Exponential growth model : from horizontal to linear asymptote

F. Dubeau and Y. Mir

Département de mathématiques Université de Sherbrooke 2500 Boulevard de l’Université Sherbrooke (Qc), Canada, J1K 2R1

francois.dubeau@usherbrooke.ca youness.mir@usherbrooke.ca

Abstract : Paper to be published in Communications in Statistics - Simulation and Computation (2014).

Keywords : Growth model, exponential model, concave curvilinear asymptote, single inflection point.

AMS Subject Classification : 26A48, 65D10, 62J02.

Summary

Growth functions with horizontal asymptotes as Gompertz, Logistics, Richards (generalized Logistics), Bridge or Michaelis-Menten are commonly and widely used in various fields, e.g. agriculture [8, 9, 11, 12, 15], biology [7, 16, 25, 26], economy [21], fishery [23], forestry [18, 28, 29], hydrology [1, 2, 4, 5, 6, 13, 17, 27], medicine [14], and other areas of applied research [19, 20, 22]. These growth functions are characterized by an horizontal asymptote and a single inflection point [3].

In some applications, for example in hydrology, it could be usefull to have growth functions with one inflection point and an increasing concave curvilinear asymptote, and even an oblique asymptote. In this paper we present one such situation. We modify an exponential function having a natural horizontal asymptote to obtain a model having an increasing concave curvilinear asymptote and, for the limiting case, having an increasing linear asymptote. The model retains the single inflection point property. We present examples on synthetic data and also apply the model to a real situation in hydrology.

References

[1] J.E. Bullard, G.H. Mctainsh, and P. Martin, Establishing stage-discharge relationships in multiple-channelled, ephemeral rivers: a case study of the Diamantina River, Australia, Geographical Research, 45 (2007), 233-245.

[2] J. Callede, P. Kosuth, and E. de Oliveira, Établissement de la relation hauteur-débit de l’Amazone à Obidos: méthode de la dénivelée normale à ”géométrie variable”, Hydrological Sciences Journal, 46 (2001), 451-463.

[3] F. Dubeau and Y. Mir, Least squares fitting with single inflection point growth curve I - The models, Mathematical Modelling and Applied Computing, to appear.

[4] F. Dubeau, Y. Mir, A.A. Assani, and A. Chalifour, Least squares fitting with single inflection point growth curve II - an application, Mathematical Modelling and Applied Computing, to appear.

[5] F. Dubeau, Y. Mir, A.A. Assani, and A. Chalifour, Modelling stage-discharge relationship with single inflection point nonlinear functions, International Journal of Hydrology Science and Technology, to appear.

[6] E.H. Habid and E.A. Meselhe, Stage-Discharge relations for low-gradient tidal streams using data-driver models, Journal of Hydraulic Engineering, ASCE (2006), 482-492.

[7] S. Huet, E. Jolivet, and A. Messéan, La régression non linéaire: méthodes et application en biologie, INRA, Paris, 1992.

[8] Z. Ismail, A. Khamis, and M.Y. Jaafar, Fitting nonlinear Gompertz curve to tobacco growth data, Pakistan Journal of Agronomy, 2 (2003), 223-236.

[9] R.C. Jain, R. Agrawal, and K.N. Singh, A whitin year growth model for crop yield forecasting, Biometrical Journal, 34 (1992), 789-799.

[10] G.J.O. Jameson, Counting zeros of generalized polynomials: Descartes’ rule of signs and Laguerre’s extensions, Mathematical Gazette, 90 (2006), 223-234.

[11] A. Khamis, Z. Ismail, K. Haron, and A.T.M. Mohammed, Nonlinear growth models for modeling oil palm yield growth, Journal of Mathematics and Statistics, 1 (2005), 225-233.

[12] H.D. Kuhi, T. Porter, S. López, E. Kebreab, A.B. Strathe, A. Dumas, J. Dijkstra, and J. France, A review of mathematical functions for the analysis of growth in poultry, World’s Poultry Science Journal, 66 (2010), 227-239.

[13] H. Liao and D.W. Knight, Analytic stage-discharge formulae for flow in straight trapezoidal open channels, Advances in Water Resources, 30 (2007), 2283-2295.

[14] M. Marusic and Z. Bajzer, Generalized two-parameter equation of growth, Journal of mathematical analysis and applications, 179 (1993), 446-462.

[15] M.M. Mischan, S.Z. Pinho, and L.R. Carvalho, Determination of a point sufficiently close to the asymptote in nonlinear growth functions, Scientia Agricola, Piracicaba, Brazil, 68 (2011) 109-114.

[16] P.H. Morgan, L.P. Mercer, and N.W. Flodin, General model for nutritional response of higher organisms, Proceedings of the National Academy Sciences USA, 72 (1975), 4327-4331.

[17] A. Petersen-Øverleir, Moelling stage-discharge relationships affected by hysteresis using the Jones formula and nonlinear regression, Hydrological Sciences Journal, 51 (2006), 365-388.

[18] M.S. Philip, Measuring trees and forests, 2nd Edition, CAB International, Wallingford, UK, 1994.

[19] D.A. Ratkowsky, Nonlinear Regression Modeling, Marcel Dekker, New York, 1983.

[20] D.A. Ratkowsky, Handbook of Nonlinear Regression Models, Marcel Dekker, New York, 1989.

[21] R. Scitovski and S. Kosanovic, Rate of change in economics research, Economics Analysis, 19 (1985), 65-73.

[22] G.A.F. Seber et C.J. Wild, Nonlinear Regression, John Wiley & Sons, New York, 1989.

[23] J. Schnute, A versatile growth model with statistically stable parameters, Canadian Journal of Fisheries and Aquatic Sciences, 38 (1981), 1128-1140.

[24] H. Späth, Numerik, Braunschweig/Wiesbaden, Vieweg, 1994.

[25] A. Tsoularis and J. Wallace, Analysis of logistic growth models, Mathematical Biosciences, 179 (2002), 21-55.

[26] L. von Bertalanffy, Quantitative laws in metabolism and growth, The Quarterly Review of Biology, 32 (1957), 217-231.

[27] J.A. Westphal, D.B. Thompson, G.T. Stevens Jr and C.N. Strauser, Stage-Discharge relations on the middle Mississippi River, Journal of Water Resources Planning and Management, 125 (1999), 48-53.

[28] Yuancai L., Marques C.P., and Macedo F.W., Comparaison of Schnute’s and Bertalanffy-Richards’ growth function, Forest Ecology and Management, 96 (1997), 283-288.

[29] L. Zhang, Cross-validation of non-linear growth functions for modelling tree height-diameter relationships, Annals of Botany, 79 (1997), 251-257.