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Mathematical Biosciences and Engineering

2013, Volume 10, Issue 2: 379-398. doi: 10.3934/mbe.2013.10.379
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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
  • Received: 01 February 2012 Accepted: 29 June 2018 Published: 01 January 2013

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

  • 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.

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

    Related Papers:

  • 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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    [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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