Methods of modelling relative growth rate
- Arne Pommerening^{1}Email author and
- Anders Muszta^{1}
DOI: 10.1186/s40663-015-0029-4
© Pommerening and Muszta; licensee Springer. 2015
Received: 1 December 2014
Accepted: 10 February 2015
Published: 10 March 2015
Abstract
Background
Analysing and modelling plant growth is an important interdisciplinary field of plant science. The use of relative growth rates, involving the analysis of plant growth relative to plant size, has more or less independently emerged in different research groups and at different times and has provided powerful tools for assessing the growth performance and growth efficiency of plants and plant populations. In this paper, we explore how these isolated methods can be combined to form a consistent methodology for modelling relative growth rates.
Methods
We review and combine existing methods of analysing and modelling relative growth rates and apply a combination of methods to Sitka spruce (Picea sitchensis (Bong.) Carr.) stem-analysis data from North Wales (UK) and British Douglas fir (Pseudotsuga menziesii (Mirb.) Franco) yield table data.
Results
The results indicate that, by combining the approaches of different plant-growth analysis laboratories and using them simultaneously, we can advance and standardise the concept of relative plant growth. Particularly the growth multiplier plays an important role in modelling relative growth rates. Another useful technique has been the recent introduction of size-standardised relative growth rates.
Conclusions
Modelling relative growth rates mainly serves two purposes, 1) an improved analysis of growth performance and efficiency and 2) the prediction of future or past growth rates. This makes the concept of relative growth ideally suited to growth reconstruction as required in dendrochronology, climate change and forest decline research and for interdisciplinary research projects beyond the realm of plant science.
Keywords
Growth efficiency Growth coefficient/multiplier Chapman-Richards growth model Standardisation Simultaneous estimationsBackground
Growth is a universal and fundamental process of life on earth. The analysis and modelling of plant growth has therefore been a particular concern in plant science as well as in production biology including forestry, agriculture and fishery to name but a few. This research has the important objective to identify growth patterns in response to environmental factors or treatments.
In this context, the concept of relative plant growth, involving the analysis and modelling of plant growth relative to plant size, has proved to be a powerful tool in comparative studies of the growth performance of plants and has a long tradition in plant science (Evans 1972, p. 190ff.; Pommerening and Muszta: Concepts of relative growth – a review, submitted). It first developed at the beginning of the 20^{th} century in what eventually became the British school of plant growth analysis, mainly at Sheffield University (Hunt 1982, p. 1, 16).
Independently of the British school another quantitative plant science group developed at Tharandt/Dresden Technical University in Germany. The Tharandt school characterised the growth of trees by using a variant of the concept of relative plant growth and on this basis eventually developed a population model and a size class model for predicting the growth of trees (Wenk et al. 1990, Wenk 1994). There is also evidence of empirical Russian work in this area (Antanaitis and Zagreev 1969) and particularly remarkable is the detailed Finnish work by Kangas (1968).
Relative growth rate is a standardised measure of growth with the benefit of avoiding, as far as possible, the inherent differences in scale between contrasting organisms so that their performances can be compared on an equitable basis (Hunt 1990, p. 6). Applications of relative growth rates include the study of dry weight, biomass, leaf area, stem volume, basal area and stem diameter. Interestingly, the concept is closely related to plant mortality (Gillner et al. 2013), i.e. low relative growth rates for extended periods of time are good indicators of imminent death. Relative growth rates are also pre-requisites for quantifying and modelling allometric relationships in plants (Gayon 2000).
As instantaneous growth rates cannot be measured in practice, the difference between growth characteristics of interest is usually studied at discrete points in time, t _{1}, t _{2}, …, t _{ n }, which for example are scheduled survey years. In this context, the period between two discrete points in time can be denoted by Δt = t _{ k } – t _{ k – 1} with k = 2, …, n. For ease of notation in the remainder of this section we set y(t _{ k }) = y _{ k } and p(t _{ k }) = p _{ k } etc. and assume equidistant time periods. However, the notation can be modified to accommodate unequal time periods (Pommerening and Muszta: Concepts of relative growth – a review, submitted).
Considering a short time period, mean relative growth rate is approximately equal to the instantaneous relative growth rate p(t). Blackman (1919) originally referred to equation (2) as “efficiency index” and “specific growth rate”, see also Causton and Venus (1981, p. 37). From the last term we can see that equation (2) can be interpreted as the logarithm of the ratio of successive size measurements divided by the corresponding time interval (Pommerening and Muszta: Concepts of relative growth – a review, submitted).
Equation (3) is also referred to as Blackman’s efficiency index which is supposed “to represent the efficiency of the plant as a producer of new material, and to give a measure of the plant’s economy in working” (Blackman 1919).
The growth coefficient or multiplier is obviously a function of relative growth rate and can also be defined as the ratio of a particular plant size characteristic at different times. Part of the fascination with M _{ k } stems from the fact that the growth multiplier plays a crucial role in predicting future growth based on relative growth rates (Kangas 1968, p. 19; Wenk et al. 1990, p. 95f.; Murphy and Pommerening 2010).
The objective of this paper is to explore how relative growth rates of individual plants as well as of plant populations can be efficiently analysed and modelled using a system of simultaneous functions of relative growth and allometric relationships developed in different research schools.
Functions of relative growth
Frequently used functions of absolute and relative plant growth rate. a , b , c are model parameters and the symbol t denotes time or age
Function name/source | Growth function | Absolute growth rate | Relative growth rate |
---|---|---|---|
Chapman-Richards (Richards 1959) | a(1 − e ^{− bt })^{ c } | abce ^{−bt }(1 − e ^{− bt })^{ c −1} | bc (1 − e ^{− bt })^{− 1} |
Gompertz (1825) | \( a{e}^{-b{e}^{-ct}} \) | \( abc{e}^{-ct}{e}^{-b{e}^{-ct}} \) | bce ^{− ct } |
Kangas (1968) | - | - | \( a{e}^{b{t}^{-c}} \) |
Korf (Zeide 1993) | \( a{e}^{-b{t}^{-c}} \) | \( abc{t}^{-c-1}{e}^{-b{t}^{-c}} \) | bct ^{− (1 + c)} |
Logistic (Verhulst 1838) | a(1 + ce ^{− bt })^{−1} | abce ^{−bt }(1 + ce ^{− bt })^{−2} | bc (c + e ^{ bt })^{− 1} |
Monomolecular (Weber 1891) | a(1 − ce ^{− bt }) | abce ^{−bt } | bc (e ^{ bt } − c)^{− 1} |
Weibull (Zeide 1993) | \( a\left(1,-,, {e}^{-b{t}^c}\right) \) | \( abc{t}^{c-1}{e}^{-b{t}^c} \) | \( bc{t}^{\left(c-1\right)}\kern0.5em {\left({e}^{b{t}^c}-1\right)}^{-1} \) |
The functions in Table 1 are based on the original growth functions and on the corresponding functions of absolute growth rate, i.e. the first derivatives of the growth functions with respect to time. Most functions of relative growth rate have the advantage that they have fewer model parameters than the corresponding functions of absolute growth rate.
Kangas (1968, p. 69) independently suggested the function listed in Table 1 next to his name for modelling the growth multiplier of equation (4) and referred to it as the growth coefficient function.
Modelling individual plant growth
The strategy of modelling individual plant growth is straightforward: 1) A suitable function of relative growth rate is selected from Table 1 or from other publications. 2) A primary plant size characteristic is identified, e.g. tree volume. 3) Secondary plant size characteristics, e.g. tree height and tree diameter, are linked to the function of relative growth rate of the primary plant size characteristic through allometric relations. 4 ) The 2–3 model parameters of the function of relative growth rate and the two allometric coefficients are estimated simultaneously through nonlinear regression. Jones et al. (2009, p. 219f.) describe how such more complex types of nonlinear regression can be calculated in R using the function optim.
Wenk et al. (1990, p. 174ff.) selected primary and secondary plant size characteristics in such a way that error propagation was effectively reduced: They identified tree volume as a three-dimensional size characteristic to be the primary characteristic and one-dimensional total tree height and stem diameter as secondary characteristics. However, since this is a generic approach, there is no need to strictly follow this recommendation. Tree volume can for example also be replaced by weight or biomass.
Characteristics and model statistics of four Sitka spruce ( Picea sitchensis (Bong.) Carr.) trees at Cefn Du (plot 1), Clocaenog Forest (North Wales, UK)
Tree # | dbh (cm) | b | c | Volume (m ^{3} ) | Total height (m) | Stem diameter ( dbh ) | ||||
---|---|---|---|---|---|---|---|---|---|---|
Min. | Max. | Bias | RMSE | m _{1} | RMSE | m _{2} | RMSE | |||
2000 | 6.4 | 32.2 | −0.05251 | 6.12877 | 0.00303 | 0.02478 | 2.82205 | 0.01309 | 1.07431 | 0.02211 |
3000 | 4.2 | 28.3 | −0.05057 | 6.44977 | 0.00307 | 0.01646 | 2.63452 | 0.00633 | 1.04791 | 0.01835 |
4000 | 4.9 | 32.6 | −0.04667 | 6.40575 | 0.00249 | 0.02401 | 2.78971 | 0.01296 | 1.13672 | 0.01294 |
5000 | 6.6 | 38.7 | −0.00822 | 3.68512 | 0.00024 | 0.01248 | 2.66006 | 0.01105 | 1.17268 | 0.01218 |
The relative growth curve of volume, p _{ v,k }, in Figure 1 (A) is based on the Chapman-Richards model (see Table 1) with model parameters b and c. The relative height (Figure 1, B) and diameter (Figure 1, C) growth rates, p _{ h,k } and p _{ d,k }, however, have not been modelled independently but linked to p _{ v,k } through the allometric coefficients m _{1} and m _{2} and equation (5).
Figure 2 demonstrates how the relative growth rates of tree # 5000 “translate” to modelling the observed growth characteristics using the growth multiplier of equation (4). Again, any other growth characteristics can be used instead of volume, total height and stem diameter, the growth analyst is completely free to choose.
As expected the three curves decline throughout growth and particularly trees # 2000, 3000 and 4000 appear to have quite similar growth patterns, as is often the case with trees of the same population growing in similar environmental conditions. Their curves form almost parallel lines in the order 4000 > 3000 > 2000, i.e. tree # 4000 has the highest relative growth rate throughout its lifetime and tree # 2000 the lowest. This trend also seems to be reflected by model parameter b, which is responsible for scaling the growth rate (Pienaar and Turnbull 1973), although this requires a more detailed study.
Tree # 5000 exhibits a growth pattern different from the other three trees: Its growth starts with a comparatively low rate, but finishes with a rate markedly higher than those of the other three trees. The curve of Tree # 5000 intersects those of the others approximately halfway through its lifetime.
The data of all four trees start at the same age (10 years) and end approximately at the same age (48–50 years). Trees # 2000 and 5000 are larger than the other two trees at the beginning, whilst trees # 5000 and 4000 are the largest at the end of their lifetime.
Many authors point out that RGR is size dependent, i.e. individuals with a smaller initial size have a larger relative growth rate (Turnbull et al. 2008; Rose et al. 2009; Rees et al. 2010). This can mask important relationships and it may be difficult to tell whether a tree grows slowly because it is large or because it is pursuing a slow growth strategy. To check up on this argument we can calculate size-standardised RGR. To this end, first the Chapman-Richards growth function (see first row in Table 1) is fitted to the observed size data of the four trees.
This conversion of time to a growth characteristic on the abscissa allows depicting size-standardised RGR in Figure 3 (B). We get a better understanding of the differences in relative volume growth rate of the four trees and see more clearly that tree # 5000 due to the lower rate of decline has a superior relative growth rate throughout most of its volume development despite the fact that its initial RGR is much lower than those of the other trees. Also the ordering of trees # 2000 and 3000 is interestingly reversed compared to non-standardised RGR (Figure 3, A). We can also see that tree # 3000 is the tree with the smallest final volume despite having a longer lifespan than all other trees.
Apparently the diameter multiplier decreases with increasing tree size relative to the population mean and levels off with very large ratios when trees are much larger than the population mean. Using the relationship between a relative tree diameter and the growth multiplier also has the advantage of predicting future growth from current size and from size dominance at the same time. Coming back to the relationship between relative growth rate and mortality (Gillner et al. 2013), trees in the lower range of M _{ d,k+1} carry a high probability of dying.
Modelling of plant populations
In the same way as it is possible to model individual plants one can also model whole plant populations. In this case, the growth rates are calculated from population summary characteristics. Details of this approach are also documented in Wenk and Nicke (1985), Nicke (1988) and in Wenk (1994).
where Q is the ratio \( {V}_{k+1}^{\mathrm{res}}/{V}_k^{\mathrm{res}}, \) i.e. the ratio of successive residual stand volumes (superscript “res” denotes “residual”). The ratio Q defines the loss and must not exceed the volume growth multiplier, as negative values are not defined. With Q = 1, p _{ V,k } = l _{ V,k } and the forest stand does not grow. The loss function is therefore a ratio of two multipliers subtracted from one and can also be estimated with the Chapman-Richards relative growth rate function.
In a first step, the volume per hectare of dead trees, i.e. the absolute loss in terms of volume, is calculated as the product of the volume of the forest stand before the disturbance, \( {V}_k^{\mathrm{prior}}, \) and the volume loss rate, l _{ V,k }. Then residual stand volume, \( {V}_k^{\mathrm{res}}, \) constitutes the difference between stand volume before disturbance, \( {V}_k^{\mathrm{prior}}, \) and the absolute volume loss, \( {V}_k^{\mathrm{lost}} \). Finally the volume of the forest stand before disturbance for the next time step, \( {V}_{k+1}^{\mathrm{prior}}, \) is calculated as the product of the residual stand volume of the current time step, \( {V}_k^{\mathrm{res}}, \) and the volume growth multiplier of the next time step, M _{ V,k + 1}.
In analogy to volume projection, the development of a density measure such as trees per hectare can be modelled. Stand height is also modelled in a similar way as outlined in the last line of equation (9) and like individual-tree heights.
f _{ H } is a static form height function. Such form height relationships are available for many species and countries.
The equations of this section can be included in a regression routine, for example in the optim function of R mentioned before (Jones et al. 2009, p. 219f.). As a function of relative growth rate we again select the Chapman-Richards function (see Table 1), but any of the other functions is suitable, too.
To illustrate this method we have used Douglas fir (Pseudotsuga menziesii (Mirb.) Franco) data from British yield tables (Edwards and Christie 1981), specifically those relating to yield class 24, initial spacing 1.7 m × 1.7 m and crown thinning. Yield tables can be interpreted as tabular summary characteristics of forest stands, whereby the yield classes represent different environmental conditions resulting in a larger or smaller carrying capacity. Naturally any other aggregated plant population data can be selected for this purpose.
The yield table data are provided for five-year intervals and as a consequence relative growth rates and growth multipliers also relate to 5-years periods.
where H is stand top height in metres.
Characteristics and model statistics of four British Douglas fir ( Pseudotsuga menziesii (Mirb.) Franco) yield tables (Edwards and Christie 1981 ) relating to initial spacing 1.7 m × 1.7 m and crown thinning
Yield class | dg (cm) | b | c | Volume (m ^{3} ∙ha ^{−1} ) | Top height (m) | Mean diameter ( d g) | |||
---|---|---|---|---|---|---|---|---|---|
Min. | Max. | Bias | RMSE | m | RMSE | RMSE | |||
24 | 12.8 | 76.7 | −0.03240 | 5.68749 | 0.00011 | 0.00185 | 3.64234 | 0.00153 | 0.00658 |
18 | 12.2 | 60.9 | −0.03147 | 5.65227 | 0.00014 | 0.00181 | 3.46283 | 0.00151 | 0.00560 |
12 | 12.4 | 43.5 | −0.02988 | 5.43637 | 0.00004 | 0.00075 | 3.20122 | 0.00124 | 0.00793 |
8 | 11.7 | 32.5 | −0.03095 | 6.16601 | −0.00003 | 0.00072 | 3.06830 | 0.00092 | 0.00757 |
Using again the Chapman-Richards growth function and equation (6) we can calculate standardised relative growth rates resulting in Figure 7 (B).
We can now clearly see that larger yield class numbers, i.e. more favourable environmental conditions, correspond to higher standardised relative growth rates. The standardisation has again helped us to better discriminate between the differences in relative growth rate of the four populations.
Final discussion and conclusions
Our examples in this paper have demonstrated that by bringing together the approaches in general plant growth analysis and in forest growth and yield science it is possible to advance the concept of relative growth. Modelling of relative growth rates is straightforward and often the corresponding growth functions have fewer model parameters than the respective versions of absolute growth rates. The recent introduction of size-standardised relative growth rates (Turnbull et al. 2008; Rose et al. 2009; Rees et al. 2010) has proved to be very effective for improving our understanding of growth efficiency and this approach can definitely be recommended.
Modelling relative growth rates serves two purposes, 1) an improved analysis of growth performance and efficiency and 2) the prediction of future or past growth rates. The methodology described in this paper also allows “backcasting” by simply dividing plant growth characteristics by the growth multiplier instead of multiplying (Murphy and Pommerening 2010).
The obvious similarity between individual-tree and population modelling of relative growth rates suggests a link between these two modelling levels (see for example Cao 2014). In short, the idea of this approach is to combine the advantages of different modelling resolutions, particularly to employ information of population models (that are mathematically more tractable and statistically more stable) for improving individual-tree models. This strategy is often referred to as disaggregation. Zhang et al. (1993) describe a method of disaggregation using relative tree sizes similar to d _{rel,k } shown in this paper. Since this relative size characteristic has also proved useful for modelling growth multipliers, a disaggregation modelling approach for linking the two modelling levels described in this paper is very likely to be successful.
The properties of the concept of relative growth make it a superb choice for growth reconstruction as required in dendrochronology, climate change and forest decline research and we hope that the methodology outlined in this paper will inspire much interdisciplinary research, as the applicability is universal to any phenomenon involving growth processes.
Declarations
Acknowledgements
Stephen T. Murphy, Jens Haufe, Owen Davies and Gareth Johnson of the Tyfiant Coed project have carefully analysed and kindly provided the time-series and stem-analysis data used as examples in this review. We are also indebted to the late Günter Wenk who inspired the first author with his enthusiasm of quantifying and modelling relative growth processes.
Authors’ Affiliations
References
- Antanaitis V, Zagreev V (1969) Прирост леса. [Forest increment.]. Издательство лесная промышленность, MoscowGoogle Scholar
- Blackman VH (1919) The compound interest law and plant growth. Ann Bot 33:353–360Google Scholar
- Bolker BM (2008) Ecological Models and Data In R. Princeton University Press, PrincetonGoogle Scholar
- Cao QV (2014) Linking individual-tree and whole-stand models for forest growth and yield prediction. Forest Ecosyst 1:1–8View ArticleGoogle Scholar
- Causton DR (1977) A Biologist’s Mathematics. Edward Arnold, LondonGoogle Scholar
- Causton DR, Venus JC (1981) The Biometry of Plant Growth. Edward Arnold, LondonGoogle Scholar
- Edwards PN, Christie JM (1981) Yield Models for Forest Management. Forestry Commission Booklet 48. Forestry Commission, EdinburghGoogle Scholar
- Evans GC (1972) The Quantitative Analysis of Plant Growth. Blackwell Scientific Publications, OxfordGoogle Scholar
- Gayon J (2000) History of the concept of allometry. Am Zool 40:748–758View ArticleGoogle Scholar
- Gillner S, Rüger N, Roloff A, Berger U (2013) Low relative growth rates predict future mortality of common beech (Fagus sylvatica L.). For Ecol Manage 302:372–378View ArticleGoogle Scholar
- 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. Phil Trans Royal Soc 115:513–585View ArticleGoogle Scholar
- Hunt R (1982) Plant Growth Curves. The Functional Approach to Plant Growth Analysis. Cambridge University Press, CambridgeGoogle Scholar
- Hunt R (1990) Basic Growth Analysis: Plant Growth Analysis for Beginners. Unwin Hyman, LondonView ArticleGoogle Scholar
- Jones O, Maillardet R, Robinson A (2009) Introduction to Scientific Programming And Simulation Using R. Chapman & Hall/CRC, Boca RatonGoogle Scholar
- Kangas Y (1968) Beschreibung des Wachstums der Bäume als Funktion ihres Alters. [Describing the growth of trees as a function of age.] Acta Forestalia Fennica, 90, Helsinki 1-136.Google Scholar
- Matthews RW, Mackie ED (2006) Forest mensuration. A handbook for practioners. Forestry Commission, EdinburghGoogle Scholar
- Michaelis M, Menten ML (1913) Die Kinetik der Invertinwirkung. [The kinetics of invertase action.]. Biochem Z 49:333–369Google Scholar
- Murphy ST, Pommerening A (2010) Modelling the growth of Sitka spruce (Picea sitchensis (Bong.) Carr.) in Wales using Wenk’s model approach. Allgemeine Forst- und Jagdzeitung 181:35–43Google Scholar
- Nicke A (1988) Aufbau eines Bestandesentwicklungsmodells. [Construction of a stand development model.]. Beiträge für die Forstwirtschaft 22:71–74Google Scholar
- Pienaar LV, Turnbull KJ (1973) The Chapman-Richards generalization of von Bertalanffy's growth model for basal area growth and yield in even-aged stands. Forest Sci 19:2–22Google Scholar
- Rees M, Osborne CP, Woodward FI, Hulme SP, Turnbull LA, Taylor SH (2010) Partitioning the components of relative growth rate: How important is plant size variation. Am Nat 176:E152–E161View ArticlePubMedGoogle Scholar
- Richards FJ (1959) A flexible growth function for empirical use. J Exp Bot 10:290–300View ArticleGoogle Scholar
- Rose KE, Atkinson RL, Turnbull LA, Rees M (2009) The costs and benefits of fast living. Ecol Lett 12:1379–1384View ArticlePubMedGoogle Scholar
- Turnbull LA, Paul-Victor C, Schmid B, Purves DW (2008) Growth rates, seed size, and physiology: Do small-seeded species really grow faster? Ecology 89:1352–1363View ArticlePubMedGoogle Scholar
- Verhulst P-F (1838) Notice sur la loi que la population pursuit dans son accroissement. [A note on population growth.]. Correspondence Mathematiques et Physiques 10:113–121Google Scholar
- Weber R (1891) Lehrbuch der Forsteinrichtung. [Textbook of forest planning.]. Springer-Verlag, BerlinGoogle Scholar
- Wenk G (1972) Zuwachsprognosen, Vorratsfortschreibung und Aufstellung bestandesindividueller Ertragstafeln mit Hilfe von Wachstumsmultiplikatoren. [Increment prognoses, projection of stand volume and stand-specific yield tables on the basis of growth multipliers.]. Wissenschaftliche Zeitschrift der Technischen Universität Dresden 21:1247–1249Google Scholar
- Wenk G (1978) Mathematische Formulierung von Wachstumsprozessen in der Forstwirtschaft. [Mathematical formulation of growth processes in forestry]. Beiträge für die Forstwirtschaft 1:25–30Google Scholar
- Wenk G (1994) A yield prediction model for pure and mixed stands. For Ecol Manage 69:259–268View ArticleGoogle Scholar
- Wenk G, Nicke A (1985) Zur Prognose der Bestandesentwicklung. [Projecting stand development.]. Beiträge für die Forstwirtschaft 19:173–176Google Scholar
- Wenk G, Antanaitis V, Smelko S (1990) Waldertragslehre. [Forest growth and yield science.]. Deutscher Landwirtschaftsverlag, BerlinGoogle Scholar
- Whitehead FH, Myerscough PJ (1962) Growth analysis of plants. The ratio of mean relative growth rate to mean relative rate of leaf area increase. New Phytologist 61:314–321View ArticleGoogle Scholar
- Zeide B (1989) Accuracy of equations describing diameter growth. Can J Forest Res 19:1283–1286View ArticleGoogle Scholar
- Zeide B (1993) Analysis of growth equations. Forest Sci 39:594–616Google Scholar
- Zhang L, Moore JA, Newberry JD (1993) Disaggregating stand volume growth to individual trees. Forest Sci 39:295–308Google Scholar
Copyright
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.