Richards-like two species population dynamics model

Ribeiro, Fabiano L.; Cabella, Brenno; Martinêz, Alexandre Souto

doi:10.1007/s12064-014-0205-z

Cited by 12 publications

(9 citation statements)

References 48 publications

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“…For more details, see Ref. [14,16,[31][32][33][34]. The solution of the generalized model can be obtained integrating both sides of (18) (see appendix (A)), which leads to…”

Section: The Generalized Model and The Bertalanffy-richards Modelmentioning

confidence: 99%

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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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“…For more details, see Ref. [14,16,[31][32][33][34]. The solution of the generalized model can be obtained integrating both sides of (18) (see appendix (A)), which leads to…”

Section: The Generalized Model and The Bertalanffy-richards Modelmentioning

confidence: 99%

“…Moreover, if q = 1 (Verhulst) and K → ∞, the generalized model also recovers the Malthus model. For more details, see [22][23][24] The solution of the generalized model can be obtained by integrating both sides of (18), as usual, which leads to the solution…”

Section: The Generalized Modelmentioning

confidence: 99%

An attempt to unify some population growth models from first principles

Ribeiro

2016

Rev. Bras. Ensino Fís.

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“…So, our model will be very useful to investigate universality and common patterns in population growth in general (tumor, animal and any other one). This subject has drawn attention of many theoretical [17,[31][32][33][34][35][36][37][38][39] and empirical [1,2,40,41] researches in the last decades. This paper is organized as follows: section (II) gives a brief description of some empirical evidences that are considered common features in the most of avascular tumors.…”

Section: Introductionmentioning

confidence: 99%

Fractal dimension and universality in avascular tumor growth

2017

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For years, the comprehension of the tumor growth process has been intriguing scientists. New research has been constantly required to better understand the complexity of this phenomenon. In this paper, we propose a mathematical model that describes the properties, already known empirically, of avascular tumor growth. We present, from an individual-level (microscopic) framework, an explanation of some phenomenological (macroscopic) aspects of tumors, such as their spatial form and the way they develop. Our approach is based on competitive interaction between the cells. This simple rule makes the model able to reproduce evidence observed in real tumors, such as exponential growth in their early stage followed by power-law growth. The model also reproduces (i) the fractal-space distribution of tumor cells and (ii) the universal growth behavior observed in both animals and tumors. Our analyses suggest that the universal similarity between tumor and animal growth comes from the fact that both can be described by the same dynamic equation-the Bertalanffy-Richards model-even if they do not necessarily share the same biological properties.

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“…(Ayala et al, 1973;Gilpin and Ayala, 1973;Pomerantz et al, 1980), and in long-term datasets on species with more complex life histories (Stubbs, 1977;Fowler, 1981;Sibly et al, 2005;Coulson et al, 2008). Models that relax the assumption of linear intraspecific density dependence have been proposed for single species (e.g., Richards, 1959;Schoener, 1973;Turchin, 2003;Sibly et al, 2005) and communities with two or more competitors (Ayala et al, 1973;Gilpin and Ayala, 1973;Schoener, 1976;Goh and Agnew, 1977;Gallagher et al, 1990), but never for mutualism (but see a generalized Verhulst-Lotka-Volterra model in Ribeiro et al 2014 and a specific facultative-obligate model in Wang 2016). Given the prevalence of nonlinear intraspecific density dependence, and its known influence on dynamics in other ecological contexts, the dearth of mutualism models that assume anything besides logistic growth suggests that our understanding of mutualistic dynamics may be quite incomplete.…”

Section: Introductionmentioning

confidence: 99%

Population dynamics of mutualism and intraspecific density dependence: How θ-logistic density dependence affects mutualistic positive feedback

Moore

Catella

Abbott

2018

Ecological Modelling

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Mutualism describes the biological phenomenon where two or more species are reciprocally beneficial, regardless of their ecological intimacy or evolutionary history. Classic theory shows that mutualistic benefit must be relatively weak, or else it overpowers the stabilizing influence of intraspecific competition and leads to unrealistic, unbounded population growth. Interestingly, the conclusion that strong positive interactions lead to runaway population growth is strongly grounded in the behavior of a single model. This model-the Lotka-Volterra competition model with a sign change to generate mutualism rather than competition between species-assumes logistic growth of each species plus a linear interaction term to represent the mutualism. While it is commonly held that the linear interaction term is to blame for the model's unrealistic behavior, we show here that a linear mutualism added to a θ-logistic model of population growth can prevent unbounded growth. We find that when density dependence is decelerating, the benefit of mutualism at equilibrium is greater than when density dependence is accelerating. Although there is a greater benefit, however, decelerating density dependence tends to destabilize populations whereas accelerating density dependence is always stable. We interpret these findings tentatively, but with promise for the understanding of the population ecology of mutualism by generating several predictions relating growth rates of mutualist populations and the strength of mutualistic interaction.

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Richards-like two species population dynamics model

Cited by 12 publications

References 48 publications

An attempt to unify some population growth models from first principles

An attempt to unify some population growth models from first principles

Fractal dimension and universality in avascular tumor growth

Population dynamics of mutualism and intraspecific density dependence: How θ-logistic density dependence affects mutualistic positive feedback

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