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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 & 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, (2008), 311. 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. Google Scholar [4] A. A. Blumberg, Logistic growth rate functions,, J. Theor. Biol., 21 (1968), 42. Google Scholar [5] R. Buis, On the generalization of the logistic law of growth,, Acta Biotheoretica, 39 (1991), 185. Google Scholar [6] A. Caneco, C. Grácio and J. L. Rocha, Kneading theory analysis of the Duffing equation,, Chaos Solit. Fract., 42 (2009), 1529. doi: 10.1016/j.chaos.2009.03.040. Google Scholar [7] M. Carrillo and J. Gonzalez, Fitting a growth model to number of tourist places in Tenerife,, Syst. Anal. Model. Simulat., 42 (2002), 895. Google Scholar [8] C. W. Clark, "Mathematical Bioeconomic: The Optimal Management of Renewable Resources,'', John Wiley and Sons, (1990). Google Scholar [9] D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Codings,", $2^{nd}$ edition, (1999). doi: 10.1017/CBO9780511626302. Google Scholar [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. doi: 10.1016/j.chaos.2008.01.015. Google Scholar [11] M. S. el Naschie, Hierarchy of kissing numbers for exceptional Lie symmetry groups in high energy physics,, Chaos Solit. Fract., 35 (2008), 420. Google Scholar [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. doi: 10.1016/j.physa.2008.06.015. Google Scholar [13] W. Melo and S. van Strien, "One-Dimensional Dynamics,", $1^{st}$ edition, (1993). Google Scholar [14] M. Misiurewicz, Horseshoes for continuous mappings of an interval,, Bull. Acad. Polish. Sci., 27 (1979), 167. Google Scholar [15] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings,, Studia Math., 67 (1980), 45. Google Scholar [16] J. Milnor and W. Thurston, On iterated maps of the interval,, Lect. Notes in Math., 1342 (1988), 465. doi: 10.1007/BFb0082847. Google Scholar [17] M. Peschel and W. Mende, "The Predator-Prey Model. Do We Leave in a Volterra World?,'', Springer, (1986). Google Scholar [18] K. M. Pruitt and M. E. Turner, A kinetic theory for analysis of complex systems,, in, (1978), 257. Google Scholar [19] J. L. Rocha, S. M. Aleixo and D. D. Pestana, Beta(p,q)-Cantor sets: Determinism and randomness,, in, (2011), 333. Google Scholar [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., (). Google Scholar [21] S. Schreiber, Chaos and population disappearances in simple ecological models,, J. Math. Biol., 42 (2001), 239. doi: 10.1007/s002850000070. Google Scholar [22] S. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models,, Theor. Popul. Biol., 64 (2003), 201. Google Scholar [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. doi: 10.1016/j.jtbi.2003.11.021. Google Scholar [24] 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 [25] M. E. Turner, E. L. Bradley, K. A. Kirk and K. M. Pruitt, A theory of growth,, Math. Biosci., 29 (1976), 367. Google Scholar

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, (2008), 311. 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. Google Scholar [4] A. A. Blumberg, Logistic growth rate functions,, J. Theor. Biol., 21 (1968), 42. Google Scholar [5] R. Buis, On the generalization of the logistic law of growth,, Acta Biotheoretica, 39 (1991), 185. Google Scholar [6] A. Caneco, C. Grácio and J. L. Rocha, Kneading theory analysis of the Duffing equation,, Chaos Solit. Fract., 42 (2009), 1529. doi: 10.1016/j.chaos.2009.03.040. Google Scholar [7] M. Carrillo and J. Gonzalez, Fitting a growth model to number of tourist places in Tenerife,, Syst. Anal. Model. Simulat., 42 (2002), 895. Google Scholar [8] C. W. Clark, "Mathematical Bioeconomic: The Optimal Management of Renewable Resources,'', John Wiley and Sons, (1990). Google Scholar [9] D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Codings,", $2^{nd}$ edition, (1999). doi: 10.1017/CBO9780511626302. Google Scholar [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. doi: 10.1016/j.chaos.2008.01.015. Google Scholar [11] M. S. el Naschie, Hierarchy of kissing numbers for exceptional Lie symmetry groups in high energy physics,, Chaos Solit. Fract., 35 (2008), 420. Google Scholar [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. doi: 10.1016/j.physa.2008.06.015. Google Scholar [13] W. Melo and S. van Strien, "One-Dimensional Dynamics,", $1^{st}$ edition, (1993). Google Scholar [14] M. Misiurewicz, Horseshoes for continuous mappings of an interval,, Bull. Acad. Polish. Sci., 27 (1979), 167. Google Scholar [15] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings,, Studia Math., 67 (1980), 45. Google Scholar [16] J. Milnor and W. Thurston, On iterated maps of the interval,, Lect. Notes in Math., 1342 (1988), 465. doi: 10.1007/BFb0082847. Google Scholar [17] M. Peschel and W. Mende, "The Predator-Prey Model. Do We Leave in a Volterra World?,'', Springer, (1986). Google Scholar [18] K. M. Pruitt and M. E. Turner, A kinetic theory for analysis of complex systems,, in, (1978), 257. Google Scholar [19] J. L. Rocha, S. M. Aleixo and D. D. Pestana, Beta(p,q)-Cantor sets: Determinism and randomness,, in, (2011), 333. Google Scholar [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., (). Google Scholar [21] S. Schreiber, Chaos and population disappearances in simple ecological models,, J. Math. Biol., 42 (2001), 239. doi: 10.1007/s002850000070. Google Scholar [22] S. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models,, Theor. Popul. Biol., 64 (2003), 201. Google Scholar [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. doi: 10.1016/j.jtbi.2003.11.021. Google Scholar [24] 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 [25] M. E. Turner, E. L. Bradley, K. A. Kirk and K. M. Pruitt, A theory of growth,, Math. Biosci., 29 (1976), 367. Google Scholar
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