Evaluation of unconstrained and constrained mathematical functions to model girth growth of rubber trees (Hevea brasiliensis) using young age measurements

Abstract

No attempt has been made to date to model growth in girth of rubber tree (Hevea brasiliansis). We evaluated the few widely used growth functions to identify the most parsimonious and biologically reasonable model for describing the girth growth of young rubber trees based on an incomplete set of young age measurements. Monthly data for girth of immature trees (age 2 to 12 years) from two locations were subjected to modelling. Re-parameterized, unconstrained and constrained growth functions of Richards (RM), Gompertz (GM) and the monomolecular model (MM) were fitted to data. Duration of growth was the constraint introduced. In the first stage, we attempted a population average (PA) model to capture the trend in growth. The best PA model was fitted as a subject specific (SS) model. We used appropriate error variance-covariance structure to account for correlation due to repeated measurements over time. Unconstrained functions underestimated the asymptotic maximum that did not reflect the carrying capacity of the locations. Underestimations were attributed to the partial set of measurements made during the early growth phase of the trees. MM proved superior to RM and GM. In the random coefficient models, both Gf and G0 appeared to be influenced by tree level effects. Inclusion of diagonal definite positive matrix removed the correlation between random effects. The results were similar at both locations. In the overall assessment MM appeared as the candidate model for studying the girth-age relationships in Hevea trees. Based on the fitted model we conclude that, in Hevea trees, growth rate is maintained at maximum value at t0, then decreases until the final state at dG/dt ≥0, resulting in yield curve with no period of accelerating growth. One physiological explanation is that photosynthetic activity in Hevea trees decreases as girth increases and constructive metabolism is larger than destructive metabolism.

Keywords

natural rubber / Hevea Brasiliensis / growth modelling / unconstrained functions / constrained functions / mixed model

References

[1] Amaro A., Reed D., Tome M., Themido I.. Modelling dominant height growth: Eucalyptus plantations in Portugal. Forest Science, 1998, 44: 37-46.

[2] Baulkwill W.J.. Webster C.C., Baulkwill W.J.. The history of natural rubber production. Rubber. 1989, UK: Longman Scientific and Technical, 1 56

[3] Caillet G.M., Smith W.D., Mollet H.F., Goldman K.J.. Age and growth studies of chondrichthyan fishes: the need for consitency in terminology, verification, validation and growth function fitting. Environmental Biology of Fishes, 2006, 77: 211-228.

[4] Causton D.R., Venus J.C.. The biometry of plant growth. 1981, UK: Edward Arnold

[5] Chandrasekhar T.R., Alice J., Varghese Y.A., Saraswathyamma C.K., Vijayakumar K.R.. Girth growth of rubber (Hevea brasiliensis) trees during the immature phase. Journal of Tropical Forest Science, 2005, 17: 399-415.

[6] Chandrashekar T.R.. Weather and growth performance of young rubber trees (Hevea brasiliensis). 2007, Kerala, India: Mahatma Gandhi University

[7] Chandrashekar T.R., Nazeer M.A., Marattukalam J.G., Prakash G.P., Annamalainathan K., Thomas J.. An analysis of growth and drought tolerance in rubber during the immature phase in a dry subhumid climate. Experimental Agriculture, 1998, 34: 287-300.

[8] Delignette-Muller ML, Baty F. 2010. Use of package nlstools to help the fit assess the quality of fit of a gaussian nonlinear model. Available at http://cran.r-project.org/web/packages/nlstools. [accessed 27 December 2010].

[9] El-Shaarawi A.H., Piegorsch W.W.. Encyclopedia of environmetrics (Vol.1). 2002, UK: Wiley, 32 41

[10] Fekedulegn D., Mac Siurtain M.P., Colbert J.J.. Parameter estimation of nonlinear growth models in forestry. Silva Fennica, 1999, 33: 327-336.

[11] France J., Thornley J.H.M.. Mathematical models in agriculture. 1984, UK: Butterworths, 223 235

[12] Gregoire T.G., Schabenberger O.. A non-linear mixed effects model to predict cumulative bole volume of standing trees. Journal of Applied Statistics, 1996, 23: 257-271.

[13] Gregoire T.G., Brillinger D.R., Diggle P.J., Russek-Cohen E., Warren W.G., Wolfinger R.D.. Modelling longitudinal and spatially correlated data. 1997, New York: Springer-Verlag

[14] Hunt R.. Plant growth curves, the functional approach to plant growth analysis. 1982, London: Edward Arnold

[15] Lei Y.C., Zhang S.Y.. Features and partial derivatives of Bertalanffy-Richards growth model in forestry. Non linear analysis: Modelling and control, 2004, 9: 65-73.

[16] Liu Z.-g., Li F.-ri.. The generalized Chapman-Ricahrds function and applications to tree and stand growth. Journal of Forestry Research, 2003, 14: 16-26.

[17] Ogle DH. 2010. Von Bertalanffy growth model vignette. Available from http://www.ncfaculty.net/dogle/fishR/gnrlex/VonBertalaffy/VonBertalanffy.pdf. [Accessed 27 December 2010].

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

[19] Pinheiro J.C., Bates D.M.. Mixed effects models in S and S-PLUS. 2000, New York: Springer-Verlag

[20] Potty S.N., Kodandaraman R., Mathew M.. Radhakrishna Pillay P.N.. Field upkeep. Hand book of Natural Rubber Production in India. 1980, Kottayam, India: Rubber Research Institute of India, 135 156

[21] Punnoose K.I., Kodandaraman R., Philip V., Jessy M.D.. George P.J., Jacob C.K.. Field upkeep and intercropping. Narural Rubber Agromanagement and Crop Processing. 2000, Kottayam, India: The Rubber Research Institute of India, 149 169

[22] Pushpadas M.V., Ahammed M.. Radhakrishna Pillay P.N.. Nutritional requirements and manorial recommendations. Hand book of Natural Rubber Production in India. 1980, Kottayam, India: Rubber Research Institute of India, 159 185

[23] R Development Core Team. 2010. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. Available from http://www.r-project.org [accessed 31 January 2011].

[24] Ratkowsky D.A.. Handbook of Nonlinear Regression Models. 1988, NewYork: Marcel Dekker

[25] Richards F.J.. A flexible growth function for empirical use. Journal of Experimental Botany, 1959, 10: 290-300.

[26] Rose D.A., Charles-Edwards D.A.. Mathematics and plant physiology. 1981, London: Academic press

[27] Salas C., Garcia O.. Modelling height development of mature Nothofagus obliqua. Forest Ecology and Management, 2006, 229: 1-6.

[28] Seber G.F.A., Wild C.J.. Nonlinear Regression. 2003, USA: John Wiley & Sons Inc.

[29] Shifley S.R., Brand G.J.. Chapman-Richards growth function constrained for maximum tree size. Forest Science, 1984, 30: 1066-1070.

[30] Shorrocks V.M., Templeton J.K., Iyer G.C.. Mineral Nutrition, growth and nutrient cycle of Hevea brasiliensis III. The relationship between girth and shoot dry weight. Journal of Rubber Research Institute of Malaya, 1965, 19: 83-92.

[31] Thornley J.H.M., France J.. Mathematical Models in Agriculture, 2007 2nd ed. Wallingford, UK: CAB International

[32] Thornley J.H.M., Johnson I.R.. Plant and crop modelling: a mathematical approach to plant and crop physiology. 1990, Oxford, UK: Oxford University Press

[33] Vanclay J.K.. Modelling forest growth and yield. 1994, Wallingford, UK: CAB International

[34] Vanclay J.K., Skovsgaard J.V.. Evaluating forest growth models. Ecological Modelling, 1997, 98: 1-12.

[35] Vonesh E.F., Chinchilli V.M.. Linear and nonlinear models for the analysis of repeated measurements. 1997, New York: Marcell Dekker

[36] Webster C.C., Paardekooper E.C.. Webster C.C., Baulkwill W.J.. The botany of the rubber tree. Rubber. 1989, England: Longman Scientific and Technical, 57 84

[37] Winsor C.P.. The Gompertz curve as a growth curve. Proceedings of the Nattionla Academy of Sciences, 1932, 16: 1-8.

[38] Zeide B.. Analysis of growth equation. Forest Science, 1993, 39: 594-616.