-
Previous Article
Boundedness vs.blow-up in a two-species chemotaxis system with two chemicals
- DCDS-B Home
- This Issue
-
Next Article
Quantitative analysis of time delays of glucose - insulin dynamics using artificial pancreas
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, 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. |
[1] |
Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643 |
[2] |
Dianmo Li, Zhen Zhang, Zufei Ma, Baoyu Xie, Rui Wang. Allee effect and a catastrophe model of population dynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 629-634. doi: 10.3934/dcdsb.2004.4.629 |
[3] |
Gheorghe Tigan. Degenerate with respect to parameters fold-Hopf bifurcations. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2115-2140. doi: 10.3934/dcds.2017091 |
[4] |
Xiaoyuan Chang, Junping Shi. Bistable and oscillatory dynamics of Nicholson's blowflies equation with Allee effect. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021242 |
[5] |
Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367 |
[6] |
Dongmei Xiao. Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 699-719. doi: 10.3934/dcdsb.2016.21.699 |
[7] |
Dan Liu, Shigui Ruan, Deming Zhu. Nongeneric bifurcations near heterodimensional cycles with inclination flip in $\mathbb{R}^4$. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1511-1532. doi: 10.3934/dcdss.2011.4.1511 |
[8] |
Chuangxia Huang, Xiaojin Guo, Jinde Cao, Ardak Kashkynbayev. Bistable dynamics on a tick population equation incorporating Allee effect and two different time-varying delays. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022122 |
[9] |
Na Min, Mingxin Wang. Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1721-1737. doi: 10.3934/dcdsb.2018073 |
[10] |
Carles Bonet-Revés, Tere M-Seara. Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3545-3601. doi: 10.3934/dcds.2016.36.3545 |
[11] |
Hicham Zmarrou, Ale Jan Homburg. Dynamics and bifurcations of random circle diffeomorphism. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 719-731. doi: 10.3934/dcdsb.2008.10.719 |
[12] |
Eduardo Liz, Alfonso Ruiz-Herrera. Delayed population models with Allee effects and exploitation. Mathematical Biosciences & Engineering, 2015, 12 (1) : 83-97. doi: 10.3934/mbe.2015.12.83 |
[13] |
Jorge Rebaza. Bifurcations and periodic orbits in variable population interactions. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2997-3012. doi: 10.3934/cpaa.2013.12.2997 |
[14] |
Antonio Pumariño, José Ángel Rodríguez, Joan Carles Tatjer, Enrique Vigil. Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 523-541. doi: 10.3934/dcdsb.2014.19.523 |
[15] |
Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129 |
[16] |
Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 |
[17] |
Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256 |
[18] |
Ale Jan Homburg, Todd Young. Intermittency and Jakobson's theorem near saddle-node bifurcations. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 21-58. doi: 10.3934/dcds.2007.17.21 |
[19] |
Nika Lazaryan, Hassan Sedaghat. Extinction and the Allee effect in an age structured Ricker population model with inter-stage interaction. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 731-747. doi: 10.3934/dcdsb.2018040 |
[20] |
Yuanshi Wang, Hong Wu, Shigui Ruan. Global dynamics and bifurcations in a four-dimensional replicator system. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 259-271. doi: 10.3934/dcdsb.2013.18.259 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]