.: A critical analysis of the Caputo–Fabrizio operator. Commun. Nonlinear Sci. Numer. Simul. 59, 608–611 (2018). https://doi.org/10.1016/j.cnsns.2017.12.001

    Article  MathSciNet  Google Scholar 

  20. Tarasov, V.E.: No nonlocality. No fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 62, 157–163 (2018). https://doi.org/10.1016/j.cnsns.2018.02.019. (arXiv:1803.00750)

    Article  MathSciNet  Google Scholar 

  21. Tarasov, V.E.: Caputo-Fabrizio operator in terms of integer derivatives: memory or distributed lag? Comput. Appl. Math. 38, 113 (2019). https://doi.org/10.1007/s40314-019-0883-8

    Article  MathSciNet  Google Scholar 

  22. Tarasova, V.V., Tarasov, V.E.: Economic interpretation of fractional derivatives. Prog. Fract. Differ. Appl. 3(1), 1–7 (2017). https://doi.org/10.18576/pfda/030101

    Article  Google Scholar 

  23. Tarasov, V.E., Tarasova, V.V.: Macroeconomic models with long dynamic memory: fractional calculus approach. Appl. Math. Comput. 338, 466–486 (2018). https://doi.org/10.1016/j.amc.2018.06.018

    MathSciNet  Google Scholar 

  24. Tarasova, V.V., Tarasov, V.E.: Dynamic intersectoral models with power-law memory. Commun. Nonlinear Sci. Numer. Simul. 54, 100–117 (2018). https://doi.org/10.1016/j.cnsns.2017.05.015

    Article  MathSciNet  Google Scholar 

  25. Tarasova, V.V., Tarasov, V.E.: Fractional dynamics of natural growth and memory effect in economics. Eur. Res. 12(23), 30–37 (2016). https://doi.org/10.20861/2410-2873-2016-23-004

    Google Scholar 

  26. Tarasova, V.V., Tarasov, V.E.: Economic growth model with constant pace and dynamic memory. Probl. Mod. Sci. Educ. 2(84), 40–45 (2017). https://doi.org/10.20861/2304-2338-2017-84-001

    Google Scholar 

  27. Tarasov, V.E., Tarasova, V.V.: Time-dependent fractional dynamics with memory in quantum and economic physics. Ann. Phys. 383, 579–599 (2017). https://doi.org/10.1016/j.aop.2017.05.017

    Article  MathSciNet  MATH  Google Scholar 

  28. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives Theory and Applications, p. 1006. Gordon and Breach, New York (1993)

    MATH  Google Scholar 

  29. Kiryakova, V.: Generalized Fractional Calculus and Applications. Longman and J. Wiley, New York, p. 360 (1994). ISBN 9780582219779

  30. Podlubny, I.: Fractional Differential Equations, p. 340. Academic Press, San Diego (1998)

    MATH  Google Scholar 

  31. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, p. 540. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  32. Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin, p. 247 (2010). https://doi.org/10.1007/978-3-642-14574-2

  33. Tarasov, V.E., Tarasova, S.S.: Fractional and integer derivatives with continuously distributed lag. Commun. Nonlinear Sci. Numer. Simul. 70, 125–169 (2019). https://doi.org/10.1016/j.cnsns.2018.10.014

    Article  MathSciNet  Google Scholar 

  34. Tarasov, V.E., Tarasova, V.V.: Phillips model with exponentially distributed lag and power-law memory. Comput. Appl. Math. (2018). Accepted 2018.09.24

  35. Tejado, I., Valerio, D., Valerio, N.: Fractional calculus in economic growth modelling. The Spanish case. In: Moreira, A.P., Matos, A., Veiga, G. (eds.) CONTROLO’2014—Proceedings of the 11th Portuguese Conference on Automatic Control. Volume 321 of the Series Lecture Notes in Electrical Engineering. Springer, pp. 449–458 (2015) https://doi.org/10.1007/978-3-319-10380-8_43

  36. Tejado, I., Valerio, D., Valerio, N.: Fractional calculus in economic growth modeling. The Portuguese case. In: Conference: International Conference on Fractional Differentiation and its Applications (FDA’14) (2014). https://doi.org/10.1109/ICFDA.2014.6967427

  37. Tejado, I., Valerio, D., Perez, E., Valerio, N.: Fractional calculus in economic growth modelling: the Spanish and Portuguese cases. Int. J. Dyn. Control 5(1), 208–222 (2015). https://doi.org/10.1007/s40435-015-0219-5

    Article  MathSciNet  Google Scholar 

  38. Tejado, I., Valerio, D., Perez, E., Valerio, N.: Fractional calculus in economic growth modelling: the economies of France and Italy. In: Spasic, D.T., Grahovac, N., Zigic, M., Rapaic, M., Atanackovic, T.M. (eds.) Proceedings of International Conference on Fractional Differentiation and its Applications. Novi Sad, Serbia, July 18–20. Novi Sad, pp. 113–123 (2016)

  39. Tejado, I., Perez, E., Valerio, D.: Fractional calculus in economic growth modelling of the group of seven. SSRN Electron. J. (2018). https://doi.org/10.2139/ssrn.3271391

    Google Scholar 

  40. Tejado, I., Perez, E., Valerio, D.: Economic growth in the European Union modelled with fractional derivatives: first results. Bull. Pol. Acad. Sci. Tech. Sci. 66(4), 455–465 (2018). https://doi.org/10.24425/124262

    Google Scholar 

  41. Luo, D., Wang, J.R., Feckan, M.: Applying fractional calculus to analyze economic growth modelling. J. Appl. Math. Stat. Inform. 14(1), 25–36 (2018). https://doi.org/10.2478/jamsi-2018-0003

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow, Russia, 119991

    Vasily E. Tarasov

  2. Faculty of Economics, Lomonosov Moscow State University, Moscow, Russia, 119991

    Valentina V. Tarasova

  3. Yandex, Ulitsa Lva Tolstogo 16, Moscow, Russia, 119021

    Valentina V. Tarasova

Authors
  1. Vasily E. Tarasov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Valentina V. Tarasova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vasily E. Tarasov.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tarasov, V.E., Tarasova, V.V. Logistic equation with continuously distributed lag and application in economics. Nonlinear Dyn 97, 1313–1328 (2019). https://doi.org/10.1007/s11071-019-05050-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-05050-1

Keywords

Mathematics Subject Classification

Search

Navigation