An attempt to unify some population growth models from first principles

Abstract

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

New formulation of the Gompertz equation to describe the kinetics of untreated tumors

“…Izquierdo-Kulich et al [4] report the fractal origin of GE (see appendix A). This fractal origin has also been reported in [5–8] but in terms only of the fractal dimension D f . Here, we have considered the one in [4] because it also takes into account the fractal structure of the boundary of the tumor.…”

“…In this study, the tumor growth in the time results of the complex interactions that happen in the tumor and between it and the surrounding healthy tissue, as in [3,14]. Nevertheless, in it does not explicitly discuss the interactions among the individuals neither the cooperative capacity of they in a population to explain its growth behavior, as in [25, 5–8]. These works confirm the fractal property of the tumors, as in this study.…”

Conformable Laplace Transform of Fractional Differential Equations

“…The von Bertalanffy equation is a logistic model widely applied to describe the growth of different types of populations [32][33][34].…”

Conformable Laplace Transform of Fractional Differential Equations

“…The Von Bertalanffy equation is a logistic model widely applied to describe growth of different types of populations [27][28][29].…”