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An extension of Gompertzian growth dynamics: Weibull and Fréchet models. (English) Zbl 1260.92100

Summary: A new probabilistic and dynamical approach to an extension of the GompIn this work aertz law is proposed. A generalized family of probability density functions, designated by \(Beta^\ast(p, q)\), 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 \(Beta^\ast(2, q)\) densities are a power of betas mixture, and that its dynamics are determined by a nonlinear 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.

MSC:

92D25 Population dynamics (general)
37B10 Symbolic dynamics
37N25 Dynamical systems in biology
92B05 General biology and biomathematics
62H10 Multivariate distribution of statistics
11S82 Non-Archimedean dynamical systems
92C50 Medical applications (general)
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