Mathematical Biosciences and Engineering, 2013, 10(2): 379-398. doi: 10.3934/mbe.2013.10.379.

Primary: 92D25, 91B62, 62H10, 11S82; Secondary: 92B05, 37H20, 37B10, 37B40.

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An extension of Gompertzian growth dynamics: Weibull and Fréchet models

1. Instituto Superior de Engenharia de Lisboa - ISEL, ADM and CEAUL, Rua Conselheiro Emídio Navarro, 1, 1959-007 Lisboa

In this work a new probabilistic and dynamical approach to an extension of the Gompertz law is proposed. A generalized family of probability density functions, designated by $Beta^*(p,q)$, which is proportional to the right hand side of the Tsoularis-Wallace model, is studied. In particular, for $p = 2$, the investigation is extended to the extreme value models of Weibull and Fréchet type. These models, described by differential equations, are proportional to the hyper-Gompertz growth model. It is proved that the $Beta^*(2,q)$ densities are a power of betas mixture, and that its dynamics are determined by a non-linear coupling of probabilities. The dynamical analysis is performed using techniques of symbolic dynamics and the system complexity is measured using topological entropy. Generally, the natural history of a malignant tumour is reflected through bifurcation diagrams, in which are identified regions of regression, stability, bifurcation, chaos and terminus.
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Keywords symbolic dynamics; q)$ densities; tumour dynamics.; $Beta^*(p; bifurcations and chaos; topological entropy; extreme value laws; Growth models

Citation: J. Leonel Rocha, Sandra M. Aleixo. An extension of Gompertzian growth dynamics: Weibull and Fréchet models. Mathematical Biosciences and Engineering, 2013, 10(2): 379-398. doi: 10.3934/mbe.2013.10.379

References

  • 1. AIP Conf. Proc. American Inst. of Physics, 1124 (2009), 3-12.
  • 2. Proc. Int. Conf. on Information Technology Interfaces, (2009), 213-218.
  • 3. in "Dynamics, Games and Science II" (eds. M. M. Peixoto, A. A. Pinto and D. A. J. Rand), Springer-Verlag (2011), 79-95.
  • 4. J. of Theoret. Biol., 21 (1968), 42-44.
  • 5. John Wiley $&$ Sons, Inc., New York, 1990.
  • 6. Math. Biosci. Eng., 6 (2009), 573-583.
  • 7. Math. Biosci., 185 (2003), 153-167.
  • 8. Br. J. Cancer, 18 (1964), 490-502.
  • 9. Growth, 29 (1965), 233-248.
  • 10. Cambridge University Press, Cambridge, 1995.
  • 11. Math. Biosci. Eng., 1 (2004), 307-324.
  • 12. Chaos, Solitons $&$ Fractals, 41 (2009), 334-347.
  • 13. Physica A, 387 (2008), 5679-5687.
  • 14. J. Math. Anal. Appl., 179 (1993), 446-462.
  • 15. Springer, New York, 1993.
  • 16. Dynamical systems (College Park, MD, 1986–87), 465-563, Lecture Notes in Math., 1342, Springer, Berlin, 1988.
  • 17. BioSystems, 92 (2008), 245-248.
  • 18. Physica D, 208 (2005), 220-235.
  • 19. Math. Biosciences, 230 (2011), 45-54.
  • 20. Fundaçāo Calouste Gulbenkian, Lisboa, 2008.
  • 21. in "Chaos Theory: Modeling, Simulation and Applications" (eds. C. H. Skiadas, Y. Dimotikalis and C. Skiadas), World Scientific Publishing Co, (2011), 309-316.
  • 22. Int. J. Math. Math. Sci., 38 (2004), 2019-2038.
  • 23. Discrete Contin. Dyn. Syst.-Ser.B, 18 (2013), 783-795.
  • 24. Ecol. Model., 205 (2007), 159-168.
  • 25. Math. Biosci. Eng., 8 (2011), 355-369.
  • 26. SIAM J. Appl. Math., 35 (1978), 260-267.
  • 27. Res. Lett. Inf. Math. Sci., 2 (2001), 23-46.
  • 28. Math. Biosci., 29 (1976), 367-373.
  • 29. Chaos Solitons $&$ Fractals, 16 (2003), 665-674.
  • 30. in "Fractals in Biology and Medicine" (eds. G. A. Losa, T. F. Nonnenmacher and E. R. Weibel), Birkhäuser, Basel, (2005), 277-286.
  • 31. Byosystems, 82 (2005), 61-73.
  • 32. J. Control Eng. and Appl. Informatics, 4 (2009), 45-52.

 

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