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Strong and weak Allee effects and chaotic dynamics in Richards' growths
1. | Instituto Superior de Engenharia de Lisboa - ISEL, ADM and CEAUL, Rua Conselheiro Emídio Navarro, 1, 1959-007 Lisboa |
2. | LAAS-CNRS, INSA, University of Toulouse, 7 Avenue du Colonel Roche, 31077 Toulouse, France |
3. | INSA, University of Toulouse, 135 Avenue du Rangueil, 31077 Toulouse, France |
References:
[1] |
S. M. Aleixo, J. L. Rocha and D. D. Pestana, Populational growth models proportional to beta densities with Allee effect, Amer. Inst. Phys., 1124 (2009), 3-12. |
[2] |
S. M. Aleixo and J. L. Rocha, Generalized models from Beta(p,2) densities with strong Allee effect: dynamical approach, Journal of Computing and Information Technology, 3 (2012), 201-207.
doi: 10.2498/cit.1002098. |
[3] |
L. Berec, E. Angulo and F. Courchamp, Multiple Allee effects and population management, Trends in Ecology & Evolution, 22 (2007), 185-191.
doi: 10.1016/j.tree.2006.12.002. |
[4] |
C. P. D. Birch, A new generalized logistic sigmoid growth equation compared with the Richards growth equation, Annals of Botany, 83 (1999), 713-723.
doi: 10.1006/anbo.1999.0877. |
[5] |
D. S. Boukal and L. Berec, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters, Journal of Theoretical Biology, 218 (2002), 375-394.
doi: 10.1006/jtbi.2002.3084. |
[6] |
C. E. Brassil, Mean time to extinction of a metapopulation with an Allee effect, Ecological Modelling, 143 (2001), 9-16.
doi: 10.1016/S0304-3800(01)00351-9. |
[7] |
J. P. Carcassès, An algorithm to determine the nature and the transitions of communication areas generated by a one-dimensional map, in Proc. European Conference on Iteration Theory (ECIT 1991), J. P Lampreia, J. Llibre et al. (Eds.), World Scientific, Singapore (1992), 27-38. |
[8] |
J. P. Carcassès, Determination of different configurations of fold and flip bifurcation curves of a one or two-dimensional map, International Journal of Bifurcation and Chaos, 3 (1993), 869-902.
doi: 10.1142/S0218127493000763. |
[9] |
C. W. Clark, "Mathematical Bioeconomics: The Optimal Management of Renewable Resources," $2^{nd}$ edition, John Wiley and Sons, 1990. |
[10] |
S. Elaydi and R. J. Sacker, Population models with Allee effect: A new model, Journal of Biological Dynamics, 4 (2009), 397-408.
doi: 10.1080/17513750903377434. |
[11] |
X. Fauvergue, J-C. Malusa, L. Giuge and F. Courchamp, Invading parasitoids suffer no Allee effect: A manipulative field experiment, Ecology, 88 (2007), 2392-2403.
doi: 10.1890/06-1238.1. |
[12] |
D. Fournier-Prunaret, The bifurcation structure of a family of degree one circle endomorphisms, International Journal of Bifurcation and Chaos, 1 (1991), 823-838.
doi: 10.1142/S0218127491000609. |
[13] |
H. Fujikawa, A. Kai and S. Morozomi, A new logistic model for Escherichia coli growth at constant and dynamic temperatures, Food Microbiology, 21 (2004), 501-509.
doi: 10.1016/j.fm.2004.01.007. |
[14] |
E. González-Olivares, B. González-Yañez, J. Mena-Lorca and J. D. Flores, Uniqueness of limit cycles and multiple attractors in a Gause-type model with nonmonotonic functional response and Allee effect on prey, Mathematical Biosciences and Engineering (MBE), 10 (2013), 345-367.
doi: 10.3934/mbe.2013.10.345. |
[15] |
M. Gyllenberg, A. V. Osipov and G. Sderbacka, Bifurcation analysis of a metapopulation model with sources and sinks, Journal of Nonlinear Science, 6 (1996), 329-366.
doi: 10.1007/BF02433474. |
[16] |
H. Kawakami, Bifurcations of periodic responses in forced dynamic nonlinear circuits: Computation of bifurcation values of the system parameters, IEEE Trans. Circuits and Systems, CAS-31 (1984), 248-260.
doi: 10.1109/TCS.1984.1085495. |
[17] |
A. M. Kramer, B. Dennis, A. M. Liebhold and J. M. Drake, The evidence for Allee effects, Population Ecology, 51 (2009), 341-354.
doi: 10.1007/s10144-009-0152-6. |
[18] |
H. D. Kuhi, E. Kebreab, S. Lopez and J. France, A comparative evaluation of functions for describing the relationship between live-weight gain and metabolizable energy intake in turkeys, J. Agricultural Sci., 142 (2004), 691-695. |
[19] |
J. P. Lampreia and J. Sousa Ramos, Symbolic dynamics of bimodal maps, Portugaliae Math., 54 (1997), 1-18. |
[20] |
M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43 (1993), 141-158.
doi: 10.1006/tpbi.1993.1007. |
[21] |
D. Li, Z. Zhang, Z. Ma, B. Xie and R. Wang, Allee effect and a catastrophe model of population dynamics, Discrete and Continuous Dynamical Systems - Series B (DCDS-B), 4 (2004), 629-634.
doi: 10.3934/dcdsb.2004.4.629. |
[22] |
D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Codings," $2^{nd}$ edition, Cambridge University Press, Cambridge, 1999.
doi: 10.1017/CBO9780511626302. |
[23] |
G. Livadiotis and S. Elaydi, General Allee effect in two-species population biology, J. Bio. Dyn., 6 (2012), 959-973.
doi: 10.1080/17513758.2012.700075. |
[24] |
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. |
[25] |
W. Melo and S. van Strien, "One-Dimensional Dynamics," $1^{nd}$ edition, Springer-Verlag, New York, 1993. |
[26] |
V. Méndez, C. Sans, I. Lopis and D. Campos, Extinction conditions for isolated populations with Allee effect, Mathematical Biosciences, 232 (2011), 78-86.
doi: 10.1016/j.mbs.2011.04.005. |
[27] |
C. Mira, "Chaotic Dynamics. From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism," World Scientific, Singapore, 1987. |
[28] |
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. |
[29] |
M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polish. Sci., 27 (1979), 167-169. |
[30] |
H. T. Odum and W. C. Allee, A note on the stable point of populations showing both intraspecific cooperation and disoperation, Ecology, 35 (1954), 95-97.
doi: 10.2307/1931412. |
[31] |
F. J. Richards, A flexible growth function for empirical use, Journal of Experimental Botany, 10 (1959), 290-301.
doi: 10.1093/jxb/10.2.290. |
[32] |
J. L. Rocha and S. M. Aleixo, Modeling Allee effect from Beta(p,2) densities, Proc. ITI 2012, 34th Int. Conf. Information Technology Interfaces, (2012), 461-466. |
[33] |
J. L. Rocha and S. M. Aleixo, An extension of Gompertzian growth dynamics: Weibull and Fréchet models, Mathematical Biosciences and Engineering (MBE), 10 (2013), 379-398.
doi: 10.3934/mbe.2013.10.379. |
[34] |
J. L. Rocha and S. M. Aleixo, Dynamical analysis in growth models: Blumberg's equation, Discrete and Continuous Dynamical Systems - Series B (DCDS-B), 18 (2013), 783-795.
doi: 10.3934/dcdsb.2013.18.783. |
[35] |
S. J. Schreiber, Chaos and population disappearances in simple ecological models, Journal of Mathematical Biology, 42 (2001), 239-260.
doi: 10.1007/s002850000070. |
[36] |
S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theoretical Population Biology, 64 (2003), 201-209.
doi: 10.1016/S0040-5809(03)00072-8. |
[37] |
O. M. Šarkovs'kiĭ, On cycles and the structure of a continuous mapping, Ukrain. Math. Ž., 17 (1965), 104-111. |
[38] |
P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185-190.
doi: 10.2307/3547011. |
[39] |
H. Thieme, T. Dhirasakdanon, Z. Han and R. Trevino, Species decline and extinction: Synergy of infectious disease and Allee effect?, Journal of Biological Dynamics, 3 (2009), 305-323.
doi: 10.1080/17513750802376313. |
[40] |
A. Tsoularis and J. Wallace, Analysis of logistic growth models, Mathematical Biosciences, 179 (2002), 21-55.
doi: 10.1016/S0025-5564(02)00096-2. |
[41] |
M. Wang and M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Mathematical Biosciences, 171 (2001), 83-97.
doi: 10.1016/S0025-5564(01)00048-7. |
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, Amer. Inst. Phys., 1124 (2009), 3-12. |
[2] |
S. M. Aleixo and J. L. Rocha, Generalized models from Beta(p,2) densities with strong Allee effect: dynamical approach, Journal of Computing and Information Technology, 3 (2012), 201-207.
doi: 10.2498/cit.1002098. |
[3] |
L. Berec, E. Angulo and F. Courchamp, Multiple Allee effects and population management, Trends in Ecology & Evolution, 22 (2007), 185-191.
doi: 10.1016/j.tree.2006.12.002. |
[4] |
C. P. D. Birch, A new generalized logistic sigmoid growth equation compared with the Richards growth equation, Annals of Botany, 83 (1999), 713-723.
doi: 10.1006/anbo.1999.0877. |
[5] |
D. S. Boukal and L. Berec, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters, Journal of Theoretical Biology, 218 (2002), 375-394.
doi: 10.1006/jtbi.2002.3084. |
[6] |
C. E. Brassil, Mean time to extinction of a metapopulation with an Allee effect, Ecological Modelling, 143 (2001), 9-16.
doi: 10.1016/S0304-3800(01)00351-9. |
[7] |
J. P. Carcassès, An algorithm to determine the nature and the transitions of communication areas generated by a one-dimensional map, in Proc. European Conference on Iteration Theory (ECIT 1991), J. P Lampreia, J. Llibre et al. (Eds.), World Scientific, Singapore (1992), 27-38. |
[8] |
J. P. Carcassès, Determination of different configurations of fold and flip bifurcation curves of a one or two-dimensional map, International Journal of Bifurcation and Chaos, 3 (1993), 869-902.
doi: 10.1142/S0218127493000763. |
[9] |
C. W. Clark, "Mathematical Bioeconomics: The Optimal Management of Renewable Resources," $2^{nd}$ edition, John Wiley and Sons, 1990. |
[10] |
S. Elaydi and R. J. Sacker, Population models with Allee effect: A new model, Journal of Biological Dynamics, 4 (2009), 397-408.
doi: 10.1080/17513750903377434. |
[11] |
X. Fauvergue, J-C. Malusa, L. Giuge and F. Courchamp, Invading parasitoids suffer no Allee effect: A manipulative field experiment, Ecology, 88 (2007), 2392-2403.
doi: 10.1890/06-1238.1. |
[12] |
D. Fournier-Prunaret, The bifurcation structure of a family of degree one circle endomorphisms, International Journal of Bifurcation and Chaos, 1 (1991), 823-838.
doi: 10.1142/S0218127491000609. |
[13] |
H. Fujikawa, A. Kai and S. Morozomi, A new logistic model for Escherichia coli growth at constant and dynamic temperatures, Food Microbiology, 21 (2004), 501-509.
doi: 10.1016/j.fm.2004.01.007. |
[14] |
E. González-Olivares, B. González-Yañez, J. Mena-Lorca and J. D. Flores, Uniqueness of limit cycles and multiple attractors in a Gause-type model with nonmonotonic functional response and Allee effect on prey, Mathematical Biosciences and Engineering (MBE), 10 (2013), 345-367.
doi: 10.3934/mbe.2013.10.345. |
[15] |
M. Gyllenberg, A. V. Osipov and G. Sderbacka, Bifurcation analysis of a metapopulation model with sources and sinks, Journal of Nonlinear Science, 6 (1996), 329-366.
doi: 10.1007/BF02433474. |
[16] |
H. Kawakami, Bifurcations of periodic responses in forced dynamic nonlinear circuits: Computation of bifurcation values of the system parameters, IEEE Trans. Circuits and Systems, CAS-31 (1984), 248-260.
doi: 10.1109/TCS.1984.1085495. |
[17] |
A. M. Kramer, B. Dennis, A. M. Liebhold and J. M. Drake, The evidence for Allee effects, Population Ecology, 51 (2009), 341-354.
doi: 10.1007/s10144-009-0152-6. |
[18] |
H. D. Kuhi, E. Kebreab, S. Lopez and J. France, A comparative evaluation of functions for describing the relationship between live-weight gain and metabolizable energy intake in turkeys, J. Agricultural Sci., 142 (2004), 691-695. |
[19] |
J. P. Lampreia and J. Sousa Ramos, Symbolic dynamics of bimodal maps, Portugaliae Math., 54 (1997), 1-18. |
[20] |
M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43 (1993), 141-158.
doi: 10.1006/tpbi.1993.1007. |
[21] |
D. Li, Z. Zhang, Z. Ma, B. Xie and R. Wang, Allee effect and a catastrophe model of population dynamics, Discrete and Continuous Dynamical Systems - Series B (DCDS-B), 4 (2004), 629-634.
doi: 10.3934/dcdsb.2004.4.629. |
[22] |
D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Codings," $2^{nd}$ edition, Cambridge University Press, Cambridge, 1999.
doi: 10.1017/CBO9780511626302. |
[23] |
G. Livadiotis and S. Elaydi, General Allee effect in two-species population biology, J. Bio. Dyn., 6 (2012), 959-973.
doi: 10.1080/17513758.2012.700075. |
[24] |
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. |
[25] |
W. Melo and S. van Strien, "One-Dimensional Dynamics," $1^{nd}$ edition, Springer-Verlag, New York, 1993. |
[26] |
V. Méndez, C. Sans, I. Lopis and D. Campos, Extinction conditions for isolated populations with Allee effect, Mathematical Biosciences, 232 (2011), 78-86.
doi: 10.1016/j.mbs.2011.04.005. |
[27] |
C. Mira, "Chaotic Dynamics. From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism," World Scientific, Singapore, 1987. |
[28] |
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. |
[29] |
M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polish. Sci., 27 (1979), 167-169. |
[30] |
H. T. Odum and W. C. Allee, A note on the stable point of populations showing both intraspecific cooperation and disoperation, Ecology, 35 (1954), 95-97.
doi: 10.2307/1931412. |
[31] |
F. J. Richards, A flexible growth function for empirical use, Journal of Experimental Botany, 10 (1959), 290-301.
doi: 10.1093/jxb/10.2.290. |
[32] |
J. L. Rocha and S. M. Aleixo, Modeling Allee effect from Beta(p,2) densities, Proc. ITI 2012, 34th Int. Conf. Information Technology Interfaces, (2012), 461-466. |
[33] |
J. L. Rocha and S. M. Aleixo, An extension of Gompertzian growth dynamics: Weibull and Fréchet models, Mathematical Biosciences and Engineering (MBE), 10 (2013), 379-398.
doi: 10.3934/mbe.2013.10.379. |
[34] |
J. L. Rocha and S. M. Aleixo, Dynamical analysis in growth models: Blumberg's equation, Discrete and Continuous Dynamical Systems - Series B (DCDS-B), 18 (2013), 783-795.
doi: 10.3934/dcdsb.2013.18.783. |
[35] |
S. J. Schreiber, Chaos and population disappearances in simple ecological models, Journal of Mathematical Biology, 42 (2001), 239-260.
doi: 10.1007/s002850000070. |
[36] |
S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theoretical Population Biology, 64 (2003), 201-209.
doi: 10.1016/S0040-5809(03)00072-8. |
[37] |
O. M. Šarkovs'kiĭ, On cycles and the structure of a continuous mapping, Ukrain. Math. Ž., 17 (1965), 104-111. |
[38] |
P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185-190.
doi: 10.2307/3547011. |
[39] |
H. Thieme, T. Dhirasakdanon, Z. Han and R. Trevino, Species decline and extinction: Synergy of infectious disease and Allee effect?, Journal of Biological Dynamics, 3 (2009), 305-323.
doi: 10.1080/17513750802376313. |
[40] |
A. Tsoularis and J. Wallace, Analysis of logistic growth models, Mathematical Biosciences, 179 (2002), 21-55.
doi: 10.1016/S0025-5564(02)00096-2. |
[41] |
M. Wang and M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Mathematical Biosciences, 171 (2001), 83-97.
doi: 10.1016/S0025-5564(01)00048-7. |
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