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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

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  • 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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