Acta Applicandae Mathematicae

, Volume 128, Issue 1, pp 39–48 | Cite as

On the Dalgaard-Strulik Model with Logistic Population Growth Rate and Delayed-Carrying Capacity



Recently Dalgaard and Strulik have proposed (in Resour. Energy Econ. 33:782–797, 2011) an energy model of capital accumulation based on the mathematical framework developed by Solow-Swan and coupled with Cobb-Douglas production function (Solow in Q. J. Economics 70:65–94, 1956; Swan in Econ. Rec. 32(63):334–361, 1956). The model is based on a constant rate of population growth assumption. The present paper, according to the analysis performed by Yukalov et al. (Physica D 238:1752–1767, 2009), improves the Dalgaard-Strulik model by introducing a logistic-type equation with delayed carrying capacity which alters the asymptotic stability of the relative steady state. Specifically, by choosing the time delay as a bifurcation parameter, it turns out that the steady state loses stability and a Hopf bifurcation occurs when time delay passes through critical values. The results are of great interest in the applied and theoretical economics.


Dalgaard-Strulik model Energy Time delay Hopf bifurcation Logistic model Nonconstant carrying capacity 


  1. 1.
    Baker, C.T.H., Bocharov, G.A., Paul, C.A.H.: Mathematical modelling of the interleukin-2 T-cell system: a comparative study of approaches based on ordinary and delay differential equations. J. Theor. Med. 2, 117–128 (1997) CrossRefGoogle Scholar
  2. 2.
    Banavar, J.R., Maritan, A., Rinaldo, A.: Size and form in efficient transportation networks. Nature 399, 130–132 (1999) CrossRefGoogle Scholar
  3. 3.
    Banks, R.B.: Growth and Diffusion Phenomena: Mathematical Frameworks and Applications. Springer, Berlin (1994) CrossRefMATHGoogle Scholar
  4. 4.
    Bianca, C.: Mathematical modelling for keloid formation triggered by virus: malignant effects and immune system competition. Math. Models Methods Appl. Sci. 21, 389–419 (2011) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bianca, C.: Kinetic theory for active particles modelling coupled to Gaussian thermostats. Appl. Math. Sci. 6, 651–660 (2012) MathSciNetMATHGoogle Scholar
  6. 6.
    Bianca, C.: An existence and uniqueness theorem to the Cauchy problem for thermostatted-KTAP models. Int. J. Math. Anal. 6, 813–824 (2012) MathSciNetMATHGoogle Scholar
  7. 7.
    Bianca, C.: Onset of nonlinearity in thermostatted active particles models for complex systems. Nonlinear Anal., Real World Appl. 13, 2593–2608 (2012) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bianca, C.: Modeling complex systems by functional subsystems representation and thermostatted-KTAP methods. Appl. Math. Inf. Sci. 6, 495–499 (2012) Google Scholar
  9. 9.
    Bianca, C.: Thermostatted kinetic equations as models for complex systems in physics and life sciences. Phys. Life Rev. 9, 359–399 (2012) CrossRefGoogle Scholar
  10. 10.
    Bianca, C.: Existence of stationary solutions in kinetic models with Gaussian thermostats. Math. Methods Appl. Sci. (2013). doi: 10.1002/mma.2722 MathSciNetGoogle Scholar
  11. 11.
    Bianca, C., Pennisi, M.: The triplex vaccine effects in mammary carcinoma: a nonlinear model in tune with simtriplex. Nonlinear Anal., Real World Appl. 13, 1913–1940 (2012) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bianca, C., Ferrara, M., Guerrini, L.: Hopf bifurcations in a delayed-energy-based model of capital accumulation. Appl. Math. Inf. Sci. 7, 139–143 (2013) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bianca, C., Ferrara, M., Guerrini, L.: The Cai model with time delay: existence of periodic solutions and asymptotic analysis. Appl. Math. Inf. Sci. 7, 21–27 (2013) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bucci, A., Guerrini, L.: Transitional dynamics in the Solow-Swan growth model with AK technology and logistic population change. B E J. Macroecon. 9, 1–16 (2009) Google Scholar
  15. 15.
    Dalgaard, C.L., Strulik, H.: Energy distribution and economic growth. Resour. Energy Econ. 33, 782–797 (2011) CrossRefGoogle Scholar
  16. 16.
    Ferrara, M.: Demographic dynamics and green economy: a sustainable growth model. Istituto Lombardo (Rend. Lett.) 143, 447–468 (2009) Google Scholar
  17. 17.
    Ferrara, M.: An AK Solow model with logistic law technology. Int. J. Pure Appl. Math. 67, 337–340 (2011) MathSciNetMATHGoogle Scholar
  18. 18.
    Ferrara, M.: A note on the Solow economic growth model with Richards population growth law. Appl. Sci. 13, 36–39 (2011) MathSciNetMATHGoogle Scholar
  19. 19.
    Ferrara, M., Guerrini, L.: On the dynamics of a three sector growth model. Int. Econ. Rev. 55, 275–283 (2008) CrossRefGoogle Scholar
  20. 20.
    Galor, O.: From stagnation to growth: unified growth theory. In: Aghion, P., Durlauf, S. (eds.) Handbook of Economic Growth, vol. 1A. North-Holland, Amsterdam (2005) Google Scholar
  21. 21.
    Guerrini, L.: The Solow-Swan model with a bounded population growth rate. J. Math. Econ. 42, 14–21 (2006) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Guerrini, L.: The Ramsey model with a bounded population growth rate. J. Macroecon. 32, 872–878 (2010) CrossRefGoogle Scholar
  23. 23.
    Guerrini, L.: The Ramsey model with AK technology and a bounded population growth rate. J. Macroecon. 32, 1178–1183 (2010) CrossRefGoogle Scholar
  24. 24.
    Guerrini, L.: A closed-form solution to the Ramsey model with logistic population growth. Econ. Model. 27, 1178–1182 (2010) CrossRefGoogle Scholar
  25. 25.
    Guerrini, L.: Transitional dynamics in the Ramsey model with AK technology and logistic population change. Econ. Lett. 109, 17–19 (2010) MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Hassard, B., Kazarino, D., Wan, Y.: Theory and Application of Hopf Bifurcation. Cambridge University Press, Cambridge (1981) Google Scholar
  27. 27.
    Hutchinson, G.E.: Circular causal systems in ecology. Ann. N.Y. Acad. Sci. 50, 221–246 (1948) CrossRefGoogle Scholar
  28. 28.
    Kühnert, C., Helbing, D., West, G.: Scaling laws in urban supply networks. Physica A 363, 96–103 (2006) CrossRefGoogle Scholar
  29. 29.
    Malthus, T.R.: An Essay on the Principle of Population. Oxford World’s Classics. Oxford University Press, Oxford (1999) Google Scholar
  30. 30.
    Ramsey, F.P.: A mathematical theory of saving. Econ. J. 38, 543–559 (1928) CrossRefGoogle Scholar
  31. 31.
    Solow, R.M.: A contribution to the theory of economic growth. Q. J. Econ. 70, 65–94 (1956) CrossRefGoogle Scholar
  32. 32.
    Swan, T.W.: Economic growth and capital accumulation. Econ. Rec. 32(63), 334–361 (1956) CrossRefGoogle Scholar
  33. 33.
    Verhulst, P.F.: Notice sur la loi que la population suit dans son accroissement. Corresp. Math. Phys. 10, 113–121 (1838) Google Scholar
  34. 34.
    Wright, E.M.: A non-linear difference-differential equation. J. Reine Angew. Math. 194, 66–87 (1955) MathSciNetMATHGoogle Scholar
  35. 35.
    Yukalov, V.I., Yukalova, E.P., Sornette, D.: Punctuated evolution due to delayed carrying capacity. Physica D 238, 1752–1767 (2009) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Dipartimento di Scienze MatematichePolitecnicoTorinoItaly
  2. 2.Dipartimento di Matematica per le Scienze Economiche e SocialiUniversità di BolognaBolognaItaly

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