Dynamical behaviour on the parameter space: New populational growth models proportional to beta densities

Aleixo, Sandra M.; Rocha, J. Leonel; Pestana, Dínis

doi:10.1109/iti.2009.5196082

Cited by 5 publications

(5 citation statements)

References 7 publications

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“…For the parameter values, such that these conditions hold, the associated dynamic behaviour is characterized by bifurcations (period doubling) and chaos, verifying Sharkovsky's Theorem, a non-decreasing function in order to the parameter r i , [1], [4] and [5]. In turn, the chaotic region ends at the designated "fullshift", whose symbolic sequence is CRL ∞ .…”

Section: Extinction Bistability Chaotic Semistability and Essentialmentioning

confidence: 53%

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Generalized Models from Beta(p, 2) Densities with Strong Allee Effect: Dynamical Approach

Aleixo

Rocha

2012

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A dynamical approach to study the behaviour of generalized populational growth models from Beta(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.

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Section: Extinction Bistability Chaotic Semistability and Essentialmentioning

confidence: 53%

“…In previous studies, the authors presented a successful approach to study population growth models, proportional to Beta(p, 2) densities, [1] and [2]. These models are natural extensions of the logistic Verhuslt growth model, originally introduced as a demographic model and often considered as the basis of modern chaos theory.…”

Section: Introductionmentioning

confidence: 99%

Generalized Models from Beta(p, 2) Densities with Strong Allee Effect: Dynamical Approach

Aleixo

Rocha

2012

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“…(1 − x n ) q−1 have been investigated by Aleixo et al [92][93][94], Brilhante et al [19][20][21], Rocha and Aleixo [95], and Gomes et al [22].…”

Section: Logistic Growth Gompertz Growth and Extensions Of The Logist...mentioning

confidence: 99%

“…, Blumberg [16] hyper-logistic DE, in (4), whose dynamics has been investigated by Aleixo et al [92][93][94] and Rocha and Aleixo [95].…”

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confidence: 99%

Generalized Beta Models and Population Growth: So Many Routes to Chaos

et al. 2023

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Logistic and Gompertz growth equations are the usual choice to model sustainable growth and immoderate growth causing depletion of resources, respectively. Observing that the logistic distribution is geo-max-stable and the Gompertz function is proportional to the Gumbel max-stable distribution, we investigate other models proportional to either geo-max-stable distributions (log-logistic and backward log-logistic) or to other max-stable distributions (Fréchet or max-Weibull). We show that the former arise when in the hyper-logistic Blumberg equation, connected to the Beta (p,q) function, we use fractional exponents p−1=1∓1/α and q−1=1±1/α, and the latter when in the hyper-Gompertz-Turner equation, the exponents of the logarithmic factor are real and eventually fractional. The use of a BetaBoop function establishes interesting connections to Probability Theory, Riemann–Liouville’s fractional integrals, higher-order monotonicity and convexity and generalized unimodality, and the logistic map paradigm inspires the investigation of the dynamics of the hyper-logistic and hyper-Gompertz maps.

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“…is proportional to the Beta(2,2) density, the authors investigated, [1], [2], [3], [10], and [11], the dynamical behaviour of functions…”

Section: An Extension Of the Beta Densities And Gompertz Growth Modelmentioning

confidence: 99%

Regular variation, Paretian distributions, and the interplay of light and heavy tails in the fractality of asymptotic models

2011

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Classical central limit theorems, culminating in the theory of infinite divisibility, accurately describe the behaviour of stochastic phenomena with asymptotically negligible components. The classical theory fails when a single component may assume an extreme protagonism. The early developments of the speculation theory didn't incorporate the pioneer work of Pareto on heavy tailed models, and the proper setup to conciliate regularity and abrupt changes, in a wide range of natural phenomena, is Karamata's concept of regular variation and the role it plays in the theory of domains of attraction, [8], and Resnick's tail equivalence leading to the importance of generalized Pareto distribution is the scope of extreme value theory, [13]. Waliszewski and Konarski discussed the applicability of the Gompertz curve and its fractal behaviour for instance in modeling healthy and neoplasic cells tissue growth, [15]. Gompertz function is the Gumbel extreme value model, whose broad domain of attraction contains intermediate tail weight laws with a wide range of behaviour. Aleixo et al. investigated fractality associated with Beta (p,q) models, [1], [2], [10] and [11]. In this work, we introduce a new family of probability density functions tied to the classical beta family, the Beta * (p,q) models, some of which are generalized Pareto, that span the possible regular variation of tails. We extend the investigation to other extreme stable models, namely Fré chet's and Weibull's types in the General Extreme Value (GEV) model.

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Dynamical behaviour on the parameter space: New populational growth models proportional to beta densities

Cited by 5 publications

References 7 publications

Generalized Models from Beta(p, 2) Densities with Strong Allee Effect: Dynamical Approach

Generalized Models from Beta(p, 2) Densities with Strong Allee Effect: Dynamical Approach

Generalized Beta Models and Population Growth: So Many Routes to Chaos

Regular variation, Paretian distributions, and the interplay of light and heavy tails in the fractality of asymptotic models

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