Generalized Mathematical Model for Biological Growths


In this paper, we present a generalization of the commonly used growth models. We introduce Koya-Goshu biological growth model, as a more general solution of the rate-state ordinary differential equation. It is shown that the commonly used growth models such as Brody, Von Bertalanffy, Richards, Weibull, Monomolecular, Mitscherlich, Gompertz, Logistic, and generalized Logistic functions are its special cases. We have constructed growth and relative growth functions as solutions of the rate-state equation. The generalized growth function is the most flexible so that it can be useful in model selection problems. It is also capable of generating new useful models that have never been used so far. The function incorporates two parameters with one influencing growth pattern and the other influencing asymptotic behaviors. The relationships among these growth models are studies in details and provided in a flow chart.

Share and Cite:

Koya, P. and Goshu, A. (2013) Generalized Mathematical Model for Biological Growths. Open Journal of Modelling and Simulation, 1, 42-53. doi: 10.4236/ojmsi.2013.14008.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] S. Brody, “Bioenergetics and Growth,” Reinhold Publishing Corporation, New York, 1945.
[2] L. von Bertalanffy, “Quantitative Laws in Metabolism and Growth,” The Quarterly Review of Biology, Vol. 3, No. 2, 1957, p. 218.
[3] F. J. Richards, “A Flexible Growth Function for Empirical Use,” Journal of Experimental Botany, Vol. 10, 1959, pp. 290-300.
[4] J. France and J. H. M. Thornley, “Mathematical Models in Agriculture,” Butterworths, London, 1984, p. 335.
[5] C. P. Winsor, “The Gompertz Curve as a Growth Curve,” Proceedings of National Academy of Science, Vol. 18, No. 1, 1932, pp. 1-8.
[6] J. A. Nelder, “The Fitting of a Generalization of the Logistic Curve,” Biometrics, Vol. 17, No.7, 1961, pp. 89-110.
[7] J. E. Brown, H. A. Fitzhugh Jr. and T. C. Cartwright, “A Comparison of Nonlinear Models for Describing Weight-Age Relationship in Cattle,” Journal of Animal Science, Vol. 42, No. 4, 1976, pp. 810-818.
[8] T. B. Robertson, “On the Normal Rate of Growth of an Individual and Its Biochemical Significance,” Archiv für Entwicklungsmechanik der Organismen, Vol. 25, No. 4, 1906, pp. 581-614.
[9] L. L. Eberhardt and J. M. Breiwick, “Models for Population Growth Curves,” ISRN Ecology, Vol. 2012, 2012, Article ID: 815016.
[10] D. Fekedulegn, M. P. Mac Siurtain and J. J. Colbert, “Parameter Estimation of Nonlinear Growth Models in Forestry,” Silva Fennica, Vol. 33 No. 4, 1999, pp. 327-336.
[11] F. J. Ayala, M. E. Gilpin and J. G. Ehrenfeld, “Competition between Species: Theoretical Models and Experimental Tests,” Theoretical Population Biology, Vol. 4, No. 3, 1973, pp. 331-356.
[12] J. O. Rawlings and W. W. Cure, “The Weibull Function as a Dose Response Model for Air Pollution Effects on Crop Yields,” Crop Science, Vol. 25, 1985, pp. 807-814.
[13] J. O. Rawlings, S. G. Pantula and D. A. Dickey, “Applied Regression Analysis: A Research Tool,” Springer, New York, 1998.
[14] W. J. Spillman and E. Lang, “The Law of Diminishing Increment,” World, Yonkers, 1924.
[15] J. France, J. Dijkstra and M. S. MDhanoa, “Growth Functions and Their Application in Animal Science,” Annales De Zootechnie, Vol. 45, Suppl. 1, 1996, pp. 165-174.
[16] I. E. Ersoy, M. Mende? and S. Keskin, “Estimation of Parameters of Linear and Nonlinear Growth Curve Models at Early Growth Stage in California Turkeys,” Archiv für Geflügelkunde, Vol. 71, No. 4, 2007, pp.175-180.
[17] Y. C. Lei and S. Y. Zhang, “Features and Partial Derivatives of Bertalanffy-Richards Growth Model in Forestry,” Nonlinear Analysis: Modelling and Control, Vol. 9, No. 1, 2004, pp. 65-73.
[18] B. Zeide, “Analysis of Growth Equations,” Forest Science, Vol. 39, No. 3, 1993, pp. 594-616.
[19] A. T. Goshu, “Simulation Study of the Commonly Used Mathematical Growth Models,” Journal of the Ethiopian Statistical Association, Vol. 17, 2008, pp. 44-53.
[20] C. H. Edwards Jr. and D. E. Penney, “Calculus with Analytic Geometry,” Printice Hall International, New Jersey, 1994.
[21] Generalised logistic function
[22] R. A. Mombiela and L. A. Nelson, “Relationships among Some Biological and Empirical Fertilizer Response Models and Use of the Power Family of Transformations to Identify an Appropriate Model,” Agronomy Journal, Vol. 73, No. 2, 1981 pp. 353-356.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.