Browsing by Author "Taha, Abdel-Kaddous"
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- Allee effect bifurcation in the γ-Ricker population model using the Lambert W functionPublication . Rocha, J. Leonel; Taha, Abdel-KaddousThe main purpose of this talk is to present the dynamical study and the bifurcation structures of the γ-Ricker population model. Resorting to the Lambert W function, the analytical solutions of the positive fixed point equation for the γ-Ricker population model are explicitly presented and conditions for the existence and stability of these fixed points are established. Another main focus of this work is the definition and characterization of the Allee effect bifurcation for the γ-Ricker population model, which is not a pitchfork bifurcation. Consequently, we prove that the phenomenon of Allee effect for the γ-Ricker population model is associated to the asymptotic behavior of the Lambert W function in a neighborhood of zero. Numerical studies are included.
- Allee's dynamics and bifurcation structures in von Bertalanffy's population size functionsPublication . Rocha, J. Leonel; Taha, Abdel-Kaddous; Fournier-Prunaret, D.The interest and the relevance of the study of the population dynamics and the extinction phenomenon are our main motivation to investigate the induction of Allee Effect in von Bertalanffy's population size functions. The adjustment or correction factor of rational type introduced allows us to analyze simultaneously strong and weak Allee's functions and functions with no Allee effect, whose classification is dependent on the stability of the fixed point x = 0. This classification is founded on the concepts of strong and weak Allee's effects to the population growth rates associated. The transition from strong Allee effect to no Allee effect, passing through the weak Allee effect, is verified with the evolution of the rarefaction critical density or Allee's limit. The existence of cusp points on a fold bifurcation curve is related to this phenomenon of transition on Allee's dynamics. Moreover, the "foliated" structure of the parameter plane considered is also explained, with respect to the evolution of the Allee limit. The bifurcation analysis is based on the configurations of fold and flip bifurcation curves. The chaotic semistability and the nonadmissibility bifurcation curves are proposed to this family of 1D maps, which allow us to define and characterize the corresponding Allee effect region.
- Allee's dynamics and bifurcation structures in von Bertalanffy's population size functionsPublication . Rocha, J. Leonel; Taha, Abdel-Kaddous; Fournier-Prunaret, D.The interest and the relevance of the study of the population dynamics and the extinction phenomenon are our main motivation to investigate the induction of Allee Effect in von Bertalanffy's population size functions. The adjustment or correction factor of rational type introduced allows us to analyze simultaneously strong and weak Allee's functions and functions with no Allee effect, whose classification is dependent on the stability of the fixed point x = 0. This classification is founded on the concepts of strong and weak Allee's effects to the population growth rates associated. The transition from strong Allee effect to no Allee effect, passing through the weak Allee effect, is verified with the evolution of the rarefaction critical density or Allee's limit. The existence of cusp points on a fold bifurcation curve is related to this phenomenon of transition on Allee's dynamics. Moreover, the "foliated" structure of the parameter plane considered is also explained, with respect to the evolution of the Allee limit. The bifurcation analysis is based on the configurations of fold and flip bifurcation curves. The chaotic semistability and the nonadmissibility bifurcation curves are proposed to this family of 1D maps, which allow us to define and characterize the corresponding Allee effect region.
- Allee's effect bifurcation in generalized logistic mapsPublication . Rocha, J. Leonel; Taha, Abdel-KaddousThis paper concerns the study of the Allee effect on the dynamical behavior of a new class of generalized logistic maps. The fundamentals of the dynamics of this 4-parameter family of one-dimensional maps are presented. A complete classification of the nature and stability of its fixed points is provided. The main results relate to the Allee effect bifurcation: a new type of bifurcation introduced for this class of unimodal maps. A necessary and sufficient condition so that the Allee fixed point is a snap-back repeller is established. In addition, in the parameters space is defined an Allee's effect region, which determines the existence of an essential extinction for the generalized logistic maps. Local and global bifurcations of generalized logistic maps are investigated.
- Bifurcation structures in a 2D exponential diffeomorphism with Allee effectPublication . Rocha, J. Leonel; Taha, Abdel-KaddousAn embedding of one-dimensional generic growth functions into a two-dimensional diffeomorphism is considered. This family of unimodal maps naturally incorporates a key item of ecological and biological research: the Allee effect. Consequently, the presence of this species extinction phenomenon leads us to a new definition of bifurcation for this two-dimensional exponential diffeomorphism: Allee’s effect bifurcation. The stability and the nature of the fixed points of the two-dimensional diffeomorphism are analyzed, by studying the corresponding contour lines. Fold and flip bifurcation structures of this exponential diffeomorphism are investigated, in which there are flip codimension-2 bifurcation points and cusp points, when some parameters evolve. Numerical studies are included.
- Big Bang Bifurcation Analysis and Allee Effect in Generic Growth FunctionsPublication . Rocha, J. Leonel; Taha, Abdel-Kaddous; Fournier-Prunaret, D.The main purpose of this work is to study the dynamics and bifurcation properties of generic growth functions, which are defined by the population size functions of the generic growth equation. This family of unimodal maps naturally incorporates a principal focus of ecological and biological research: the Allee effect. The analysis of this kind of extinction phenomenon allows to identify a class of Allee’s functions and characterize the corresponding Allee’s effect region and Allee’s bifurcation curve. The bifurcation analysis is founded on the performance of fold and flip bifurcations. The dynamical behavior is rich with abundant complex bifurcation structures, the big bang bifurcations of the so-called “box-within-a-box” fractal type being the most outstanding. Moreover, these bifurcation cascades converge to different big bang bifurcation curves with distinct kinds of boxes, where for the corresponding parameter values several attractors are associated. To the best of our knowledge, these results represent an original contribution to clarify the big bang bifurcation analysis of continuous 1D maps.
- Big Bang bifurcations and allee effect in Blumberg’s dynamicsPublication . Rocha, J. Leonel; Fournier-Prunaret, Danièle; Taha, Abdel-KaddousThis paper concerns dynamics and bifurcations properties of a class of continuous-defined one-dimensional maps, in a three-dimensional parameter space: Blumberg's functions. This family of functions naturally incorporates a major focus of ecological research: the Allee effect. We provide a necessary condition for the occurrence of this phenomenon, associated with the stability of a fixed point. A central point of our investigation is the study of bifurcations structure for this class of functions. We verified that under some sufficient conditions, Blumberg's functions have a particular bifurcations structure: the big bang bifurcations of the so-called "box-within-a-box" type, but for different kinds of boxes. Moreover, it is verified that these bifurcation cascades converge to different big bang bifurcation curves, where for the corresponding parameter values are associated distinct attractors. This work contributes to clarify the big bang bifurcation analysis for continuous maps. To support our results, we present fold and flip bifurcations curves and surfaces, and numerical simulations of several bifurcation diagrams.
- Big bang bifurcations in von Bertalanffy’s dynamics with strong and weak Allee effectsPublication . Rocha, J. Leonel; Taha, Abdel-Kaddous; Fournier-Prunaret, DanièleThe 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.
- Dynamical analysis and Big Bang bifurcations of 1D and 2D Gompertz’s growth functionsPublication . Rocha, J. Leonel; Taha, Abdel-Kaddous; Fournier-Prunaret, D.In this paper, we study the dynamics and bifurcation properties of a three-parameter family of 1D Gompertz’s growth functions, which are defined by the population size functions of the Gompertz logistic growth equation. The dynamical behavior is complex leading to a diversified bifurcation structure, leading to the big bang bifurcations of the so-called “box-within-a-box” fractal type. We provide and discuss sufficient conditions for the existence of these bifurcation cascades for 1D Gompertz’s growth functions. Moreover, this work concerns the description of some bifurcation properties of a Hénon’s map type embedding: a “continuous” embedding of 1D Gompertz’s growth functions into a 2D diffeomorphism. More particularly, properties that characterize the big bang bifurcations are considered in relation with this coupling of two population size functions, varying the embedding parameter. The existence of communication areas of crossroad area type or swallowtails are identified for this 2D diffeomorphism.
- Explosion birth and extinction: double big bang bifurcations and Allee effect in Tsoularis-Wallace's growth modelsPublication . Rocha, J. Leonel; Taha, Abdel-Kaddous; Fournier-Prunaret, DanièleThis work concerns dynamics and bifurcations properties of a new class of continuous-defined one-dimensional maps: Tsoularis-Wallace's functions. This family of functions naturally incorporates a major focus of ecological research: the Allee effect. We provide a necessary condition for the occurrence of this phenomenon of extinction. To establish this result we introduce the notions of Allee's functions, Allee's effect region and Allee's bifurcation curve. Another central point of our investigation is the study of bifurcation structures for this class of functions, in a three-dimensional parameter space. We verified that under some sufficient conditions, Tsoularis-Wallace's functions have particular bifurcation structures: the big bang and the double big bang bifurcations of the so-called "box-within-a-box" type. The double big bang bifurcations are related to the existence of flip codimension-2 points. Moreover, it is verified that these bifurcation cascades converge to different big bang bifurcation curves, where for the corresponding parameter values are associated distinct kinds of boxes. This work contributes to clarify the big bang bifurcation analysis for continuous maps and understand their relationship with explosion birth and extinction phenomena.