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Explosion birth and extinction: Double big bang bifurcations and Allee effect in Tsoularis-Wallace's growth models
1. | Instituto Superior de Engenharia de Lisboa - ISEL, ADM and CEAUL, Rua Conselheiro Emídio Navarro, 1, 1959-007 Lisboa |
2. | INSA, University of Toulouse, 135 Avenue du Rangueil, 31077 Toulouse |
3. | LAAS-CNRS, INSA, University of Toulouse, 7 Avenue du Colonel Roche, 31077 Toulouse |
References:
[1] |
W. C. Allee, Animal Aggregations,, University of Chicago Press, (1931). Google Scholar |
[2] |
S. M. Aleixo, J. L. Rocha and D. D. Pestana, Populational growth models proportional to beta densities with Allee effect,, Am. Inst. of Phys., 1124 (2009), 3.
|
[3] |
S. M. Aleixo and J. L. Rocha, Generalized models from Beta(p,2) densities with strong Allee effect: Dynamical approach,, J. Comput. Inf. Technol., 20 (2012), 201.
doi: 10.2498/cit.1002098. |
[4] |
V. Avrutin, G. Wackenhut and M. Schanz, On dynamical systems with piecewise defined system functions,, in Proc. Int. Conf. Tools for Mathematical Modelling (Mathtols'99), 4 (1999), 4.
|
[5] |
V. Avrutin and M. Schanz, Multi-parametric bifurcations in a scalar piecewise-linear map,, Nonlinearity, 19 (2006), 531.
doi: 10.1088/0951-7715/19/3/001. |
[6] |
V. Avrutin, M. Schanz and S. Banerjee, Multi-parametric bifurcations in a piecewise-linear discontinuous map,, Nonlinearity, 19 (2006), 1875.
doi: 10.1088/0951-7715/19/8/007. |
[7] |
V. Avrutin, A. Granados and M. Schanz, Sufficient conditions for a period incrementing big bang bifurcation in one-dimensional map,, Nonlinearity, 24 (2011), 2575.
doi: 10.1088/0951-7715/24/9/012. |
[8] |
J.-P. Carcasses, Sur Quelques Structures Complexes de Bifurcations de Systemes Dynamiques,, Doctorat de L'Universite Paul Sabatier, (1990). Google Scholar |
[9] |
B. Dennis, Allee effects: Population growth, critical density and the chance of extinction,, Nat. Res. Mod., 3 (1989), 481.
|
[10] |
S. Elaydi and R. J. Sacker, Population models with Allee effect: A new model,, J. Bio. Dyn., 4 (2010), 397.
doi: 10.1080/17513750903377434. |
[11] |
D. Fournier-Prunaret, The bifurcation structure of a family of degree one circle endomorphisms,, Int. J. Bifurc. Chaos, 1 (1991), 823.
doi: 10.1142/S0218127491000609. |
[12] |
H. Fujikawa, A. Kai and S. Morozomi, A new logistic model for Escherichia coli growth at constant and dynamic temperatures,, Food Microbiol., 21 (2004), 501.
doi: 10.1016/j.fm.2004.01.007. |
[13] |
L. Gardini, U. Merlone and F. Tramontana, Inertia in binary choices: Continuity breaking and big-bang bifurcation points,, J. Econ. Behav. Organ, 80 (2011), 153.
doi: 10.1016/j.jebo.2011.03.004. |
[14] |
L. Gardini, V. Avrutin and I. Sushko, Codimension-2 border collision bifurcations in one-dimensional discontinuous piecewise smooth maps,, Int. J. Bifurc. Chaos, 24 (2014).
doi: 10.1142/S0218127414500242. |
[15] |
M. Gyllenberg, A. V. Osipov and G. Soderbacka, Bifurcation analysis of a metapopulation model with sources and sinks,, J. Nonlinear Sci., 6 (1996), 329.
doi: 10.1007/BF02433474. |
[16] |
A. M. Kramer, B. Dennis, A. M. Liebhold and J. M. Drake, The evidence for Allee effects,, Popul. Ecol., 51 (2009), 341.
doi: 10.1007/s10144-009-0152-6. |
[17] |
A. K. Laird, Dynamics of tumour growth,, Br. J. Cancer, 18 (1964), 490. Google Scholar |
[18] |
A. K. Laird, S. A. Tyler and A. D. Barton, Dynamics of normal growth,, Growth, 29 (1965), 233. Google Scholar |
[19] |
M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms,, Theor. Popul. Biol., 43 (1993), 141.
doi: 10.1006/tpbi.1993.1007. |
[20] |
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.
doi: 10.1016/j.physa.2008.06.015. |
[21] |
M. Marusić and Z. Bajzer, Generalized two-parameter equation of growth,, J. Math. Anal. Appl., 179 (1993), 446.
doi: 10.1006/jmaa.1993.1361. |
[22] |
W. Melo and S. van Strien, One-Dimensional Dynamics,, Springer, (1993).
doi: 10.1007/978-3-642-78043-1. |
[23] |
C. Mira, Chaotic Dynamics. From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism,, World Scientific, (1987).
doi: 10.1142/0413. |
[24] |
C. Mira, On some codimension three bifurcations occuring in maps. Spring area-crossroad area transitions,, in Proc. European Conference on Iteration Theory (ECIT 1991), (1991), 168. Google Scholar |
[25] |
C. Mira, L. Gardini, A. Barugola and J.-C. Cathala, Chaotic Dynamics in Two-Dimensional Noninvertible Maps,, World Scientific, (1996).
doi: 10.1142/9789812798732. |
[26] |
C. Mira and L. Gardini, From the box-within-a-box bifurcation organization to the Julia set. Part I: Revisited properties of the sets generated by a quadratic complex map with a real parameter},, Int. J. Bifurc. Chaos, 19 (2009), 281.
doi: 10.1142/S0218127409022877. |
[27] |
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.
doi: 10.1016/j.mbs.2011.01.001. |
[28] |
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, (2011), 309.
doi: 10.1142/9789814350341_0036. |
[29] |
J. L. Rocha and S. M. Aleixo, An extension of gompertzian growth dynamics: Weibull and Fréchet models,, Math. Biosci. Eng., 10 (2013), 379.
doi: 10.3934/mbe.2013.10.379. |
[30] |
J. L. Rocha, D. Fournier-Prunaret and A.-K. Taha, Strong and weak Allee effects and chaotic dynamics in Richards' growths,, Discrete Contin. Dyn. Syst.-Ser.B, 18 (2013), 2397.
doi: 10.3934/dcdsb.2013.18.2397. |
[31] |
J. L. Rocha and S. M. Aleixo, Dynamical analysis in growth models: Blumberg's equation,, Discrete Contin. Dyn. Syst.-Ser.B, 18 (2013), 783.
doi: 10.3934/dcdsb.2013.18.783. |
[32] |
J. L. Rocha, D. Fournier-Prunaret and A.-K. Taha, Big bang bifurcations and Allee effect in Blumberg's dynamics,, Nonlinear Dyn., 77 (2014), 1749.
doi: 10.1007/s11071-014-1415-0. |
[33] |
J. L. Rocha, A.-K. Taha and D. Fournier-Prunaret, Symbolic dynamics and big bang bifurcation in Weibull-Gompertz-Fréchet's growth models,, Appl. Math. Inf. Sci., 9 (2015), 2377.
doi: 10.12785/amis/090520. |
[34] |
S. Schreiber, Chaos and population disappearances in simple ecological models,, J. Math. Biol., 42 (2001), 239.
doi: 10.1007/s002850000070. |
[35] |
S. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models,, Theor. Popul. Biol., 64 (2003), 201.
doi: 10.1016/S0040-5809(03)00072-8. |
[36] |
A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps,, Kluwer Academic Publishers, (1997).
doi: 10.1007/978-94-015-8897-3. |
[37] |
D. Singer, Stable orbits and bifurcations of maps of the interval,, SIAM J. Appl. Math., 35 (1978), 260.
doi: 10.1137/0135020. |
[38] |
A. Tsoularis and J. Wallace, Analysis of logistic growth models,, Math. Biosci., 179 (2002), 21.
doi: 10.1016/S0025-5564(02)00096-2. |
[39] |
M. E. Turner, E. L. Bradley, K. A. Kirk and K. M. Pruitt, A theory of growth,, Math. Biosci., 29 (1976), 367.
doi: 10.1016/0025-5564(76)90112-7. |
[40] |
E. Uleri, S. Beltrami, J. Gordon, A. Dolei and I. K. Sariyer, Extinction of tumor antigen expression by SF2/ASF in JCV-transformed cells,, Genes Cancer, 2 (2011), 728.
doi: 10.1177/1947601911424578. |
show all references
References:
[1] |
W. C. Allee, Animal Aggregations,, University of Chicago Press, (1931). Google Scholar |
[2] |
S. M. Aleixo, J. L. Rocha and D. D. Pestana, Populational growth models proportional to beta densities with Allee effect,, Am. Inst. of Phys., 1124 (2009), 3.
|
[3] |
S. M. Aleixo and J. L. Rocha, Generalized models from Beta(p,2) densities with strong Allee effect: Dynamical approach,, J. Comput. Inf. Technol., 20 (2012), 201.
doi: 10.2498/cit.1002098. |
[4] |
V. Avrutin, G. Wackenhut and M. Schanz, On dynamical systems with piecewise defined system functions,, in Proc. Int. Conf. Tools for Mathematical Modelling (Mathtols'99), 4 (1999), 4.
|
[5] |
V. Avrutin and M. Schanz, Multi-parametric bifurcations in a scalar piecewise-linear map,, Nonlinearity, 19 (2006), 531.
doi: 10.1088/0951-7715/19/3/001. |
[6] |
V. Avrutin, M. Schanz and S. Banerjee, Multi-parametric bifurcations in a piecewise-linear discontinuous map,, Nonlinearity, 19 (2006), 1875.
doi: 10.1088/0951-7715/19/8/007. |
[7] |
V. Avrutin, A. Granados and M. Schanz, Sufficient conditions for a period incrementing big bang bifurcation in one-dimensional map,, Nonlinearity, 24 (2011), 2575.
doi: 10.1088/0951-7715/24/9/012. |
[8] |
J.-P. Carcasses, Sur Quelques Structures Complexes de Bifurcations de Systemes Dynamiques,, Doctorat de L'Universite Paul Sabatier, (1990). Google Scholar |
[9] |
B. Dennis, Allee effects: Population growth, critical density and the chance of extinction,, Nat. Res. Mod., 3 (1989), 481.
|
[10] |
S. Elaydi and R. J. Sacker, Population models with Allee effect: A new model,, J. Bio. Dyn., 4 (2010), 397.
doi: 10.1080/17513750903377434. |
[11] |
D. Fournier-Prunaret, The bifurcation structure of a family of degree one circle endomorphisms,, Int. J. Bifurc. Chaos, 1 (1991), 823.
doi: 10.1142/S0218127491000609. |
[12] |
H. Fujikawa, A. Kai and S. Morozomi, A new logistic model for Escherichia coli growth at constant and dynamic temperatures,, Food Microbiol., 21 (2004), 501.
doi: 10.1016/j.fm.2004.01.007. |
[13] |
L. Gardini, U. Merlone and F. Tramontana, Inertia in binary choices: Continuity breaking and big-bang bifurcation points,, J. Econ. Behav. Organ, 80 (2011), 153.
doi: 10.1016/j.jebo.2011.03.004. |
[14] |
L. Gardini, V. Avrutin and I. Sushko, Codimension-2 border collision bifurcations in one-dimensional discontinuous piecewise smooth maps,, Int. J. Bifurc. Chaos, 24 (2014).
doi: 10.1142/S0218127414500242. |
[15] |
M. Gyllenberg, A. V. Osipov and G. Soderbacka, Bifurcation analysis of a metapopulation model with sources and sinks,, J. Nonlinear Sci., 6 (1996), 329.
doi: 10.1007/BF02433474. |
[16] |
A. M. Kramer, B. Dennis, A. M. Liebhold and J. M. Drake, The evidence for Allee effects,, Popul. Ecol., 51 (2009), 341.
doi: 10.1007/s10144-009-0152-6. |
[17] |
A. K. Laird, Dynamics of tumour growth,, Br. J. Cancer, 18 (1964), 490. Google Scholar |
[18] |
A. K. Laird, S. A. Tyler and A. D. Barton, Dynamics of normal growth,, Growth, 29 (1965), 233. Google Scholar |
[19] |
M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms,, Theor. Popul. Biol., 43 (1993), 141.
doi: 10.1006/tpbi.1993.1007. |
[20] |
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.
doi: 10.1016/j.physa.2008.06.015. |
[21] |
M. Marusić and Z. Bajzer, Generalized two-parameter equation of growth,, J. Math. Anal. Appl., 179 (1993), 446.
doi: 10.1006/jmaa.1993.1361. |
[22] |
W. Melo and S. van Strien, One-Dimensional Dynamics,, Springer, (1993).
doi: 10.1007/978-3-642-78043-1. |
[23] |
C. Mira, Chaotic Dynamics. From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism,, World Scientific, (1987).
doi: 10.1142/0413. |
[24] |
C. Mira, On some codimension three bifurcations occuring in maps. Spring area-crossroad area transitions,, in Proc. European Conference on Iteration Theory (ECIT 1991), (1991), 168. Google Scholar |
[25] |
C. Mira, L. Gardini, A. Barugola and J.-C. Cathala, Chaotic Dynamics in Two-Dimensional Noninvertible Maps,, World Scientific, (1996).
doi: 10.1142/9789812798732. |
[26] |
C. Mira and L. Gardini, From the box-within-a-box bifurcation organization to the Julia set. Part I: Revisited properties of the sets generated by a quadratic complex map with a real parameter},, Int. J. Bifurc. Chaos, 19 (2009), 281.
doi: 10.1142/S0218127409022877. |
[27] |
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.
doi: 10.1016/j.mbs.2011.01.001. |
[28] |
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, (2011), 309.
doi: 10.1142/9789814350341_0036. |
[29] |
J. L. Rocha and S. M. Aleixo, An extension of gompertzian growth dynamics: Weibull and Fréchet models,, Math. Biosci. Eng., 10 (2013), 379.
doi: 10.3934/mbe.2013.10.379. |
[30] |
J. L. Rocha, D. Fournier-Prunaret and A.-K. Taha, Strong and weak Allee effects and chaotic dynamics in Richards' growths,, Discrete Contin. Dyn. Syst.-Ser.B, 18 (2013), 2397.
doi: 10.3934/dcdsb.2013.18.2397. |
[31] |
J. L. Rocha and S. M. Aleixo, Dynamical analysis in growth models: Blumberg's equation,, Discrete Contin. Dyn. Syst.-Ser.B, 18 (2013), 783.
doi: 10.3934/dcdsb.2013.18.783. |
[32] |
J. L. Rocha, D. Fournier-Prunaret and A.-K. Taha, Big bang bifurcations and Allee effect in Blumberg's dynamics,, Nonlinear Dyn., 77 (2014), 1749.
doi: 10.1007/s11071-014-1415-0. |
[33] |
J. L. Rocha, A.-K. Taha and D. Fournier-Prunaret, Symbolic dynamics and big bang bifurcation in Weibull-Gompertz-Fréchet's growth models,, Appl. Math. Inf. Sci., 9 (2015), 2377.
doi: 10.12785/amis/090520. |
[34] |
S. Schreiber, Chaos and population disappearances in simple ecological models,, J. Math. Biol., 42 (2001), 239.
doi: 10.1007/s002850000070. |
[35] |
S. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models,, Theor. Popul. Biol., 64 (2003), 201.
doi: 10.1016/S0040-5809(03)00072-8. |
[36] |
A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps,, Kluwer Academic Publishers, (1997).
doi: 10.1007/978-94-015-8897-3. |
[37] |
D. Singer, Stable orbits and bifurcations of maps of the interval,, SIAM J. Appl. Math., 35 (1978), 260.
doi: 10.1137/0135020. |
[38] |
A. Tsoularis and J. Wallace, Analysis of logistic growth models,, Math. Biosci., 179 (2002), 21.
doi: 10.1016/S0025-5564(02)00096-2. |
[39] |
M. E. Turner, E. L. Bradley, K. A. Kirk and K. M. Pruitt, A theory of growth,, Math. Biosci., 29 (1976), 367.
doi: 10.1016/0025-5564(76)90112-7. |
[40] |
E. Uleri, S. Beltrami, J. Gordon, A. Dolei and I. K. Sariyer, Extinction of tumor antigen expression by SF2/ASF in JCV-transformed cells,, Genes Cancer, 2 (2011), 728.
doi: 10.1177/1947601911424578. |
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