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Revisiting and redefining return rate for determination of the precise growth status of a species

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Abstract

Growth curve models play an instrumental role in quantifying the growth of biological processes and have immense practical applications across all disciplines. The most popular growth metric to capture the species fitness is the “Relative Growth Rate” in this domain. The different growth laws, such as exponential, logistic, Gompertz, power, and generalized Gompertz or generalized logistic, can be characterized based on the monotonic behavior of the relative growth rate (RGR) to size or time. Thus, in this case, species fitness can be determined truly through RGR. However, in nature, RGR is often non-monotonic and specifically bell-shaped, especially in the situation when a species is adapting to a new environment [1]. In this case, species may experience with the same fitness (RGR) for two different time points. The species precise growth and maturity status cannot be determined from this RGR function. The instantaneous maturity rate (IMR), as proposed by [2], helps to determine the correct maturity status of the species. Nevertheless, the metric IMR suffers from severe drawbacks; (i) IMR is intractable for all non-integer values of a specific parameter. (ii) The measure depends on a model parameter. The mathematical expression of IMR possesses the term “carrying capacity” which is unknown to the experimenter. (iii) Note that for identifying the precise growth status of a species, it is also necessary to understand its response when the populations are deflected from their equilibrium position at carrying capacity. This is an established concept in population biology, popularly known as the return rate. However, IMR does not provide information on the species deflection rate at the steady state. Hence, we propose a new growth measure connected with the species return rate, termed the “reverse of relative of relative growth rate” (henceforth, RRRGR), which is treated as a proxy for the IMR, having similar mathematical properties. Finally, we introduce a stochastic RRRGR model for specifying precise species growth and status of maturity. We illustrate the model through numerical simulations and real fish data. We believe that this study would be helpful for fishery biologists in regulating the favorable conditions of growth so that the species can reach a steady state with optimum effort.

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Acknowledgements

The author Ayan Paul is thankful to the Department of Science and Technology, Government of India, i.e., DST-INSPIRE (Grant Number: IF180793), for supporting the fellowship. We must acknowledge Md Aktar Ul Karim, CSIR Senior Research Fellow from Institute of Chemical Technology, India, and Selim Reja, UGC Senior Research Fellow from the Indian Statistical Institute, Kolkata, for the technical help in preparing the revised version of the manuscript. We are also thankful to Dr. Soumalya Mukhopadhyay, Dr. Amiya Ranjan Bhowmick, Dr. Biman Chakraborty from the Visva-Bharati University, Institute of Chemical Technology, the Aliah University of India, respectively, and the anonymous reviewers for their valuable suggestions to improve the quality of the manuscript.

Funding

The author Ayan Paul is thankful to the Department of Science and Technology, Government of India, i.e., DST-INSPIRE (Grant Number: IF180793), for supporting the fellowship.

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Authors and Affiliations

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

    Ayan Paul & Sabyasachi Bhattacharya

  2. Department of Mathematical Sciences, University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH, 45221, USA

    Neelakshi Chatterjee

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  1. Ayan Paul
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Correspondence to Sabyasachi Bhattacharya.

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Paul, A., Chatterjee, N. & Bhattacharya, S. Revisiting and redefining return rate for determination of the precise growth status of a species. J Biol Phys 49, 195–234 (2023). https://doi.org/10.1007/s10867-023-09628-0

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