#### 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.

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 *f*_{t′}″) and the error in the estimation of the first derivative at that point (*SE − f*′_{t′}) are computed, the error in the estimation of the local maximum (*SE − t*′) can be directly computed as:

- (
*SE* − *t*′) = (*SE* − *f*′_{t′}) / ?*f*_{t′}″ 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:

- (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.