American Institute of Mathematical Sciences

Advanced
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 & 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, (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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

W. Melo and S. van Strien, One-Dimensional Dynamics,, Springer, (1993).  doi: 10.1007/978-3-642-78043-1.  Google Scholar

[23]

C. Mira, Chaotic Dynamics. From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism,, World Scientific, (1987).  doi: 10.1142/0413.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[37]

D. Singer, Stable orbits and bifurcations of maps of the interval,, SIAM J. Appl. Math., 35 (1978), 260.  doi: 10.1137/0135020.  Google Scholar

[38]

A. Tsoularis and J. Wallace, Analysis of logistic growth models,, Math. Biosci., 179 (2002), 21.  doi: 10.1016/S0025-5564(02)00096-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

W. Melo and S. van Strien, One-Dimensional Dynamics,, Springer, (1993).  doi: 10.1007/978-3-642-78043-1.  Google Scholar

[23]

C. Mira, Chaotic Dynamics. From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism,, World Scientific, (1987).  doi: 10.1142/0413.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[37]

D. Singer, Stable orbits and bifurcations of maps of the interval,, SIAM J. Appl. Math., 35 (1978), 260.  doi: 10.1137/0135020.  Google Scholar

[38]

A. Tsoularis and J. Wallace, Analysis of logistic growth models,, Math. Biosci., 179 (2002), 21.  doi: 10.1016/S0025-5564(02)00096-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 & 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 & Continuous Dynamical Systems - A, 2017, 37 (4) : 2115-2140. doi: 10.3934/dcds.2017091
[4]

Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367
[5]

Dongmei Xiao. Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 699-719. doi: 10.3934/dcdsb.2016.21.699
[6]

Dan Liu, Shigui Ruan, Deming Zhu. Nongeneric bifurcations near heterodimensional cycles with inclination flip in $\mathbb{R}^4$. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1511-1532. doi: 10.3934/dcdss.2011.4.1511
[7]

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 & Continuous Dynamical Systems - B, 2018, 23 (4) : 1721-1737. doi: 10.3934/dcdsb.2018073
[8]

Hicham Zmarrou, Ale Jan Homburg. Dynamics and bifurcations of random circle diffeomorphism. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 719-731. doi: 10.3934/dcdsb.2008.10.719
[9]

Carles Bonet-Revés, Tere M-Seara. Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3545-3601. doi: 10.3934/dcds.2016.36.3545
[10]

Jorge Rebaza. Bifurcations and periodic orbits in variable population interactions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2997-3012. doi: 10.3934/cpaa.2013.12.2997
[11]

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
[12]

Antonio Pumariño, José Ángel Rodríguez, Joan Carles Tatjer, Enrique Vigil. Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 523-541. doi: 10.3934/dcdsb.2014.19.523
[13]

Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129
[14]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[15]

Ale Jan Homburg, Todd Young. Intermittency and Jakobson's theorem near saddle-node bifurcations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 21-58. doi: 10.3934/dcds.2007.17.21
[16]

Nika Lazaryan, Hassan Sedaghat. Extinction and the Allee effect in an age structured Ricker population model with inter-stage interaction. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 731-747. doi: 10.3934/dcdsb.2018040
[17]

Yuanshi Wang, Hong Wu, Shigui Ruan. Global dynamics and bifurcations in a four-dimensional replicator system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 259-271. doi: 10.3934/dcdsb.2013.18.259
[18]

Moitri Sen, Malay Banerjee, Yasuhiro Takeuchi. Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model. Mathematical Biosciences & Engineering, 2018, 15 (4) : 883-904. doi: 10.3934/mbe.2018040
[19]

Yujing Gao, Bingtuan Li. Dynamics of a ratio-dependent predator-prey system with a strong Allee effect. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2283-2313. doi: 10.3934/dcdsb.2013.18.2283
[20]

Yuying Liu, Yuxiao Guo, Junjie Wei. Dynamics in a diffusive predator-prey system with stage structure and strong allee effect. Communications on Pure & Applied Analysis, 2020, 19 (2) : 883-910. doi: 10.3934/cpaa.2020040

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences