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- Generalized models from Beta(p,2) densities with strong allee effect: dynamical approachPublication . Aleixo, Sandra; Rocha, J. LeonelA dynamical approach to study the behaviour of generalized populational growth models from Bets(p, 2) densities, with strong Allee effect, is presented. The dynamical analysis of the respective unimodal maps is performed using symbolic dynamics techniques. The complexity of the correspondent discrete dynamical systems is measured in terms of topological entropy. Different populational dynamics regimes are obtained when the intrinsic growth rates are modified: extinction, bistability, chaotic semistability and essential extinction.
- 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.
- An Extension of Gompertzian Growth Dynamics Weibull and Frechet ModelsPublication . Rocha, J. Leonel; Aleixo, SandraIn this work a new probabilistic and dynamical approach to an extension of the Gompertz law is proposed. A generalized family of probability density functions, designated by Beta* (p, q), which is proportional to the right hand side of the Tsoularis-Wallace model, is studied. In particular, for p = 2, the investigation is extended to the extreme value models of Weibull and Frechet type. These models, described by differential equations, are proportional to the hyper-Gompertz growth model. It is proved that the Beta* (2, q) densities are a power of betas mixture, and that its dynamics are determined by a non-linear coupling of probabilities. The dynamical analysis is performed using techniques of symbolic dynamics and the system complexity is measured using topological entropy. Generally, the natural history of a malignant tumour is reflected through bifurcation diagrams, in which are identified regions of regression, stability, bifurcation, chaos and terminus.
- Information theory, synchronization and topological order in complete dynamical networks of discontinuous mapsPublication . Rocha, J. Leonel; Carvalho, S.This paper is dedicated to the study of information measures, synchronization and a topological order in complete dynamical networks of discontinuous piecewise linear maps with different slopes. It stands out that the networks topologies are characterized by circulant matrices and the conditional Lyapunov exponents are explicitly determined. Some properties of the mutual information rate and the Kolmogorov–Sinai entropy, depending on the synchronization interval, are discussed. A topological order between the complete dynamical networks is presented, which is characterized by the monotony of the network topological entropy. It is proved that if the network topological entropy increases, then the mutual information rate and the Kolmogorov–Sinai entropy increase or decrease, according to the variation of the coupling parameter. Furthermore, various types of computer simulations show the experimental applications of these results and techniques.
- Synchronization in Von Bertalanffy’s modelsPublication . Rocha, J. Leonel; Aleixo, Sandra; Caneco, AcilinaMany data have been useful to describe the growth of marine mammals, invertebrates and reptiles, seabirds, sea turtles and fishes, using the logistic, the Gom-pertz and von Bertalanffy's growth models. A generalized family of von Bertalanffy's maps, which is proportional to the right hand side of von Bertalanffy's growth equation, is studied and its dynamical approach is proposed. The system complexity is measured using Lyapunov exponents, which depend on two biological parameters: von Bertalanffy's growth rate constant and the asymptotic weight. Applications of synchronization in real world is of current interest. The behavior of birds ocks, schools of fish and other animals is an important phenomenon characterized by synchronized motion of individuals. In this work, we consider networks having in each node a von Bertalanffy's model and we study the synchronization interval of these networks, as a function of those two biological parameters. Numerical simulation are also presented to support our approaches.
- 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.
- Dynamical behaviour on the parameter space: new populational growth models proportional to beta densitiesPublication . Aleixo, Sandra; Rocha, J. Leonel; Pestana, Dinis D.We present new populational growth models, generalized logistic models which are proportional to beta densities with shape parameters p and 2, where p > 1, with Malthusian parameter r. The complex dynamical behaviour of these models is investigated in the parameter space (r, p), in terms of topological entropy, using explicit methods, when the Malthusian parameter r increases. This parameter space is split into different regions, according to the chaotic behaviour of the models.
- Von Bertalanffy's dynamics under a polynomial correction: Allee effect and big bang bifurcationPublication . Rocha, J. Leonel; Taha, A. K.; Fournier-Prunaret, D.In this work we consider new one-dimensional populational discrete dynamical systems in which the growth of the population is described by a family of von Bertalanffy's functions, as a dynamical approach to von Bertalanffy's growth equation. The purpose of introducing Allee effect in those models is satisfied under a correction factor of polynomial type. We study classes of von Bertalanffy's functions with different types of Allee effect: strong and weak Allee's functions. Dependent on the variation of four parameters, von Bertalanffy's functions also includes another class of important functions: functions with no Allee effect. The complex bifurcation structures of these von Bertalanffy's functions is investigated in detail. We verified that this family of functions has particular bifurcation structures: the big bang bifurcation of the so-called "box-within-a-box" type. The big bang bifurcation is associated to the asymptotic weight or carrying capacity. This work is a contribution to the study of the big bang bifurcation analysis for continuous maps and their relationship with explosion birth and extinction phenomena.
- Generalized r-Lambert function in the analysis of fixed points and bifurcations of homographic 2-Ricker mapsPublication . Rocha, J. Leonel; TAHA, Abdel-KaddousThis paper aims to study the nonlinear dynamics and bifurcation structures of a new mathematical model of the γ-Ricker population model with a Holling type II per-capita birth function, where the Allee effect parameter is γ = 2. A generalized r-Lambert function is defined on the 3D parameters space to determine the existence and variation of the number of nonzero fixed points of the homographic 2-Ricker maps considered. The singularity points of the generalized r-Lambert function are identified with the cusp points on a fold bifurcation of the homographic 2-Ricker maps. In this approach, the application of the transcendental generalized r-Lambert function is demonstrated based on the analysis of local and global bifurcation structures of this three-parameter family of homographic maps. Some numerical studies are included to illustrate the theoretical results.
- 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.