# American Institute of Mathematical Sciences

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

## 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  February 2012 Revised  October 2012 Published  January 2013

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. Mathematical Biosciences & Engineering, 2013, 10 (2) : 379-398. doi: 10.3934/mbe.2013.10.379
##### References:
 [1] S. M. Aleixo, J. L. Rocha and D. D. Pestana, Populational growth models proportional to beta densities with Allee effect, AIP Conf. Proc. American Inst. of Physics, 1124 (2009), 3-12. [2] S. M. Aleixo, J. L. Rocha and D. D. Pestana, Dynamical behavior on the parameter space: new populational growth models proportional to beta densities, Proc. Int. Conf. on Information Technology Interfaces, (2009), 213-218. [3] S. M. Aleixo, J. L. Rocha and D. D. Pestana, Probabilistic methods in dynamical analysis: populations growths associated to models Beta$(p,q)$ with Allee effect, in "Dynamics, Games and Science II" (eds. M. M. Peixoto, A. A. Pinto and D. A. J. Rand), Springer-Verlag (2011), 79-95. doi: 10.1007/978-3-642-14788-3_5. [4] A. A. Blumberg, Logistic growth rate functions, J. of Theoret. Biol., 21 (1968), 42-44. [5] C. W. Clark, "Mathematical Bioeconomics: The Optimal Management of Renewable Resources," John Wiley & Sons, Inc., New York, 1990. [6] D. Kirschner and A. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy, Math. Biosci. Eng., 6 (2009), 573-583. doi: 10.3934/mbe.2009.6.573. [7] F. Kozusko and Z. Bajzer, Combining gompertzian growth and cell population dynamics, Math. Biosci., 185 (2003), 153-167. doi: 10.1016/S0025-5564(03)00094-4. [8] A. K. Laird, Dynamics of tumour growth, Br. J. Cancer, 18 (1964), 490-502. [9] A. K. Laird, S. A. Tyler and A. D. Barton, Dynamics of normal growth, Growth, 29 (1965), 233-248. [10] D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Codings," Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302. [11] R. López-Ruiz and D. Fournier-Prunaret, Complex behavior in a discrete coupled logistic model for the symbiotic interaction of two species, Math. Biosci. Eng., 1 (2004), 307-324. doi: 10.3934/mbe.2004.1.307. [12] R. López-Ruiz and D. Fournier-Prunaret, Periodic and chaotic events in a discrete model of logistic type for the competitive interaction of two species, Chaos, Solitons & Fractals, 41 (2009), 334-347. doi: 10.1016/j.chaos.2008.01.015. [13] A. S. Martinez, R. S. González and C. A. S. Terçariol, Continuous growth models in terms of generalized logarithm and exponential functions, Physica A, 387 (2008), 5679-5687. doi: 10.1016/j.physa.2008.06.015. [14] M. Marušić and Ž. Bajzer, Generalized two-parameter equation of growth, J. Math. Anal. Appl., 179 (1993), 446-462. doi: 10.1006/jmaa.1993.1361. [15] W. Melo and S. van Strien, "One-Dimensional Dynamics," Springer, New York, 1993. [16] J. Milnor and W. Thurston, On iterated maps of the interval, Dynamical systems (College Park, MD, 1986–87), 465-563, Lecture Notes in Math., 1342, Springer, Berlin, 1988. doi: 10.1007/BFb0082847. [17] M. Molski and J. Konarsky, On the Gompertzian growth in the fractal space-time, BioSystems, 92 (2008), 245-248. [18] A. d'Onofrio, A general framework for modeling tumor-imune system competition and immunotherapy: Matematical analysis and biomedical inferences, Physica D, 208 (2005), 220-235. doi: 10.1016/j.physd.2005.06.032. [19] A. d'Onofrio, A. Fasano and B. Monechi, A generalization of Gompertz law compatible with the Gyllenberg-Webb theory for tumour growth, Math. Biosciences, 230 (2011), 45-54. doi: 10.1016/j.mbs.2011.01.001. [20] D. D. Pestana and S.Velosa, "Introduçāo à Probabilidade e à Estatística," Fundaçāo Calouste Gulbenkian, Lisboa, 2008. [21] D. D. Pestana, S. M. Aleixo and J. L. Rocha, Regular variation, paretian distributions, and the interplay of light and heavy tails in the fractality of asymptotic models, in "Chaos Theory: Modeling, Simulation and Applications" (eds. C. H. Skiadas, Y. Dimotikalis and C. Skiadas), World Scientific Publishing Co, (2011), 309-316. [22] J. L. Rocha and J. Sousa Ramos, Weighted kneading theory of one-dimensional maps with a hole, Int. J. Math. Math. Sci., 38 (2004), 2019-2038. doi: 10.1155/S016117120430428X. [23] J. L. Rocha and S. M. Aleixo, Dynamical analysis in growth models: Blumberg's equation, Discrete Contin. Dyn. Syst.-Ser.B, 18 (2013), 783-795. [24] S. Sakanoue, Extended logistic model for growth of single-species populations, Ecol. Model., 205 (2007), 159-168. [25] H. Schättler, U. Ledzewicz and B. Cardwell, Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis, Math. Biosci. Eng., 8 (2011), 355-369. doi: 10.3934/mbe.2011.8.355. [26] D. Singer, Stable orbits and bifurcations of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267. [27] A. Tsoularis , Analysis of logistic growth models, Res. Lett. Inf. Math. Sci., 2 (2001), 23-46. [28] M. E. Turner Jr., E. L. Bradley Jr., K. A. Kirk and K. M. Pruitt, A theory of growth, Math. Biosci., 29 (1976), 367-373. [29] P. Waliszewski and J. Konarski, The gompertzian curve reveals fractal properties of tumour growth, Chaos Solitons & Fractals, 16 (2003), 665-674. [30] P. Waliszewski and J. Konarski, A mystery of the Gompertz function, in "Fractals in Biology and Medicine" (eds. G. A. Losa, T. F. Nonnenmacher and E. R. Weibel), Birkhäuser, Basel, (2005), 277-286. [31] P. Waliszewski, A principle of fractal-stochastic dualism and Gompertzian dynamics of growth and self-organization, Byosystems, 82 (2005), 61-73. [32] P. Waliszewski, A principle of fractal-stochastic dualism, couplings, complementarity growth, J. Control Eng. and Appl. Informatics, 4 (2009), 45-52.

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##### References:
 [1] S. M. Aleixo, J. L. Rocha and D. D. Pestana, Populational growth models proportional to beta densities with Allee effect, AIP Conf. Proc. American Inst. of Physics, 1124 (2009), 3-12. [2] S. M. Aleixo, J. L. Rocha and D. D. Pestana, Dynamical behavior on the parameter space: new populational growth models proportional to beta densities, Proc. Int. Conf. on Information Technology Interfaces, (2009), 213-218. [3] S. M. Aleixo, J. L. Rocha and D. D. Pestana, Probabilistic methods in dynamical analysis: populations growths associated to models Beta$(p,q)$ with Allee effect, in "Dynamics, Games and Science II" (eds. M. M. Peixoto, A. A. Pinto and D. A. J. Rand), Springer-Verlag (2011), 79-95. doi: 10.1007/978-3-642-14788-3_5. [4] A. A. Blumberg, Logistic growth rate functions, J. of Theoret. Biol., 21 (1968), 42-44. [5] C. W. Clark, "Mathematical Bioeconomics: The Optimal Management of Renewable Resources," John Wiley & Sons, Inc., New York, 1990. [6] D. Kirschner and A. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy, Math. Biosci. Eng., 6 (2009), 573-583. doi: 10.3934/mbe.2009.6.573. [7] F. Kozusko and Z. Bajzer, Combining gompertzian growth and cell population dynamics, Math. Biosci., 185 (2003), 153-167. doi: 10.1016/S0025-5564(03)00094-4. [8] A. K. Laird, Dynamics of tumour growth, Br. J. Cancer, 18 (1964), 490-502. [9] A. K. Laird, S. A. Tyler and A. D. Barton, Dynamics of normal growth, Growth, 29 (1965), 233-248. [10] D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Codings," Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302. [11] R. López-Ruiz and D. Fournier-Prunaret, Complex behavior in a discrete coupled logistic model for the symbiotic interaction of two species, Math. Biosci. Eng., 1 (2004), 307-324. doi: 10.3934/mbe.2004.1.307. [12] R. López-Ruiz and D. Fournier-Prunaret, Periodic and chaotic events in a discrete model of logistic type for the competitive interaction of two species, Chaos, Solitons & Fractals, 41 (2009), 334-347. doi: 10.1016/j.chaos.2008.01.015. [13] A. S. Martinez, R. S. González and C. A. S. Terçariol, Continuous growth models in terms of generalized logarithm and exponential functions, Physica A, 387 (2008), 5679-5687. doi: 10.1016/j.physa.2008.06.015. [14] M. Marušić and Ž. Bajzer, Generalized two-parameter equation of growth, J. Math. Anal. Appl., 179 (1993), 446-462. doi: 10.1006/jmaa.1993.1361. [15] W. Melo and S. van Strien, "One-Dimensional Dynamics," Springer, New York, 1993. [16] J. Milnor and W. Thurston, On iterated maps of the interval, Dynamical systems (College Park, MD, 1986–87), 465-563, Lecture Notes in Math., 1342, Springer, Berlin, 1988. doi: 10.1007/BFb0082847. [17] M. Molski and J. Konarsky, On the Gompertzian growth in the fractal space-time, BioSystems, 92 (2008), 245-248. [18] A. d'Onofrio, A general framework for modeling tumor-imune system competition and immunotherapy: Matematical analysis and biomedical inferences, Physica D, 208 (2005), 220-235. doi: 10.1016/j.physd.2005.06.032. [19] A. d'Onofrio, A. Fasano and B. Monechi, A generalization of Gompertz law compatible with the Gyllenberg-Webb theory for tumour growth, Math. Biosciences, 230 (2011), 45-54. doi: 10.1016/j.mbs.2011.01.001. [20] D. D. Pestana and S.Velosa, "Introduçāo à Probabilidade e à Estatística," Fundaçāo Calouste Gulbenkian, Lisboa, 2008. [21] D. D. Pestana, S. M. Aleixo and J. L. Rocha, Regular variation, paretian distributions, and the interplay of light and heavy tails in the fractality of asymptotic models, in "Chaos Theory: Modeling, Simulation and Applications" (eds. C. H. Skiadas, Y. Dimotikalis and C. Skiadas), World Scientific Publishing Co, (2011), 309-316. [22] J. L. Rocha and J. Sousa Ramos, Weighted kneading theory of one-dimensional maps with a hole, Int. J. Math. Math. Sci., 38 (2004), 2019-2038. doi: 10.1155/S016117120430428X. [23] J. L. Rocha and S. M. Aleixo, Dynamical analysis in growth models: Blumberg's equation, Discrete Contin. Dyn. Syst.-Ser.B, 18 (2013), 783-795. [24] S. Sakanoue, Extended logistic model for growth of single-species populations, Ecol. Model., 205 (2007), 159-168. [25] H. Schättler, U. Ledzewicz and B. Cardwell, Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis, Math. Biosci. Eng., 8 (2011), 355-369. doi: 10.3934/mbe.2011.8.355. [26] D. Singer, Stable orbits and bifurcations of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267. [27] A. Tsoularis , Analysis of logistic growth models, Res. Lett. Inf. Math. Sci., 2 (2001), 23-46. [28] M. E. Turner Jr., E. L. Bradley Jr., K. A. Kirk and K. M. Pruitt, A theory of growth, Math. Biosci., 29 (1976), 367-373. [29] P. Waliszewski and J. Konarski, The gompertzian curve reveals fractal properties of tumour growth, Chaos Solitons & Fractals, 16 (2003), 665-674. [30] P. Waliszewski and J. Konarski, A mystery of the Gompertz function, in "Fractals in Biology and Medicine" (eds. G. A. Losa, T. F. Nonnenmacher and E. R. Weibel), Birkhäuser, Basel, (2005), 277-286. [31] P. Waliszewski, A principle of fractal-stochastic dualism and Gompertzian dynamics of growth and self-organization, Byosystems, 82 (2005), 61-73. [32] P. Waliszewski, A principle of fractal-stochastic dualism, couplings, complementarity growth, J. Control Eng. and Appl. Informatics, 4 (2009), 45-52.
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