American Institute of Mathematical Sciences

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November  2015, 20(9): 3131-3163. doi: 10.3934/dcdsb.2015.20.3131

Explosion birth and extinction: Double big bang bifurcations and Allee effect in Tsoularis-Wallace's growth models

J. Leonel Rocha 1, , Abdel-Kaddous Taha 2, and  Danièle Fournier-Prunaret 3,

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

Received  October 2014 Revised  June 2015 Published  September 2015

This work concerns dynamics and bifurcations properties of a new class of continuous-defined one-dimensional maps: Tsoularis-Wallace's functions. This family of functions naturally incorporates a major focus of ecological research: the Allee effect. We provide a necessary condition for the occurrence of this phenomenon of extinction. To establish this result we introduce the notions of Allee's functions, Allee's effect region and Allee's bifurcation curve. Another central point of our investigation is the study of bifurcation structures for this class of functions, in a three-dimensional parameter space. We verified that under some sufficient conditions, Tsoularis-Wallace's functions have particular bifurcation structures: the big bang and the double big bang bifurcations of the so-called ``box-within-a-box'' type. The double big bang bifurcations are related to the existence of flip codimension--2 points. Moreover, it is verified that these bifurcation cascades converge to different big bang bifurcation curves, where for the corresponding parameter values are associated distinct kinds of boxes. This work contributes to clarify the big bang bifurcation analysis for continuous maps and understand their relationship with explosion birth and extinction phenomena.
Keywords: Population dynamics, bib bang bifurcations, Allee effect., Tsoularis-Wallace's growth models, fold and flip bifurcations.
Mathematics Subject Classification: Primary: 92D25, 37H20; Secondary: 37E0.
Citation: J. Leonel Rocha, Abdel-Kaddous Taha, Danièle Fournier-Prunaret. Explosion birth and extinction: Double big bang bifurcations and Allee effect in Tsoularis-Wallace's growth models. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3131-3163. doi: 10.3934/dcdsb.2015.20.3131
References:
[1]

W. C. Allee, Animal Aggregations, University of Chicago Press, Chicago, 1931.

[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), 3-12.

[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), 201-207. 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), St. Petersburg, 4 (1999), 4-20.

[5]

V. Avrutin and M. Schanz, Multi-parametric bifurcations in a scalar piecewise-linear map, Nonlinearity, 19 (2006), 531-552. 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), 1875-1906. 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), 2575-2598. 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, INSA, Toulouse, 1990.

[9]

B. Dennis, Allee effects: Population growth, critical density and the chance of extinction, Nat. Res. Mod., 3 (1989), 481-538.

[10]

S. Elaydi and R. J. Sacker, Population models with Allee effect: A new model, J. Bio. Dyn., 4 (2010), 397-408. 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), 823-838. 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), 501-509. 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), 153-167. 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), 1450024, 30pp. 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), 329-366. 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), 341-354. doi: 10.1007/s10144-009-0152-6.

[17]

A. K. Laird, Dynamics of tumour growth, Br. J. Cancer, 18 (1964), 490-502.

[18]

A. K. Laird, S. A. Tyler and A. D. Barton, Dynamics of normal growth, Growth, 29 (1965), 233-248.

[19]

M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, Theor. Popul. Biol., 43 (1993) 141-158. 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), 5679-5687. 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), 446-462. doi: 10.1006/jmaa.1993.1361.

[22]

W. Melo and S. van Strien, One-Dimensional Dynamics, Springer, New York, 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, Singapore, 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), J.P. Lampreia, J. Llibre et al. (Eds.), World Scientific, Singapore, 1992, 168-177.

[25]

C. Mira, L. Gardini, A. Barugola and J.-C. Cathala, Chaotic Dynamics in Two-Dimensional Noninvertible Maps, World Scientific, Singapore, 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), 281-327. 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), 45-54. 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, Simulation and Applications (eds. C. H. Skiadas, Y. Dimotikalis and C. Skiadas), World Scientific Publishing Co, 2011, 309-316. 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), 379-398. 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), 2397-2425. 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), 783-795. 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), 1749-1771. 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), 2377-2388. doi: 10.12785/amis/090520.

[34]

S. Schreiber, Chaos and population disappearances in simple ecological models, J. Math. Biol., 42 (2001), 239-260. doi: 10.1007/s002850000070.

[35]

S. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209. 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, Netherlands, 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), 260-267. doi: 10.1137/0135020.

[38]

A. Tsoularis and J. Wallace, Analysis of logistic growth models, Math. Biosci., 179 (2002), 21-55. 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), 367-373. 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), 728-736. doi: 10.1177/1947601911424578.

show all references

References:
[1]

W. C. Allee, Animal Aggregations, University of Chicago Press, Chicago, 1931.

[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), 3-12.

[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), 201-207. 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), St. Petersburg, 4 (1999), 4-20.

[5]

V. Avrutin and M. Schanz, Multi-parametric bifurcations in a scalar piecewise-linear map, Nonlinearity, 19 (2006), 531-552. 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), 1875-1906. 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), 2575-2598. 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, INSA, Toulouse, 1990.

[9]

B. Dennis, Allee effects: Population growth, critical density and the chance of extinction, Nat. Res. Mod., 3 (1989), 481-538.

[10]

S. Elaydi and R. J. Sacker, Population models with Allee effect: A new model, J. Bio. Dyn., 4 (2010), 397-408. 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), 823-838. 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), 501-509. 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), 153-167. 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), 1450024, 30pp. 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), 329-366. 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), 341-354. doi: 10.1007/s10144-009-0152-6.

[17]

A. K. Laird, Dynamics of tumour growth, Br. J. Cancer, 18 (1964), 490-502.

[18]

A. K. Laird, S. A. Tyler and A. D. Barton, Dynamics of normal growth, Growth, 29 (1965), 233-248.

[19]

M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, Theor. Popul. Biol., 43 (1993) 141-158. 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), 5679-5687. 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), 446-462. doi: 10.1006/jmaa.1993.1361.

[22]

W. Melo and S. van Strien, One-Dimensional Dynamics, Springer, New York, 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, Singapore, 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), J.P. Lampreia, J. Llibre et al. (Eds.), World Scientific, Singapore, 1992, 168-177.

[25]

C. Mira, L. Gardini, A. Barugola and J.-C. Cathala, Chaotic Dynamics in Two-Dimensional Noninvertible Maps, World Scientific, Singapore, 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), 281-327. 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), 45-54. 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, Simulation and Applications (eds. C. H. Skiadas, Y. Dimotikalis and C. Skiadas), World Scientific Publishing Co, 2011, 309-316. 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), 379-398. 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), 2397-2425. 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), 783-795. 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), 1749-1771. 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), 2377-2388. doi: 10.12785/amis/090520.

[34]

S. Schreiber, Chaos and population disappearances in simple ecological models, J. Math. Biol., 42 (2001), 239-260. doi: 10.1007/s002850000070.

[35]

S. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209. 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, Netherlands, 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), 260-267. doi: 10.1137/0135020.

[38]

A. Tsoularis and J. Wallace, Analysis of logistic growth models, Math. Biosci., 179 (2002), 21-55. 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), 367-373. 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), 728-736. doi: 10.1177/1947601911424578.

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