Big bang bifurcations in von Bertalanffy's dynamics with strong and weak Allee effects

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• 作者：J. Leonel Rocha ; Abdel-Kaddous Taha ; Danièle Fournier-Prunaret
• 关键词：Von Bertalanffy’s dynamics ; Strong and weak Allee effects ; Big bang bifurcation ; Extinction
• 刊名：Nonlinear Dynamics
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：84
• 期：2
• 页码：607-626
• 全文大小：2,906 KB
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  35.Rocha, J.L., Taha, A.-K., Fournier-Prunaret, D.: Explosion birth and extinction: double big bang bifurcations and Allee effect in Tsoularis–Wallace’s growth models. Discret. Contin. Dyn. Syst. Ser. B 20(9), 3131–3163 (2015)MathSciNet CrossRef MATH
  
• 作者单位：J. Leonel Rocha (1)
  Abdel-Kaddous Taha (2)
  Danièle Fournier-Prunaret (3)
  
  1. ADM, ISEL-Instituto Sup. de Engenharia de Lisboa, Instituto Politécnico de Lisboa, Rua Conselheiro Emídio Navarro 1, 1959-007, Lisboa, Portugal
  2. INSA, University of Toulouse, 135 Avenue du Rangueil, 31077, Toulouse, France
  3. LAAS-CNRS, INSA, University of Toulouse, 7 Avenue du Colonel Roche, 31077, Toulouse, France
  
• 刊物类别：Engineering
• 刊物主题：Vibration, Dynamical Systems and Control
  Mechanics
  Mechanical Engineering
  Automotive and Aerospace Engineering and Traffic
  
• 出版者：Springer Netherlands
• ISSN：1573-269X

文摘

The main purpose of this work was to study population dynamic discrete models in which the growth of the population is described by generalized von Bertalanffy’s functions, with an adjustment or correction factor of polynomial type. The consideration of this correction factor is made with the aim to introduce the Allee effect. To the class of generalized von Bertalanffy’s functions is identified and characterized subclasses of strong and weak Allee’s functions and functions with no Allee effect. This classification is founded on the concepts of strong and weak Allee’s effects to population growth rates associated. A complete description of the dynamic behavior is given, where we provide necessary conditions for the occurrence of unconditional and essential extinction types. The bifurcation structures of the parameter plane are analyzed regarding the evolution of the Allee limit with the aim to understand how the transition from strong Allee effect to no Allee effect, passing through the weak Allee effect, is realized. To generalized von Bertalanffy’s functions with strong and weak Allee effects is identified an Allee’s effect region, to which is associated the concepts of chaotic semistability curve and Allee’s bifurcation point. We verified that under some sufficient conditions, generalized von Bertalanffy’s functions have a particular bifurcation structure: the big bang bifurcations of the so-called box-within-a-box type. To this family of maps, the Allee bifurcation points and the big bang bifurcation points are characterized by the symmetric of Allee’s limit and by a null intrinsic growth rate. The present paper is also a significant contribution in the framework of the big bang bifurcation analysis for continuous 1D maps and unveil their relationship with the explosion birth and the extinction phenomena.