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Mathematical Study of the Verhulst and Gompertz Growth Functions and Their Contemporary Applications

Year 2021, Issue: 34, 73 - 102, 26.07.2021

Abstract

This study examines the mathematical characteristics of the logistic, the generalized logistic and the Gompertz growth function used in human population analysis. When a population growth is mathematically modeled, it starts with differential equations considered as a preliminary study. Then, a general solution equation is derived. This is the method followed by the mathematicians who developed these models. To prepare for the study, I used the framework of the objectives and adhered to the resources and approaches outlined by mathematicians who developed the growth function. In addition, I wanted to evaluate the methodologies that remain valid in contemporary applications using the current perspectives. Mathematician and actuary Benjamin Gompertz developed the first survivors’ function in 1825 which was used later as a population growth function while systematizing life tables. In his three published articles, the mathematician Pierre-François Verhulst developed a logistic human population growth function based on his economic analysis. In addition, he searched for test opportunities using the limited population statistics of France, Belgium, England, the USA, and Russia. Contemporary authors Richards and, ‘Ricketts and Head’ made very invaluable contributions to logistic growth function.

Project Number

Yok

References

  • Allen R.G.D. (1938). Mathematical Analysis for Economists. 1969 reprint. London. Macmillan and Co. Ltd.
  • Fekedulegn, D., Mac Siurtain, M. P., & Colbert, J. J. (1999). Parameter Estimation of Nonlinear Growth Models in Forestry.” Silva Fennica, 33(4). 327–336.
  • Gompertz, B. (1825). On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies (Chapter I and II). Philosophical Transactions of the Royal Society of London, Vol. 115, 513–583. http://www.jstor.org/stable/107756.
  • Gray, P. (1857). On Mr. Gompertz's Method for the Adjustment of Tables of Mortality. The Assurance Magazine, and Journal of the Institute of Actuaries 7(3): 121–130. https://www.jstor.org/stable/41134787.
  • Heinen, M. (1999). Analytical Growth Equations and Their Genstat 5 Equivalents. Netherlands Journal of Agricultural Science, 47, 67–89. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.694.6993&rep=rep1&type=pdf
  • İskender, C. (2018). Türkiye Nüfus Büyümesi ve Tahminleri: Matematiksel Büyüme Modelleri ve İstatistiksel Analiz ile Kuramsal ve Uygulamalı Bir Yaklaşım. Istanbul University, Econometrics and Statistics e-Journal, 14(28), 75-141. Retrieved from https://dergipark.org.tr/tr/pub/iuekois/issue/39224
  • İskender, C. (2019). Türkiye 2014-2016 Hayat Tablolarında Doğrusal-olmayan Büyüme Fonksiyonları Uygulaması. Ekoist: Journal of Econometrics and Statistics, 14(29),151-168. Retrieved from https://dergipark.org.tr/tr/pub/ekoist/issue/47194
  • King, G. (1902). Text Book of The Principles of Interest, Life Annuities, and Assurances, and Their Practical Application Part II. London: Charles & Edwin Layton, 56, Farringdon Street. E.G.
  • Matsui, K. (2009). Gompertzian Model. Kanzo 50, 324-338. Retrieved from https://www.jstage.jst.go.jp/article/kanzo/50/6/50_6_324/_pdf.
  • Matsui, K. (2009). Gompertz-Matsui Model for HCV Kinetics. 1-5. Retrieved from http://gompertz-matsui.la.coocan.jp/Gompertz-Matsui_model/Gompertz-Matsui_model_files/Gompertz-Matsui%20model.pdf
  • Matsui, K. (1999). Gompertz Curve. Retrieved from http://gompertz-matsui.la.coocan.jp/Gompertz/English.html.
  • Paine, C. E. T., Marthews, T. R., Vogt, D. R., Purves, D., Rees, M., Hector, A., & Turnbull, L. A., (2012). How to Fit Nonlinear Plant Growth Models and Calculate Growth Rates: An Update for Ecologists. Methods in Ecology and Evolution, 3, 245–256. https://doi.org/10.1111/j.2041-210X.2011.00155.x
  • Richards, F. J. (1959). A Flexible Growth Function for Empirical Use. Journal of Experimental Botany, 10(2), 290–300. https://doi.org/10.1093/jxb/10.2.290
  • Ricketts, J. H., & Head, G. A. (1999). A Five-Parameter Logistic Equation for Investigating Asymmetry of Curvature in Baroreflex Studies. American Journal of Physiology-Regulatory, Integrative and Comparative Physiology, 277(2), R441–454. https://doi.org/10.1152/ajpregu.1999.277.2.R441
  • Sit, V., & Poulin-Costello, M. (1994). Catalog of Curves for Curve Fitting. Victoria: Ministry of Forests Canada.
  • Skiadas, C. H. (2010). Exact Solutions of Stochastic Differential Equations: Gompertz, Generalized Logistic and Revised Exponential. Methodology and Computing in Applied Probability 12(2): 261–270. DOI 10.1007/s11009-009-9145-3 http://www.cmsim.net/sitebuildercontent/sitebuilderfiles/exact_solutions_of_stochastic_differential_equations.pdf.
  • Sprague, T. B. (1861). On Mr. Gompertz's Law of Human Mortality, and Mr. Edmonds's Claims to its Independent Discovery and Extension. The Assurance Magazine, and Journal of the Institute of Actuaries, 9(5), 288–295. https://www.jstor.org/stable/41135117.
  • Tjørve, E., & Tjørve, K. M. C. 2010. A Unified Approach to the Richards-Model Family for Use in Growth Analyses: Why We Need Only Two Model Forms. Journal of Theoretical Biology, 267, 417–425.
  • Tjørve, K. M. C., & Tjørve, E. (2017). “The Use of Gompertz Models in Growth Analyses, and New Gompertz-Model Approach: An Addition to the Unified-Richards Family.” PLoS ONE 12(6): 1–17. https://doi.org/10.1371/journal.pone.0178691
  • Verhulst, P.-F. (1838). Notice Sur La Loi Que La Population Suit Dans Son Accroissement. 113-121. In book of Jean Guillaume Garnier, Adolpho Quetelet Correspondance Mathématique et Physique de L’observatoire de Bruxelles, Tome Quatrième, Bruxelles.
  • Verhulst, P.-F. (1844). Recherches Mathématiques Sur La Loi d'accroissement de La Population. Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles, 18, 1–42.
  • Verhulst, P.-F. (1847). Deuxième Mémoire Sur La Loi d'accroissement de La Population. Mémoires de l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique, 20, 1–32.
  • Winsor, C. P. (1932). The Gompertz Curve as a Growth Curve. Proceedings of the National Academy of Sciences, 18(1), 1–8. http://www.pnas.org/cgi/doi/10.1073/pnas.18.1.1.
Year 2021, Issue: 34, 73 - 102, 26.07.2021

Abstract

Supporting Institution

Yok

Project Number

Yok

References

  • Allen R.G.D. (1938). Mathematical Analysis for Economists. 1969 reprint. London. Macmillan and Co. Ltd.
  • Fekedulegn, D., Mac Siurtain, M. P., & Colbert, J. J. (1999). Parameter Estimation of Nonlinear Growth Models in Forestry.” Silva Fennica, 33(4). 327–336.
  • Gompertz, B. (1825). On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies (Chapter I and II). Philosophical Transactions of the Royal Society of London, Vol. 115, 513–583. http://www.jstor.org/stable/107756.
  • Gray, P. (1857). On Mr. Gompertz's Method for the Adjustment of Tables of Mortality. The Assurance Magazine, and Journal of the Institute of Actuaries 7(3): 121–130. https://www.jstor.org/stable/41134787.
  • Heinen, M. (1999). Analytical Growth Equations and Their Genstat 5 Equivalents. Netherlands Journal of Agricultural Science, 47, 67–89. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.694.6993&rep=rep1&type=pdf
  • İskender, C. (2018). Türkiye Nüfus Büyümesi ve Tahminleri: Matematiksel Büyüme Modelleri ve İstatistiksel Analiz ile Kuramsal ve Uygulamalı Bir Yaklaşım. Istanbul University, Econometrics and Statistics e-Journal, 14(28), 75-141. Retrieved from https://dergipark.org.tr/tr/pub/iuekois/issue/39224
  • İskender, C. (2019). Türkiye 2014-2016 Hayat Tablolarında Doğrusal-olmayan Büyüme Fonksiyonları Uygulaması. Ekoist: Journal of Econometrics and Statistics, 14(29),151-168. Retrieved from https://dergipark.org.tr/tr/pub/ekoist/issue/47194
  • King, G. (1902). Text Book of The Principles of Interest, Life Annuities, and Assurances, and Their Practical Application Part II. London: Charles & Edwin Layton, 56, Farringdon Street. E.G.
  • Matsui, K. (2009). Gompertzian Model. Kanzo 50, 324-338. Retrieved from https://www.jstage.jst.go.jp/article/kanzo/50/6/50_6_324/_pdf.
  • Matsui, K. (2009). Gompertz-Matsui Model for HCV Kinetics. 1-5. Retrieved from http://gompertz-matsui.la.coocan.jp/Gompertz-Matsui_model/Gompertz-Matsui_model_files/Gompertz-Matsui%20model.pdf
  • Matsui, K. (1999). Gompertz Curve. Retrieved from http://gompertz-matsui.la.coocan.jp/Gompertz/English.html.
  • Paine, C. E. T., Marthews, T. R., Vogt, D. R., Purves, D., Rees, M., Hector, A., & Turnbull, L. A., (2012). How to Fit Nonlinear Plant Growth Models and Calculate Growth Rates: An Update for Ecologists. Methods in Ecology and Evolution, 3, 245–256. https://doi.org/10.1111/j.2041-210X.2011.00155.x
  • Richards, F. J. (1959). A Flexible Growth Function for Empirical Use. Journal of Experimental Botany, 10(2), 290–300. https://doi.org/10.1093/jxb/10.2.290
  • Ricketts, J. H., & Head, G. A. (1999). A Five-Parameter Logistic Equation for Investigating Asymmetry of Curvature in Baroreflex Studies. American Journal of Physiology-Regulatory, Integrative and Comparative Physiology, 277(2), R441–454. https://doi.org/10.1152/ajpregu.1999.277.2.R441
  • Sit, V., & Poulin-Costello, M. (1994). Catalog of Curves for Curve Fitting. Victoria: Ministry of Forests Canada.
  • Skiadas, C. H. (2010). Exact Solutions of Stochastic Differential Equations: Gompertz, Generalized Logistic and Revised Exponential. Methodology and Computing in Applied Probability 12(2): 261–270. DOI 10.1007/s11009-009-9145-3 http://www.cmsim.net/sitebuildercontent/sitebuilderfiles/exact_solutions_of_stochastic_differential_equations.pdf.
  • Sprague, T. B. (1861). On Mr. Gompertz's Law of Human Mortality, and Mr. Edmonds's Claims to its Independent Discovery and Extension. The Assurance Magazine, and Journal of the Institute of Actuaries, 9(5), 288–295. https://www.jstor.org/stable/41135117.
  • Tjørve, E., & Tjørve, K. M. C. 2010. A Unified Approach to the Richards-Model Family for Use in Growth Analyses: Why We Need Only Two Model Forms. Journal of Theoretical Biology, 267, 417–425.
  • Tjørve, K. M. C., & Tjørve, E. (2017). “The Use of Gompertz Models in Growth Analyses, and New Gompertz-Model Approach: An Addition to the Unified-Richards Family.” PLoS ONE 12(6): 1–17. https://doi.org/10.1371/journal.pone.0178691
  • Verhulst, P.-F. (1838). Notice Sur La Loi Que La Population Suit Dans Son Accroissement. 113-121. In book of Jean Guillaume Garnier, Adolpho Quetelet Correspondance Mathématique et Physique de L’observatoire de Bruxelles, Tome Quatrième, Bruxelles.
  • Verhulst, P.-F. (1844). Recherches Mathématiques Sur La Loi d'accroissement de La Population. Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles, 18, 1–42.
  • Verhulst, P.-F. (1847). Deuxième Mémoire Sur La Loi d'accroissement de La Population. Mémoires de l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique, 20, 1–32.
  • Winsor, C. P. (1932). The Gompertz Curve as a Growth Curve. Proceedings of the National Academy of Sciences, 18(1), 1–8. http://www.pnas.org/cgi/doi/10.1073/pnas.18.1.1.
There are 23 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Cemil İskender 0000-0003-2841-5964

Project Number Yok
Publication Date July 26, 2021
Submission Date February 8, 2021
Published in Issue Year 2021 Issue: 34

Cite

APA İskender, C. (2021). Mathematical Study of the Verhulst and Gompertz Growth Functions and Their Contemporary Applications. EKOIST Journal of Econometrics and Statistics(34), 73-102.