Sandra M. Aleixo

Populational Growth Models Proportional to Beta Densities with Allee Effect

We consider populations growth models with Allee effect, proportional to beta densities with shape parameters p and 2, where the dynamical complexity is related with the Malthusian parameter r. For p>2, these models exhibit a population dynamics with natural Allee effect. However, in the case of 1<p⩽2, the proposed models do not include this effect. In order to inforce it, we present some alternative models and investigate their dynamics, presenting some important results.

An Extension of Gompertzian Growth Dynamics Weibull and Frechet Models

In 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.

Populational growth models in the light of symbolic dynamics

Using symbolic dynamic techniques, populational growth models proportional to beta densities, are investigated. Our results give explicit methods to investigate the chaotic behaviour of populational growth models, when the malthusean parameter increases. The chaotic complexity is measured in terms of the topological entropy.

Generalized models from Beta(p, 2) densities with strong Allee effect: Dynamical approach

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.

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

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 Fre chet’s and Weibull’s types in the General Extreme Value (GEV) model.

Beta(p,q)-Cantor sets: Determinism and randomness

Usually randomness appears as a sophisticated extension of deterministic models, that are then presented as expectation of some class of random models (this approach is exceedingly well managed in the classical Barucha-Reid’s treatise on random functions and stochastic processes). The works [1], [2], [3] and [5] summarize previous studies by the authors, using stochastic definitions of extensions of Cantor’s fractal to put forward appropriate deterministic models, that in a precise sense are the expectation of a structured class of models, and investigated bifurcations, Allee’ effect, and the Hausdorff dimension.   q p, Beta models, with either 1  p or 1  q , or the classical Verhulsts model   2   q p , proportionate interesting computable models for which computations both of Hausdorff dimension and probabilities can be explicitly evaluated, either analytically or using the Monte Carlo method. The present extension, axed on arbitrary symbolic dynamical systems, further develops new fundamental classes of geometric constructions, and exploits the interplay of determinism and randomness on the richness of the limit fractal set, in a recursive construction. This sheds new light on the concept of Hausdorff dimensionality. We show that the dependence of the random order statistics is at the core of the apparent anomaly of consistently smaller Hausdorff dimensions of the random sets, when compared with the corresponding “expected” deterministic counterparts. We also recover Falconner’s, Pesin’s and Weiss’ (among others) ideas on recursive geometric constructions as a straightforward approach to important issues in fractality and chaos.

Modeling Allee effect from Beta(p, 2) densities

In this work we develop and investigate generalized populational growth models, adjusted from Beta(p, 2) densities, with Allee effect. The use of a positive parameter C leads the presented generalization, which yields some more flexible models with variable extinction rates. An Allee limit is incorporated so that the models under study have strong Allee effect.

Probabilistic Methods in Dynamical Analysis: Populations Growths Associated to Models Beta(p, q) with Allee Effect

New populational growth models, proportional to beta densities, with shape parameters p and 2, where p > 1, and Malthusian parameter r, are developed. For p > 2, these models exhibit natural Allee effect. However, in the case of 1 < p ≤ 2, the proposed models do not include this effect. In order to inforce it, we deduce alternative models and investigate their dynamical behaviour. The Verhulst Model, which is a cornerstone of modern chaos theory, is a special case of those models. The complex dynamical behaviour of these models is analysed in the parameter space (r, p), in terms of topological entropy, using explicit methods of dynamical systems. We emphasize some particular disjoint regions in these parameter space, according to the chaotic behaviour of the models, the main result being the characterization of those disjoint regions. We also present some important results about these modified models.

Hausdorff dimension of the random middle third Cantor set

The iterative elimination of the middle spacing in the random division of intervals with two points “at random” — in the narrow sense of uniformly distributed — generates a random middle Cantor set. We compute the Hausdorff dimension (which intuitively evaluates how “dense” a set is) of the random middle third Cantor set, and we verify that although the deterministic middle third Cantor set is the expectation of the random middle third Cantor set, it is more dense than its stochastic counterpart. This can be explained by the dependence of order statistics

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

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.