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May  2013, 18(3): 783-795. doi: 10.3934/dcdsb.2013.18.783

Dynamical analysis in growth models: Blumberg's equation

1. 

Instituto Superior de Engenharia de Lisboa - ISEL, ADM and CEAUL, Rua Conselheiro Emídio Navarro, 1, 1959-007 Lisboa, Portugal, Portugal

Received  October 2011 Revised  November 2012 Published  December 2012

We present a new dynamical approach to the Blumberg's equation, a family of unimodal maps. These maps are proportional to $Beta(p,q)$ probability densities functions. Using the symmetry of the $Beta(p,q)$ distribution and symbolic dynamics techniques, a new concept of mirror symmetry is defined for this family of maps. The kneading theory is used to analyze the effect of such symmetry in the presented models. The main result proves that two mirror symmetric unimodal maps have the same topological entropy. Different population dynamics regimes are identified, when the intrinsic growth rate is modified: extinctions, stabilities, bifurcations, chaos and Allee effect. To illustrate our results, we present a numerical analysis, where are demonstrated: monotonicity of the topological entropy with the variation of the intrinsic growth rate, existence of isentropic sets in the parameters space and mirror symmetry.
Citation: J. Leonel Rocha, Sandra M. Aleixo. Dynamical analysis in growth models: Blumberg's equation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 783-795. doi: 10.3934/dcdsb.2013.18.783
References:
[1]

S. M. Aleixo, J. L. Rocha and D. D. Pestana, Populational growth models in the light of symbolic dynamics, in "Proc. of the ITI 2008, 30th Int. Conf. on Information Technology Interfaces" (eds. V. Luzar-Stiffler, V.H. Dobric and Z. Bekic), IEEE, (2008), 311-316.

[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.

[4]

A. A. Blumberg, Logistic growth rate functions, J. Theor. Biol., 21 (1968), 42-44.

[5]

R. Buis, On the generalization of the logistic law of growth, Acta Biotheoretica, 39 (1991), 185-195.

[6]

A. Caneco, C. Grácio and J. L. Rocha, Kneading theory analysis of the Duffing equation, Chaos Solit. Fract., 42 (2009), 1529-1538. doi: 10.1016/j.chaos.2009.03.040.

[7]

M. Carrillo and J. Gonzalez, Fitting a growth model to number of tourist places in Tenerife, Syst. Anal. Model. Simulat., 42 (2002), 895-906.

[8]

C. W. Clark, "Mathematical Bioeconomic: The Optimal Management of Renewable Resources,'' John Wiley and Sons, 1990.

[9]

D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Codings," $2^{nd}$ edition, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511626302.

[10]

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 Solit. Fract., 41 (2009), 334-347 doi: 10.1016/j.chaos.2008.01.015.

[11]

M. S. el Naschie, Hierarchy of kissing numbers for exceptional Lie symmetry groups in high energy physics, Chaos Solit. Fract., 35 (2008), 420-422.

[12]

A. S. Martinez, R. S. González and C. A. S. Terriol, 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.

[13]

W. Melo and S. van Strien, "One-Dimensional Dynamics," $1^{st}$ edition, Springer-Verlag, New York, 1993.

[14]

M. Misiurewicz, Horseshoes for continuous mappings of an interval, Bull. Acad. Polish. Sci., 27 (1979), 167-169.

[15]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.

[16]

J. Milnor and W. Thurston, On iterated maps of the interval, Lect. Notes in Math., 1342 (1988), 465-563. doi: 10.1007/BFb0082847.

[17]

M. Peschel and W. Mende, "The Predator-Prey Model. Do We Leave in a Volterra World?,'' Springer, Wien, 1986.

[18]

K. M. Pruitt and M. E. Turner, A kinetic theory for analysis of complex systems, in "Biomolecular Structure and Function" (ed. P. F. Agris), Academic Press, New York, (1978), 257-265.

[19]

J. L. Rocha, S. M. Aleixo and D. D. Pestana, Beta(p,q)-Cantor sets: Determinism and randomness, in "Chaos Theory: Modeling, Simulation and Applications" (eds. C. H. Skiadas, Y. Dimotikalis and C. Skiadas), World Scientific Publishing Co., (2011), 333-340.

[20]

J. L. Rocha and S. M. Aleixo, An extension of Gompertzian growth dynamics: Weibull and Fr\'echet models,, to appear in Math. Biosci. Eng., (). 

[21]

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

[22]

S. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209.

[23]

E. Skubica, M. Taborskya, J. M. McNamarac and A. I. Houston, When to parasitize? A dynamic optimization model of reproductive strategies in a cooperative breeder, J. Theor. Biol., 227 (2004), 487-501. doi: 10.1016/j.jtbi.2003.11.021.

[24]

A. Tsoularis and J. Wallace, Analysis of logistic growth models, Math. Biosci., 179 (2002), 21-55. doi: 10.1016/S0025-5564(02)00096-2.

[25]

M. E. Turner, E. L. Bradley, K. A. Kirk and K. M. Pruitt, A theory of growth, Math. Biosci., 29 (1976), 367-373.

show all references

References:
[1]

S. M. Aleixo, J. L. Rocha and D. D. Pestana, Populational growth models in the light of symbolic dynamics, in "Proc. of the ITI 2008, 30th Int. Conf. on Information Technology Interfaces" (eds. V. Luzar-Stiffler, V.H. Dobric and Z. Bekic), IEEE, (2008), 311-316.

[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.

[4]

A. A. Blumberg, Logistic growth rate functions, J. Theor. Biol., 21 (1968), 42-44.

[5]

R. Buis, On the generalization of the logistic law of growth, Acta Biotheoretica, 39 (1991), 185-195.

[6]

A. Caneco, C. Grácio and J. L. Rocha, Kneading theory analysis of the Duffing equation, Chaos Solit. Fract., 42 (2009), 1529-1538. doi: 10.1016/j.chaos.2009.03.040.

[7]

M. Carrillo and J. Gonzalez, Fitting a growth model to number of tourist places in Tenerife, Syst. Anal. Model. Simulat., 42 (2002), 895-906.

[8]

C. W. Clark, "Mathematical Bioeconomic: The Optimal Management of Renewable Resources,'' John Wiley and Sons, 1990.

[9]

D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Codings," $2^{nd}$ edition, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511626302.

[10]

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 Solit. Fract., 41 (2009), 334-347 doi: 10.1016/j.chaos.2008.01.015.

[11]

M. S. el Naschie, Hierarchy of kissing numbers for exceptional Lie symmetry groups in high energy physics, Chaos Solit. Fract., 35 (2008), 420-422.

[12]

A. S. Martinez, R. S. González and C. A. S. Terriol, 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.

[13]

W. Melo and S. van Strien, "One-Dimensional Dynamics," $1^{st}$ edition, Springer-Verlag, New York, 1993.

[14]

M. Misiurewicz, Horseshoes for continuous mappings of an interval, Bull. Acad. Polish. Sci., 27 (1979), 167-169.

[15]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.

[16]

J. Milnor and W. Thurston, On iterated maps of the interval, Lect. Notes in Math., 1342 (1988), 465-563. doi: 10.1007/BFb0082847.

[17]

M. Peschel and W. Mende, "The Predator-Prey Model. Do We Leave in a Volterra World?,'' Springer, Wien, 1986.

[18]

K. M. Pruitt and M. E. Turner, A kinetic theory for analysis of complex systems, in "Biomolecular Structure and Function" (ed. P. F. Agris), Academic Press, New York, (1978), 257-265.

[19]

J. L. Rocha, S. M. Aleixo and D. D. Pestana, Beta(p,q)-Cantor sets: Determinism and randomness, in "Chaos Theory: Modeling, Simulation and Applications" (eds. C. H. Skiadas, Y. Dimotikalis and C. Skiadas), World Scientific Publishing Co., (2011), 333-340.

[20]

J. L. Rocha and S. M. Aleixo, An extension of Gompertzian growth dynamics: Weibull and Fr\'echet models,, to appear in Math. Biosci. Eng., (). 

[21]

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

[22]

S. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209.

[23]

E. Skubica, M. Taborskya, J. M. McNamarac and A. I. Houston, When to parasitize? A dynamic optimization model of reproductive strategies in a cooperative breeder, J. Theor. Biol., 227 (2004), 487-501. doi: 10.1016/j.jtbi.2003.11.021.

[24]

A. Tsoularis and J. Wallace, Analysis of logistic growth models, Math. Biosci., 179 (2002), 21-55. doi: 10.1016/S0025-5564(02)00096-2.

[25]

M. E. Turner, E. L. Bradley, K. A. Kirk and K. M. Pruitt, A theory of growth, Math. Biosci., 29 (1976), 367-373.

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