New Zealand Journal of Agricultural Research, 1998, Vol. 41: /98/ $7.00/0 The Royal Society of New Zealand 1998

Transcription

1 New Zealand Journal of Agricultural Research, 1998, Vol. 41: /98/ $7.00/0 The Royal Society of New Zealand 1998 Determination of tiller and root appearance in perennial ryegrass (Lolium perenne) swards by observation of the tiller axis, and potential application in mechanistic modelling C. MATTHEW J. Z. YANG* Department of Plant Science Massey University Private Bag Palmerston North, New Zealand J. F. POTTER New Zealand Institute for Crop & Food Research Limited Private Bag 4005 Levin, New Zealand *Present address: Department of Agronomy, Shanxi Agricultural University, Taigu, Shanxi, People's Republic of China. Abstract Recording of bud status at successive phytomers on the tiller axis in perennial ryegrass was shown to provide information similar to that obtained from conventional recording of tiller appearance by monitoring marked tillers. The technique can be extended to monitor the number of roots formed. These observations lead to the establishment of a notation which describes tiller and root appearance, respectively, on the basis of probability or frequency per phytomer. These statistics are analogous to but less ambiguous in their interpretation than site filling statistics currently used to define tiller, and sometimes root, appearance. Potential modelling applications of quantifying sward dynamics at the phytomer level are discussed. In particular, nodal frequency statistics allow development of mass flow equations to describe leaf, tiller, and root production, and seasonal variation in allocation of photosynthate to developing tillers and roots. A96054 Received 10 July 1996; accepted 21 October 1997 Keywords mass flow; nodal probability; nodal frequency; perennial ryegrass; root appearance; site filling; phytomer; tiller appearance INTRODUCTION In past studies of grass sward behaviour, processes of leaf, tiller, or root formation have usually been examined as distinct entities, e.g., Bircham & Hodgson (1983) and Parsons et al. (1983) (leaf formation and death); Langer (1963), Langer et al. (1964), Garwood (1969), and Korte (1986) (tiller population demography); and Garwood (1967), Caradus & Evans (1977), and Matthew (1992) (root dynamics). Although the reasons for studying these different processes separately have probably been logistical rather than philosophical, the practice of focusing on a particular sward process in major studies has resulted in a lack of attention to understanding the ways these processes interrelate. Such understanding is of increasing importance, in order to both understand the grass tiller as a functional unit and provide an accurate representation for modelling purposes. The most commonly used framework for evaluating the balance between respective processes of leaf, tiller, and root production has been to calculate a so-called site filling ratio (F s ). This concept was first introduced by Davies (1974) in order to gain a measure of comparative tiller production following differing defoliation regimes, and has since been extended to cover other ratios including a root:leaf (F r ) and a root:tiller (F rt ) ratio (Hunt & Thomas 1985). More recently,'there has been detailed mathematical analysis of properties of site-filling ratios (Neuteboom & Lantinga 1989) and application of the concept in modelling plant development (Van Loo 1992). Morphologically, the grass tiller has a segmental organisation, new segments known as phytomers being formed at regular intervals at the apical meristem (Sharman 1947; Davies 1977; Jewiss

2 1981). A few authors have recognised and explored the link between segmental morphology of grasses and their behaviour. Notable examples are structural diagrams of Poa pratensis plants (Etter 1951) and the work of Silsbury (1970). Matthew et al. (1991) attempted to integrate information about leaf, shoot, and root formation and these authors considered the tiller axis to be the integrating principle. When leaf, tiller, and root appearance data are to be expressed as events on the tiller axis, the logical approach is to develop a phytomer-based notation. Thus, Matthew (1992) suggested calculation of nodal probability of formation of a tiller or a root. The term probability implies that values range between 0 and 1, and is appropriate when describing the fate of the single axillary bud at each phytomer. However, there are often several roots per phytomer. For example, there are up to four roots per phytomer in wheat (Klepper et al. 1984) and sometimes more than ten in maize (Demotes-Mainard & Pellerin 1992). Therefore, an alternative term, nodal frequency, is adopted here when referring to root formation. Nodal probability of tiller formation or frequency of root formation is designated T n or R n, respectively, and is formally defined in Materials and Methods, below. These entities are analogous to the equivalent site-filling ratios, F s or F r i, but are intrinsically more logical since the theoretical maximum for T n is 1.00 rather than 0.69 as determined for F s by Neuteboom & Lantinga (1989). Similarly, the theoretical maximum for R n is the number of root primordia per node. Another advantage of nodal frequency notation over sitefilling notation is that site-filling ratios are ambiguous in their interpretation. For example, an increase in F s will arise either where there is an increase in the nodal probability of tiller formation or a decrease in the number of phytomers (or delay) between site of leaf formation and site of tiller formation on the tiller axis (Neuteboom & Lantinga 1989). Also, the theoretical maximum for F s may be 0.41 or 0.69 depending on whether or not tillers form in prophyll axes. Finally, these maxima of 0.41 and 0.69 assume that tillers on successive phytomers on the tiller axis form in strict succession. They would be exceeded in conditions where tillers develop more or less simultaneously on two or more adjacent phytomers. Although the properties of F r have been less thoroughly explored in the literature, the ambiguities in interpretation of F s also apply to the interpretation of F r - A concept very similar to New Zealand Journal of Agricultural Research, 1998, Vol. 41 our nodal frequency concept, namely site usage, was proposed by Skinner & Nelson (1992). However, site usage as defined by Skinner & Nelson (1992) is a whole plant average for a period of time, rather than for a particular phytomer or node number on the tiller axis. Two potential applications for nodal frequency data are evident. First, to the extent that the rate of phytomer formation and development is constant for different tillers in a population, nodal frequency data for numbers of tillers or roots present at particular node numbers comprise a record of tiller and root appearance. In this case, historical tiller or root appearance could be obtained by a single examination of harvested tillers in the laboratory as an alternative to sequential monitoring of tagged tillers in the field. Secondly, to the extent that phytomer formation and development can be assumed to be a steady state process, mass flow equations for tissue turnover of particular organs are readily generated. This paper reports investigations into the development of the methodology. Measurements were made in association with three separate experiments already in progress. A first data set was collected to test the feasibility of obtaining agronomically useful information on tiller appearance rate by observation of bud status at successive nodes on the tiller axis. Results proved promising and further measurements were carried out to compare nodal probability data with that obtained by the conventional method of observing tagged tillers, and to test the extension of the technique to collect root appearance information simultaneously. We also develop an equation to calculate the nodal probability or frequency equivalent of a given sitefilling ratio and briefly discuss the formulation of mass flow equations. Yang et al. (1998) describe a small but more systematic study set up to map the number of phytomers and their state of development from formation at the apical meristem to senescence following root formation and death, for tillers in swards of perennial ryegrass {Lolium perenne) and tall fescue (Festuca arundinacea). MATERIALS AND METHODS Tiller appearance determination Two initial data sets were collected in Britain, both from within the Cae Ruel field, adjacent to the Institute for Grassland and Environmental Research

3 Matthew et al. Determination of tiller and root appearance Station near Aberystwyth, Wales. A brief description of the site is given in Matthew et al. (1995). Two separate experiments were in progress in that field, a study of characteristics of contrasting grass, clover, and urine patch microsites (Experiment 1), and a study of sward gap-filling behaviour (Experiment 2). Differences in patch type and in gap size were expected to affect tiller appearance. To test whether agronomically useful information on tiller appearance could be collected by observation of tiller axes in the laboratory, tillers from different patch types in Experiment 1 were examined. Ryegrass tillers from ryegrass dominant, clover dominant, or urine patch microsites were harvested in June 1993 after completion of leaf tissue turnover determinations reported in Matthew et al. (1995, table 1). Eight tillers were harvested from each of eight replicate microsites of each patch type, making a total of 192 tillers examined. When these data revealed highly significant differences in tiller appearance for different patch types, a second data set was collected. In the same field, gaps of differing sizes (20, 40, and 60 mm) were created in the sward, in late August 1993, by removing plugs of grass with corers of the appropriate diameters and filling the resulting holes with sieved soil. The experiment was primarily to determine gap-filling behaviour of white clover stolons (W. D. Kemball pers. comm.). In early September, two ryegrass tillers at the edge of each gap were tagged with plastic rings (10 replicates with four gaps of each size per replicate, giving 80 tillers in total for each gap size). Eighty tillers were also tagged at appropriate points in undisturbed swards as a control. Tiller appearance on tagged tillers was counted three weeks after tagging, and ten weeks later, in mid November, tagged tillers were harvested and values of T n determined in the laboratory. Root appearance An opportunity to test the extension of this methodology for assessing differences in root number per node arose in August 1995 when it was hypothesised that cultivar differences in "pulling" during grazing might be a consequence of differences in root production. Thirty tillers each of two unnamed experimental ryegrass cultivars (N.Z. Agriseeds Al and A3) were collected from No. 2 Dairy farm, Ruakura, and sent to Massey University for analysis. On this occasion both bud status and numbers of roots at each node were determined. Counting procedures To record nodal bud status or root count data, a binocular dissecting microscope fitted with a zoom lens was used. Typically, 15x magnification was found to be the most convenient. The reference point chosen for numbering of nodes in the three studies reported here was the oldest live leaf, the point of attachment of this leaf being classed as node zero, with counting of node number proceeding in a basipetal direction. Tiller bud status was determined for nodes 1 to 5 of tillers in Experiment 1, and for nodes 1 to 8 for Experiment 2 and the Ruakura data. It was found that the status of the bud at each node (dormant bud, aborted tiller, developing, or developed tiller) could easily be determined after practice, although the determination became less certain with older deteriorating buds at the base of the tiller axis. Some buds were found to be hidden behind remains of dead leaf bases, but interruption of the distichous arrangement of successive buds on opposite sides of the tiller axis was an indication of the presence of a hidden bud. Counting of roots attached at each node was a slower process and was more prone to error, partly because of the larger number of roots per node and partly because leaf scars were often no longer visible after the first four or five nodes. This made assignment of roots to node numbers difficult. To improve accuracy when counting roots, we found it helpful to slice tiller axes, prior to counting, into two-node lengths, using bud position as the dissection criterion. A nodal probability coefficient for tiller appearance at a given node number in a population (T n ) was defined as n = n t /T (1) where n t is the number of nodes at that node number observed to bear a daughter tiller and T is the total number of tiller axes examined. By analogy, nodal probabilities for the number of undeveloped buds (B n ) or the number of dead tillers at a node may also be recorded. Similarly, nodal frequency for root appearance (R n ) is defined as R n = n r /T (2) where n r is the total number of roots observed at that node number and T is as above. Data analysis For data from Experiment 1, T n and B n values were obtained from the sample of 8 tillers collected at each microsite, making 24 observations (8 replicates

4 New Zealand Journal of Agricultural Research, 1998, Vol Residual Fig. 1 Normal probability plot for T n data, node 3, Experiment 1 conducted at Cae Ruel field, near Aberystwyth. of the 3 patch types) for analysis. Since there appears to be little precedent in the literature for presentation of data of this type, three approaches to data analysis were explored and compared. First, data were subjected to analysis of variance, a separate analysis being performed for each node number and for each bud status category. Data were checked for normality using the %normplot command of the Minitab statistical package (version 10.5Xtra). Second, using the Genstat statistical program, a Generalised Linear Model (GLM) was fitted with a logit link function, assuming a binomial error distribution. This is equivalent to Logistic Regression and is theoretically more appropriate for discrete counts of "successes" out of a fixed number of observational units, as in the present data set. Hereafter in this paper, this method is referred to as "analysis of deviance". Finally, a multivariate canonical discriminant analysis (CDA, as used by Matthew et al. 1994) was performed for data from the first five recorded node numbers. In all three analyses, data for node 6 were omitted because of missing values. Also, some microsites initially characterised as grass and clover patches had obviously been urinated on by animals after initial characterisation but before the harvesting of tillers for bud status determination. Such patches were assigned to a fourth patch type for the purposes of statistical analysis, but not reported here. For data from Experiment 2, analysis of deviance was perfomed as above, but using the statistical program S-Plus (version 3.4), fort n data of nodes 1 to 8. CDA was performed for a subset of the data, nodes 2 to 6. Since there were four gap sizes in this experiment there were three treatment * 0.28 degrees of freedom and these were partitioned into linear response (1 d.f.) and lack of fit (2 d.f.) components. Lack of fit effects were not significant in any of the analyses, and are therefore not reported further. For the smaller data set from ryegrass tillers harvested at Ruakura, T n and R n data were subjected to chi-square analysis to test for differences between the two cultivars. Tiller appearance on tagged tillers, not being a count of successful development events from a known number of buds, was analysed by conventional ANOVA. RESULTS Comparison of statistical analyses In Experiment 1 at Cae Ruel, only two of ten variables tested (T n and B n values for the five node numbers) showed significant departure from normality, apparently because of the presence of numbers of zeros rather than because of skew or the presence of outliers. A typical normal probability plot is shown (Fig. 1). In general, analysis of variance and analysis of deviance returned similar significance levels (Table 1). Patch-type characteristics To avoid repetition, only selected results are presented, namely a summary of bud status data presented in graphical form (Fig. 2) and the Table 1 Comparison of significance levels from analysis of variance and analysis of residual deviance for data from Experiment 1 at Cae Ruel. Statistics relate to T n data for a given node number analysed as a univariate variable. Normality test values are the probabilities obtained from an Anderson-Darling normality test using the %normplot command of the Minitab statistical package. If residual mean deviance < 1, deviance ratios are based on a denominator of 1, if > 1 the denominator is the residual mean deviance thereby giving a conservative test. For Tables 1, 2, & 4: + = P<0.l;*=P<0.05; **=/><0.01; *** = P< Analysis of variance Normality test F-value Significance level Analysis of deviance Deviance ratio Significance level ** Node number *** *** * *** *

5 Matthew et al. Determination of tiller and root appearance Fig. 2 T n and B n values for Experiment 1, for microsites either grass dominant, clover dominant, or urine patch sites. Standard error bars show average SEM for B n and T n = 0.6 CO n o 0.4 a. Patch type Urine Clover Node Number ] Dormant Bud 1 Tiller [gj Other (Elongated/dead Bud) coefficients and scores for the CDA (Tables 2 and 3, respectively). T n for ryegrass tillers taken from grass and clover patch microsites in early summer was typically around 0.2 but was greatly elevated (P < 0.001), and approached or exceeded 0.7 for nodes 3 and 4 of tillers harvested at urine patch microsites (Fig. 2), indicating detection of large differences in past tiller appearance and good statistical resolution of those differences. At node 1, T n at urine patches and more especially at clover patches was reduced (P = 0.001, Fig. 2) compared with ryegrass patches. CDA also identified these two features in the data set. High T n values at nodes 3, 4, and 5 resulted in increased scores for discriminant function 1 (DF 1) for urine patch microsites (P < 0.001, Tables 2 and 3), while low T n values of urine patch and clover microsites (compared with grass microsites) at nodes 1 and 2 resulted in increased scores for DF 2, compared with grass patch microsites (P = 0.02, Tables 2 and 3) Gap filling behaviour For tillers adjacent to gaps, tiller appearance determined by counting newly emerged tillers on tagged tillers after 21 days, yielded tiller appearance ranging from 0.013, to (s.e. ± 0.004, P < 0.05) tillers tiller" 1 day^1, with tiller appearance in undisturbed swards comparable with that adjacent to 60 mm gaps (Table 4). After harvesting of the same tillers near the end of November, values of T n were determined for the uppermost 8 nodes (Fig. 3). The time scale is based on a measured leaf appearance interval of 12 days in mid September, with increase for October and November deemed proportionate to data in Davies (1977, table 1). A trend towards higher daughter tiller appearance on Table 2 Canonical structure for the first two canonical discriminant functions (DF) distinguishing patch types on the basis of nodal probability of tiller appearance. Coefficients less than 0.2 have been suppressed. P mv denotes proportion of mutivariate dispersion explained. DF 1 DF Node number * mv Significance *** **

6 New Zealand Journal of Agricultural Research, 1998, Vol. 41 Fig. 3 T n values for nodes 1 to 8 of tillers adjacent to gaps in Experiment 2 conducted at Cae Ruel field, near Aberystwyth. SEM for nodes 1 to 8, respectively, 0.07, 0.04, 0.05, 0.05, 0.06, 0.05, 0.05, Node number Aug Sep Oct Nov Gap size: I no gap N 20 mm 340 mm I 60 mm tillers adjacent to larger gaps was evident over a three month period from September to November (Fig. 3) but was statistically significant only for the more recently active nodes 3 and 4. When data for nodes 2 to 6 were analysed by CDA, this feature Table 3 Patch-type mean scores for canonical discriminant functions (DF) reported in Table 1. CDF1 CDF 2 Urine Clover Grass accounted for 70% of the multivariate dispersion, but was not statistically significant. Comparatively high tiller appearance for the "no gap" treatment (undisturbed swards) at node 5 coincides with tagging of tillers and may represent a disturbance effect (Matthew 1992). Where leaf appearance interval is known, T n data may be used to calculate tiller appearance rate for the period in question. For example, the 12 day leaf appearance interval determined by observation of tagged tillers in September, with the T n data for node 5 (Fig. 3), gives tiller appearance values similar to those Table 4 Comparison of tiller appearance values (tillers nr 2 day ') obtained from counting number of new daughter tillers appearing on tagged tillers with values obtained by counting the proportion of axillary bud sites occupied by tillers 10 weeks later. SEMs reported were obtained by ANOVA, as was P value for tagged tillers, these data not being expected to have binomial error distribution. P value for tiller axis observation was obtained by analysis of deviance. Method Tagged tillers (Sept 93) Tiller axis observation (Nov 93, node number 5) Gap size (mm) SEM P

7 Matthew et al. Determination of tiller and root appearance obtained above by conventional methods (Table 4). Correlations between number of daughter tillers counted on tagged tillers in September (Table 4) and nodal probability for successive nodes were 0.00, 0.28, 0.31 (P = 0.05), 0.27, 0.12, and for nodes 3 to 8, respectively. Ruakura data For ryegrass swards of N.Z. Agriseeds lines A1 and A3 at Ruakura No. 2 Dairy in August, T n for mature nodes ranged from 0.30 to 0.69 and R n values ranged from 1.60 to 2.27 (Table 5). Cultivar differences based on analysis of counts for node numbers two to five were significant (x 2, 3 d.f. = 8.4, P < 0.05), values for A1 being generally higher than those for A3 (Table 5). DISCUSSION Statistical analysis Analysis of deviance is theoretically more appropriate for data of this type than conventional analysis of variance, though statistical significance was similar for both methods (Table 1). Although statistical significance was a little greater for ANOVA than for analysis of deviance in Table 1, this trend was reversed for nodal probability data in Fig. 3 (ANOVA, P = and 0.054; Analysis of deviance, P = and 0.02 for nodes 3 and 4, respectively). It is arguably more correct to consider the tiller axis as one entity in a single analysis than to build up a profile from the combined results of separate univariate analyses for each node number, yet multivariate analysis of deviance techniques are in their infancy and their properties not yet well understood, hence the choice of CDA as a multivariate technique for these data sets. Further Table 5 Values of T n and R n for perennial ryegrass (N.Z. Agriseeds unnamed cultivars Al and A3) tillers collected at Ruakura number 2 dairy farm (mean of 30 tillers). T n Al A3 Rn Al A Node number development of analytical methodology for data of this type would therefore be helpful. In particular, there would be interest in appropriate multivariate analysis of deviance methodology and in examining whether the presence of a tiller at a particular node is correlated with the number of roots at the same position or with the presence of a tiller or number of roots at neighbouring positions. Tiller and root appearance Examination of shoot axes to determine historical population dynamics is not a new technique. For example, Callaghan (1976) was able to obtain data for tiller appearance and death over 12 generations for a species of Carex, and Newton & Hay (1993) have used a similar approach in studying population dynamics of white clover. Some recent studies of reed canary grass (Phalaris arundinacea) in Japan have documented events at the phytomer level (Ito etal. 1993). The aim of Experiment 1 at Cae Ruel field, Aberystwyth, was to ascertain whether or not the approach of mapping events at particular nodes could provide meaningful agronomic comparisons of swards with different histories. There would be no point in developing a body of theory that lacked practical application. In each of the data sets presented here, agronomically useful information has been obtained. For Experiment 1 at Cae Ruel, determination of T n for the five most recent nodes on the tiller axis allowed characterisation of historical differences in tiller appearance for grass, clover, and urine patch microsites within a sward. Experiment 2 at the same site provided statistically significant evidence of increased daughter tiller formation adjacent to gaps artificially created in that sward, and there also appears to be evidence for an increase in tillering of undisturbed swards following tiller tagging. The Ruakura data show that R n values vary less between nodes than those for T n and confirm that R n is typically around 2 (Matthew 1992). There was also evidence of cultivar differences in R n. Comparison of methods for tiller appearance measurement Monitoring of tagged tillers requires considerable time input in the field, often in adverse weather conditions. By contrast, recording of bud status involved a single destructive harvest of tillers (which need not necessarily be previously tagged), minimising time spent in the field. From

8 New Zealand Journal of Agricultural Research, 1998, Vol. 41 comparison of tiller appearance data obtained by counting new daughter tillers on tagged tillers (Table 4) and the nodal probability data (Fig. 3), it appears that the nodal probability method is more likely to give statistically significant results when applied to more recently active nodes higher on the tiller axis. On tillers harvested some three months after creation of gaps, the nodal probability method was able to detect differences in tiller appearance associated with different gap sizes and occurring some two months after gaps were created, with a high level of statistical significance (nodes 3 and 4, Fig. 3). Tiller appearance observed by conventional methods, for tillers tagged in September, showed highly significant treatment differences but with anomalously high values for undisturbed swards (Table 4). Nodal probabilities determined in November for node 5 on the same tillers, were significantly correlated with the earlier data and also showed the same ranking of treatments with high tiller appearance in undisturbed swards, but treatment differences were not significant for these older node numbers (Table 4, Fig. 3). Two reasons for the lower statistical significance of T n data at older nodes are evident. First, the pattern of correlation between T n and conventional tiller appearance data for successive nodes (see above) and the decrease in treatment variation relative to the overall mean for T n data compared with conventional data (Table 4) both suggest that definition of a transient treatment effect is partially lost at older nodes because of differences between tillers in leaf appearance interval. Second, the T n data for node 5 (Table 3) would have yielded statistically significant gap size effects (P < 0.05) if two unusual observations with a large residual had been omitted from the analysis. The fact that tiller appearance on tagged tillers in previously undisturbed swards was similar to that of tillers adjacent to 60 mm gaps is suggestive of a disturbance effect as reported by Matthew (1992), and other workers. From Fig. 3, it would seem that the suggested disturbance effect was transitory, and that T n data for nodes 3 and 4 (Fig. 3) reflect the tiller appearance in undisturbed swards in the absence of such effects. In summary, the nodal probability method of determining tiller appearance can give a result comparable with that obtained by conventional methods (Table 4), and can give data free from spikes of tiller appearance induced by the physical disturbance of tagging (nodes 3 and 4, Fig. 3 compared with monitoring tagged tillers, Table 4 and node 5, Fig. 3), but is less likely to give statistical significance when applied to older, less recently active node numbers. Recording of bud status for harvested tillers typically required about five minutes per tiller once a person had learned the technique. Therefore, recording of nodal probability data, while requiring different skills, is probably marginally faster than tagging a tiller and making one or two subsequent field observations on that tiller, although the need to collect information on leaf appearance interval for interpretation of nodal probability or frequency data should also be considered. Relationship between F s and T n or R n It is not desirable to compare the site filling and nodal probability approaches directly. The former, particularly in its recent application (see, e.g., Van Loo 1992), is more applicable to describing growth of individual plants, particularly during the establishment phase when increase in leaf and tiller number per plant is more or less exponential, where there are no deaths occurring within the population and where tiller appearance at successive nodes follows a regular sequential pattern. Site filling methodology is likely to remain an important analytical tool for experiments on seedling plants where such conditions apply. By contrast, the nodal probability approach is most applicable where birth rates are low and balanced by death rates, allowing assumption of steady state population dynamics and constant mass flows. However, in order to reconcile the two approaches, it can be shown from Van Loo (1992, equations A1 to A5) that T n =(e F --l/e F - (I - d) (3) where d is the delay (expressed in leaf appearance intervals, designated n by Neuteboom & Lantinga 1989) between leaf and tiller production at a particular phytomer. Equivalently, if applied on a single plant basis, d is the number of node numbers on the tiller axis not bearing a daughter tiller. Apparent site filling coefficients for a range of values of T n are given in Table 6. Equation 3 and Table 6 show that previously published values for F r i of approximately 1 (Hunt & Thomas 1985) are consistent with values for R n (Table 5) of 2 to 2.5, providing d for root formation was close to 1. Potential modelling applications An immediate attraction of the nodal mapping approach to describing sward dynamics is the ease

9 Matthew et al. Determination of tiller and root appearance with which mass flow equations for leaf, tiller, or root production may be developed. In the simplest case where the tiller axis is regarded as producing a steady state flow of segments, then dp/dt = p. dl/dt (4) where dp/dt is the flow rate of phytomers per unit area per unit time, p is the tiller population density per unit area, and dl/dt is the rate of leaf appearance per unit time. Multiplying the flow rate of segments by the mass of leaf, tiller, or root produced per segment we get M o = dp/dt. X n.m 0 (5) where M o is the tissue mass flow for the organ (leaf, tiller, or root) of interest, X n is the nodal probability or frequency, and m 0 is the final coverage mass of the individual organ in question. For example, to estimate individual root weight, in a ryegrass sward of tiller density 5000 m~ 2, with a leaf appearance interval often days, a P r of 2.0, and a root production of 5 g DM m~ 2 day" 1, there must be a final root mass per individual root of g. With further data on mean mass per unit length, calculation of length per root would also be possible. Alternatively, if the mean weight of an individual root were known, the root production (g DM irr 2 day" 1 ) could be calculated. More complex equations are also possible, for example m / P r. m r. It-d/It = shootroot partitioning ratio (6) where nodal probability of leaf formation is assumed to be unity and disappears from the equation, mi and m r are mass of leaf and root at nodes t and t-d, respectively, I denotes leaf appearance interval, and the ratio I n _d/it is as explained below. Table 6 Apparent site filling ratios (F s or F r i) which would be observed for selected values of nodal probability of tiller or root appearance (T n or R n ) and delay (d) between leaf and tiller appearance at a particular node number. T n /R n Number of nodes delay (d) Morphogenetic variation in seasonal allocation to roots These equations also predict change in seasonal shoot:root allocation purely from morphogenetic considerations. This arises because of seasonal difference in leaf appearance interval (Davies 1977) and the delay between leaf and root formation at a given phytomer. Thus, a correction term is needed to allow for the fact that a smaller or larger number of leaves would be active during the formation of roots at a particular phytomer. Such a correction term would have the form I^d/It. where I designates leaf appearance interval at times t and t-d, d being the number of nodes delay between leaf and root formation or, by analogy, tiller formation. This correction term implies a decline in root (or tiller) production per node in autumn and an increase in spring. The magnitude of It-d/It for ryegrass may be estimated from the tiller axis map of Yang et al. (1998). Let it be assumed that a leaf in its lifetime exports approximately the same amount of photosynthate to the tiller axis regardless of season, that seasonal change in leaf appearance interval follows a sine curve with a minimum of 8 days and maximum of 24 days (cf. Davies 1977), and that an elongating root at phytomer 13 receives substrate mainly from the second live leaf, three nodes higher at phytomer 10 (Yang et al. 1998, fig. 3). On this basis, It-d/It has a value of about 1.45 in spring and 0.7 in autumn, and photosynthate allocation per phytomer for root formation is expected to vary more than twofold on a seasonal basis as a result of the delay between leaf and root formation at a given phytomer. Further study of these mass flow relationships and verification for field conditions would be the next step in development of this work. ACKNOWLEDGMENTS This work was initiated while the senior author was on sabbatical leave at IGER, Aberystwyth, UK, and partially supported by an IGER fellowship. We thank H. J. Bos of Wageningen Agricultural University for assistance with Equation 3, W. Kemball for allowing us access to his Cae Ruel experiment, and N.Z. Agriseeds Ltd. and E. R. Thom of the Dairying Research Corporation for provision of Al and A3 ryegrass tillers, and G. C. Arnold, Department of Statistics, Massey University, for assistance with analyses and helpful comments. We thank Shan Xi Agricultural University, People's Republic of China, for support to the second author in undertaking study leave.

10 10 New Zealand Journal of Agricultural Research, 1998, Vol. 41 REFERENCES Bircham, J. S.; Hodgson, J. 1983: The influence of sward condition on rates of herbage growth and senescence in mixed swards under continuous stocking management. Grass and forage science 38: Caradus, J. R.; Evans, P. S. 1977: Seasonal root formation of white clover, ryegrass, and cocksfoot in New Zealand. New Zealand journal of agricultural research 20: Callaghan, T. V. 1976: Growth and population dynamics of Carex bigelowii in an alpine environment. Oikos 27: Davies, A. 1974: Leaf tissue remaining after cutting and regrowth in perennial ryegrass. Journal of agricultural science, Cambridge 82: Davies, A. 1977: Structure of the grass sward. In: Gilsenan B. ed. Proceedings of an international meeting on animal production from temperate grassland. An Foras Taluntais, Dublin. Pp Demotes-Mainard, S.; Pellerin, S. 1992: Effect of mutual shading on the mergence of nodal roots and the root/ shoot ratio of maize. Plant and soil 147: Etter, A. G. 1951: How Kentucky bluegrass grows. Annals of the Missouri Botanical Garden 38: Garwood, E. A. 1967: Seasonal variation in appearance and growth of grass roots. Journal of the, British Grassland Society 22: Garwood, E. A. 1969: Seasonal tiller populations of grass and grass/clover swards with and without irrigation. Journal of the British Grassland Society 24: Hunt, W. F.; Thomas V. J. 1985: Growth and developmental responses of perennial ryegrass sown at constant temperature. II. Influence of light and temperature on leaf, tiller, and root appearance. Australian journal of plant physiology 12: Ito, M.; Kuwakino, H.; Yamashina, K. 1993: Relationships between the development of dormant tiller buds and the successive leaf and internode formation of mother shoots in reed canary grass (Phalaris arundinacea L.) grown under a sward condition. Journal of the Japanese Society of Grassland Science 38: Jewiss, O. R. 1981: Shoot development and number. In: Hodgson, J.; Baker R. D.; Davies, A.; Laidlaw A. S.; Leaver, J. D. ed. Sward measurement handbook. Hurley, British Grassland Society. Klepper, E.; Belford, R. K.; Rickman, R. W. 1984: Root and shoot development in winter wheat. Agronomy journal 76: Korte, C. J. 1986: Tillering in 'Grasslands Nui' perennial ryegrass swards. 2. Seasonal pattern of tillering and age of flowering tillers with two mowing frequencies. New Zealand journal of agricultural research 29: Langer, R. H. M. 1963: Tillering in herbage grasses. Herbage abstracts 33: Langer, R. H. M.; Ryle, S. M.; Jewiss, O. R. 1964: The changing plant and tiller populations of timothy and meadow fescue swards. 1. Plant survival and the pattern of tillering. Journal of applied ecology 1: Matthew, C. 1992: A study of seasonal root and tiller dynamics in swards of perennial ryegrass (Lolium perenne L.). Unpublished PhD thesis, Massey University, Palmerston North, New Zealand. 247 p. Matthew, C; Cresswell, A.; Haggar, R. J. 1995: Characteristics of microsites with and without clover in two field swards. Grass and forage science 50: Matthew, C; Lawoko, C. R. O.; Korte, C. J.; Smith, D. 1994: Application of canonical discriminant analysis, principal component analysis, and canonical correlation analysis as tools for evaluating differences in pasture botanical composition. New Zealand journal of agricultural research 37: Matthew, C; Xia, J. X.; Chu, A. C. P.; Mackay, A. D.; Hodgson, J; 1991: Relationship between root production and tiller appearance rates in perennial ryegrass (Lolium perenne L.). In: Atkinson, D. ed. Plant root growth, an ecological perspective. British Ecological Society special publication no. 10. Oxford, Blackwell Scientific Publications. Pp Neuteboom, J. H.; Lantinga, E. A. 1989: Tillering potential and relationship between leaf and tiller production in perennial ryegrass. Annals of botany 63: Newton, P. C. D.; Hay, M. J. M. 1993: Inherent and realised potential of axillary buds of white clover in field populations. Proceedings XVII International Grasslands Congress. Pp Parsons, A. J.; Leafe, E. L.; Collett, B.; Penning, P. D.; Lewis, J. 1983: The physiology of grass production under grazing. II. Photosynthesis, crop growth and animal intake of continuously-grazed swards. Journal of applied ecology 20: Sharman, B. C. 1947: The biology and developmental morphology of the shoot apex in the Gramineae. New phytologist 46: Silsbury, J. H. 1970: Leaf growth in pasture grasses. Tropical grasslands 4: Skinner, R. H.; Nelson, C. J. 1992: Estimation of potential tiller production and site usage during tall fescue canopy development. Annals ofbotany 70: Van Loo, E. N. 1992: Tillering, leaf expansion and growth of plants of two cultivars of perennial ryegrass grown using hydroponics at two water potentials. Annals of botany 70: Yang, J. Z.; Matthew, C; Rowland, R. E Tiller axis observations for perennial ryegrass (Lolium perenne) and tall fescue (Festuca arundinacea): number of active phytomers, probability of tiller appearance, and frequency of root appearance per phytomer for three cutting heights. New Zealand journal of agricultural research 41: