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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 |
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. |
show all references
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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