Use of germination curves to describe variation in germination characteristics in three turfgrass species.
When comparing germination characteristics of cultivars and seed lots, it is desirable to assess both the FGP and the distribution of the timing of germination. In standard germination tests, germination is usually evaluated by counting the number of normal plants at an interim first count and at a final count at the end of the test (International Seed Testing Association, 1999), but in practical testing of grass seeds, the germination percentage usually differs very little between first and final count (e.g., Nienhuis and Baltjes, 1985). The first count, hence, provides very limited information about differences in the timing and uniformity of germination, and more frequent counting of germination during the germination test is more likely to detect such differences.
Fitted curves can efficiently summarize information from germination time courses, provided they fit the observed data sufficiently closely (Brown and Mayer, 1988b). As an alternative to nonlinear regression, Hunter et al. (1984) analyzed germination frequency using a multinomial distribution, assuming that the seeds germinated independently of each other and that the numbers of seed that germinated in each time interval followed a multinomial distribution. The model, however, required transformation of the time scale before fitting a normal distribution to the germination times, and the applied transformation was of major importance for the fit of the model. Since time to germination is not always normally distributed, often causing positively skewed germination time courses (Nichols and Heydecker, 1968; Campbell and Sorensen, 1979; Cheng and Gordon, 2000), it is relevant to take this into account when analyzing germination data. A generalized hyperbolic distribution is very flexible and has been successfully used to describe asymmetric and heavy-tailed behavior of, for example, turbulence and financial data (Bibby and Sorensen, 2003). A function based on this distribution can also describe the variation in germination times even when the distribution is very skewed, and by fitting this function to germination data, it is possible to summarize germination time courses into characteristics of biological relevance. Such parameters may comprise FGP as an expression of the proportion of seeds with germination capacity, MGT as an inverse expression of overall germination speed or rate for the whole population, and [T.sub.25-75] as an expression of spread of germination times or germination uniformity (Bewley and Black, 1994).
The aim of this study was (i) to estimate biologically relevant germination characteristics from germination time courses with a curve fitting procedure based on an asymmetric multinomial distribution, and (ii) to study the variation in germination percentage, speed, and uniformity within and among cultivars of red rescue, perennial ryegrass, and Kentucky bluegrass.
MATERIALS AND METHODS
Twenty seed lots of red rescue, 19 of perennial ryegrass, and 16 of Kentucky bluegrass were chosen for the study. The species were represented by four, four, and five cultivars, respectively, and each cultivar was represented by three to eight seed lots, except for the Kentucky bluegrass 'Pepaya', which was represented by only one seed lot. Cultivars and seed lots were randomly chosen and supplied by the seed company DLF-Trifolium, Denmark. All cultivars are amenity grass types and all cultivars have been or are being tested in cultivar trials in either Denmark or the UK for commercial registration. All seed lots were harvested in Denmark in 2000. From each of the 55 seed lots, a sample of [approximately equal to] 250 g was extracted from which subsamples of 100 seeds were selected for germination.
Germination tests were performed in December 2000 to January 2001. Each seed lot was germinated according to standard rules for germination testing (International Seed Testing Association, 1999), with four replicates of 100 seeds. Seeds were germinated on top of filter paper (AGF725, Frisenette, Denmark) in small plastic germination boxes with a small water reservoir in each box (Jacobsen apparatus; International Seed Testing Association, 1999). At the beginning of the test, the filter paper was saturated in a 0.2% solution of KN[O.sub.3], whereas pure water was used in the reservoir (International Seed Testing Association, 1999). The germination boxes were placed in two germination chambers (KBP 6395 LL, Termaks, Norway) with cool white fluorescent light (14 000 to 30 000 lux, 400-700 nm) for 8 h per day at 25 [+ or -] 1[degrees]C and darkness for 16 h per day at 15 [+ or -] 1[degrees]C. For each species, two replicates were randomized on one shelf and placed in one of the germination chambers, and two replicates were randomized on one shelf in the other chamber. To diminish the potential effect of small temperature gradients within and between the germination chambers, the germination boxes were rearranged daily.
Seeds were considered germinated when the radicle had protruded [approximately equal to] 2 mm. Germination was recorded once or twice daily for 21,20, and 23 d for red fescue, perennial ryegrass, and Kentucky bluegrass, respectively. Germination was recorded again at the end of the test at Day 31, 30, and 33, respectively. Germination was recorded for seed lots in the same order as the germination tests were started, and the exact time for inspection of the seeds was noted for each individual seed lot. To ensure that none of the seed lots possessed dormant seeds, seeds that had not germinated at the end of the germination test were exposed to a cold treatment by transferring to darkness and 5[degrees]C constantly for 14 d and then retransferring to 15/25[degrees]C and darkness/light for 16/8 h [d.sup.-1] for 7 d. Only five seeds out of the total of 22 000 seeds germinated after the cold treatment, indicating that none of the seed lots contained dormant seeds.
Fitting of Germination Time Courses
To derive biologically relevant information from the germination experiment, germination curves were fitted to the germination data. For each replicate from each of the 55 seed lots, an individual germination curve was fitted using an asymmetric multinomial distribution of the number of germinating seeds, giving a total of 220 fitted germination curves. The multinomial probabilities were calculated as the integral across the appropriate time interval of the density of the germination times as described by Hunter et al. (1984). The distribution of germination times within replicates was assumed to be generally hyperbolic (see Bibby and Sorensen, 2003). The fitting of germination curves and the estimation of parameters were accomplished using the software package R (The R Foundation for Statistical Computing, 2003). On the basis of the fitted function for each replicate within each seed lot, FGP, MGT, and [T.sub.25-75] were estimated. The FGP and MGT were estimated using maximum likelihood with FGP on a logit scale. Estimated standard errors for these estimates were obtained from the observed information matrix. An estimate for the parameter [T.sub.25-75] was subsequently calculated as
 [T.sub.25-75] = [F.sup.-1](0.75) - [F.sup.-1] (0.25)
where F is the estimated distribution function of the underlying generalized hyperbolic distribution of the germination time. No standard error was obtained for this estimate.
For illustration of the variation in germination characteristics between cultivars, a mean germination curve was fitted to each cultivar with the average result of all seed lots and replicates within the cultivar. Similarly, the variation between seed lots within a cultivar was illustrated by fitting a germination curve to the average germination result of individual seed lots within a cultivar. This was done for one cultivar within each species, choosing the cultivar with largest variation in MGT among seed lots.
Analysis of Variance
To examine the variation in germination characteristics among seed lots, cultivars, and species, the estimates from the curve-fitting procedure were used in an ANOVA. The estimated values of FGP (percentage), MGT (days), and [T.sub.25-75] (days), respectively, from each of the fitted curves were analyzed as a nested factorial design with replicates nested within seed lots, seed lots nested within cultivars, and cultivars nested within species. Cultivar differences were analyzed in a model with seed lot variation as a random effect. Species differences were analyzed in a model with cultivar variation as a random effect. The ANOVA was performed with the mixed procedure of the SAS package (SAS Institute, Cary, NC).
To obtain homogeneous variance in the ANOVA, the response variables were transformed and, where possible, the values of the response variables were weighted according to the reliability of the curve fitting. Estimates of FGP were analyzed with logit-transformed values of the FGP, and the estimates were weighted by the reciprocal standard error of the estimate from the curve fitting. Estimates of MGT were analyzed with square root transformed values of mean time to germination, and the estimates were weighted by the reciprocal standard error. Estimates of [T.sub.25-75] were analyzed with square root transformed values of time from 25 to 75% of final germination, and the estimates were not weighted in the ANOVA.
Pair-wise comparisons of cultivars and species, respectively, were performed by t tests. To diminish the risk of erroneous conclusions in the multiple pair-wise comparisons, comparisons of cultivars were only done within species and not among species. The predicted values and 95% confidence limits were presented on the natural, untransformed scale. To compare the relative contribution of the variance components corresponding to cultivar, seed lot, and replicate for each of the three species, separate analyses of variance were performed for each species. Each variance component is presented on the transformed scale; that is, FGP is presented on a logit scale whereas variances of MGT and [T.sub.25-75] are presented on a square-root time scale.
In the analysis of MGT, one observation was excluded as an outlier, whereas two observations were excluded as outliers in the analysis of [T.sub.25-75]. During the germination experiment it was noticed that the replicates in question were severely contaminated by fungi, which is likely to have delayed the germination considerably. All other analyses were performed both with and without the contaminated replicates, but the results were unchanged.
Fitting of Germination Time Courses
The applied curve-fitting procedure proved successful in describing the germination time courses. As an illustration of the ability of the curves to describe the observed germination, the estimated germination curves with the lowest and the highest standard error, respectively, of the estimate of MGT within each species are shown as the best and the poorest fit, respectively (Fig. 1). The standard error of MGT ranged from 0.01 to 2.0 d in red rescue, from 0.06 to 2.5 d in perennial ryegrass, and from 0.1 to 21.5 d in Kentucky bluegrass. Visual inspection of curves in Fig. 1 demonstrates that overall, the fitted curves described the observed germination very well, and in all three species even the poorest fit provided a satisfying description of the germination time course.
[FIGURE 1 OMITTED]
Variation in FGP, MGT, and [T.sub.25-75]
The three species differed significantly in FGP (p < 0.0001) when tested against cultivar differences. Estimated mean values of FGP were 91.5% for red rescue, 96.0% for perennial ryegrass, and 87.4% for Kentucky bluegrass with FGP being significantly higher for perennial ryegrass than for red rescue (p = 0.0094) and Kentucky bluegrass (p < 0.0001) but with no significant differences between red rescue and Kentucky bluegrass (p = 0.1468).
There were significant differences in FGP among cultivars within red fescue (p = 0.0271), perennial ryegrass (p = 0.0009), and Kentucky bluegrass (p = 0.0004) when tested against seed lot variation within cultivars, indicating that there were true cultivar differences in FGP (Table 1). In red fescue, 'Napoli' had significantly lower FGP than 'Cinderella', and in Kentucky bluegrass, 'Mardona' had lower FGP than all other cultivars (Table 1). In perennial ryegrass, 'Taya' had higher FGP than 'Merci' and 'Allegro', and 'Figaro' also had higher FGP than Merci. The FGP differed least among cultivars of perennial ryegrass (from 93.3 to 97.7%) and most among cultivars of Kentucky bluegrass (from 76.3 to 91.6%).
There were significant differences in MGT between the three species (p < 0.0001) when tested against cultivar differences. Estimated mean values of MGT were 4.7 d for red fescue, 3.6 d for perennial ryegrass, and 7.5 d for Kentucky bluegrass, with all species differing from all other species (p < 0.0001).
There were significant differences in MGT among cultivars within red fescue (p < 0.0001) and Kentucky bluegrass (p = 0.0089), but not within perennial ryegrass (p = 0.41), indicating that there were true cultivar differences in MGT in two of the species when tested against seed lot variation within cultivars (Table 2). In red rescue, Napoli and 'Smirna' germinated more slowly (higher MGT) than Cinderella and 'Symphony', and in Kentucky bluegrass, 'Conni' germinated more slowly than 'Broadway', 'Andante', and Mardona (Table 2). The MGT differed most among cultivars of Kentucky bluegrass (from 7.2 to 8.1 d) and least among cultivars of perennial ryegrass (from 3.5 to 3.7 d) and with an intermediate range between red rescue cultivars (from 4.4 to 5.2 d) (Table 2).
The [T.sub.25-75] differed significantly among the three species (p < 0.0001) when tested against cultivar differences. Estimated mean values of T25-75 were 1.1 d for red fescue, 0.8 d for perennial ryegrass, and 2.0 d for Kentucky bluegrass, with [T.sub.25-75] being significantly higher for Kentucky bluegrass than for red fescue and perennial ryegrass (p < 0.0001). No significant difference was found between red fescue and perennial ryegrass (p = 0.0556).
The [T.sub.25-75] also varied significantly among cultivars within red fescue (p < 0.0001) and Kentucky bluegrass (p < 0.0001), but not within perennial ryegrass (p = 0.64), indicating that there were true cultivar differences in [T.sub.25-75] in two of the species when tested against seed lot variation within cultivars (Table 3). The [T.sub.25-75] differed most among cultivars of Kentucky bluegrass (from 1.5 to 2.3 d), whereas it ranged least among cultivars of perennial ryegrass (from 0.8 to 0.9 d), and with an intermediate range among red rescue cultivars (from 0.8 to 1.4 d) (Table 3).
Relative Contribution of the Variance Components
The relative contributions of the variance components to the variance of FGP, MGT, and [T.sub.25-75], respectively, are shown in Table 4 for the overall analysis of all three species and for the individual analysis of each species. Overall for all three species, cultivars accounted for 24% of the total variance in FGP, 15% in MGT, and 37% in [T.sub.25-75], whereas seed lots accounted for 14, 10, and 19%, respectively, indicating that the cultivar factor generally contributes more than the seed lot factor to the variation in germination characteristics. For all three germination characteristics, however, replicate variance was larger than cultivar variance and seed lot variance, and the replicate factor accounted for 44 to 76% of the total variance. A similar contribution of the variance components was seen within individual species, with cultivar variance being larger than seed lot variance in most but not all cases. Again, the replicate variance was considerably larger than the cultivar variance, except for [T.sub.25-75]. In three cases, an estimated variance component was very small and negative, and in Table 4 these estimates are set to 0.
Mean Germination Curves
The predicted mean germination curve for each cultivar (Fig. 2) illustrates how cultivars differ in their germination characteristics. The variation is smallest among cultivars of perennial ryegrass and there are almost no cultivar differences in the timing of germination in this species. In red fescue and Kentucky bluegrass, differences in the timing of germination are larger, and particularly in Kentucky bluegrass there are considerable differences in FGP.
[FIGURE 2 OMITTED]
There is also considerable variation in germination characteristics among seed lots within cultivars. For each species, Fig. 2 shows the fitted germination curves for each individual seed lot within the cultivar with the largest seed lot variation. In red fescue cv. Smirna, MGT ranged from 4.6 to 5.6 d, in perennial ryegrass cv. Taya from 3.2 to 4.3 d, and in Kentucky bluegrass from 7.4 to 13.6 d.
Species differed significantly in germination characteristics when tested against cultivar differences, although FGP did not differ between red fescue and Kentucky bluegrass, and [T.sub.25-75] did not differ between red fescue and perennial ryegrass. These results and the components of variance illustrate that the species differ more in germination characteristics than cultivars within the species. Germination was slowest, least uniform, and reached the lowest final percentage in Kentucky bluegrass. Germination of perennial ryegrass was slightly superior to red rescue in all three germination characteristics (Tables 1,2,3; Fig. 2). This is an acceptable explanation of why seedlings of Kentucky bluegrass may emerge >10 d later than seedlings of perennial ryegrass when sown in field conditions (Skirde, 1967; Pommer, 1972; Bo, 1989). In the field, seeds are often exposed to unfavorable environmental conditions which may restrain and delay germination (Happ et al., 1993).
Cultivar Differences and Seed Lot Differences
When cultivars are only represented by one seed lot each, variation between cultivars cannot be distinguished from variation between seed lots within cultivars, and if a cultivar happens to be represented by a particularly poor germinating seed lot, false conclusions may be drawn about the general germination characteristics of that cultivar. Gooding et al. (1989) found that the time to 50% germination varied from 6 to 20 d between 30 cultivars of Kentucky bluegrass, and Newell and Bludau (1993) found a corresponding variation from 5 to 14 d between 44 cultivars. These variations are much larger than those found among cultivars in this study (Table 2). In both of the mentioned studies, each cultivar was represented by only one seed lot each, and the slowest germinating cultivars generally had very low FGPs. Since low germination percentage is often accompanied by slow germination (Roberts, 1986; Culleton et al., 1991), the slow germination of certain cultivars is most likely to be partly explained by low seed vigor and quality of the applied seed lot. Considering the MGT ranging from 7.4 to 13.6 d among seed lots of Kentucky bluegrass cv. Conni, at least some of the variation in germination time found by Gooding et al. (1989) and Newell and Bludau (1993) was presumably due to seed lot differences in seed quality which were not representative for the cultivars. Use of more than one seed lot per cultivar and testing cultivar differences against seed lot differences would give more realistic information about true cultivar differences and possibly prevent false conclusions.
Naylor (1982) compared eight perennial ryegrass cultivars, each represented by one to five seed lots, and found significant cultivar differences in MGT but not in FGP. The present study, on the other hand, did not detect cultivar differences in MGT in perennial ryegrass but demonstrated such differences in red rescue and Kentucky bluegrass. The study confirms that true cultivar differences in germination characteristics do occur and can be detected by detailed registration of germination time courses and by summarizing with a flexible function.
The MGT is generally inversely related to FGP (Roberts, 1986) and in this study, the perennial ryegrass and red rescue cultivars with the lowest FGP (Table 1) also had the highest MGT (Table 2). Conversely, in Kentucky bluegrass, Mardona had the lowest FGP and also the lowest MGT whereas Conni had one of the highest FGP values and the highest MGT. Despite the low FGP of Mardona, which would normally indicate low quality, this cultivar germinated relatively fast, supporting the hypothesis that MGT is, in fact, partially under genetic control.
Variance Components and Implications for Experimental Design
Although the relative variation depended on the species and germination characteristic considered, cultivars accounted for 15 to 37% of the total variation, whereas seed lots accounted for 10 to 19% (Table 4). In comparison, Naylor (1981) found that differences between seed lots accounted for 16% of the overall variation in field emergence of Italian ryegrass (Lolium multiflorum Lain.), whereas cultivar differences accounted for 42% of the variation. Together, these estimates suggest that cultivars generally differ more than seed lots in germination characteristics. This conclusion can, to some extent, justify that cultivars are often represented by only one seed lot when searching for cultivar differences (e.g., Ellis et al., 1987); that is, true cultivar differences are likely to be detected because seed lot variation is generally smaller. On the other hand, given the occasionally very large seed lot differences within a cultivar (Fig. 2.), there is a risk of failing to detect cultivar differences if only one seed lot is used, especially if this seed lot happens to be of a particularly poor quality. Therefore, studies of cultivar differences should be based on more than one seed lot per cultivar or, alternatively, preliminary experiments of scud lot variation should ensure that a representative seed lot of high quality is chosen from each cultivar.
Considering the variance contribution of the replicate factor, it is striking that this factor accounts for much more of the variation than the scud lot factor and the cultivar factor (Table 4). This may reflect the heterogeneity of seeds within a population, which can affect the variation between replicates. Additionally, the large variation of the replicate factor, especially for MGT, may reflect the difficulties of achieving homogeneous experimental conditions in germination studies. It is difficult to control temperature precisely without small gradients within a germination chamber (Ellis and Roberts, 1980), and even small temperature differences may affect MGT. Regular rearrangement of germination boxes can diminish the effect of temperature gradients, but despite application of four replicates of 100 seeds and daily rearrangement within and between germination chambers, there was still considerable replicate variation. Thus, in addition to considering the number of seed lots per cultivar, the number of replicates per seed lot should also be considered when conducting germination experiments. The choice of experimental design may, however, often depend on the scope of the experiment, and decisions may be aided by previous knowledge about the relative variance contribution of the factors considered.
Abbreviations: FGP, final germination percentage; MGT, mean germination time; [T.sub.25-75], time from 25 to 75% germination.
Table 1. Parameter estimates for the final germination percentage (FGP) and 95% confidence limits for species and for cultivars within species. Cultivars within species are sorted by ascending FGP. 95% conf. FGP limits % Red fescue 91.5a ([dagger]) [87.5, 94.3] Perennial ryegrass 96.0b [94.0, 97.4] Kentucky bluegrass 87.4a [82.3, 91.2] 95% conf. Cultivar FGP limits % Red fescue Napoli 88.6a [85.8, 91.0] Symphony 90.6ab [87.0, 93.3] Smirna 92.3ab [89.2, 94.5] Cinderella 93.9b [91.3, 95.7] Perennial Merci 93.3a [90.9, 95.2] ryegrass Allegro 95.6ab [93.9, 96.8] Figaro 96.4bc [94.8, 97.6] Taya 97.7c [96.7, 98.5] Kentucky Mardona 76.3a [68.3, 82.9] bluegrass Pepaya 83.3ab [71.1, 91.1] Andante 89.6b [85.7, 92.6] Conni 90.5b [86.8, 93.3] Broadway 91.6b [88.2, 94.0] ([dagger]) Species and cultivars within a species, respectively, followed by the same letter are not significantly different at the 0.05 probability level. The letter grouping is based on pairwise comparisons using t tests. Table 2. Parameter estimates for the mean germination time (MGT) and 95% confidence limits for species and for cultivars within species. Cultivars within species are sorted by descending MGT. 95% conf. Species MGT limits d Red fescue 4.76b ([dagger]) [4.46, 5.03] Perennial ryegrass 3.59c [3.35, 3.84] Kentucky bluegrass 7.54a [7.19, 7.90] 95% conf. Species Cultivar MGT limits d Red fescue Napoli 5.17a [4.99, 5.35] Smirna 4.93a [4.69, 5.19] Cinderella 4.47b [4.24, 4.71] Symphony 4.37b [4.14, 4.60] Perennial ryegrass Merci 3.70a [3.51, 3.90] Allegro 3.63a [3.44, 3.83] Taya 3.51a [3.33, 3.70] Figaro 3.51a [3.31, 3.71] Kentucky bluegrass Conni 8.12a [7.74, 8.51] Pepaya 7.83ab [7.12, 8.56] Broadway 7.36b [6.99, 7.74] Andante 7.30b [6.95, 7.65] Mardona 7.18b [6.76, 7.62] ([dagger]) Species and cultivars within a species, respectively, followed by the same letter are not significantly different at the 0.05 probability level. The letter grouping is based on pairwise comparisons using t tests. Table 3. Parameter estimates for the time from 25 to 75% of final germination percentage ([T.sub.25-75]) and 95% confidence limits for species and for cultivars within species. Cultivars within species are sorted by descending [T.sub.25-75]. 95% conf. Species [T.sub.25-75] limits d Red fescue 1.08b ([dagger]) [0.90, 1.29] Perennial ryegrass 0.82b [0.67, 1.01] Kentucky bluegrass 1.96a [1.72, 2.22] 95% conf. Species Cultivar [T.sub.25-75] limits d Red fescue Napoli 1.35a [1.22, 1.49] Smirna 1.16ab [0.99, 1.34] Cinderella 0.98bc [0.83, 1.15] Symphony 0.83c [0.69, 0.98] Perennial ryegrass Merci 0.89a [0.76, 1.04] Allegro 0.81a [0.69, 0.95] Figaro 0.80a [0.66, 0.95] Taya 0.79a [0.66, 0.92] Kentucky bluegrass Conni 2.33a [2.09, 2.59] Mardona 2.12ab [1.86, 2.40] Pepaya 2.03ab [1.59, 2.52] Broadway 1.90b [1.68, 2.14] Andante 1.49c [1.30, 1.69] ([dagger]) Species and cultivars within a species, respectively, followed by the same letter are not significantly different at the 0.05 probability level. The letter grouping is based on pairwise comparisons using t tests. Table 4. Relative contribution of the variance components expressed as variance of final germination percentage (FGP), mean germination time (MGT), and time from 25 to 75% germination ([T.sub.25-75]) overall for all species and within red fescue, perennial ryegrass, and Kentucky bluegrass, respectively. Variance Variance Species component FGP MGT [T.sub.25-75] logit % [square root of (d)] All species Cultivar 0.1588 0.0037 0.0080 Seed lot 0.0944 0.0024 0.0040 Replicate 0.4143 0.0190 0.0095 Red fescue Cultivar 0.0671 0.0073 0.0116 Seed lot 0.0688 0.0012 0.0000 Replicate 0.3354 0.0275 0.0108 Perennial ryegrass Cultivar 0.1853 0.0000 0.0000 Seed lot 0.0739 0.0034 0.0063 Replicate 0.4859 0.0131 0.0073 Kentucky bluegrass Cultivar 0.2077 0.0041 0.0120 Seed lot 0.1496 0.0036 0.0072 Replicate 0.4276 0.0149 0.0094
The authors wish to thank the Royal Veterinary and Agricultural University, Danish Centre for Forest, Landscape, and Planning, and The Danish Research Agency for the financial support. The supply of seed samples from DLF-Trifolium A/S is gratefully acknowledged.
Adams, W.A., and R.J. Gibbs. 1994. Natural turf for sport and amenity: Science and practice. CABI, Wallingford, UK
Arens, R. 1962. Auswirkungen der Saatstarke auf das Konkurrenzverhalten der Arten und die erste Bestandsbildung bei Weideansaaten. J. Agron. Crop Sci. (Z. Acker-Pflanzenbau) 115:357-374.
Bewley, J.D., and M. Black. 1994. Seeds: Physiology of development and germination. 2nd ed. Plenum Press, New York.
Bibby, B.M., and M. Sorensen. 2003. Hyperbolic processes in finance. p. 211-248. In S. Rachev (ed.) Handbook of heavy tailed distributions in finance. Elsevier Science, Amsterdam.
Bo, S. 1989. Utviklinga hos plengras i saningsaret. Norsk Landbruksforskning 3:39-48.
Brown, R.F., and D.G. Mayer. 1988b. Representing cumulative germination. 2. The use of the Weibull function and other empirically derived curves. Ann. Brit. (London) 61:127-138.
Campbell, R.K., and F.C. Sorensen. 1979. A new basis for characterizing germination. J. Seed Technol. 4:24-34.
Cheng, C., and I.L. Gordon. 2000. The Richards function and quantitative analysis of germination and dormancy in meadowfoam (Limnanthes alba). Seed Sci. Res. 10:265-277.
Culleton, N., V. McCarthy, and D. McGilloway. 1991. A note on the germinability and early seedling growth of L. perenne. Irish J. Agric. Res. 30:159-161.
Ellis, R.H., and E.H. Roberts. 1980. Towards a rational basis for testing seed quality, p. 605-635. In P.D. Hebblethwaite (ed.) Seed production. Butterworths, London.
Ellis, R.H., G. Simon, and S. Covell. 1987. The influence of temperature on seed germination rate in grain legumes. III. A comparison of five faba bean genotypes at constant temperatures using a new screening method. J. Exp. Bot. 38:1033-1043.
Ene. B.N., and E.W. Bean. 1975. Variations in seed quality between certified seed lots of perennial ryegrass, and their relationship to nitrogen supply and moisture status during seed development. J. Br. Grassl. Soc. 30:195-199.
Gooding, M.J., N.A. Baldwin, and J.R. Bennett. 1989. Post-emergence damping-off in Poa pratensis and its relationship with seed weight and seedling establishment. J. Sports Turf Res. Inst. 65:91-101.
Happ, K., M.B. McDonald, and T.K. Danneberger. 1993. Vigour testing in perennial ryegrass (Lolium perenne L.) seeds. Seed Sci. Technol. 21:375-381.
Hunter, E.A., C.A. Glasbey, and R.E.L. Naylor. 1984. The analysis of data from germination tests. J. Agric. Sci. (Cambridge) 102:207-213. International Seed Testing Association. 1999. International rules for scud testing: Rules 1999. Adopted at the twenty-fifth International Seed Testing Congress, South Africa 1998, to become effective in 1 July 1999. Seed Sci. Technol. 27(suppl.):1-333.
Lush, W.M., and J.A. Birkenhead. 1987. Establishment of turf using advanced ("pregerminated") seeds. Aust. J. Exp. Agric. 27:323-327.
Maguire, J.D. 1962. Speed of germination--Aid in selection and evaluation for seedling emergence and vigour. Crop Sci. 2:176-177.
Naylor, R.E.L. 1980. Effects of seed size and emergence time on subsequent growth of perennial ryegrass. New Phytol. 84:313-318.
Naylor, R.E.L. 1981. An evaluation of various germination indices for predicting differences in seed vigour in Italian ryegrass. Seed Sci. Technol. 9:593-600.
Naylor, R.E.L. 1982. Differences between cultivars of perennial ryegrass in laboratory germination and field emergence. Ann. Appl. Biol. 100(suppl.):106-107.
Naylor, R.E.L., and H.J.A. Hutcheson. 1986. The germination behaviour in soil and compost of different seed lots of perennial ryegrass. Crop Res. 25:123-132.
Newell, A.J., and N.K. Bludau. 1993. Variation in total germination, rate of germination and seed weight among cultivars of Poa pratensis. J. Sports Turf Res. Inst. 69:83-89.
Nichols, M.A., and W. Heydecker. 1968. Two approaches to the study of germination data. Proc. Int. Seed Test. Assoc. 33:531-540.
Nienhuis, K.H., and H.J. Baltjes. 1985. Seed storage and germination in testing varieties for distinctness, uniformity and stability. Seed Sci. Technol. 13:19-25.
Pommer, G. 1972. Art-und sortenbedingte Variation von Rasengrasern. Rasen, Torf, Gazon 3:89-93.
Roberts, E.H. 1986. Quantifying seed deterioration, p. 101-123. In M.B. McDonald, Jr., and C.J. Nelson (ed.) Physiology of seed deterioration. CSSA Spec. Publ. 11. CSSA, Madison, WI.
Ross, M.A., and J.L. Harper. 1972. Occupation of biological space during seedling establishment. J. Ecol. 60:77-88.
Shildrick, J.P., and R.W. Laycock. 1979. Correlations between seed characters and initial establishment in Lolium perenne. J. Sports Turf Res. Inst. 55:69-81.
Skirde, W. 1967. Ergebnisse eins Versuches mit verschiedenen Saatmengen und Saatzeiten von Rasengrasern. Rasen und Rasengraser 1:28-44.
The R Foundation for Statistical Computing. 2003. The R Project for Statistical Computing [Online]. Available at: http://www.r-project. org [modified 7 Apr. 2003; verified 15 Jan. 2004]. The R Foundation for Statistical Computing, Vienna, Austria.
Soren Ugilt Larsen * and Bo Martin Bibby
S.U. Larsen, Danish Centre for Forest, Landscape, and Planning, Horsholm Kongevej 11, DK-2970 Horsholm, Denmark; B.M. Bibby, Dep. of Mathematics and Physics, The Royal Veterinary and Agricultural Div., Thorvaldsensvej 40, DK-1871 Frederiksberg C, Denmark. Received 10 June 2003. * Corresponding author (firstname.lastname@example.org).
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|Title Annotation:||Turfgrass Science|
|Author:||Larsen, Soren Ugilt; Bibby, Bo Martin|
|Date:||May 1, 2004|
|Previous Article:||Genetic differentiation of tetraploid creeping bentgrass and hexaploid redtop bentgrass genotypes by AFLP and their use in turfgrass breeding.|
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