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.
Highlights
The use of mathematical modeling to describe population growth behaviors has been of great success in the last decades
With the exception of Malthus model, all the models covered by the generalized model present a per capita growth rate which decreases as a function of time
The microscopic model allows stating that the generalization parameter q is a relation between the repulsive potential range among the cells and the dimension of the structure formed by the population
Summary
The use of mathematical modeling to describe population growth behaviors has been of great success in the last decades. Mathematical models are addressed throughout this paper, from the simplest model to the more complete ones, always in an attempt to explain the population growth and using these yeast data as a validity test The efficiency of these models, measured by the coefficient of determination r2, are shown in table (1). This kind of model is built to achieve macroscopic phenomena as a consequence, or better, an emergent property, of the interactions between individuals In this way, microscopic models allow describing population growth from first principles. In section (6), the generalized model is deduced from first principles, i.e., from the interaction between the individuals This deduction allows to explain population growth (macroscopic level) from the way cells interact (microscopic level)
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