An extension of Gompertzian growth dynamics: Weibull and Fréchet models
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...
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AIMS Press
2012-12-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.379 |
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| author | J. Leonel Rocha Sandra M. Aleixo |
| author_facet | J. Leonel Rocha Sandra M. Aleixo |
| author_sort | J. Leonel Rocha |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-0ae9f28a8ecb4ef287f64c498b5c1b08 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2012-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-0ae9f28a8ecb4ef287f64c498b5c1b082025-01-24T02:25:53ZengAIMS PressMathematical Biosciences and Engineering1551-00182012-12-0110237939810.3934/mbe.2013.10.379An extension of Gompertzian growth dynamics: Weibull and Fréchet modelsJ. Leonel Rocha0Sandra M. Aleixo1Instituto Superior de Engenharia de Lisboa - ISEL, ADM and CEAUL, Rua Conselheiro Emídio Navarro, 1, 1959-007 LisboaInstituto Superior de Engenharia de Lisboa - ISEL, ADM and CEAUL, Rua Conselheiro Emídio Navarro, 1, 1959-007 LisboaIn 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.https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.379symbolic dynamicsq)$ densitiestumour dynamics.$beta^*(pbifurcations and chaostopological entropyextreme value lawsgrowth models |
| spellingShingle | J. Leonel Rocha Sandra M. Aleixo An extension of Gompertzian growth dynamics: Weibull and Fréchet models Mathematical Biosciences and Engineering symbolic dynamics q)$ densities tumour dynamics. $beta^*(p bifurcations and chaos topological entropy extreme value laws growth models |
| title | An extension of Gompertzian growth dynamics: Weibull and Fréchet models |
| title_full | An extension of Gompertzian growth dynamics: Weibull and Fréchet models |
| title_fullStr | An extension of Gompertzian growth dynamics: Weibull and Fréchet models |
| title_full_unstemmed | An extension of Gompertzian growth dynamics: Weibull and Fréchet models |
| title_short | An extension of Gompertzian growth dynamics: Weibull and Fréchet models |
| title_sort | extension of gompertzian growth dynamics weibull and frechet models |
| topic | symbolic dynamics q)$ densities tumour dynamics. $beta^*(p bifurcations and chaos topological entropy extreme value laws growth models |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.379 |
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