- Email: [email protected]
L-DONAX, a growth model of the invasive weed species, Arundo donax L.
Aquatic Botany 87 (2007) 275–284 www.elsevier.com/locate/aquabot
L-DONAX, a growth model of the invasive weed species, Arundo donax L. David Thornby a,*, David Spencer a, Jim Hanan b, Anna Sher a,c,d a b
USDA-ARS Exotic and Invasive Weeds Research Unit, Robbins Hall, 1 Shields Avenue, University of California, Davis, CA 95616, USA Advanced Computational Modelling Centre, ARC Centre for Complex Systems, University of Queensland, St Lucia, QLD 4072, Australia c Department of Biological Sciences, University of Denver, 2190 E. Illif Avenue, Denver, CO 80208, USA d Department of Research, Herbaria and Records, Denver Botanic Gardens, 909 York St., Denver, CO 80206, USA Received 14 September 2005; received in revised form 14 June 2007; accepted 22 June 2007 Available online 12 July 2007
Abstract Arundo donax L. is a perennial reed and is an invasive weed of riparian systems in North America. A structural model (L-DONAX) of the species was constructed using L-system modelling in order to assist in understanding and demonstrating the complexities of the plant’s development and structure. The model produces a realistic number of plant components from a single rhizome segment over the course of the first year of growth, using empirical relationships derived from outdoor experiments. Biomass production is also simulated, through the use of relationships found between aerial plant portion sizes and masses. L-DONAX demonstrates that control of A. donax clumps is likely to require more than annual biomass removal, due to the bulk of biomass being present underground, and the ability of remaining rhizome or stem segments to produce large clumps quickly. The model extrapolates to years of growth beyond the first, but is found to require some re-parameterisation to improve accuracy. Published by Elsevier B.V. Keywords: Arundo donax; L-DONAX; Giant reed; L-system; Structural model; Invasive species
1. Introduction Giant reed (Arundo donax L.) is a successful invasive, nonnative weed of riparian zones in California, as well as other areas in North America. A. donax is a clonal plant that expands by rhizome expansion and, in California, propagates solely through shooting of separated rhizome pieces or stem pieces (Decruyenaere and Holt, 2001; DiTomaso and Healey, 2003). A. donax is difficult to control for several reasons. Firstly, it produces large amounts of biomass rapidly, both above and below ground (Sharma et al., 1998). Secondly, it propagates very readily vegetatively, by regrowth from rhizome and stem pieces (Decruyenaere and Holt, 2001), which are easily dispersed by river currents. Finally, because a large proportion of its biomass is sequestered underground in the rhizome,
* Corresponding author. Present address: Queensland Department of Primary Industries and Fisheries, P.O. Box 2282, Toowoomba, QLD 4350, Australia. Tel.: +61 7 46398811; fax: +61 7 46398800. E-mail address: [email protected] (D. Thornby). 0304-3770/$ – see front matter. Published by Elsevier B.V. doi:10.1016/j.aquabot.2007.06.012
removal of whole clumps is difficult. In order to more successfully manage infestations of A. donax in a given site, information is required regarding its potential invasiveness and its responses to environmental conditions and stimuli, including temperature, light, soil moisture, nutrition, and the activities of other organisms in the plant’s environment. In particular, information is needed on the relationships between environmental conditions and the plant’s rate of growth, including the rate of new shoot production (from rhizome segments), the rate of expansion of the new shoots, the production of branches, and the longevity of plant parts. The plant’s light environment is a key factor in the ecophysiology of the plant. Recent work in structural modelling using computational methods can help in the analysis of the effects of light and other factors on plant structures, and vice versa. Structural models have been produced for a number of significant crop plants—for example, cotton (Hanan and Hearn, 2003; Thornby et al., 2003; Hanan et al., 2005) and maize (Fournier and Andrieu, 1999). However, morphogenetic models of plants that have less immediate economic value (such as weeds) have been less frequently produced. Empirical
276
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284
models of plant morphogenesis have been used as a basis for further physiologically explicit research and to demonstrate various implications and aspects of growth, development, and environmental interactions in rice (Watanabe et al., 2005), sorghum (Kaitaniemi et al., 1999), and wheat, (Evers et al., 2005). Spencer and Ksander (2006) produced a set of equations that relate morphological or structural growth to accumulated degree days (DD). These equations are used as the developmental basis for the L-system morphogenetic model described here. Our approach is to produce an empirical structural model of A. donax growth and development, which we have denoted L-DONAX, using established growth relationships. Our study attempts to elucidate morphogenetic and architectural parameters of A. donax growth from a component level up to the scale of whole stands, with the ability subsequently to add mechanistic environmental functional aspects, such as interactions between the canopy and light or biocontrol agents. 2. Methods 2.1. Structural modelling with L-systems: background The L-system formalism for modelling the modular growth of organisms (including chains and branched chains of bacteria, and plants) was devised initially by Lindenmayer (Lindenmayer, 1968; Prusinkiewicz and Lindenmayer, 1990; Hanan, 1997; Prusinkiewicz et al., 2000a). The formalism is inherently suitable for modelling plant morphogenesis, because it simulates modular organism components (such as leaves, internodes, meristems, flowers and fruits) as a set of symbols arranged in a branching topological string, in much the same way that plants, being modular (Room et al., 1994), consist of a set of modules or organs arranged in predictable topologies. Plant development and growth are captured through a set of replacement rules describing how and when components are added, and how they change over time. Computer software used to interpret L-system rules, such as L-Studio (University of Calgary; Prusinkiewicz et al., 2000b) used in this work, allow complicated, interacting sets of rules to produce realistically complex plant growth behaviours, and to express the results visually, in two- or three-dimensional representations of plant forms. Structural models of plant growth, including our model of A. donax, have a number of advantages. The visualisation of the performance of a plant can provide insight and can assist in the transfer of knowledge to non-researchers. Structural plant models can also be used as a base model for investigating interactions between the plant canopy and other biotic and abiotic factors – for example, insect movement or light interactions – in a spatially explicit way. Modelling plant growth at a component level (that is, ‘bottom up’) allows the model a high level of flexibility in extrapolating or interpolating to smaller or larger plant structures using established growth rules and different sets of simulated conditions.
2.2. Data collection We collected morphological data from a set of A. donax plants using digitising techniques (detailed below as Section 2.2.1), and used other morphological data from previous studies of A. donax morphology (Spencer et al., 2005; Spencer and Ksander, 2006) in order to provide morphological parameters for L-DONAX. We used further datasets to develop relationships between component sizes and biomass (see Section 2.5 below). Much of this data is as presented in Spencer et al. (2005, 2006); however, we also used data from a 2002 study of A. donax responses to nutrient levels, referred to below as Section 2.2.2 For the digitising studies, we used a magnetic digitiser (Polhemus Fastrack) in conjunction with Floradig software (www.cpai.uq.edu.au, University of Queensland). The digitiser collects point data from a magnetic field with a known reference point. The point data is sent to Floradig which collates the data into a set of points representing the three dimensional structure of the plant. Each point collected is designated (for A. donax digitising) either a node or one of 5 points on a leaf lamina surface. Floradig constructs a hierarchy of points that resembles a map of the plant’s topology, and uses the 3D coordinates of each point to calculate lengths, angles, and areas of each plant organ. 2.2.1. Experiment 1 The data used in this study were part of a larger data set from A. donax morphology experiments. The digitised plants were grown outdoors at Davis, California, in large fibreglass tubs 116 cm wide 189 cm long 30 cm deep, filled with local sandy topsoil. The plants were grown from rhizome sections 3.5 cm long, each with a single bud, planted in 2001 (see Spencer and Ksander, 2006). At the end of 2001, the stems were removed. The rhizomes remained in situ and were allowed to sprout in 2002. The digitising study from which the present data were taken occurred in September 2002, at the end of the plants’ second growing season. 2.2.2. Experiment 2 Data used to relate above-ground to below-ground biomass were taken from an experiment in which rhizome pieces, each with a single bud, were collected from sites at Cache Creek in northern California and planted into fibreglass tubs (similar to those used in Section 2.2.1) filled with sandy loam topsoil in February 2002. Plants were destructively harvested periodically from those planted five times during the growing season. We used data from the fourth harvest, in October 2002, in which 10 plants were harvested. Soil was washed from the roots, and we then separated the plants into several categories of aboveground and below-ground biomass (leaves, stems, rhizomes, coarse roots, fine roots), though for this study we amalgamated above-ground biomass into a single category, and used two categories of below-ground biomass: roots and rhizomes. Linear relationships between above-ground biomass and each of the below-ground biomass categories were developed. We used leaf morphology data collected and stored as digital photographs from a previous experiment (Spencer et al., 2006) to
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284
277
analyse relationships between leaf position and lamina length, lamina width, and collar width (and hence stem diameter). Data were compiled and analysed using SAS (SAS Institute Inc., 1999), Access 2000 (Microsoft Corporation, Redmond WA, USA), Excel 2000 (Microsoft Corporation, Redmond WA, USA), and R (R Foundation for Statistical Computing, 2003). 2.3. Validation data In order to validate the model’s predictions for stem emergence and leaf area, we have used data taken from a further experiment (referred to here as Experiment 3) conducted in 2002. Twelve rhizome pieces with a single bud per piece similar to those in Sections 2.2.1 and 2.2.2 were planted in sand in 1 m 1 m 1 m plywood cubes in February 2002. The plants were watered either with deionised water or with well water (higher in N and P than the deionised water) to which extra nutrients were added in the form of 12.6 g of Miracle-Gro1 soluble fertiliser per 6 L of water, once per month. The plants were harvested in March 2003. Digitiser measurements of plant morphology were taken at regular intervals from plant emergence until November 2002. Two plants from the withnutrient treatment were discarded as they were over-watered following a leak in the irrigation system, leaving a validation dataset of four plants with nutrient and six without. 2.4. Morphological data The data used to parameterise L-DONAX are presented below. Main stem leaf lamina lengths collected by Spencer et al. (2006) are shown in Fig. 1. A linear approximation of the relationship between lamina length and position was produced for stem positions one to ten (Eq. (1) below). As the relationship between position and lamina length was weak above position ten, we elected to use a single mean value (70.2 cm) for lamina length for all leaves above that position. Lengths and widths of all organs are calculated in cm in all cases. Llamina ðpositionÞ ¼ 7:11 position 4:85
Fig. 2. Relationship between Arundo donax main stem mean, leaf sheath lengths and position on the stem. Error bars represent one standard error of the mean. Solid line represents a linear relationship between sheath length and position for sheaths 1–29 (length = 0.61*position + 23.74; R2 = 0.89).
A linear approximation of the relationship between sheath length and position at the end of the season (Fig. 2) was produced for all sheaths measured (Eq. (2)). Sheaths above position 29 use the value for position 29, to allow the model to be used to extrapolate to higher numbers of phytomers per shoot, without producing unreasonably short leaf sheaths. Lsheath ðpositionÞ ¼ 0:61 position þ 23:74
(2)
Three linear relationships were developed between position number and internode length, for three cohorts of internodes: internodes 1–4, 5–15, and 16–29 (Fig. 3; Eqs. (3)–(5) respectively). Internodes above position 29 currently use the value for position 29. Linternodes 14 ðpositionÞ ¼ 0:37 position þ 4:64
(3)
Linternodes 515 ðpositionÞ ¼ 0:31 position þ 8:62
(4)
Linternodes 1629 ðpositionÞ ¼ 0:05 position þ 2:58
(5)
(1)
Fig. 1. Relationship between Arundo donax main stem mean, leaf lamina lengths and position on the stem. Error bars represent one standard error of the mean. Solid line represents a linear relationship between lamina length and leaf position for leaves 1–10 (length = 7.11*position 4.85, R2 = 0.97).
Fig. 3. Relationship between Arundo donax main stem mean, internode lengths and position on the stem. Error bars represent one standard error of the mean. Solid lines represent separate linear regressions between internode length and stem position for internodes 1–4 (length = 0.37*position + 4.64; R2 = 0.20), 5– 15 (length = 0.31*position + 8.62; R2 = 0.96), and 16–29 (length = 0.05*position + 2.58; R2 = 0.30).
278
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284
Fig. 4. Relationship between Arundo donax main stem mean, lamina widths and position on the stem. Error bars represent one standard error of the mean. Solid line represents a power law relationship between lamina width and position for leaves 1–29 (width = 1.38*position0.30; R2 = 0.93).
Parameterising leaf lamina width proved more difficult. We expected a relationship between lamina width and shoot diameter to provide the most accurate estimation of width; however, we did not observe a strong relationship in the digitiser data (data not shown). We used a power law regression between main stem leaf position and lamina width (Fig. 4; Eq. (6)), as this provided a significantly better fit to the data than a linear relationship in this case. W lamina ðpositionÞ ¼ 1:38 position0:30
(6)
A. donax produces branches from each main stem, usually in second and subsequent years. However, the dimensions of internodes and leaves on these branches are much reduced compared to those on the main stem. There is also markedly less variation in branch leaf and internode sizes compared to those on the main stem (data not shown). Accordingly, we use single values for each all branch internode lengths (3.1 cm), lamina length (14.0 cm), and lamina width (2.5 cm)—these are equal to mean values derived from the digitiser data. 2.5. Implementation of the L-system model of A. donax The structural model L-DONAX consists of a set of Lsystem symbols (Table 1), a set of rules (called productions) that modify the structure iteratively, and a set of drawing instructions that produce a dynamic visual simulation of the growth of an A. donax clone. Thus, a string of components representing a short A. donax stem might be: Base I [Sheath Lamina] [Bud] I [Sheath Lamina] [Bud] Apex. As noted above, the square brackets indicate the start and end of branches in the string. Each symbol in the L-system has a number of parameters associated with it. The parameters are used to keep track of the age (in thermal time) of components, the current size and maximum size of a given component, a number index indicating a component’s position of attachment on a shoot, and the year in which a component was produced. Each symbol is associated with a subset of these parameters (Table 1); each
symbol retains values only for those parameters used in growth rules applicable to that symbol. The starting state of the string for the simulation (the axiom) is a single rhizome segment with a terminal meristem: Rseg(1) Rhiz(0,1) In order to simulate growth dynamically, L-systems apply a set of productions to the current string of symbols, and the resulting string replaces the current one at the end of the step. These productions may include programming statements and conditions. The productions used in L-DONAX are summarised in the statements in Table 1, although for simplicity the programming syntax used to implement the model is not reproduced in its original form. Predictions from the model, in terms of numbers of components, shoots, biomass, and accumulated degree days, are output daily by L-Studio to a log file. The rhizome and shoot apex development patterns are modelled in a top–down fashion, with the rules of Spencer and Ksander (2006) acting as overarching control mechanisms. New shoots develop in three groups, or cohorts. There is a fixed number of shoots in each cohort (though the number of shoots per cohort can also be varied randomly), and it takes a predetermined number of accumulated DD for all the shoots in a cohort to be produced, one after the other. The first shoot cohort is produced between accumulated DD 0 and 525. The second shoot cohort is produced between accumulated DD 335 and 1210. The third shoot cohort is produced between accumulated DD 1070 and 2500 (Spencer and Ksander, 2006). Similarly, the phytomers of each shoot are separated into two cohorts. The first cohort of phytomers is produced between accumulated DD 0 and 1000; the second cohort of phytomers is produced between accumulated DD 1001 and 2000. Once the predetermined number of phytomers for a given shoot have been produced, no further development is possible for that shoot during the thermal time predetermined as the limits for the current cohort, regardless of the number of days left before the cohort’s time is up, or the amount of thermal time experienced by the shoot following the cessation of development. Similarly, once the plant has produced its allotted number of shoots in the current cohort, development for that cohort ceases regardless of the number of days or DD subsequently experienced by the plant. At the beginning of each step, the model determines the number of shoots present for the day by reading the day’s maximum and minimum temperatures from a temperature data file, determining the day’s DD, and calculating the number of shoots predicted to be present from the appropriate equation following Spencer and Ksander (2006). There are three basic meristematic development rules, which control morphogenesis of the virtual A. donax plant. The first is for rhizome meristem development, simulating the extension of the branching underground rhizome and thereby expanding the ground area covered by the A. donax clump. The second is for shoot (or culm) apical meristem development, simulating the upward development of Arundo shoots. Each shoot develops in the same way and at the same rate over thermal time. The third is for branch apical meristem development, simulating the
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284
279
Table 1 Symbols used in the Arundo donax L-system and the plant components they represent Symbol
Component represented
L-system production rules description
Associated parameters
Lamina
Leaf lamina
I
Main stem internode
Age in DD, length, width, max length, shoot position, year produced Age in DD, length, max length, diameter, year produced
Apex
Main stem apical meristem
Bud
Axillary shoot meristem
Calculate new organ size for current step, using a sigmoid function relating component expansion to the age of the organ in DD; calculate the current biomass of the component and add the value to the day’s biomass value Calculate new organ size for current step, using a sigmoid function relating component expansion to the age of the organ in DD; calculate the current biomass of the component and add the value to the day’s biomass value Determine today’s predicted number of phytomers for this shoot; if the difference between the current number of phytomers and the predicted number of phytomers is greater than or equal to one, calculate the maximum length of a new lamina, sheath, and internode by stem position, and produce a new I, Bud, Sheath, Lamina, and Apex If the bud is greater than 1 year old, and its position on the stem is within this stem’s branch cluster, transform the bud into a Bapex
Sheath
Leaf sheath
Bapex
Branch apical meristem
Bint Rhiz
Branch internode Rhizome terminal meristem
Rseg
Rhizome somatic segment Junction between shoot and rhizome segment
Base
If current day is the last of the year, and this sheath’s position is below the number of leaves that survive into the next year, remove this sheath and the attached lamina; otherwise, calculate the new size of the sheath and accumulate biomass as for internode or leaf lamina If the number of DD accumulated to date by this meristem is greater than the value set for the branch apex plastochron, produce a new Bint, Bsheath, Lamina and Bapex; otherwise, accumulate today’s DD to the value of DD so far experienced by the apex None required; no changes other than appearance are made to these modules If the difference between the number of stems predicted by the model and the current number of stems present is greater than or equal to one, and this meristem is identified as the one due next to produce a new set of organs, produce a new Rseg, Base, I, Bud, Sheath, and Lamina, and a new pair of Rhiz terminal meristems None required; no changes other than appearance are made to these modules If the current day is the last of the year, and the age of this Base is greater than the longevity (in years) of stems (derived from Perdue (1958)), remove this Base and all symbols attached distally to it; if the Base has not reached maximum age, check against the annual random chance of this Base being removed
ability of A. donax’s primary shoots to produce secondary branches, usually in their second year. It should be noted that the rhizome undergoes dichotomous branching in L-DONAX, although in real rhizomes the rhizome occasionally produces one or three new segments from a rhizome terminal, instead of two. The rhizome meristems produce new internode segments sequentially based on age; that is, the oldest currently undeveloped rhizome segment is the one that next produces a new shoot. The equations for determining the current number of shoots per clump given by Spencer and Ksander (2006) return a real number value, and this is compared with the current number of shoots in the model. Since the difference between the two values must be greater than or equal to one, the model produces a new rhizome segment and shoot every time the model ‘ticks over’ from one integer value to the next. The extension of shoots, rhizomes, and branches is obligate once begun, with the rate of development determined by temperature. The model includes the facility to alter the temperatures read from the data file upwards or downwards by a user-determined factor, to simulate warmer or cooler years, with corresponding increase and decrease in the real rate of growth. It is also possible to substitute an entirely different set of daily temperatures, in order to simulate growth in different climates.
Shoot position, shoot number, age in DD, year produced
Year produced, shoot position, position of first branch on parent shoot Age in DD, length, max length, diameter, shoot position, year produced Age in DD, shoot position, year produced Shoot position, year produced Rhizome segment number, year produced
Year produced Rhizome segment number, year produced
Expansion of leaf laminae, leaf sheaths, and internodes is modelled as a sigmoid curve. Although we did not have sufficiently detailed data on organ expansion in A. donax to determine definitively that the expansion of these components is in fact sigmoid, expansion data for organs in related species such as maize (Birch et al., 2002) indicates that it is reasonable to assume a sigmoid curve for expansion of leaves and internodes in A. donax. L-studio allows the use of functions which can be scaled using component-specific or global parameters (Fig. 5; compare with Fig. 7 in Birch et al., 2002), as an alternative to developing individual logistic equations to describe sheath, lamina, and internode expansion. Each main shoot is able to produce a random number of branches in its second year, which is largely the case for real Californian A. donax plants (DiTomaso and Healey, 2003), starting at a randomly selected position on the stem and proceeding distally until all branches for the stem have been produced. The parameters of the random numbers generally cause the starting position to be in the upper part of the stem. This produces a randomly selected cluster of secondary branches on each main stem. The ‘Base’ modules are used to introduce a survivorship quotient for the whole clump (in which a certain percentage of shoots die each year), as well as providing for the death of rhizome segments at the end of their normal lifespan. The
280
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284
considered separately by the model, biomass for these two components must be likewise determined separately. However, the equation describing the relationship between dry weight and leaf lamina area is also used to determine the biomass of each leaf sheath, and can be expressed as: M lamina ðaÞ ðor M sheath Þ ¼ a 0:011
(8)
where Mlamina is calculated in g, and a is lamina or sheath area, measured in cm2 (R2 = 0.94; p < 0.001). Whole-clump shoot portion biomass is calculated in each step as the sum of the dry weights for each internode, lamina, and sheath present in the model at their current sizes on any given day. 3. Model results
Fig. 5. L-studio’s visual function editor showing sigmoid curve used to model organ expansion in Arundo donax.
default value of the yearly survivorship quotient is 10%, based on a study in which mortality of existing shoots was observed to be approximately 10% in sites with adequate soil nitrogen and 16% in sites with low soil nitrogen (Decruyenaere and Holt, 2005). The Rseg present in the model remains so that the spacing between live rhizome segments remains consistent from 1 year to the next, although these ‘dead’ segments can be identified in the visualisation through appearance changes implemented in the drawing rules for the model. For simplicity, the removal of a number of stems occurs in a single step, whereas in real plants, stems that do die tend to do so over a number of weeks or months towards the end of the growing season.
L-DONAX produces a visual model of an Arundo clump that includes stems of varying thickness and length, leaves of varying lengths, and a simple representation of the rhizome (Fig. 6). The visualisation is three-dimensional and is useful in assisting stakeholders, particularly those who are not scientifically trained, to understand the plant’s rate and pattern of growth. Predicted number of stems present at different numbers of accumulated degree days agree with the versions presented by Spencer and Ksander (2006). 3.1. Validation Data from Experiment 3 are compared to the model’s predictions of whole-plant leaf area and shoot number (Fig. 7a and b) in order to validate the model. For this model run,
2.6. Using the model to estimate biomass L-DONAX produces a realistic number of leaves and internodes at a realistic rate over the whole year. Given that the sizes of these parts is also realistic, and given that consistent relationships between organ size and biomass can be established, the model can be used to estimate the aboveground biomass produced during a year by an A. donax clump starting from a single rhizome segment. A number of relationships were established between leaf and stem size and biomass, as well as between leaf and internode size and their positions on the stem and in the clump (following Spencer et al., 2005 and using data collected by Spencer et al., 2005, 2006, and Spencer and Ksander, 2006). The relationship between dry weight and stem dimensions is multivariate, and can be expressed as: M stem ðd; lÞ ¼ ð0:25 dÞ þ ð0:18 lÞ
(7)
where Mstem is calculated in g; d and l are stem diameter and stem length, respectively, and are both measured in mm (R2 = 0.86; p < 0.01). Since leaf laminae and sheaths are
Fig. 6. Visualisation of 1 year’s growth of Arundo donax.
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284
281
Table 2 Number of shoots produced over time under different theoretical increases in growth rate Year
1 2 3 4
Value of mod increased annually by 0
+0.25
+0.5
1.2
1.3
18 36 54 72
18 41 69 102
18 46 84 131
18 41 70 107
18 42 74 116
the model in one or more cohorts may need to be revised slightly, in light of this result. 3.2. Clump size in number of shoots
Fig. 7. (a) Comparison of mean observed leaf area produced per Arundo donax plant during 2003 from a single rhizome segment, and L-DONAX prediction of leaf area produced. Filled circles represent with-nutrient treatment; empty circles represent without-nutrient treatment; solid line represents the model’s daily predicted stem number. Error bars represent one standard error of the mean. The arrow refers to the point or time slice in the model represented by the visualisation. (b) Comparison of mean observed number of stems produced per Arundo donax plant during 2003 from a single rhizome segment, and LDONAX prediction of number of stems produced. Filled circles represent with-nutrient treatment; empty circles represent without-nutrient treatment; solid line represents the model’s daily predicted stem number. Error bars represent one standard error of the mean.
climate data for Davis, CA, in 2002 were used to simulate A. donax growth. Leaf area predictions (Fig. 7a) are close to the measured leaf areas produced by the plants with nutrient added, showing that the model predicts leaf area development with a good level of accuracy ( p < 0.001 for Pearson’s moment correlation test between predicted and observed values for each of with-, without-nutrient, and combined data). Stem number predictions (Fig. 7b) are somewhat less accurate, falling between the stem numbers observed in the with- and without-nutrient treatments, but nevertheless the model is statistically a good fit ( p < 0.001 for Pearson’s moment correlation test between predicted and observed values for each of with-, without-nutrient, and combined data). Stem development predicted by the model is some days in advance of that observed for the two treatments studied here. The eventual number of stems predicted is somewhat lower than the number of stems observed in the plants with nutrient added. The number of stems predicted by
The data used in the model’s construction were for 1 year’s development from a single rhizome piece (Spencer and Ksander, 2006). However, the model can simulate several years’ growth, either by resetting the temperature data file to the beginning after 365 time steps have been processed, or by using a multiple-year temperature data file, and by assuming that the rates of shoot production and shoot development do not change from 1 year to the next. However, mature clumps produced by the model have somewhat fewer stems than would be expected from observational evidence. The model predicts that a 4-yearold clump will have produced 73 stems and, given an estimated 10% mortality rate of stems each winter, would consist of approximately 52 living stems at the end of the fourth year. This accumulation pattern is premised on the hypothesis that real growth rate (expressed as number of new stems produced per year) does not change. This is intuitively not the case, and is not supported by available data, in which 4-year-old clumps of Arundo consisted of approximately 114 shoots (data collected by Spencer and Ksander, 2006), and in which real growth rates rose in each year. L-DONAX can be used to begin to address this issue in an accessible way. A variable (‘mod’) is added to the model to increase the number of stems produced in each cohort over time; as the number of simulated years increases, so
Fig. 8. Above-ground Arundo donax biomass predicted by the model each day for the first year in four different counties in California (temperature data for 2000).
282
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284 Table 3 Proportion of below-ground parts of Arundo donax to biomass of the shoot portion
Fig. 9. Dry weight of the Arundo donax clump produced by the model each day for 5 years, using weather data collected at Davis, California, between 1970 and 1974.
the value of mod increases, and the number of shoots produced per year increases accordingly. In Table 2, the number of stems produced under different values of linear and exponential increase in the number of shoots produced per year are shown. Alternative relationships between years of growth and numbers of shoots produced are equally possible—and indeed, sufficiently detailed experimentation may show that the relationship is a logistic one. 3.3. Accumulated biomass of clump The dry weight (biomass) calculations performed in each step (Eqs. (1)–(3)) are returned by the model as output to the log file, and were used to produce graphs of the accumulation of biomass over 1 year (Fig. 8, Yolo data) and 5 years (Fig. 9). We then used the model to extrapolate A. donax biomass production from the original location in Davis to three other locations that experience different climatic conditions (Camarillo, Longbeach, and Siskiyou data, Fig. 8). After 1 year in three ‘warm’ counties in California (typical Californian habitats for A. donax), the shoot portion of the clump is predicted to have a dried mass of approximately 1.3 kg. After 1 year in a ‘cool’
Fig. 10. Biomass accumulated per month in below- and above-ground portions of the simulated Arundo donax plant.
Plant component
Proportion of component’s biomass to shoot biomass
Standard error
Roots Rhizome Belowground parts
0.84 0.70 1.54
0.04 0.02 0.14
county in California (less typical A. donax habitat), the shoot portion of the clump is predicted to have a dry weight of approximately 600 g. After 5 years’ growth in Davis, California, the shoot portion of the clump is predicted to have a dried mass of approximately 7 kg (Fig. 10). The 5-year biomass accumulation pattern in Fig. 9 consists of multiple sigmoid curves (for each year’s accumulated growth) punctuated by the loss of increasingly large (but sizeproportional) masses of leaf and stem at the end of each year, as specified in the growth rule for end of year shoot mortality rates. Correlations between the biomass produced in the shoot portion and that produced in the root and rhizome (Table 3) of A. donax plants, produced using data from Section 2.2.2, were used to make predictions of root and rhizome biomasses after 1 year of the model (Fig. 10). It shows that the majority of biomass is sequestered below ground, and that the majority of this below-ground biomass has been produced by mid-July. 4. Discussion L-DONAX allows us to demonstrate the yearly pattern of growth of A. donax in an effective, understandable, and succinct way. It is clear both from viewing the simulation and reviewing the data produced by the model that the majority of the rapid growth phase of Arundo clumps in central California occurs between July and October, and the timing of potential control methods needs to take this information into account. The rapid accumulation of biomass below ground, and the ability of the plant to regrow a substantial biomass in 1 year from a single broken-off rhizome segment, has important consequences for control tactics used on this weed. The construction of the model allows us to identify a number of morphological parameters and characteristics about which we have insufficient information. One missing piece of information concerns the longevity of stems and rhizome segments. No published data are currently available on the longevity of A. donax stems or rhizome segments in California, although we are in the process of collecting this information. The model allows us to investigate the morphological effects of a range of different possible values for stem and rhizome longevity, and provides visual feedback on the results. Also, the activity of the stem apex in years beyond the first is currently unknown. Mature stems in wild plants in California display dead, dried distal leaves, implying that the stem apex in these plants has died, but it is not known
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284
whether this occurs at the end of the first year, or after several years’ growth, and how this varies under different conditions. Stems in India flower in their first or second year, depending on their time of emergence (Sharma et al., 1998), and die soon after flowering. Most stems on these plants produce an inflorescence. In California, flower production is rare (DiTomaso and Healey, 2003) and flowers that do appear do not produce viable seed. Therefore, it is not possible to state that stem longevity (and stem apex longevity) in Californian A. donax is, for the majority of stems, connected with flowering. Stems usually produce branches during the growing season of their second and subsequent years. The death of the apex may be a stimulus for the production of the lateral branches in later years. Thus, in order to improve the model’s depiction of Arundo growth, we need more information on the growth behaviour and longevity of older stems that do not flower. Much of the published data on A. donax’s biomass production per year concerns mature, closed stands of Arundo expressed as g of dry weight per m2 of stand area per year (Sharma et al., 1998), rather than the productivity of a newly established clump expressed at a component level. The model’s biomass predictions are calculated using individual component dimensions, rather than the area covered by a population of stems. However, given more data concerning the morphological development of mature clumps, we anticipate being able to model biomass productivity in Californian A. donax at both a component level and a stand area level. Several new directions are possible for L-DONAX. It is currently being used to provide information on the light and shade characteristics of the clump, by introducing a Monte Carlo-style light model. Through the addition of leaf nitrogen content data, the model could provide useful simulations of likely insect feeding behaviours for leaves with different characteristics (such as age, position on the stem, stem number in the clump, and whether the leaf is growing in a sunny or shaded position). Another potential use of the model is as a base for simulating A. donax’s responses to a variety of environmental conditions and cues not currently present in the model, such as nutrition, water levels, and shoot damage. Further work in this direction would include exploiting the model’s potential for assisting in decision-making for prioritising agents for biological control. L-DONAX already indicates that an agent or agents affecting both shoot and rhizome survival would be the best candidate. These potential uses indicate that the model has both exploratory potential – that is, potential to assist scientists investigate properties of the plant, including, as mentioned above, assisting in developing avenues for real-plant experimentation – and expositive potential – that is, potential to demonstrate A. donax’s growth behaviours to stakeholders and therefore to provide better information to those making management decisions. Acknowledgements Dr. John Goolsby and Dr. Birgit Loch read an earlier version of the manuscript and provided helpful comments. The support
283
of Dr. Ray Carruthers of the U.S. Department of Agriculture for this work is gratefully acknowledged. Mention of a manufacturer or product does not constitute a warranty or guarantee of the product by the U.S. Department of Agriculture nor an endorsement over other products not mentioned. This work was supported by the U.S. Department of Agriculture and the Australian Research Council. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.aquabot.2007.06.012. References Birch, C.J., Andrieu, B., Fournier, C., 2002. Dynamics of internode and stem elongation in three cultivars of maize. Agronomie 22, 511–524. Decruyenaere, J.G., Holt, J.S., 2001. Seasonality of clonal propagation in giant reed. Weed Sci. 49, 760–767. Decruyenaere, J.G., Holt, J.S., 2005. Ramet demography of a clonal invader, Arundo donax (Poaceae), in southern California. Plant Soil 277, 41–52. DiTomaso, J.M., Healey, E.A., 2003. Aquatic and Riparian Weeds of the West. University of California Agriculture and Natural Resources. Evers, J.B., Vos, J., Fournier, C., Andrieu, B., Chelle, M., Struik, P.C., 2005. Towards a generic architectural model of tillering in Gramineae, as exemplified by spring wheat (Triticum aestivum). New Phytol. 166, 801–812. Fournier, C., Andrieu, B., 1999. ADEL-maize: an L-system based model for the integration of growth processes from the organ to the canopy. Application to regulation of morphogenesis by light availability. Agronomie 19, 313–327. Hanan, J., 1997. Virtual plants—integrating architectural and physiological models. Environ. Model. Softw. 12, 35–42. Hanan, J.S., Hearn, A.B., 2003. Linking physiological and architectural models of cotton. Agric. Syst. 75, 47–77. Hanan, J., Thornby, D., Adkins, S., 2005. Modelling cotton plant development with L-systems: a template model for incorporating physiology. In: Zerger, A., Argent, R.M., (Eds.), MODSIM 2005 International Congress on Modelling and Simulation. Modelling and Simulation Society of Australia and New Zealand, Melbourne, December 2005. Kaitaniemi, P., Hanan, J.S., Room, P.M., 1999. Architecture and morphogenesis of grain sorghum, Sorghum bicolor (L.). Moench. Field Crops Res. 61, 51– 60. Lindenmayer, A., 1968. Mathematical models for cellular interactions in development, parts I and II. J. Theor. Biol. 18, 280–315. Perdue, R.E., 1958. Arundo donax—source of musical reeds and industrial cellulose. Econ. Bot. 12, 368–404. Prusinkiewicz, P., Lindenmayer, A., 1990. In: Hanan, J.S., Fracchia, F.D., Fowler, D.R., de Boer, M.J.M., Mercer, L. (Eds.), The Algorithmic Beauty of Plants. Springer-Verlag, New York. Prusinkiewicz, P., Hanan, J.S., Mech, R., 2000a. An L-system-based plant modeling language. Lect. Notes Comput. Sci. 1779: Applications of graph transformation with industrial relevance, 395–410. Prusinkiewicz, P., Hanan, J.S., Mech, R., Karwowski, R., 2000b. L-studio/ cpfg: A software system for modeling plants. Lect. Notes Comput. Sci. 1779: Applications of graph transformation with industrial relevance, 457–464. Room, P.M., Maillette, L., Hanan, J.S., 1994. Module and metamer dynamics and virtual plants. Adv. Ecol. Res. 25, 105–157. SAS Institute Inc. 1999. SAS/Stat User’s Guide, Version 8. Cary, NC. pp. 3884. Sharma, K.P., Kushwaha, S.P.S., Gopal, B., 1998. A comparative study of stand structure and standing crops of two wetland species, Arundo donax and Phragmites karka, and primary production in Arundo donax with observations on the effect of clipping. Trop. Ecol. 39, 3–14.
284
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284
Spencer, D.F., Ksander, G.G., 2006. Estimating Arundo donax ramet recruitment using degree-day based equations. Aquat. Bot. 85, 282–288. Spencer, D.F., Ksander, G.G., Whitehand, L.C., 2005. Spatial and temporal variation in RGR and leaf quality of a clonal riparian plant: Arundo donax. Aquat. Bot. 81, 27–36. Spencer, D.F., Liow, P.S., Chan, W.K., Ksander, G.G., Getsinger, K.D., 2006. Estimating Arundo donax shoot biomass. Aquat. Bot. 84, 272–276.
Thornby, D., Renton, M., Hanan, J., 2003. Using computational plant science tools to investigate morphological aspects of compensatory growth. Lect. Notes Comput. Sci. 2660, 708–717. Watanabe, T., Hanan, J.S., Room, P.M., Hasegawa, T., Nakagawa, H., Takahashi, W., 2005. Rice morphogenesis and plant architecture: measurement, specification and the reconstruction of structural development by 3D architectural modelling. Ann. Bot. 95, 1131–1143.
L-DONAX, a growth model of the invasive weed species, Arundo donax L. David Thornby a,*, David Spencer a, Jim Hanan b, Anna Sher a,c,d a b
USDA-ARS Exotic and Invasive Weeds Research Unit, Robbins Hall, 1 Shields Avenue, University of California, Davis, CA 95616, USA Advanced Computational Modelling Centre, ARC Centre for Complex Systems, University of Queensland, St Lucia, QLD 4072, Australia c Department of Biological Sciences, University of Denver, 2190 E. Illif Avenue, Denver, CO 80208, USA d Department of Research, Herbaria and Records, Denver Botanic Gardens, 909 York St., Denver, CO 80206, USA Received 14 September 2005; received in revised form 14 June 2007; accepted 22 June 2007 Available online 12 July 2007
Abstract Arundo donax L. is a perennial reed and is an invasive weed of riparian systems in North America. A structural model (L-DONAX) of the species was constructed using L-system modelling in order to assist in understanding and demonstrating the complexities of the plant’s development and structure. The model produces a realistic number of plant components from a single rhizome segment over the course of the first year of growth, using empirical relationships derived from outdoor experiments. Biomass production is also simulated, through the use of relationships found between aerial plant portion sizes and masses. L-DONAX demonstrates that control of A. donax clumps is likely to require more than annual biomass removal, due to the bulk of biomass being present underground, and the ability of remaining rhizome or stem segments to produce large clumps quickly. The model extrapolates to years of growth beyond the first, but is found to require some re-parameterisation to improve accuracy. Published by Elsevier B.V. Keywords: Arundo donax; L-DONAX; Giant reed; L-system; Structural model; Invasive species
1. Introduction Giant reed (Arundo donax L.) is a successful invasive, nonnative weed of riparian zones in California, as well as other areas in North America. A. donax is a clonal plant that expands by rhizome expansion and, in California, propagates solely through shooting of separated rhizome pieces or stem pieces (Decruyenaere and Holt, 2001; DiTomaso and Healey, 2003). A. donax is difficult to control for several reasons. Firstly, it produces large amounts of biomass rapidly, both above and below ground (Sharma et al., 1998). Secondly, it propagates very readily vegetatively, by regrowth from rhizome and stem pieces (Decruyenaere and Holt, 2001), which are easily dispersed by river currents. Finally, because a large proportion of its biomass is sequestered underground in the rhizome,
* Corresponding author. Present address: Queensland Department of Primary Industries and Fisheries, P.O. Box 2282, Toowoomba, QLD 4350, Australia. Tel.: +61 7 46398811; fax: +61 7 46398800. E-mail address: [email protected] (D. Thornby). 0304-3770/$ – see front matter. Published by Elsevier B.V. doi:10.1016/j.aquabot.2007.06.012
removal of whole clumps is difficult. In order to more successfully manage infestations of A. donax in a given site, information is required regarding its potential invasiveness and its responses to environmental conditions and stimuli, including temperature, light, soil moisture, nutrition, and the activities of other organisms in the plant’s environment. In particular, information is needed on the relationships between environmental conditions and the plant’s rate of growth, including the rate of new shoot production (from rhizome segments), the rate of expansion of the new shoots, the production of branches, and the longevity of plant parts. The plant’s light environment is a key factor in the ecophysiology of the plant. Recent work in structural modelling using computational methods can help in the analysis of the effects of light and other factors on plant structures, and vice versa. Structural models have been produced for a number of significant crop plants—for example, cotton (Hanan and Hearn, 2003; Thornby et al., 2003; Hanan et al., 2005) and maize (Fournier and Andrieu, 1999). However, morphogenetic models of plants that have less immediate economic value (such as weeds) have been less frequently produced. Empirical
276
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284
models of plant morphogenesis have been used as a basis for further physiologically explicit research and to demonstrate various implications and aspects of growth, development, and environmental interactions in rice (Watanabe et al., 2005), sorghum (Kaitaniemi et al., 1999), and wheat, (Evers et al., 2005). Spencer and Ksander (2006) produced a set of equations that relate morphological or structural growth to accumulated degree days (DD). These equations are used as the developmental basis for the L-system morphogenetic model described here. Our approach is to produce an empirical structural model of A. donax growth and development, which we have denoted L-DONAX, using established growth relationships. Our study attempts to elucidate morphogenetic and architectural parameters of A. donax growth from a component level up to the scale of whole stands, with the ability subsequently to add mechanistic environmental functional aspects, such as interactions between the canopy and light or biocontrol agents. 2. Methods 2.1. Structural modelling with L-systems: background The L-system formalism for modelling the modular growth of organisms (including chains and branched chains of bacteria, and plants) was devised initially by Lindenmayer (Lindenmayer, 1968; Prusinkiewicz and Lindenmayer, 1990; Hanan, 1997; Prusinkiewicz et al., 2000a). The formalism is inherently suitable for modelling plant morphogenesis, because it simulates modular organism components (such as leaves, internodes, meristems, flowers and fruits) as a set of symbols arranged in a branching topological string, in much the same way that plants, being modular (Room et al., 1994), consist of a set of modules or organs arranged in predictable topologies. Plant development and growth are captured through a set of replacement rules describing how and when components are added, and how they change over time. Computer software used to interpret L-system rules, such as L-Studio (University of Calgary; Prusinkiewicz et al., 2000b) used in this work, allow complicated, interacting sets of rules to produce realistically complex plant growth behaviours, and to express the results visually, in two- or three-dimensional representations of plant forms. Structural models of plant growth, including our model of A. donax, have a number of advantages. The visualisation of the performance of a plant can provide insight and can assist in the transfer of knowledge to non-researchers. Structural plant models can also be used as a base model for investigating interactions between the plant canopy and other biotic and abiotic factors – for example, insect movement or light interactions – in a spatially explicit way. Modelling plant growth at a component level (that is, ‘bottom up’) allows the model a high level of flexibility in extrapolating or interpolating to smaller or larger plant structures using established growth rules and different sets of simulated conditions.
2.2. Data collection We collected morphological data from a set of A. donax plants using digitising techniques (detailed below as Section 2.2.1), and used other morphological data from previous studies of A. donax morphology (Spencer et al., 2005; Spencer and Ksander, 2006) in order to provide morphological parameters for L-DONAX. We used further datasets to develop relationships between component sizes and biomass (see Section 2.5 below). Much of this data is as presented in Spencer et al. (2005, 2006); however, we also used data from a 2002 study of A. donax responses to nutrient levels, referred to below as Section 2.2.2 For the digitising studies, we used a magnetic digitiser (Polhemus Fastrack) in conjunction with Floradig software (www.cpai.uq.edu.au, University of Queensland). The digitiser collects point data from a magnetic field with a known reference point. The point data is sent to Floradig which collates the data into a set of points representing the three dimensional structure of the plant. Each point collected is designated (for A. donax digitising) either a node or one of 5 points on a leaf lamina surface. Floradig constructs a hierarchy of points that resembles a map of the plant’s topology, and uses the 3D coordinates of each point to calculate lengths, angles, and areas of each plant organ. 2.2.1. Experiment 1 The data used in this study were part of a larger data set from A. donax morphology experiments. The digitised plants were grown outdoors at Davis, California, in large fibreglass tubs 116 cm wide 189 cm long 30 cm deep, filled with local sandy topsoil. The plants were grown from rhizome sections 3.5 cm long, each with a single bud, planted in 2001 (see Spencer and Ksander, 2006). At the end of 2001, the stems were removed. The rhizomes remained in situ and were allowed to sprout in 2002. The digitising study from which the present data were taken occurred in September 2002, at the end of the plants’ second growing season. 2.2.2. Experiment 2 Data used to relate above-ground to below-ground biomass were taken from an experiment in which rhizome pieces, each with a single bud, were collected from sites at Cache Creek in northern California and planted into fibreglass tubs (similar to those used in Section 2.2.1) filled with sandy loam topsoil in February 2002. Plants were destructively harvested periodically from those planted five times during the growing season. We used data from the fourth harvest, in October 2002, in which 10 plants were harvested. Soil was washed from the roots, and we then separated the plants into several categories of aboveground and below-ground biomass (leaves, stems, rhizomes, coarse roots, fine roots), though for this study we amalgamated above-ground biomass into a single category, and used two categories of below-ground biomass: roots and rhizomes. Linear relationships between above-ground biomass and each of the below-ground biomass categories were developed. We used leaf morphology data collected and stored as digital photographs from a previous experiment (Spencer et al., 2006) to
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284
277
analyse relationships between leaf position and lamina length, lamina width, and collar width (and hence stem diameter). Data were compiled and analysed using SAS (SAS Institute Inc., 1999), Access 2000 (Microsoft Corporation, Redmond WA, USA), Excel 2000 (Microsoft Corporation, Redmond WA, USA), and R (R Foundation for Statistical Computing, 2003). 2.3. Validation data In order to validate the model’s predictions for stem emergence and leaf area, we have used data taken from a further experiment (referred to here as Experiment 3) conducted in 2002. Twelve rhizome pieces with a single bud per piece similar to those in Sections 2.2.1 and 2.2.2 were planted in sand in 1 m 1 m 1 m plywood cubes in February 2002. The plants were watered either with deionised water or with well water (higher in N and P than the deionised water) to which extra nutrients were added in the form of 12.6 g of Miracle-Gro1 soluble fertiliser per 6 L of water, once per month. The plants were harvested in March 2003. Digitiser measurements of plant morphology were taken at regular intervals from plant emergence until November 2002. Two plants from the withnutrient treatment were discarded as they were over-watered following a leak in the irrigation system, leaving a validation dataset of four plants with nutrient and six without. 2.4. Morphological data The data used to parameterise L-DONAX are presented below. Main stem leaf lamina lengths collected by Spencer et al. (2006) are shown in Fig. 1. A linear approximation of the relationship between lamina length and position was produced for stem positions one to ten (Eq. (1) below). As the relationship between position and lamina length was weak above position ten, we elected to use a single mean value (70.2 cm) for lamina length for all leaves above that position. Lengths and widths of all organs are calculated in cm in all cases. Llamina ðpositionÞ ¼ 7:11 position 4:85
Fig. 2. Relationship between Arundo donax main stem mean, leaf sheath lengths and position on the stem. Error bars represent one standard error of the mean. Solid line represents a linear relationship between sheath length and position for sheaths 1–29 (length = 0.61*position + 23.74; R2 = 0.89).
A linear approximation of the relationship between sheath length and position at the end of the season (Fig. 2) was produced for all sheaths measured (Eq. (2)). Sheaths above position 29 use the value for position 29, to allow the model to be used to extrapolate to higher numbers of phytomers per shoot, without producing unreasonably short leaf sheaths. Lsheath ðpositionÞ ¼ 0:61 position þ 23:74
(2)
Three linear relationships were developed between position number and internode length, for three cohorts of internodes: internodes 1–4, 5–15, and 16–29 (Fig. 3; Eqs. (3)–(5) respectively). Internodes above position 29 currently use the value for position 29. Linternodes 14 ðpositionÞ ¼ 0:37 position þ 4:64
(3)
Linternodes 515 ðpositionÞ ¼ 0:31 position þ 8:62
(4)
Linternodes 1629 ðpositionÞ ¼ 0:05 position þ 2:58
(5)
(1)
Fig. 1. Relationship between Arundo donax main stem mean, leaf lamina lengths and position on the stem. Error bars represent one standard error of the mean. Solid line represents a linear relationship between lamina length and leaf position for leaves 1–10 (length = 7.11*position 4.85, R2 = 0.97).
Fig. 3. Relationship between Arundo donax main stem mean, internode lengths and position on the stem. Error bars represent one standard error of the mean. Solid lines represent separate linear regressions between internode length and stem position for internodes 1–4 (length = 0.37*position + 4.64; R2 = 0.20), 5– 15 (length = 0.31*position + 8.62; R2 = 0.96), and 16–29 (length = 0.05*position + 2.58; R2 = 0.30).
278
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284
Fig. 4. Relationship between Arundo donax main stem mean, lamina widths and position on the stem. Error bars represent one standard error of the mean. Solid line represents a power law relationship between lamina width and position for leaves 1–29 (width = 1.38*position0.30; R2 = 0.93).
Parameterising leaf lamina width proved more difficult. We expected a relationship between lamina width and shoot diameter to provide the most accurate estimation of width; however, we did not observe a strong relationship in the digitiser data (data not shown). We used a power law regression between main stem leaf position and lamina width (Fig. 4; Eq. (6)), as this provided a significantly better fit to the data than a linear relationship in this case. W lamina ðpositionÞ ¼ 1:38 position0:30
(6)
A. donax produces branches from each main stem, usually in second and subsequent years. However, the dimensions of internodes and leaves on these branches are much reduced compared to those on the main stem. There is also markedly less variation in branch leaf and internode sizes compared to those on the main stem (data not shown). Accordingly, we use single values for each all branch internode lengths (3.1 cm), lamina length (14.0 cm), and lamina width (2.5 cm)—these are equal to mean values derived from the digitiser data. 2.5. Implementation of the L-system model of A. donax The structural model L-DONAX consists of a set of Lsystem symbols (Table 1), a set of rules (called productions) that modify the structure iteratively, and a set of drawing instructions that produce a dynamic visual simulation of the growth of an A. donax clone. Thus, a string of components representing a short A. donax stem might be: Base I [Sheath Lamina] [Bud] I [Sheath Lamina] [Bud] Apex. As noted above, the square brackets indicate the start and end of branches in the string. Each symbol in the L-system has a number of parameters associated with it. The parameters are used to keep track of the age (in thermal time) of components, the current size and maximum size of a given component, a number index indicating a component’s position of attachment on a shoot, and the year in which a component was produced. Each symbol is associated with a subset of these parameters (Table 1); each
symbol retains values only for those parameters used in growth rules applicable to that symbol. The starting state of the string for the simulation (the axiom) is a single rhizome segment with a terminal meristem: Rseg(1) Rhiz(0,1) In order to simulate growth dynamically, L-systems apply a set of productions to the current string of symbols, and the resulting string replaces the current one at the end of the step. These productions may include programming statements and conditions. The productions used in L-DONAX are summarised in the statements in Table 1, although for simplicity the programming syntax used to implement the model is not reproduced in its original form. Predictions from the model, in terms of numbers of components, shoots, biomass, and accumulated degree days, are output daily by L-Studio to a log file. The rhizome and shoot apex development patterns are modelled in a top–down fashion, with the rules of Spencer and Ksander (2006) acting as overarching control mechanisms. New shoots develop in three groups, or cohorts. There is a fixed number of shoots in each cohort (though the number of shoots per cohort can also be varied randomly), and it takes a predetermined number of accumulated DD for all the shoots in a cohort to be produced, one after the other. The first shoot cohort is produced between accumulated DD 0 and 525. The second shoot cohort is produced between accumulated DD 335 and 1210. The third shoot cohort is produced between accumulated DD 1070 and 2500 (Spencer and Ksander, 2006). Similarly, the phytomers of each shoot are separated into two cohorts. The first cohort of phytomers is produced between accumulated DD 0 and 1000; the second cohort of phytomers is produced between accumulated DD 1001 and 2000. Once the predetermined number of phytomers for a given shoot have been produced, no further development is possible for that shoot during the thermal time predetermined as the limits for the current cohort, regardless of the number of days left before the cohort’s time is up, or the amount of thermal time experienced by the shoot following the cessation of development. Similarly, once the plant has produced its allotted number of shoots in the current cohort, development for that cohort ceases regardless of the number of days or DD subsequently experienced by the plant. At the beginning of each step, the model determines the number of shoots present for the day by reading the day’s maximum and minimum temperatures from a temperature data file, determining the day’s DD, and calculating the number of shoots predicted to be present from the appropriate equation following Spencer and Ksander (2006). There are three basic meristematic development rules, which control morphogenesis of the virtual A. donax plant. The first is for rhizome meristem development, simulating the extension of the branching underground rhizome and thereby expanding the ground area covered by the A. donax clump. The second is for shoot (or culm) apical meristem development, simulating the upward development of Arundo shoots. Each shoot develops in the same way and at the same rate over thermal time. The third is for branch apical meristem development, simulating the
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284
279
Table 1 Symbols used in the Arundo donax L-system and the plant components they represent Symbol
Component represented
L-system production rules description
Associated parameters
Lamina
Leaf lamina
I
Main stem internode
Age in DD, length, width, max length, shoot position, year produced Age in DD, length, max length, diameter, year produced
Apex
Main stem apical meristem
Bud
Axillary shoot meristem
Calculate new organ size for current step, using a sigmoid function relating component expansion to the age of the organ in DD; calculate the current biomass of the component and add the value to the day’s biomass value Calculate new organ size for current step, using a sigmoid function relating component expansion to the age of the organ in DD; calculate the current biomass of the component and add the value to the day’s biomass value Determine today’s predicted number of phytomers for this shoot; if the difference between the current number of phytomers and the predicted number of phytomers is greater than or equal to one, calculate the maximum length of a new lamina, sheath, and internode by stem position, and produce a new I, Bud, Sheath, Lamina, and Apex If the bud is greater than 1 year old, and its position on the stem is within this stem’s branch cluster, transform the bud into a Bapex
Sheath
Leaf sheath
Bapex
Branch apical meristem
Bint Rhiz
Branch internode Rhizome terminal meristem
Rseg
Rhizome somatic segment Junction between shoot and rhizome segment
Base
If current day is the last of the year, and this sheath’s position is below the number of leaves that survive into the next year, remove this sheath and the attached lamina; otherwise, calculate the new size of the sheath and accumulate biomass as for internode or leaf lamina If the number of DD accumulated to date by this meristem is greater than the value set for the branch apex plastochron, produce a new Bint, Bsheath, Lamina and Bapex; otherwise, accumulate today’s DD to the value of DD so far experienced by the apex None required; no changes other than appearance are made to these modules If the difference between the number of stems predicted by the model and the current number of stems present is greater than or equal to one, and this meristem is identified as the one due next to produce a new set of organs, produce a new Rseg, Base, I, Bud, Sheath, and Lamina, and a new pair of Rhiz terminal meristems None required; no changes other than appearance are made to these modules If the current day is the last of the year, and the age of this Base is greater than the longevity (in years) of stems (derived from Perdue (1958)), remove this Base and all symbols attached distally to it; if the Base has not reached maximum age, check against the annual random chance of this Base being removed
ability of A. donax’s primary shoots to produce secondary branches, usually in their second year. It should be noted that the rhizome undergoes dichotomous branching in L-DONAX, although in real rhizomes the rhizome occasionally produces one or three new segments from a rhizome terminal, instead of two. The rhizome meristems produce new internode segments sequentially based on age; that is, the oldest currently undeveloped rhizome segment is the one that next produces a new shoot. The equations for determining the current number of shoots per clump given by Spencer and Ksander (2006) return a real number value, and this is compared with the current number of shoots in the model. Since the difference between the two values must be greater than or equal to one, the model produces a new rhizome segment and shoot every time the model ‘ticks over’ from one integer value to the next. The extension of shoots, rhizomes, and branches is obligate once begun, with the rate of development determined by temperature. The model includes the facility to alter the temperatures read from the data file upwards or downwards by a user-determined factor, to simulate warmer or cooler years, with corresponding increase and decrease in the real rate of growth. It is also possible to substitute an entirely different set of daily temperatures, in order to simulate growth in different climates.
Shoot position, shoot number, age in DD, year produced
Year produced, shoot position, position of first branch on parent shoot Age in DD, length, max length, diameter, shoot position, year produced Age in DD, shoot position, year produced Shoot position, year produced Rhizome segment number, year produced
Year produced Rhizome segment number, year produced
Expansion of leaf laminae, leaf sheaths, and internodes is modelled as a sigmoid curve. Although we did not have sufficiently detailed data on organ expansion in A. donax to determine definitively that the expansion of these components is in fact sigmoid, expansion data for organs in related species such as maize (Birch et al., 2002) indicates that it is reasonable to assume a sigmoid curve for expansion of leaves and internodes in A. donax. L-studio allows the use of functions which can be scaled using component-specific or global parameters (Fig. 5; compare with Fig. 7 in Birch et al., 2002), as an alternative to developing individual logistic equations to describe sheath, lamina, and internode expansion. Each main shoot is able to produce a random number of branches in its second year, which is largely the case for real Californian A. donax plants (DiTomaso and Healey, 2003), starting at a randomly selected position on the stem and proceeding distally until all branches for the stem have been produced. The parameters of the random numbers generally cause the starting position to be in the upper part of the stem. This produces a randomly selected cluster of secondary branches on each main stem. The ‘Base’ modules are used to introduce a survivorship quotient for the whole clump (in which a certain percentage of shoots die each year), as well as providing for the death of rhizome segments at the end of their normal lifespan. The
280
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284
considered separately by the model, biomass for these two components must be likewise determined separately. However, the equation describing the relationship between dry weight and leaf lamina area is also used to determine the biomass of each leaf sheath, and can be expressed as: M lamina ðaÞ ðor M sheath Þ ¼ a 0:011
(8)
where Mlamina is calculated in g, and a is lamina or sheath area, measured in cm2 (R2 = 0.94; p < 0.001). Whole-clump shoot portion biomass is calculated in each step as the sum of the dry weights for each internode, lamina, and sheath present in the model at their current sizes on any given day. 3. Model results
Fig. 5. L-studio’s visual function editor showing sigmoid curve used to model organ expansion in Arundo donax.
default value of the yearly survivorship quotient is 10%, based on a study in which mortality of existing shoots was observed to be approximately 10% in sites with adequate soil nitrogen and 16% in sites with low soil nitrogen (Decruyenaere and Holt, 2005). The Rseg present in the model remains so that the spacing between live rhizome segments remains consistent from 1 year to the next, although these ‘dead’ segments can be identified in the visualisation through appearance changes implemented in the drawing rules for the model. For simplicity, the removal of a number of stems occurs in a single step, whereas in real plants, stems that do die tend to do so over a number of weeks or months towards the end of the growing season.
L-DONAX produces a visual model of an Arundo clump that includes stems of varying thickness and length, leaves of varying lengths, and a simple representation of the rhizome (Fig. 6). The visualisation is three-dimensional and is useful in assisting stakeholders, particularly those who are not scientifically trained, to understand the plant’s rate and pattern of growth. Predicted number of stems present at different numbers of accumulated degree days agree with the versions presented by Spencer and Ksander (2006). 3.1. Validation Data from Experiment 3 are compared to the model’s predictions of whole-plant leaf area and shoot number (Fig. 7a and b) in order to validate the model. For this model run,
2.6. Using the model to estimate biomass L-DONAX produces a realistic number of leaves and internodes at a realistic rate over the whole year. Given that the sizes of these parts is also realistic, and given that consistent relationships between organ size and biomass can be established, the model can be used to estimate the aboveground biomass produced during a year by an A. donax clump starting from a single rhizome segment. A number of relationships were established between leaf and stem size and biomass, as well as between leaf and internode size and their positions on the stem and in the clump (following Spencer et al., 2005 and using data collected by Spencer et al., 2005, 2006, and Spencer and Ksander, 2006). The relationship between dry weight and stem dimensions is multivariate, and can be expressed as: M stem ðd; lÞ ¼ ð0:25 dÞ þ ð0:18 lÞ
(7)
where Mstem is calculated in g; d and l are stem diameter and stem length, respectively, and are both measured in mm (R2 = 0.86; p < 0.01). Since leaf laminae and sheaths are
Fig. 6. Visualisation of 1 year’s growth of Arundo donax.
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284
281
Table 2 Number of shoots produced over time under different theoretical increases in growth rate Year
1 2 3 4
Value of mod increased annually by 0
+0.25
+0.5
1.2
1.3
18 36 54 72
18 41 69 102
18 46 84 131
18 41 70 107
18 42 74 116
the model in one or more cohorts may need to be revised slightly, in light of this result. 3.2. Clump size in number of shoots
Fig. 7. (a) Comparison of mean observed leaf area produced per Arundo donax plant during 2003 from a single rhizome segment, and L-DONAX prediction of leaf area produced. Filled circles represent with-nutrient treatment; empty circles represent without-nutrient treatment; solid line represents the model’s daily predicted stem number. Error bars represent one standard error of the mean. The arrow refers to the point or time slice in the model represented by the visualisation. (b) Comparison of mean observed number of stems produced per Arundo donax plant during 2003 from a single rhizome segment, and LDONAX prediction of number of stems produced. Filled circles represent with-nutrient treatment; empty circles represent without-nutrient treatment; solid line represents the model’s daily predicted stem number. Error bars represent one standard error of the mean.
climate data for Davis, CA, in 2002 were used to simulate A. donax growth. Leaf area predictions (Fig. 7a) are close to the measured leaf areas produced by the plants with nutrient added, showing that the model predicts leaf area development with a good level of accuracy ( p < 0.001 for Pearson’s moment correlation test between predicted and observed values for each of with-, without-nutrient, and combined data). Stem number predictions (Fig. 7b) are somewhat less accurate, falling between the stem numbers observed in the with- and without-nutrient treatments, but nevertheless the model is statistically a good fit ( p < 0.001 for Pearson’s moment correlation test between predicted and observed values for each of with-, without-nutrient, and combined data). Stem development predicted by the model is some days in advance of that observed for the two treatments studied here. The eventual number of stems predicted is somewhat lower than the number of stems observed in the plants with nutrient added. The number of stems predicted by
The data used in the model’s construction were for 1 year’s development from a single rhizome piece (Spencer and Ksander, 2006). However, the model can simulate several years’ growth, either by resetting the temperature data file to the beginning after 365 time steps have been processed, or by using a multiple-year temperature data file, and by assuming that the rates of shoot production and shoot development do not change from 1 year to the next. However, mature clumps produced by the model have somewhat fewer stems than would be expected from observational evidence. The model predicts that a 4-yearold clump will have produced 73 stems and, given an estimated 10% mortality rate of stems each winter, would consist of approximately 52 living stems at the end of the fourth year. This accumulation pattern is premised on the hypothesis that real growth rate (expressed as number of new stems produced per year) does not change. This is intuitively not the case, and is not supported by available data, in which 4-year-old clumps of Arundo consisted of approximately 114 shoots (data collected by Spencer and Ksander, 2006), and in which real growth rates rose in each year. L-DONAX can be used to begin to address this issue in an accessible way. A variable (‘mod’) is added to the model to increase the number of stems produced in each cohort over time; as the number of simulated years increases, so
Fig. 8. Above-ground Arundo donax biomass predicted by the model each day for the first year in four different counties in California (temperature data for 2000).
282
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284 Table 3 Proportion of below-ground parts of Arundo donax to biomass of the shoot portion
Fig. 9. Dry weight of the Arundo donax clump produced by the model each day for 5 years, using weather data collected at Davis, California, between 1970 and 1974.
the value of mod increases, and the number of shoots produced per year increases accordingly. In Table 2, the number of stems produced under different values of linear and exponential increase in the number of shoots produced per year are shown. Alternative relationships between years of growth and numbers of shoots produced are equally possible—and indeed, sufficiently detailed experimentation may show that the relationship is a logistic one. 3.3. Accumulated biomass of clump The dry weight (biomass) calculations performed in each step (Eqs. (1)–(3)) are returned by the model as output to the log file, and were used to produce graphs of the accumulation of biomass over 1 year (Fig. 8, Yolo data) and 5 years (Fig. 9). We then used the model to extrapolate A. donax biomass production from the original location in Davis to three other locations that experience different climatic conditions (Camarillo, Longbeach, and Siskiyou data, Fig. 8). After 1 year in three ‘warm’ counties in California (typical Californian habitats for A. donax), the shoot portion of the clump is predicted to have a dried mass of approximately 1.3 kg. After 1 year in a ‘cool’
Fig. 10. Biomass accumulated per month in below- and above-ground portions of the simulated Arundo donax plant.
Plant component
Proportion of component’s biomass to shoot biomass
Standard error
Roots Rhizome Belowground parts
0.84 0.70 1.54
0.04 0.02 0.14
county in California (less typical A. donax habitat), the shoot portion of the clump is predicted to have a dry weight of approximately 600 g. After 5 years’ growth in Davis, California, the shoot portion of the clump is predicted to have a dried mass of approximately 7 kg (Fig. 10). The 5-year biomass accumulation pattern in Fig. 9 consists of multiple sigmoid curves (for each year’s accumulated growth) punctuated by the loss of increasingly large (but sizeproportional) masses of leaf and stem at the end of each year, as specified in the growth rule for end of year shoot mortality rates. Correlations between the biomass produced in the shoot portion and that produced in the root and rhizome (Table 3) of A. donax plants, produced using data from Section 2.2.2, were used to make predictions of root and rhizome biomasses after 1 year of the model (Fig. 10). It shows that the majority of biomass is sequestered below ground, and that the majority of this below-ground biomass has been produced by mid-July. 4. Discussion L-DONAX allows us to demonstrate the yearly pattern of growth of A. donax in an effective, understandable, and succinct way. It is clear both from viewing the simulation and reviewing the data produced by the model that the majority of the rapid growth phase of Arundo clumps in central California occurs between July and October, and the timing of potential control methods needs to take this information into account. The rapid accumulation of biomass below ground, and the ability of the plant to regrow a substantial biomass in 1 year from a single broken-off rhizome segment, has important consequences for control tactics used on this weed. The construction of the model allows us to identify a number of morphological parameters and characteristics about which we have insufficient information. One missing piece of information concerns the longevity of stems and rhizome segments. No published data are currently available on the longevity of A. donax stems or rhizome segments in California, although we are in the process of collecting this information. The model allows us to investigate the morphological effects of a range of different possible values for stem and rhizome longevity, and provides visual feedback on the results. Also, the activity of the stem apex in years beyond the first is currently unknown. Mature stems in wild plants in California display dead, dried distal leaves, implying that the stem apex in these plants has died, but it is not known
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284
whether this occurs at the end of the first year, or after several years’ growth, and how this varies under different conditions. Stems in India flower in their first or second year, depending on their time of emergence (Sharma et al., 1998), and die soon after flowering. Most stems on these plants produce an inflorescence. In California, flower production is rare (DiTomaso and Healey, 2003) and flowers that do appear do not produce viable seed. Therefore, it is not possible to state that stem longevity (and stem apex longevity) in Californian A. donax is, for the majority of stems, connected with flowering. Stems usually produce branches during the growing season of their second and subsequent years. The death of the apex may be a stimulus for the production of the lateral branches in later years. Thus, in order to improve the model’s depiction of Arundo growth, we need more information on the growth behaviour and longevity of older stems that do not flower. Much of the published data on A. donax’s biomass production per year concerns mature, closed stands of Arundo expressed as g of dry weight per m2 of stand area per year (Sharma et al., 1998), rather than the productivity of a newly established clump expressed at a component level. The model’s biomass predictions are calculated using individual component dimensions, rather than the area covered by a population of stems. However, given more data concerning the morphological development of mature clumps, we anticipate being able to model biomass productivity in Californian A. donax at both a component level and a stand area level. Several new directions are possible for L-DONAX. It is currently being used to provide information on the light and shade characteristics of the clump, by introducing a Monte Carlo-style light model. Through the addition of leaf nitrogen content data, the model could provide useful simulations of likely insect feeding behaviours for leaves with different characteristics (such as age, position on the stem, stem number in the clump, and whether the leaf is growing in a sunny or shaded position). Another potential use of the model is as a base for simulating A. donax’s responses to a variety of environmental conditions and cues not currently present in the model, such as nutrition, water levels, and shoot damage. Further work in this direction would include exploiting the model’s potential for assisting in decision-making for prioritising agents for biological control. L-DONAX already indicates that an agent or agents affecting both shoot and rhizome survival would be the best candidate. These potential uses indicate that the model has both exploratory potential – that is, potential to assist scientists investigate properties of the plant, including, as mentioned above, assisting in developing avenues for real-plant experimentation – and expositive potential – that is, potential to demonstrate A. donax’s growth behaviours to stakeholders and therefore to provide better information to those making management decisions. Acknowledgements Dr. John Goolsby and Dr. Birgit Loch read an earlier version of the manuscript and provided helpful comments. The support
283
of Dr. Ray Carruthers of the U.S. Department of Agriculture for this work is gratefully acknowledged. Mention of a manufacturer or product does not constitute a warranty or guarantee of the product by the U.S. Department of Agriculture nor an endorsement over other products not mentioned. This work was supported by the U.S. Department of Agriculture and the Australian Research Council. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.aquabot.2007.06.012. References Birch, C.J., Andrieu, B., Fournier, C., 2002. Dynamics of internode and stem elongation in three cultivars of maize. Agronomie 22, 511–524. Decruyenaere, J.G., Holt, J.S., 2001. Seasonality of clonal propagation in giant reed. Weed Sci. 49, 760–767. Decruyenaere, J.G., Holt, J.S., 2005. Ramet demography of a clonal invader, Arundo donax (Poaceae), in southern California. Plant Soil 277, 41–52. DiTomaso, J.M., Healey, E.A., 2003. Aquatic and Riparian Weeds of the West. University of California Agriculture and Natural Resources. Evers, J.B., Vos, J., Fournier, C., Andrieu, B., Chelle, M., Struik, P.C., 2005. Towards a generic architectural model of tillering in Gramineae, as exemplified by spring wheat (Triticum aestivum). New Phytol. 166, 801–812. Fournier, C., Andrieu, B., 1999. ADEL-maize: an L-system based model for the integration of growth processes from the organ to the canopy. Application to regulation of morphogenesis by light availability. Agronomie 19, 313–327. Hanan, J., 1997. Virtual plants—integrating architectural and physiological models. Environ. Model. Softw. 12, 35–42. Hanan, J.S., Hearn, A.B., 2003. Linking physiological and architectural models of cotton. Agric. Syst. 75, 47–77. Hanan, J., Thornby, D., Adkins, S., 2005. Modelling cotton plant development with L-systems: a template model for incorporating physiology. In: Zerger, A., Argent, R.M., (Eds.), MODSIM 2005 International Congress on Modelling and Simulation. Modelling and Simulation Society of Australia and New Zealand, Melbourne, December 2005. Kaitaniemi, P., Hanan, J.S., Room, P.M., 1999. Architecture and morphogenesis of grain sorghum, Sorghum bicolor (L.). Moench. Field Crops Res. 61, 51– 60. Lindenmayer, A., 1968. Mathematical models for cellular interactions in development, parts I and II. J. Theor. Biol. 18, 280–315. Perdue, R.E., 1958. Arundo donax—source of musical reeds and industrial cellulose. Econ. Bot. 12, 368–404. Prusinkiewicz, P., Lindenmayer, A., 1990. In: Hanan, J.S., Fracchia, F.D., Fowler, D.R., de Boer, M.J.M., Mercer, L. (Eds.), The Algorithmic Beauty of Plants. Springer-Verlag, New York. Prusinkiewicz, P., Hanan, J.S., Mech, R., 2000a. An L-system-based plant modeling language. Lect. Notes Comput. Sci. 1779: Applications of graph transformation with industrial relevance, 395–410. Prusinkiewicz, P., Hanan, J.S., Mech, R., Karwowski, R., 2000b. L-studio/ cpfg: A software system for modeling plants. Lect. Notes Comput. Sci. 1779: Applications of graph transformation with industrial relevance, 457–464. Room, P.M., Maillette, L., Hanan, J.S., 1994. Module and metamer dynamics and virtual plants. Adv. Ecol. Res. 25, 105–157. SAS Institute Inc. 1999. SAS/Stat User’s Guide, Version 8. Cary, NC. pp. 3884. Sharma, K.P., Kushwaha, S.P.S., Gopal, B., 1998. A comparative study of stand structure and standing crops of two wetland species, Arundo donax and Phragmites karka, and primary production in Arundo donax with observations on the effect of clipping. Trop. Ecol. 39, 3–14.
284
D. Thornby et al. / Aquatic Botany 87 (2007) 275–284
Spencer, D.F., Ksander, G.G., 2006. Estimating Arundo donax ramet recruitment using degree-day based equations. Aquat. Bot. 85, 282–288. Spencer, D.F., Ksander, G.G., Whitehand, L.C., 2005. Spatial and temporal variation in RGR and leaf quality of a clonal riparian plant: Arundo donax. Aquat. Bot. 81, 27–36. Spencer, D.F., Liow, P.S., Chan, W.K., Ksander, G.G., Getsinger, K.D., 2006. Estimating Arundo donax shoot biomass. Aquat. Bot. 84, 272–276.
Thornby, D., Renton, M., Hanan, J., 2003. Using computational plant science tools to investigate morphological aspects of compensatory growth. Lect. Notes Comput. Sci. 2660, 708–717. Watanabe, T., Hanan, J.S., Room, P.M., Hasegawa, T., Nakagawa, H., Takahashi, W., 2005. Rice morphogenesis and plant architecture: measurement, specification and the reconstruction of structural development by 3D architectural modelling. Ann. Bot. 95, 1131–1143.