On the Gompertzian growth in the fractal space–time

Molski, Marcin; Konarski, Jerzy

doi:10.1016/j.biosystems.2008.03.003

Cited by 3 publications

(1 citation statement)

References 14 publications

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“…which is, in fact, more usually found in the literature [18][19][20][21]. Note that if we integrate the Eq.…”

Section: The Gompertz Modelsupporting

confidence: 58%

An attempt to unify some population growth models from first principles

Ribeiro

2016

Rev. Bras. Ensino Fís.

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In this work, some phenomenological growth models based only on the population information (macroscopic level) are deduced in an intuitive way. These models, for instance Verhulst, Gompertz and Bertalanffy-Richards models, are introduced in such a way that all the parameters involved have a physical interpretation. A model based on the interaction (distance dependent) between the individuals (microscopic level) is also presented. This microscopic model have some phenomenological models as particular cases. In this approach, the Verhulst model represents the situation in which all the individuals interact in the same way, regardless of the distance between them (mean field approach). Other phenomenological models are retrieved from the microscopic model according to two quantities: i) the way that the interaction decays as a function the distance between two individuals and ii) the dimension of the spatial structure formed by the individuals of the population. This microscopic model allows understanding population growth by first principles, because it predicts that some phenomenological models can be seen as a consequence of interaction at individual level. The microscopic model discussed here paves the way to finding universal patterns that are common to all types of growth, even in systems of very different nature. Keywords: Population growth, Complex Systems, Mathematical Modelling Neste trabalho, alguns modelos fenomenológicos de crescimento populacional são deduzidos de uma forma intuitiva. Estes modelos, por exemplo os modelos de Verhulst, Gompertz e Bertalanffy-Richards, são introduzidos de maneira que todos os parâmetros envolvidos tenham uma interpretação física. Um modelo baseado na interação (dependente da distância) entre os indivíduos (nível microscópico) tambéḿ e apresentado.É mostrado que alguns modelos fenomenológicos são casos particulares deste modelo microscópico, de acordo com: i) a forma com que a interação competitiva entre indivíduos decai com a distância que os separam; e ii) a dimensão euclidiana da estrutura espacial formada pelos indivíduos da população. Nestas circunstâncias, o modelo de Verhulst, por exemplo, decorre quando todos os indivíduos interagem de forma independente da distância que os separam: uma situação tipo campo-médio. Este modelo microscópico permite compreender o crescimento populacional por primeiros princípios, uma vez que mostra que alguns modelos fenomenológicos são explicados a partir de interações no nível do indivíduo. Dessa forma, o modelo microscópico discutido aqui abre caminho para encontrar padrões universais que sejam comuns a todos os tipos de crescimento. Palavras-chave: Crescimento Populacional, Sistemas Complexos, Modelagem Matemática

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“…which is, in fact, more usually found in the literature [18][19][20][21]. Note that if we integrate the Eq.…”

Section: The Gompertz Modelsupporting

confidence: 58%

An attempt to unify some population growth models from first principles

Ribeiro

2016

Rev. Bras. Ensino Fís.

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Cellular automata models of tumour natural shrinkage

Naumov

Hoekstra

Sloot

2011

Physica A: Statistical Mechanics and its Applications

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An extension of Gompertzian growth dynamics: Weibull and Fréchet models

Rocha

Aleixo

2013

Mathematical Biosciences and Engineering

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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.

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On the Gompertzian growth in the fractal space–time

Cited by 3 publications

References 14 publications

An attempt to unify some population growth models from first principles

An attempt to unify some population growth models from first principles

Cellular automata models of tumour natural shrinkage

An extension of Gompertzian growth dynamics: Weibull and Fréchet models

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