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A Novel Unification Method to Characterize a Broad Class of Growth Curve Models Using Relative Growth Rate

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Abstract

Growth curve models serve as the mathematical framework for the quantitative studies of growth in many areas of applied science. The evolution of novel growth curves can be categorized in two notable directions, namely generalization and unification. In case of generalization, a modeler starts with a simple mathematical form to describe the behavior of the data and increases the complexity of the equation by incorporating more parameters to obtain a more flexible shape. The unification refers to the process of obtaining a compact representation of a large number of growth equations. An enormous number of growth equations are made available in the literature by means of the generalization of existing growth laws. However, the unification of growth equations has received relatively less attention from the researchers. Two significant unification functions are available in the literature, namely the Box–Cox transformation by Garcia (For Biometry Model Inf Sci 1:63–68, 2005) and generalized logarithmic and exponential functions by Martinez et al. (Phys A 387:5679–5687, 2008; Phys A 388:2922–2930, 2009). Existing unification approaches are found to have limited applications if the growth equation is characterized by the relative growth rate (RGR). RGR has immense practical value in biological growth curve analysis, which has been amplified by the construction of size and time covariate models, in which; RGR is represented either as a function of size or time or both. The present study offers a unification function for the RGR growth curves. The proposed function combines a broad class of the growth curves and possesses a greater generality than the existing unification functions. We also propose the notion of generalized RGR, which is capable of making interrelations among the unifying functions. Our proposed method is expected to enhance the generality of software and may aid in choosing an optimal model from a set of competitor growth equations.

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Acknowledgements

A significant development of this work was carried out during the academic visit July 11–13th, 2018 by the second author ARB in the Agricultural and Ecological Research Unit, Indian Statistical Institute, Kolkata, India. ARB thanks the Technical Education Quality Improvement Programme (TEQIP, Phase-III), Institute of Chemical Technology, Mumbai, for the financial support. We are thankful to the two anonymous reviewers for their suggestions that greatly improved the revised version of the manuscript from its earlier version. We sincerely thank the Editor-in-Chief Prof. Alan Hastings for his valuable suggestions.

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Authors and Affiliations

  1. Department of Statistics, Aliah University, IIA/27 Newtown, Kolkata, 700156, India

    Biman Chakraborty

  2. Department of Mathematics, Institute of Chemical Technology, Mumbai, 400019, India

    Amiya Ranjan Bhowmick

  3. Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata, 700108, India

    Joydev Chattopadhyay & Sabyasachi Bhattacharya

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  1. Biman Chakraborty
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  2. Amiya Ranjan Bhowmick
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  3. Joydev Chattopadhyay
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  4. Sabyasachi Bhattacharya
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Correspondence to Sabyasachi Bhattacharya.

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Chakraborty, B., Bhowmick, A.R., Chattopadhyay, J. et al. A Novel Unification Method to Characterize a Broad Class of Growth Curve Models Using Relative Growth Rate. Bull Math Biol 81, 2529–2552 (2019). https://doi.org/10.1007/s11538-019-00617-w

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  • DOI: https://doi.org/10.1007/s11538-019-00617-w

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