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Keywords:

  • Cistaceae;
  • nonlinear;
  • pollination;
  • reproductive biology;
  • synchrony

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

1. This paper presents a method for modelling unimodal flowering phenology based on fitting data to an exponential sine function with a minimum number of parameters.

2. Such parameters have the advantage of being direct surrogates for the most relevant features of any phenology (start, duration, intensity, skewness and length of the tails).

3. The use of the proposed function is exemplified using flowering data collected from populations of five Cistaceae species (four from the genus Cistus and one from Halimium).

4. The fitted models account for a large part (> 90%) of the variance in the data, and their parameters are easily interpreted in ecological terms.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

Phenological issues have always been important in pollination studies (Frankie 1975; Rathcke & Lacey 1985; Waser 1983), although there are few analytical tools with which to model flowering phenology. This contrasts with the many sophisticated models developed for other aspects of plant reproductive ecology (Bishop & Schemske 1998; Kiester et al. 1984; Morgan & Schoen 1997; Murren & Ellison 1996). Most studies of flowering phenology remain limited to graphical presentations of raw data, followed by statements about initial and final flowering dates as well as the maximum number of flowers recorded (Aizen 2001; Galetto et al. 2000). Such analyses, although valid for descriptive purposes, have limitations whenever comparisons or generalizations are sought (Rathcke & Lacey 1985).

This paper describes a new method for modelling flowering phenology in terms of the duration and start date of flowering, the date and value of the flowering peak, skewness, and the degree of tailing at the beginning and end of flowering.

Materials and methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

The model

The basic equation proposed for modelling flowering phenology combines the sine function with a minimum number of parameters that describe the main characteristics of unimodal data:

  • f(t) = a{sin[π(t/c)d]}e     eqn 1

Equation 1 produces a wave along the x-axis whose maximum is determined by a, its length by c, its asymmetry is controlled by d, and the length of the tails by e. From equation 1 it is also possible to derive the date of the phenological peak (when f(t), the number of flowers per plant, is maximized) and the total number of flower-days during the flowering period.

Figure 1 shows the curves obtained by different combinations of parameter values. Figure 1(a) presents the basic wave produced by the function in a case of a temporally symmetrical phenology with an abrupt start and end (d = 1, e = 1). The capacity of the parameter d to produce left-skewed (d < 1) and right-skewed (d > 1) curves is shown in Fig. 1(b). The two curves in Fig. 1(c) exemplify the effect of e on tail length. Finally, Fig. 1(d) shows two combinations of asymmetry and tailing.

image

Figure 1. Examples of phenologies produced by equation 1 for different values of d and e. (a) The basic form of a temporally symmetrical and tailless phenology; (b) temporally asymmetrical phenologies produced by varying d; (c,d) show how e > 1 affects the tails of the distributions. a = maximum number of flowers produced; b = day at which flowering starts; b + c= day at which flowering stops.

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Such curves can be fitted to experimental data with nonlinear estimation procedures using many statistical packages. In this case, the nonlinear estimation module of statistica 5·1 (Statsoft 1995) with the least-squares residuals minimization procedure used to estimate parameters (Scales 1985). The fit of models was assessed using the proportion of variance explained, and the statistical significance of each parameter tested with a t-test based on standard errors of parameters. These are directly computed by the statistical package through a differential approximation, and test results are presented as part of the nonlinear estimation module output.

The date of peak flowering (t′) is solved by setting the first derivative of equation 1 to zero, which gives:

  • t′ = b + c(1/2)1/d     eqn 2

Then the standard error can be estimated using the delta method, a standard procedure to estimate the variance of a function about a point (Cook & Weisberg 1999; Cresswell 2000; Rice 1995). In short, this procedure, when applied to the error in the location of the local maximum of a function, gives an estimation that is directly related to the uncertainty in the model regarding the position of the peak (mathematically represented by the standard error in the estimation of the point where the first derivative is zero). It is inversely related to the value of the second derivative about the maximum (a measure of the peak’s sharpness). However, this estimation cannot be done directly on the modelled function because it is not possible to rearrange the terms in its first derivative to isolate t′. Instead, it must be approximated using a Taylor polynomial fitted about t′ (Rice 1995), and the error in the estimation of the first derivative has to be computed using modelling software. Once the value of the second derivative at t′ (represented by ft′″) and the error in the estimation of the first derivative at that point (SE − ft) are computed, the error in the estimation of the local maximum (SE − t′) can be directly computed as:

  • (SE − t′) = (SE − ft) / ?ft′″     eqn 3

The total number of flower-days is calculated by integrating equation 1. This can be done directly with mathematical or modelling software (e.g. ModelMaker), or it can be estimated by calculating the predicted number of flowers for each day in the phenology with a simple spreadsheet.

The estimation of a, c, d, e and t′ allows phenologies to be compared statistically in terms of ecologically relevant features. The estimation of the parameters’ standard errors allows for model-to-model com-parisons. However, comparisons will usually involve parameter estimation for individual plants in order to make comparisons among populations using estimated parameters as raw variables. Such comparisons are done using standard parametric methods such as t-test or anova.

To avoid the problem of the model-estimation routine becoming stuck at local minima, which occurs if data contain dates with zero flower observations, it is advisable to proceed using equation 4:

  • image(eqn 4)

Equation 4 applies equation 1 only during the period b to b + c, b being the date when flowering begins, and defines zero flowers for the rest of the time.

Most common statistical packages cannot apply equation 4 directly, although the problem can be overcome by trial and error based on the a priori selection of several effective flowering periods (b to b + c). The procedure in this case is to fit equation 1 iteratively with the statistical package to different subsets of data. At each step the equation is fitted considering only the data collected between b and b + c. This is repeated until the solution with the smallest residual is found. To do it, residuals of models must be computed by adding to the least-squares deviation provided by the statistical package the sum-of-squares values left out of the analysis (Scales 1985). By this means, we include in our whole model (stated by equation 4) the error corresponding to the observed flowering outside b to b + c, as such values are modelled as zeros.

Although this procedure may seem protracted because of the many possible b and c values to be selected, it is simple in practice. Approximate start and end dates of flowering are usually obvious, and this restricts the number of combinations to be tested. To speed up the process, it is advisable to begin by adjusting equation 1 to the mid-flowering period, and to expand the selected data subset progressively to a wider period in an iterative search for the best fit. The additive nature of least-squares deviations means that the manual computations are simple and efficient.

However, the problem can be solved more efficiently using modelling software such as ModelMaker 3·0 (Walker & Crout 1997). ModelMaker can handle and optimize equation 4 to directly estimate the values and standard errors of b and c. Afterwards, all functions for the ‘effective flowering period’ that were defined by the modelling software were fitted with statistica 5·1. This allows the use of standard outputs (graphs, parameter significance, residual analysis, etc.) provided by the statistical package.

Equation 4 introduces a subtlety to the assessment of the significance of b and c. As the procedure does not restrict the number of observations with zero flowers included in the analysis, the degrees of freedom used in the t-test could be inflated artificially by including zero-flower observations taken long before or after the flowering period. To avoid this, the conservative option of computing the degrees of freedom for observations made between the first and last flowering dates plus one previous and one posterior zero-observation is recommended. The degrees of freedom for all parameters other than b and c are based on the number of observations made during the ‘effective flowering’ period.

Applying the model

To test the procedure, flowering data collected from five populations of Cistaceae species (one population per species) growing at the Universidad Autónoma de Madrid, Spain were used. Plants were introduced to the site and some species are outside their natural distribution, although they are uncultivated. Thus collected data (and models) should be taken as methodological examples, but not used as descriptions of the species’ natural phenologies. Accordingly, only a subset of the possible tests among parameters are shown in the present paper, and they should be taken as only examples of potential applications of the method.

Plants were monitored during spring and summer 2000. Three counts were made each week (Mondays, Wednesdays and Fridays) of the number of open flowers at mid-morning. The selected species were Cistus albidus L. (n = 6 plants); C. ladanifer L. (n = 13); C. laurifolius L. (n = 3); C. salviifolius L. (n = 3); and Halimium atriplicifolium (Lam.) Spach (n = 2).

The significance of parameters was computed by the nonlinear estimation module of statistica 5·1 with a t-test based on the estimation of their values and standard errors. In all cases, the null hypotheses tested was that a, b, c, d and e = 0. The parameter comparison between models was carried out with the ‘difference between two means’ command in the basic statistics module of statistica 5·1. This procedure assumes the normal distribution of estimates, and is based on the value of parameters, their standard deviations, and their number of degrees of freedom. The total number of flower-days in each phenology was computed by integration of the corresponding model with ModelMaker 3·0.

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

Models fitted to the flowering data accounted for between 90 and 98% of the variance (Table 1), and predicted values for number of flowers close to the observed data in most instances (Figs 2 and 3). The flowering phenologies of the four Cistus species in Fig. 2 were visually similar. In contrast, H. atriplicifolium had a longer flowering period (108 days; Fig. 3), and one that lasted for almost 2 months after the flowering peak.

Table 1.  Parameter estimates and goodness-of-fit statistics for the models fitted to flowering data for five Cistaceae species
 Estimated parameters
abcde
  1. Each parameter estimation is followed by its standard error estimated by differential approximation by the statistical package. The residual of the model for each species is shown with the proportion of explained variance (r2) and the peak flowering date (for which SE was always 0·0, computed using the delta method). The units of parameters are: a, number of flowers; b, day of the year (b = 0 corresponding to 19 March); c, days; d and e are dimensionless. The cases in which the degrees of freedom for the significance of b and c differ from those for the model for the flowering period are indicated with superscripts (see text for details): *20 df; †22 df; ‡76 df.

Cistus albidus (model residual: 19 121; r2 = 0·897; day of peak flowering: 50·7)
Estimate (± SE)196 ± 19·6 24·8 ± 0·3 36·6 ± 0·4  2·01 ± 0·29  0·82 ± 0·27
t (12 df) 10·01 74·5*103·85*  6·90  3·02
P< 0·001< 0·001< 0·001< 0·001  0·011
Cistus ladanifer (model residual: 6838; r2 = 0·935; day of peak flowering: 51·8)
Estimate (± SE)161 ± 9·9  7·0 ± 2·0 52·2 ± 2·0  4·49 ± 0·46  0·76 ± 0·16
t (19 df) 16·34  3·56 25·52  9·84  4·80
P< 0·001< 0·001< 0·001< 0·001< 0·001
Cistus laurifolius (model residual: 159·6; r2 = 0·983; day of peak flowering: 72·4)
Estimate (± SE) 65·4 ± 4·6 61·5 ± 0·9 16·7 ± 1·0  1·63 ± 0·15  1·34 ± 0·36
t (4 df) 14·08 66·40 16·41 10·72  3·71
P< 0·001< 0·001< 0·001< 0·001  0·021
Cistus salviifolius (model residual: 15 148; r2 = 0·909; day of peak flowering: 52·7)
Estimate (± SE)200 ± 17·6 17·9 ± 1·0 43·4 ± 1·0  3·12 ± 0·39  0·99 ± 0·29
t (15 df) 11·33 18·43 43·74  8·07  3·37
P< 0·001< 0·001< 0·001< 0·001  0·004
Halimium atriplicifolium (model residual: 3183; r2 = 0·970; day of peak flowering: 71·5)
Estimate (± SE)178 ± 5·5 45·7 ± 6·0108 ± 54·2  0·48 ± 0·00  44·2 ± 3·48
t (42 df) 32·61  7·62  1·99177·11  12·70
P< 0·001< 0·001  0·050< 0·001< 0·001
image

Figure 2. Original data and phenological models fitted to the flowering of four Cistus species (Table 1). Each x-axis shows the times of the start, peak and end of flowering as estimated by the model. (a) Cistus albidus; (b) Cistus ladanifer; (c) Cistus laurifolius; (d) Cistus salviifolius.

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image

Figure 3. Original data and model fitted to Halimium atriplicifolium flowering (Table 1). The x-axis shows the times of the start, peak and end of flowering as estimated by the model. The arrow indicates when the last flower was produced.

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The estimated parameters in Table 1 would be the basis for any statistical comparisons among species’ phenologies. It would thus be possible to state that the flowering period of C. laurifolius (c = 16·7 ± 1·0 days) was significantly shorter than that of the other Cistus species (P < 0·001 in all cases, see values in Table 1). The left-skewness of H. atriplicifolium’s flowering period (d = 0·48 ± 0·02) was significantly different from the right-skewness of the other species (P < 0·001 in all cases, d values in Table 1). Moreover, right-skewness was significantly more marked in C. salvifolius (d = 3·12 ± 1·69) than in C. albidus (d = 2·01 ± 1·30, P = 0·026) and C. laurifolius (d = 1·63 ± 0·48, P = 0·012), but not statistically different between the latter two species (P = 0·387). Similar comparisons could be done in this way for whichever set of parameters was considered relevant in each study.

Finally, from the fitted functions the total number of flower-days for each species were calculated to be: 3895 in C. albidus, 2955 in C. ladanifer, 542 in C. laurifolius, 3426 in C. salviifolius, and 2264 in H. atriplicifolium.

Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

The exponential sine equation has several advantages over existing phenology models.

First, equation 4 can be fitted easily to flowering data, especially if modelling software is used. The alternative trial-and-error procedure with statistical packages is not complex, although it is somewhat tedious.

Second, the parameters of the exponential sine model have straightforward, ecologically meaningful interpretations. This contrasts with other approaches, such as the approximation of flowering records to a Gaussian distribution to obtain mean, skewness and kurtosis estimates (Bosch 1992; Rabinowitz et al. 1981; Thompson 1980). That approach has a more complex interpretation of the estimated parameters, and the infinite length of the distribution’s tails is unrealistic.

Third, it is possible to compare statistically flowering parameters derived from different species, or individuals and populations of the same species. Analyses of this kind are common in plant ecology, including comparisons among years, populations subjected to different environmental conditions, and plants or genders of a species (Rathcke & Lacey 1985). There has been much interest in the dates when flowering begins and when it peaks; the number of flowers produced; or the overlap and symmetry of phenologies (Aizen 2001; Dafni 1992; Galetto et al. 2000; Ollerton & Lack 1998; Rabinowitz et al. 1981). It is possible to estimate the number of flowers produced from the total number of flower-days if the longevity of individual flowers is known. For each species studied here, the number of flowers produced was approximately the same as the number of flower-days, because flowers of C. ladanifer, C. salviifolius, C. albidus, Halimium halimifolium and other species in the Cistaceae last for only 1 day (Blasco & Mateu 1995; Bosch 1992; Brandt & Gottenberger 1988, Herrera 1987a). Flower longevity should be verified for each study, however, as the duration of C. ladanifer flowers has also been reported to be 2–3 days (Talavera et al. 1993).

Fourth, the proposed method could be used to analyse animal–plant interactions such as pollination or florivory, or those involving nectar production, provided these processes have roughly bell-shaped temporal patterns (Arizaga et al. 2000; Pilson 2000).

Although the exponential sine model is robust and simulates natural patterns accurately, two points require caution. The first is to remember that nonlinear estimation procedures may get stuck at local minima, and solutions do not necessarily reach the best possible fit (Scales 1985). To counter this problem, it is recommended that parameter estimations be repeated for each data set using different initial values (Statsoft 1995; Walker & Crout 1997).

The second cautionary point is that the exponential sine equation can be fitted properly only to unimodal processes, and it is inappropriate to describe multimodal or irregular flowering phenologies (Ruiz et al. 2000). Equally, its application to asymmetric phenologies that combine a steep peak with long tails may also be problematic, as in the H. atriplicifolium example (Fig. 3). Although equation 4 achieved a good overall fit of the model (r2 = 0·97), it underestimated the number of flowers produced after the peak date. Such an asymmetric pattern was caused in this case by the incompatibility between the two H. atriplicifolium plants in the area (Herrera 1987a; Herrera 1987b), leading to their fruiting failure and the secondary growth of flowering branches. If modelling the pattern of flowering along the tails of the phenological curves is of particular interest, the exponential sine model may be unsuitable.

In conclusion, the exponential sine model has several distinct advantages for analysing many ecologically important phenological patterns. Its use allows such patterns to be compared objectively, a development that has been long overdue (Lloyd & Webb 1977; Rathcke & Lacey 1985; Waser 1983).

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

Comments and help provided by Adrián Escudero and two anonymous referees were crucial during model development and manuscript preparation. This work was carried out along with Project I + D AMB-99 0382 from the Spanish Ministry of Science and Technology.

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
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