Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 901014, 10 pages
http://dx.doi.org/10.1155/2013/901014
Research Article

The Time Delays’ Effects on the Qualitative Behavior of an Economic Growth Model

1Dipartimento di Scienze Matematiche, Politecnico, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
2Department of Law and Economics, University Mediterranea of Reggio Calabria and CRIOS University Bocconi of Milan, Via dei Bianchi 2, 89127 Reggio Calabria, Italy
3Department of Management, Polytechnic University of Marche, 60121 Ancona, Italy

Received 14 August 2013; Accepted 1 November 2013

Academic Editor: Constantin Udriste

Copyright © 2013 Carlo Bianca et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. T. Erneux, Applied Delay Differential Equations, Springer, New York, NY, USA, 2009. View at MathSciNet
  2. P. J. Cunningham and W. J. Wangersky, Time Lag in Population Models, Yale, 1958.
  3. G. I. Marchuk, Mathematical Modelling of Immune Response in Infectious Diseases, Kluwer Academic, Dordrecht, Germany, 1997. View at MathSciNet
  4. U. Foryś, “Interleukin mathematical model of an immune system,” Journal of Biological Systems, vol. 3, pp. 889–902, 1995. View at Publisher · View at Google Scholar
  5. U. Foryś, “Global analysis of Marchuks model in case of strong immune system,” Journal of Biological Systems, vol. 8, pp. 331–346, 2000. View at Publisher · View at Google Scholar
  6. M. Bodnar and U. Foryś, “A model of immune system with time-dependent immune reactivity,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 2, pp. 1049–1058, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. U. Foryś and M. Bodnar, “Time delays in proliferation process for solid avascular tumour,” Mathematical and Computer Modelling, vol. 37, no. 11, pp. 1201–1209, 2003. View at Publisher · View at Google Scholar · View at Scopus
  8. H. M. Byrne, “The effect of time delays on the dynamics of avascular tumor growth,” Mathematical Biosciences, vol. 144, no. 2, pp. 83–117, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. J. Piotrowska, “Hopf bifurcation in a solid avascular tumour growth model with two discrete delays,” Mathematical and Computer Modelling, vol. 47, no. 5-6, pp. 597–603, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. Bodnar and U. Foryś, “Three types of simple DDE’s describing tumour growth,” Journal of Biological Systems, vol. 15, pp. 1–19, 2007. View at Google Scholar
  11. M. Bodnar and U. Foryś, “Global stability and the Hopf bifurcation for some class of delay differential equation,” Mathematical Methods in the Applied Sciences, vol. 31, no. 10, pp. 1197–1207, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  12. B. Shi, F. Zhang, and S. Xu, “Hopf bifurcation of a mathematical model for growth of tumors with an action of inhibitor and two time delays,” Abstract and Applied Analysis, vol. 2011, Article ID 980686, 10 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. N. Bielczyk, U. Foryś, and T. Platkowski, “Dynamical models of Dyadic interactions with delay,” The Journal of Mathematical Sociology, vol. 37, no. 4, 2013. View at Google Scholar
  14. S. Invernizzi and A. Medio, “On lags and chaos in economic dynamic models,” Journal of Mathematical Economics, vol. 20, no. 6, pp. 521–550, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. P. J. Zak, “Kaleckian lags in general equilibrium,” Review of Political Economy, vol. 11, no. 3, pp. 321–330, 1999. View at Publisher · View at Google Scholar
  16. G. Dibeh, “Speculative dynamics in a time-delay model of asset prices,” Physica A, vol. 355, no. 1, pp. 199–208, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  17. M. Szydlowski, A. Krawiec, and J. Tobola, “Nonlinear oscillations in business cycle model with time lags,” Chaos, Solitons and Fractals, vol. 12, no. 3, pp. 505–517, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  18. L. Zhou and Y. Li, “A dynamic IS-LM business cycle model with two time delays in capital accumulation equation,” Journal of Computational and Applied Mathematics, vol. 228, no. 1, pp. 182–187, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. C. Bianca and L. Guerrini, “On the Dalgaard-Strulik model with logistic population growth rate and delayed-carrying capacity,” Acta Applicandae Mathematicae, vol. 128, pp. 39–48, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  20. C. Bianca, M. Ferrara, and L. Guerrini, “Hopf bifurcations in a delayed-energy-based model of capital accumulation,” Applied Mathematics & Information Sciences, vol. 7, no. 1, pp. 139–143, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  21. C. Bianca, M. Ferrara, and L. Guerrini, “The Cai model with time delay: existence of periodic solutions and asymptotic analysis,” Applied Mathematics & Information Sciences, vol. 7, no. 1, pp. 21–27, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  22. Y. Ma, “Global Hopf bifurcation in the Leslie-Gower predator-prey model with two delays,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 370–375, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Y. Song, J. Wei, and M. Han, “Local and global Hopf bifurcation in a delayed hematopoiesis model,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 14, no. 11, pp. 3909–3919, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. J. Wei, “Bifurcation analysis in a scalar delay differential equation,” Nonlinearity, vol. 20, no. 11, pp. 2483–2498, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. E. Burger, “On the stability of certain economic systems,” Econometrica, vol. 24, pp. 488–493, 1956. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. N. D. Hayes, “Roots of the transcendental equation associated with a certain difference-differential equation,” Journal of the London Mathematical Society, vol. 25, pp. 226–232, 1950. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. J. K. Hale and W. Z. Huang, “Global geometry of the stable regions for two delay differential equations,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 344–362, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. X. Li, S. Ruan, and J. Wei, “Stability and bifurcation in delay-differential equations with two delays,” Journal of Mathematical Analysis and Applications, vol. 236, no. 2, pp. 254–280, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. J. Wei and Y. Yuan, “Synchronized Hopf bifurcation analysis in a neural network model with delays,” Journal of Mathematical Analysis and Applications, vol. 312, no. 1, pp. 205–229, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. J. K. Hale and W. Z. Huang, “Global geometry of the stable regions for two delay differential equations,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 344–362, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. M. J. Piotrowska, “A remark on the ODE with two discrete delays,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 664–676, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. R. M. Solow, “A contribution to the theory of economic growth,” Quarterly Journal of Economics, vol. 70, pp. 65–94, 1956. View at Publisher · View at Google Scholar
  33. S. Ruan and J. Wei, “On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 10, no. 6, pp. 863–874, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Springer, New York, NY, USA, 1993. View at MathSciNet
  35. B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, 1981. View at MathSciNet
  36. C. Bianca and M. Pennisi, “The triplex vaccine effects in mammary carcinoma: a nonlinear model in tune with SimTriplex,” Nonlinear Analysis: Real World Applications, vol. 13, no. 4, pp. 1913–1940, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. C. Bianca, “Thermostatted kinetic equations as models for complex systems in physicsand life sciences,” Physics of Life Reviews, vol. 9, pp. 359–399, 2012. View at Publisher · View at Google Scholar
  38. C. Bianca, “Kinetic theory for active particles modelling coupled to Gaussian thermostats,” Applied Mathematical Sciences, vol. 6, no. 13-16, pp. 651–660, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. C. Bianca, “Modeling complex systems by functional subsystems representation and thermostatted-KTAP methods,” Applied Mathematics & Information Sciences, vol. 6, pp. 495–499, 2012. View at Google Scholar
  40. C. Bianca, “An existence and uniqueness theorem to the Cauchy problem for thermostatted-KTAP models,” International Journal of Mathematical Analysis, vol. 6, no. 17–20, pp. 813–824, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. C. Bianca, M. Ferrara, and L. Guerrini, “High-order moments conservation in thermostatted kinetic models,” Journal of Global Optimization, 2013. View at Publisher · View at Google Scholar
  42. C. Bianca, “Onset of nonlinearity in thermostatted active particles models for complex systems,” Nonlinear Analysis: Real World Applications, vol. 13, no. 6, pp. 2593–2608, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. C. Bianca, “Controllability in hybrid kinetic equations modeling nonequilibrium multicellular systems,” The Scientific World Journal, vol. 2013, Article ID 274719, 6 pages, 2013. View at Publisher · View at Google Scholar
  44. C. Bianca, “Existence of stationary solutions in kinetic models with Gaussian thermostats,” Mathematical Methods in the Applied Sciences, vol. 36, pp. 1768–1775, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  45. P. Degond and B. Wennberg, “Mass and energy balance laws derived from high-field limits of thermostatted Boltzmann equations,” Communications in Mathematical Sciences, vol. 5, no. 2, pp. 355–382, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  46. V. Bagland, B. Wennberg, and Y. Wondmagegne, “Stationary states for the noncutoff Kac equation with a Gaussian thermostat,” Nonlinearity, vol. 20, no. 3, pp. 583–604, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  47. B. Wennberg and Y. Wondmagegne, “Stationary states for the Kac equation with a Gaussian thermostat,” Nonlinearity, vol. 17, no. 2, pp. 633–648, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  48. B. Wennberg and Y. Wondmagegne, “The Kac equation with a thermostatted force field,” Journal of Statistical Physics, vol. 124, no. 2–4, pp. 859–880, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet