American Institute of Mathematical Sciences

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November  2013, 18(9): 2397-2425. doi: 10.3934/dcdsb.2013.18.2397

Strong and weak Allee effects and chaotic dynamics in Richards' growths

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

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

Received  April 2013 Revised  June 2013 Published  September 2013

In this paper we define and investigate generalized Richards' growth models with strong and weak Allee effects and no Allee effect. We prove the transition from strong Allee effect to no Allee effect, passing through the weak Allee effect, depending on the implicit conditions, which involve the several parameters considered in the models. New classes of functions describing the existence or not of Allee effect are introduced, a new dynamical approach to Richards' populational growth equation is established. These families of generalized Richards' functions are proportional to the right hand side of the generalized Richards' growth models proposed. Subclasses of strong and weak Allee functions and functions with no Allee effect are characterized. The study of their bifurcation structure is presented in detail, this analysis is done based on the configurations of bifurcation curves and symbolic dynamics techniques. Generically, the dynamics of these functions are classified in the following types: extinction, semi-stability, stability, period doubling, chaos, chaotic semistability and essential extinction. We obtain conditions on the parameter plane for the existence of a weak Allee effect region related to the appearance of cusp points. To support our results, we present fold and flip bifurcations curves and numerical simulations of several bifurcation diagrams.
Keywords: Richards' equation, fold and flip bifurcations, strong and weak Allee effects, Population dynamics, symbolic dynamics..
Mathematics Subject Classification: Primary: 92D25, 37H20; Secondary: 37B1.
Citation: J. Leonel Rocha, Danièle Fournier-Prunaret, Abdel-Kaddous Taha. Strong and weak Allee effects and chaotic dynamics in Richards' growths. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2397-2425. doi: 10.3934/dcdsb.2013.18.2397
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. Google Scholar

[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. doi: 10.2498/cit.1002098. Google Scholar

[3]

L. Berec, E. Angulo and F. Courchamp, Multiple Allee effects and population management,, Trends in Ecology & Evolution, 22 (2007), 185. doi: 10.1016/j.tree.2006.12.002. Google Scholar

[4]

C. P. D. Birch, A new generalized logistic sigmoid growth equation compared with the Richards growth equation,, Annals of Botany, 83 (1999), 713. doi: 10.1006/anbo.1999.0877. Google Scholar

[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. doi: 10.1006/jtbi.2002.3084. Google Scholar

[6]

C. E. Brassil, Mean time to extinction of a metapopulation with an Allee effect,, Ecological Modelling, 143 (2001), 9. doi: 10.1016/S0304-3800(01)00351-9. Google Scholar

[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), (1992), 27. Google Scholar

[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. doi: 10.1142/S0218127493000763. Google Scholar

[9]

C. W. Clark, "Mathematical Bioeconomics: The Optimal Management of Renewable Resources,", $2^{nd}$ edition, (1990). Google Scholar

[10]

S. Elaydi and R. J. Sacker, Population models with Allee effect: A new model,, Journal of Biological Dynamics, 4 (2009), 397. doi: 10.1080/17513750903377434. Google Scholar

[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. doi: 10.1890/06-1238.1. Google Scholar

[12]

D. Fournier-Prunaret, The bifurcation structure of a family of degree one circle endomorphisms,, International Journal of Bifurcation and Chaos, 1 (1991), 823. doi: 10.1142/S0218127491000609. Google Scholar

[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. doi: 10.1016/j.fm.2004.01.007. Google Scholar

[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. doi: 10.3934/mbe.2013.10.345. Google Scholar

[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. doi: 10.1007/BF02433474. Google Scholar

[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. doi: 10.1109/TCS.1984.1085495. Google Scholar

[17]

A. M. Kramer, B. Dennis, A. M. Liebhold and J. M. Drake, The evidence for Allee effects,, Population Ecology, 51 (2009), 341. doi: 10.1007/s10144-009-0152-6. Google Scholar

[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. Google Scholar

[19]

J. P. Lampreia and J. Sousa Ramos, Symbolic dynamics of bimodal maps,, Portugaliae Math., 54 (1997), 1. Google Scholar

[20]

M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms,, Theoretical Population Biology, 43 (1993), 141. doi: 10.1006/tpbi.1993.1007. Google Scholar

[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. doi: 10.3934/dcdsb.2004.4.629. Google Scholar

[22]

D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Codings,", $2^{nd}$ edition, (1999). doi: 10.1017/CBO9780511626302. Google Scholar

[23]

G. Livadiotis and S. Elaydi, General Allee effect in two-species population biology,, J. Bio. Dyn., 6 (2012), 959. doi: 10.1080/17513758.2012.700075. Google Scholar

[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, 41 (2009), 334. doi: 10.1016/j.chaos.2008.01.015. Google Scholar

[25]

W. Melo and S. van Strien, "One-Dimensional Dynamics,", $1^{nd}$ edition, (1993). Google Scholar

[26]

V. Méndez, C. Sans, I. Lopis and D. Campos, Extinction conditions for isolated populations with Allee effect,, Mathematical Biosciences, 232 (2011), 78. doi: 10.1016/j.mbs.2011.04.005. Google Scholar

[27]

C. Mira, "Chaotic Dynamics. From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism,", World Scientific, (1987). Google Scholar

[28]

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

[29]

M. Misiurewicz, Horseshoes for mappings of the interval,, Bull. Acad. Polish. Sci., 27 (1979), 167. Google Scholar

[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. doi: 10.2307/1931412. Google Scholar

[31]

F. J. Richards, A flexible growth function for empirical use,, Journal of Experimental Botany, 10 (1959), 290. doi: 10.1093/jxb/10.2.290. Google Scholar

[32]

J. L. Rocha and S. M. Aleixo, Modeling Allee effect from Beta(p,2) densities,, Proc. ITI 2012, (2012), 461. Google Scholar

[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. doi: 10.3934/mbe.2013.10.379. Google Scholar

[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. doi: 10.3934/dcdsb.2013.18.783. Google Scholar

[35]

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

[36]

S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models,, Theoretical Population Biology, 64 (2003), 201. doi: 10.1016/S0040-5809(03)00072-8. Google Scholar

[37]

O. M. Šarkovs'kiĭ, On cycles and the structure of a continuous mapping,, Ukrain. Math. Ž., 17 (1965), 104. Google Scholar

[38]

P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?,, Oikos, 87 (1999), 185. doi: 10.2307/3547011. Google Scholar

[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. doi: 10.1080/17513750802376313. Google Scholar

[40]

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

[41]

M. Wang and M. Kot, Speeds of invasion in a model with strong or weak Allee effects,, Mathematical Biosciences, 171 (2001), 83. doi: 10.1016/S0025-5564(01)00048-7. Google Scholar

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. Google Scholar

[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. doi: 10.2498/cit.1002098. Google Scholar

[3]

L. Berec, E. Angulo and F. Courchamp, Multiple Allee effects and population management,, Trends in Ecology & Evolution, 22 (2007), 185. doi: 10.1016/j.tree.2006.12.002. Google Scholar

[4]

C. P. D. Birch, A new generalized logistic sigmoid growth equation compared with the Richards growth equation,, Annals of Botany, 83 (1999), 713. doi: 10.1006/anbo.1999.0877. Google Scholar

[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. doi: 10.1006/jtbi.2002.3084. Google Scholar

[6]

C. E. Brassil, Mean time to extinction of a metapopulation with an Allee effect,, Ecological Modelling, 143 (2001), 9. doi: 10.1016/S0304-3800(01)00351-9. Google Scholar

[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), (1992), 27. Google Scholar

[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. doi: 10.1142/S0218127493000763. Google Scholar

[9]

C. W. Clark, "Mathematical Bioeconomics: The Optimal Management of Renewable Resources,", $2^{nd}$ edition, (1990). Google Scholar

[10]

S. Elaydi and R. J. Sacker, Population models with Allee effect: A new model,, Journal of Biological Dynamics, 4 (2009), 397. doi: 10.1080/17513750903377434. Google Scholar

[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. doi: 10.1890/06-1238.1. Google Scholar

[12]

D. Fournier-Prunaret, The bifurcation structure of a family of degree one circle endomorphisms,, International Journal of Bifurcation and Chaos, 1 (1991), 823. doi: 10.1142/S0218127491000609. Google Scholar

[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. doi: 10.1016/j.fm.2004.01.007. Google Scholar

[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. doi: 10.3934/mbe.2013.10.345. Google Scholar

[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. doi: 10.1007/BF02433474. Google Scholar

[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. doi: 10.1109/TCS.1984.1085495. Google Scholar

[17]

A. M. Kramer, B. Dennis, A. M. Liebhold and J. M. Drake, The evidence for Allee effects,, Population Ecology, 51 (2009), 341. doi: 10.1007/s10144-009-0152-6. Google Scholar

[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. Google Scholar

[19]

J. P. Lampreia and J. Sousa Ramos, Symbolic dynamics of bimodal maps,, Portugaliae Math., 54 (1997), 1. Google Scholar

[20]

M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms,, Theoretical Population Biology, 43 (1993), 141. doi: 10.1006/tpbi.1993.1007. Google Scholar

[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. doi: 10.3934/dcdsb.2004.4.629. Google Scholar

[22]

D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Codings,", $2^{nd}$ edition, (1999). doi: 10.1017/CBO9780511626302. Google Scholar

[23]

G. Livadiotis and S. Elaydi, General Allee effect in two-species population biology,, J. Bio. Dyn., 6 (2012), 959. doi: 10.1080/17513758.2012.700075. Google Scholar

[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, 41 (2009), 334. doi: 10.1016/j.chaos.2008.01.015. Google Scholar

[25]

W. Melo and S. van Strien, "One-Dimensional Dynamics,", $1^{nd}$ edition, (1993). Google Scholar

[26]

V. Méndez, C. Sans, I. Lopis and D. Campos, Extinction conditions for isolated populations with Allee effect,, Mathematical Biosciences, 232 (2011), 78. doi: 10.1016/j.mbs.2011.04.005. Google Scholar

[27]

C. Mira, "Chaotic Dynamics. From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism,", World Scientific, (1987). Google Scholar

[28]

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

[29]

M. Misiurewicz, Horseshoes for mappings of the interval,, Bull. Acad. Polish. Sci., 27 (1979), 167. Google Scholar

[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. doi: 10.2307/1931412. Google Scholar

[31]

F. J. Richards, A flexible growth function for empirical use,, Journal of Experimental Botany, 10 (1959), 290. doi: 10.1093/jxb/10.2.290. Google Scholar

[32]

J. L. Rocha and S. M. Aleixo, Modeling Allee effect from Beta(p,2) densities,, Proc. ITI 2012, (2012), 461. Google Scholar

[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. doi: 10.3934/mbe.2013.10.379. Google Scholar

[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. doi: 10.3934/dcdsb.2013.18.783. Google Scholar

[35]

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

[36]

S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models,, Theoretical Population Biology, 64 (2003), 201. doi: 10.1016/S0040-5809(03)00072-8. Google Scholar

[37]

O. M. Šarkovs'kiĭ, On cycles and the structure of a continuous mapping,, Ukrain. Math. Ž., 17 (1965), 104. Google Scholar

[38]

P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?,, Oikos, 87 (1999), 185. doi: 10.2307/3547011. Google Scholar

[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. doi: 10.1080/17513750802376313. Google Scholar

[40]

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

[41]

M. Wang and M. Kot, Speeds of invasion in a model with strong or weak Allee effects,, Mathematical Biosciences, 171 (2001), 83. doi: 10.1016/S0025-5564(01)00048-7. Google Scholar

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