UF, J.
Annals of Botany 90: 405-415, 2002
© 2002 Annals of Botany Company
A Stochastic Flowering Model Describing an Asynchronically Flowering Set of Trees
UF31 CIRAD-FLHOR, BP 180, 97455 Saint Pierre cedex, Réunion Island, France, 2 INRA, Plante et Systèmes de Culture Horticoles, Bât. A, Domaine St Paul, Site Agroparc, F-84914 Avignon cedex 9, France and 3 INRA, Biométrie, Domaine St Paul, Site Agroparc, F-84914 Avignon cedex 9, France
* For correspondence. Fax: +262 2 62 50 58 44, e-mail normand{at}cirad.fr
Received: 11 October 2001; Returned for revision: 18 March 2002; Accepted: 14 June 2002
 |
ABSTRACT |
|---|
 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
|---|
A general stochastic model is presented that simulates the time course of flowering of individual trees and populations, integrating the synchronization of flowering both between and within trees. Making some hypotheses, a simplified expression of the model, called the ‘shoot’ model, is proposed, in which the synchronization of flowering both between and within trees is characterized by specific parameters. Two derived models, the ‘tree’ model and the ‘population’ model, are presented. They neglect the asynchrony of flowering, respectively, within trees, and between and within trees. Models were fitted and tested using data on flowering of Psidium cattleianum observed at study sites at elevations of 200, 520 and 890 m in Réunion Island. The ‘shoot’ model fitted the data best and reproduced the strong irregularities in flowering shown by empirical data. The asynchrony of flowering in P. cattleianum was more pronounced within than between trees. Simulations showed that various flowering patterns can be reproduced by the ‘shoot’ model. The use of different levels of organization of the general model is discussed.
Key words: Phenology, flowering asynchrony, flowering model, stochastic model, degree days, Psidium cattleianum.
|
INTRODUCTION |
|---|
|
TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
|---|
The time course of flowering of individual trees and of their population, i.e. the number of flowers open daily for the duration of flowering, has important consequences for the reproductive success and the genetic structure of a population (Bawa, 1983; Ims, 1990; Murawski and Hamrick, 1991; Hof et al., 1999). In particular, seed and fruit production of self-incompatible species rely on flowering times of compatible genotypes overlapping, with important economic consequences for agriculture. The time course of flowering of a population can also be considered in relation to the sensitivity of flowers to pathogens in order to aid the development of integrated pest and disease management (Dodd et al., 1992), or for human health when the prediction of airborne pollen is used to forecast allergic risks (Frenguelli et al., 1989; Ickovic et al., 1989; Andersen, 1991; Belmonte and Roure, 1991).
Plants, in particular tropical trees and shrubs, display a large variety of flowering patterns (Gentry, 1974; Bawa, 1983). Asynchronous flowering is widespread (Primack, 1980; Bawa, 1983; Ims, 1990; Carthew, 1993), even in mass-flowering species (Augspurger, 1983). The time course of the population flowering, as the superposition of the time course of individual flowerings, is affected by the time and duration of individual flowering, as well as by the individual number of flowers. Two main components can be distinguished in the synchronization of flowering (Hof et al., 1999): the synchronization between plants and the synchronization within plants. Flowering phenology is mainly under genetic control (Nienstaedt, 1974; Primack, 1980; Mosseler and Papadopol, 1989; Pors and Werner, 1989; Carthew, 1993; O’Brien and Calder, 1993; Boudry et al., 1994; Mitchell-Olds, 1996; Hof et al., 1999), implying that a detailed study of flowering phenology must consider plants at the individual level (e.g. Augspurger, 1983; Fripp et al., 1987; Carthew, 1993). Improving synchronization of flowering and the flowering period have been important objectives for breeding and selection programmes (Janick and Moore, 1975; Hof et al., 1999; Citadin et al., 2001). Flowering phenology is also affected by the environment (Murfet, 1977; Primack, 1980).
Temperature is recognized as being the main variable driving the timing of budburst or flowering in woody plants, and different models have been proposed to predict the date of flowering of a population. They all assume that budburst, or flowering, occurs when a critical development threshold a is reached, the stage of development being a sum of daily rates of development. The models differ in the expression of the rate of development as a function of temperature (e.g. Hänninen, 1987; Chuine et al., 1998, 1999), and in the way they integrate chilling requirements for temperate trees (Cannell and Smith, 1983; Hänninen, 1987; Murray et al., 1989; Kramer, 1994; Chuine et al., 1998, 1999). The predicted date of flowering is the day on which the critical development threshold is reached. It generally corresponds to the onset of bloom, to the mid-bloom date, or to the date of maximum concentration of airborne pollen (Boyer, 1973; Chuine et al., 1998, 1999).
However, very little attention has been paid to modelling of the time course of flowering of a tree or a population. Agostini et al. (1999) proposed a stochastic flowering model for an orchard of kiwifruit [Actinidia deliciosia (A. Chev.) C.F. Liang & A.R. Ferguson] female vines (see Materials and Methods). In such a clonal population, the time course of individual flowering is expected to be reasonably similar, with a strong synchronization of flowering between plants. If flowering is a regular process on individuals and the environment is homogeneous, then the time course of flowering for the population and for individuals should not differ, except for the number of flowers open per day. More generally, in a non-clonal population, such a model would be inappropriate due to asynchronous flowering and differences in the time course of individual flowerings (e.g. Primack, 1980; Augspurger, 1983; Fripp et al., 1987; Carthew, 1993). Simple empirical functions have been proposed to describe the time course of flowering at the individual level using plant-specific parameters (Fripp et al., 1987; Medan and Bartoloni, 1998; Hof et al., 1999), but without integration at the population level.
With regard to classical phenological models, it appears necessary to deal with flowering variability both within and between trees to model the time course of flowering at the tree and at the population level. The within-tree flowering variability, i.e. the source of the time course of flowering at the tree level, may be related to genetic factors, to physiological factors, to the time-lag in flower induction or in bud break, or to the buds’ effective temperature. The between-trees variability is related to genetic and local environmental factors. The study and modelling of flowering phenological processes is a way to build an explanatory model to simulate the time course of flowering. However, results may be specific to the species studied and such models will probably need inputs that are difficult to obtain and will therefore be of limited use, in particular for prediction. Moreover, data on these processes are lacking. Another approach, that we chose to adopt, is to model the components of flowering variability in order to take them into account explicitly in the flowering model. The objective was not to explain the processes underlying the time course of flowering, but to simulate their effects with a model of general use.
This paper proposes a theoretical model to simulate the time course of flowering of individual trees and their population which integrates the flowering variability between and within trees. A general model is presented and hypotheses are proposed to simplify it and reduce the number of parameters. Derived models neglecting the within-tree or the within- and between-trees variability are also presented. A procedure to estimate the model parameters is proposed. The models are then fitted and tested using experimental data obtained in the subtropical Réunion Island on strawberry guava (Psidium cattleianum Sabine) populations growing in natural conditions at three elevations. The simplifying assumptions used for the model construction are tested on P. cattleianum. Model structure and uses are then discussed.
|
MATERIALS AND METHODS |
|---|
|
TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
|---|
Model description
The heart of the model was developed by Osawa et al. (1983) to describe bud phenology of balsam fir [Abies balsamea (L.) Mill.]. Dennis et al. (1986) adapted it to insect development, and Agostini et al. (1999) to kiwifruit flowering. We recall here its assumptions and formulation. It assumes that the development of a given bud, here a flower bud, is a stochastic process consisting of the accumulation of small increments of development ud. This process is supposed to begin with the flower bud at a given stage at time t = 0. The process S(t) is defined as the amount of development time a flower bud has accumulated by actual time t:
The amount of development time, S(t), and t are expressed in degree days. If the increment of time 
d is small and the increments of development ud are independent and identically distributed with an expectation E(ud) = d, then S(t) is normally distributed with mean t and variance 
2t (Osawa et al., 1983). Flowering occurs when the amount of development S(t) passes through a threshold a. So the probability that a given flower bud is flowering at time t is:
where
is the cumulated probability between u and +
of the standard normal distribution.
The probability that a given flower bud is flowering between ti – 1 and ti is:
The implicit assumption of this model is that all the flower buds follow the same stochastic process, i.e. they accumulate increments of development in the same way and have the same flowering threshold.
Let us now consider a population of K trees, and a development time scale calculated at the population level. The flowering variability between trees is related to different individual flowering patterns linked to the genetic variability and also, with reference to a common development time scale, to the differences of effective temperature at the level of each tree. We assume that the flowering variability within trees is related to different flowering patterns of the tree flowering units. We define the flowering unit as a level of organization whose flower buds are in a similar stage of development and experience a similar environment. Consequently, they have a similar pattern of development. For convenience, we will later call the flowering units flowering shoots. However, the architectural level of a flowering unit needs to be determined for each species (e.g. inflorescence for some species, etc.). The within-tree flowering variability is defined by two components: (1) the flowering variability of the flower buds on a flowering unit, described by the stochastic process [eqn (2)]; and (2) the variability in flowering time between flowering units. The flowering variability can be formulated as a variable flowering threshold a and a variable parameter of variance relative to flowering 2 at the shoot and at the tree levels. Increasing values of the parameter of variance relative to flowering are expected when the level of organization increases (shoot < tree < population).
In the population of K trees, each tree, p, has FSp flowering shoots. A flowering shoot, denoted fsj,p (j 
[1,FSp]), bears Nj,p flower buds that are supposed to follow the same stochastic process of development. Equation (2) can then be applied at the flowering shoot level with the specific parameters aj,p and j,p2 common to all its flower buds. Let us consider the expectation of parameters aj,p, ap = E(aj,p) (j [1,FSp]), as the tree flowering threshold for tree p. The difference between the flowering threshold of the flowering shoot and the tree flowering threshold, dj,p = aj,p – ap, expressed in thermal time units, represents the advance (dj,p < 0) or the delay (dj,p > 0) of the mid-bloom on the shoot fsj,p compared with the mid-bloom on all the shoots of the tree. By construction, the expectation of dj,p is zero.
Therefore, the probability that a flower bud on shoot fsj,p, j [1,FSp], p [1,K] will flower between ti – 1 and ti is:
This is the general form of the flowering model, where ap, dj,p and j,p2 are parameters. All flower buds on shoot fsj,p have the same probability of flowering between ti – 1 and ti. This probability depends only on time and on the model parameters. Therefore, the model assumes that flowering of a given flower bud on shoot fsj,p is not affected by the other flower buds on the shoot, i.e. the flower buds are independent with regard to their probability of flowering.
The model parameters are numerous and their number
depends on the number of shoots and trees. This general formulation is not easy to use for prediction, and parameter values are specific to the shoots and trees upon which they are estimated. These parameters are the values of the tree flowering threshold ap for each tree and of the two parameters of the stochastic flowering process (aj,p = ap + dj,p and j,p) for each flowering shoot. To simplify the model and extend its use, in particular for prediction, we propose to model the variability of these parameters using statistical distributions whose parameters are estimated from the data. The components of the flowering variability are thus quantified. We propose three hypotheses for the distributions of ap, dj,p and j,p: (1) the differences dj,p follow the same normal distribution N(0,
2) for all trees, i.e. the tree has no influence on the distribution of shoot flowering thresholds around the tree flowering threshold; (2) the parameter of variance relative to flowering at the shoot level, j,p2, is independent of the shoot and the tree, and is called s2; and (3) the tree flowering thresholds, ap, follow a normal distribution N(
s,ßs2).
Given the first and second hypotheses, we can calculate the probability of flowering between ti – 1 and ti for a flower bud on tree p, unconditional on the shoot, using eqn (3):
is the density function of the normal distribution N(0,2).
Given the third hypothesis, we can calculate the probability of a flower bud flowering between ti – 1 and ti, unconditional on the shoot and the tree, using eqn (4):
where
is the density function of the normal distribution N(s,ßs2). Or:
where g(u) and f(a) are as in eqns (4) and (5), respectively.
This formulation of the model is of general use and has only four parameters related to the components of the flowering variability: s2 is the parameter of variance relative to flowering at the shoot level and expresses the time course of the flowering process at the shoot level; 2 is the variance of the central normal distribution of the differences dj,p on the tree and expresses the synchronization of shoots flowering within a tree (the higher the value of 2, the more asynchronous is flowering among shoots on a tree); s is the mean of the normal distribution of the tree flowering thresholds, i.e. the mean flowering threshold of the flowering shoot population taking into account the flowering of individual shoots; and ßs2 is the variance of the normal distribution of the tree flowering thresholds and expresses the flowering synchronization between trees (the lower the value of ßs2, the more synchronous is flowering between trees).
To evaluate the relevance of this model, called hereafter the ‘shoot’ model, two simpler models have been derived from it for comparison. The first, called the ‘tree’ model, considers that the within-tree variability can be neglected (i.e. the flowering units of a tree are quasi-synchronous): dj,p = 0. The effect of trees on flowering is then characterized by their respective tree flowering threshold ap', distributed according to a normal distribution N(t, ßt2). We hypothesize that the parameter of variance relative to flowering, t2, is independent of trees. This model has three parameters: t, ßt2 and t2, and the probability of a flower bud flowering between ti – 1 and ti, unconditional on the tree, is:
where
is the density function of the normal distribution N(t, ßt2).
The second simpler model, termed the ‘population’ model, considers that there is no effect of the trees on flowering, i.e. that their respective flowering is quasi-synchronous at the population level. Trees have the same flowering threshold a, and there is a parameter of variance relative to the flowering of the whole population: 2. This model, similar to that of Agostini et al. (1999), has two parameters, a and 2, and is described by eqn (2).
Model simulation
The number of flower buds on each flowering shoot must be known for simulations using the ‘shoot’ model, so that their respective weight in the total flowering can be taken into account. A development time scale is also required. Simulations are realized as follows. First, a flowering threshold, ap, is randomly sampled for each tree in the normal distribution N(s,ßs2). Then, a difference, dj,p, is randomly sampled for each flowering shoot of each tree in the normal distribution N(0,2). Given ap, the flowering threshold aj,p of each flowering shoot is calculated: aj,p = ap + dj,p. Given aj,p, the flowering probability distribution (FPD) of the Nj,p flower buds of the shoot is determined using eqn (3), using j,p or a common s according to the model used. A flowering date is randomly sampled within FPD for each of the Nj,p flower buds of the shoot. Simulated flowering dates are then aggregated at the shoot, tree or population level.
To illustrate the effect of between- and within-tree synchronization on the flowering pattern, flowering of a five tree population has been simulated with four distinct cases of synchronization: A, asynchrony strong between trees and weak within trees (ßs = 250 °Cd, = 2 °Cd); B, strong asynchrony between and within trees (ßs = 150 °Cd, = 150 °Cd); C, asynchrony weak between trees and medium within trees (ßs = 20 °Cd, = 100 °Cd); D, weak asynchrony between and within trees (ßs = 20 °Cd, = 2 °Cd). The other parameters were set at s = 1000 °Cd and s = 0·7 °Cd. The development time scale ran from 500 to 1500 °Cd, with a 10 °Cd daily increment. Each tree had the same weight: 100 flowering shoots, each with 15 flower buds, i.e. 1500 flower buds per tree.
Data
Flowering of Psidium cattleianum (strawberry guava tree) was recorded from November 1998 to January 1999 at three sites at elevations of 200, 520 and 890 m on the windward east coast of the subtropical Réunion Island, Indian Ocean (21°06'S, 55°32'E). The sites at elevations of 200 and 890 m were fallow lands invaded by feral strawberry guava trees. The trees at 520 m were from an experimental 5-year-old seedling orchard which had not received any cultural care for the previous year and were thus considered feral. All individuals were genetically different on all the sites.
On each site, ten trees of similar size (1·5–2 m tall) were sampled. Flower buds are borne on new shoots emerging from the terminal branches at the end of the cool, dry season. Generally, one to three shoots emerge quasi-simultaneously from the same terminal branch (Normand and Habib, 2001). These shoots and their flower buds are therefore the same age and experience the same micro-environment. Consequently, we considered all the flowering shoots emerging from a terminal branch as the flowering unit for the purposes of the model, termed hereafter the ‘flowering shoot’. Terminal branches were sampled randomly in the tree canopy before bud burst: 20 branches were sampled on eight trees and 100 on two trees at the site at 520 m and 20 per tree at the sites at 200 and 890 m. On each tree on which 100 shoots were observed, 20 shoots were randomly sampled among the 100 shoots, and parameters were estimated using these 20 sampled shoots to avoid a higher weight of these trees in the likelihood estimation due to more flowers being observed. On the other hand, the complete 100-shoot data sets were used to estimate the distribution of parameters dj,p and j,p on a large sample to test the simplifying hypotheses made to derive the ‘shoot’ model. Each flower stays open for 1 d: the flower bud bursts in the morning (0700–0900), and anthers and petals dry and fall in the late afternoon. At each site, open flowers were counted daily on the shoots that had emerged from the terminal branches sampled. The data set obtained from the site at 520 m was used for parameter estimation and model comparison, and the data sets from the sites at 200 and 890 m were used for model validation.
Mean daily air temperatures, (Tmax + Tmin)/2, were estimated at the sites at 520 and 890 m using the daily thermic gradient between temperatures recorded at 40 and 1025 m, as described by Normand and Habib (2001). Mean daily air temperatures were determined directly at the site at 200 m. The development time scale used was a thermal time scale (Cannell and Smith, 1983). Data collected in orchards over several years and at several elevations showed that the delay between the triggering of a flowering shoot upon fertilization and mid-bloom was 914 °Cd with a base temperature, Tb, of 8·1 °C. Although flowering data for this study were collected on natural, non-triggered flowers, heat units were calculated using 8·1 °C as the base temperature, and the starting date, t0, for the sum of heat units was determined at each site so that mid-bloom occurred on the day when calculated degree days were the closest to 914 °Cd.
Empirical and simulated time courses of flowering were expressed as flowering frequency distribution (FFD). The flowering frequency on day ti was:
where n(ti) was the number of flowers open on day ti, and N was the total number of flowers observed.
Statistical analysis
For each model, parameters were estimated by the maximum likelihood method (Dacünha-Castelle and Duflo, 1982) on data giving the time course of the number of open flowers on a sample of flowering shoots observed on a sample of M trees. Each tree, p, has FSOp flowering shoots, denoted fsoj,p, each bearing NOj,p flower buds (j [1,FSOp], p [1,M]). Observations were made from t1 to tD and covered the whole flowering period of the M trees. At time ti (i [1,D]), nj,p(ti), flower buds burst into bloom on shoot fsoj,p. Equation (3) assumes that flower buds are independent with regard to their probability of flowering, so that the probability of the observed time course of flowering from t1 to tD on shoot fsoj,p follows a multinomial distribution and is
The multinomial coefficient depends only on the sample size and does not influence the parameter estimation and the maximum likelihood tests. It is then removed from the following calculations.
Furthermore, we hypothesize that flowering shoots on a tree are independent with regard to their probability of flowering. Then, the likelihood of the sample for the shoot model is given by:
where
For the ‘tree’ and the ‘population’ models, the likelihood of a sample is a simpler expression.
The likelihood estimates of the parameters s2, 2, s and ßs2 are the values which maximize L, or minimize –log(L). Calculations were performed with a minimization program for non-linear functions (Splus 2000 statistical package) using a general quasi-Newton optimizer (Mathsoft, 1999) on the M = 10 trees (2370 flower buds) observed at the site at 520 m.
Computing limitations did not permit the minimization of –log(L) for the ‘shoot’ model including two nested integrals [eqn (8)]. We estimated the parameters (s, ßs, , s) by selecting the value that minimized –log(L) in a range of values centred on a first estimation derived from the data (s, and s were the means of these parameters estimated at the tree level, and ßs was the standard deviation of the tree flowering thresholds). Although not optimal, this method leads to asymptotically unbiased parameters whose variance tends to zero when the number of flowers tends to infinity.
The maximum likelihood test (Lehman, 1983) was used to compare the three models of increasing complexity (‘population’, ‘tree’ and ‘shoot’), and to test the pertinence of some of the hypotheses made in model construction. The test evaluates the increase in likelihood between two models compared with the difference in their numbers of parameters. For a pair comparison of n models, twice the difference of log-likelihood, calculated with the estimated parameters of a data set, was compared with a 
2 value for P = 5/n % and a number of degrees of freedom equal to the difference between their respective parameters numbers.
Distributions were compared using a bilateral Kolmogorov– Smirnov’s test (Sprent, 1989).
|
RESULTS |
|---|
|
TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
|---|
Model simulations
Simulations performed with the ‘shoot’ model in Fig. 1 show four levels of between- and within-tree synchronization. The main flowering patterns were well represented by the model (Gentry, 1974; Bawa, 1983), in particular, the case of a strong asynchrony between trees, which is not reproduced by classical flowering models (Fig. 1A). Moreover, these patterns may be modified by differing numbers of flower buds on the shoots and trees. Flowering of individual trees was also simulated (data not shown), and can be useful to quantify the degree of overlap in flowering among trees (Primack, 1980; Augspurger, 1983).
|
Model fitting and model comparison
The data set collected at the site at 520 m is summarized in Table 1. Not all the terminal branches studied produced flowering shoots. The number of flower buds per flowering shoot varied from one to 50. The total number of flower buds observed was 4151, and 2370 flower buds were taken into account for parameter estimation following a 20-shoot sampling among the 100 shoots of two trees (Table 1). At the sites at 200 and 890 m, 695 and 972 flower buds were observed. Differences were related to different flushing and flowering intensities of the trees among the sites. Only eight trees flowered at the site at 200 m.
|
Parameters of the three models fitted on the 520 m data set were: ‘shoot’ model
s = 927·3 °Cd, ßs = 40·6 °Cd, = 95·4 °Cd, s = 0·96 °Cd; ‘tree’ model t = 923·7 °Cd, ßt = 50·8 °Cd, t = 2·61 °Cd; ‘population’ model a = 914·4 °Cd, = 3·12 °Cd.
As expected, the value of the parameter of variance relative to flowering increased with the level of organization of the model (shoot < tree < population). It reflected the dispersion of flowering at each level. ßt and ßs were high, indicating the asynchrony of flowering between trees (Table 1). Individual tree flowering thresholds, ap', estimated by the ‘tree’ model, varied from 837·7 to 998·9 °Cd. The difference represented 14 d. Likewise, was high, indicating the asynchrony between shoots on each tree. The largest difference between shoot flowering thresholds, aj,p, on a tree was 445 °Cd, or 38 d.
The sample log-likelihood was high for each model because of the large sample size. The test of maximum likelihood was highly significant for all pair comparisons of the three models (Table 2). The large gain of likelihood between the ‘shoot’ model and the two other models indicated that the former fitted the data in a more likely manner than the other models, i.e. taking into account the shoot level brought a large gain of accuracy in describing the data.
|
To study the behaviour of each model at the population level, 1000 simulations were run, using shoot and/or tree flowering thresholds randomly sampled in estimated distributions, and using the thermal time scale and the number of flower buds per shoot and tree observed at the site at 520 m. The 0·025 and 0·975 quantiles and the median of the simulated flowering frequencies were determined for each date, ti. The empirical data were then graphically compared with the range of 95 % of the simulated values belted by these quantiles (Fig. 2). Empirical FFD showed strong irregularities, particularly during peak flowering, indicating that flowering was not a smooth process. These irregularities were due to the simultaneous flowering of several shoots bearing many flower buds. The three models gave a good general fit of the flowering pattern, in particular the mid-bloom date and the overall flowering duration. The median FFDs simulated by each model were not significantly different (Kolmogorov–Smirnov’s test for pair comparison, P > 0·84), indicating that, on average, the three models gave similar FFDs (data not shown). The models differed in the width of the inter-quantiles band, which traduces the capacity of the model to simulate irregularities in the time course of flowering as those of the empirical FFD. The band was narrower as the level of organization of the model increased (Fig. 2). Consequently, the empirical FFD was more satisfactorily included in the ‘shoot’ and ‘tree’ model inter-quantile bands than in that of the population model. The ‘tree’ model band-width was as large as that of the ‘shoot’ model during peak flowering, but was narrower at the onset and the end of flowering. The ‘tree’ model was thus as efficient as the ‘shoot’ model at representing the strong irregularities of empirical FFDs during peak flowering, but was less efficient at the onset and the end of flowering. This was probably the consequence of the random attribution, by the ‘shoot’ model, of a flowering threshold to each flowering unit whose weight in the tree flowering is its number of flower buds. Thus, flowering units with numerous flower buds could flower early or late, and increased the simulated flowering frequency at the onset or at the end of overall flowering.
|
Model validation
When expressed in thermal time, the duration of overall empirical flowering at the three sites was similar (592·8 °Cd at 200 m, 648·9 °Cd at 520 m and 472·2 °Cd at 890 m), indicating that flowering was mainly driven by temperature, which was our basic hypothesis. However, flowering patterns were different, with a straight flowering peak during mid-bloom at the site at 200 m (Fig. 3), and just before mid-bloom at the 890 m site (Fig. 4). These peaks were related to heavily flowering trees with similar flowering thresholds. At the 200 m site, four trees (of eight) bore 71·1 % of the observed flower buds and had very similar flowering thresholds (902·7–918·7 °Cd). At the site at 890 m, three trees (of ten) bore 56·8 % of the observed flower buds and also had similar flowering thresholds (861·6–933·4 °Cd).
|
|
The ‘shoot’ model was validated for temperature and site variation at the 200 and 890 m sites using the previous method. Simulations were run using parameter values estimated from the data set collected at the 520 m site, and the empirical thermal time scale and number of flower buds per shoot relating to each site (Figs 3 and 4). Empirical FFDs were satisfactorily included in the range of 95 % of the simulated values, except for the flowering peaks observed at these sites. The model gave a good representation of overall flowering duration and mid-bloom date at both sites. The ‘tree’ and ‘population’ models were validated in the same way. At each site, results of model behaviour were similar to those of the model comparison at 520 m (data not shown).
Hypothesis testing
Three hypotheses were made to simplify the general form of the flowering model given by eqn (3) and to derive the shoot model [eqn (6)]. These hypotheses were tested on the 520 m data set to verify their biological relevance for P. cattleianum.
Kolmogorov–Smirnov’s tests indicated that the differences, dj,p, followed a central normal distribution for all trees, but with different standard deviations, p (Table 3). The hypothesis of the same normal distribution of the differences dj,p on all trees was tested using a maximum likelihood test comparing a model (ap',p2,) with the same for all trees with a model (ap',p2,p) with p estimated for each tree. Each tree was characterized by the estimated values of its flowering threshold, ap', and its parameter of variance relative to the flowering of its shoots p2. The test was not significant at the 5 % level (2lL = 16·6, d.f. = 9, 2 = 16·9, P = 0·06), indicating that this hypothesis was verified by our sample. However, the test’s probability was closed to the significance threshold, suggesting that further work is necessary to verify, and eventually modify, this hypothesis for P. cattleianum.
|
The parameter of variance relative to flowering at the shoot level,
j,p2 [eqn (3)], was hypothesized to have a unique value, s2, independently of the shoot and the tree. We first tested its independence with the shoot using a test of maximum likelihood comparing a model (aj,p, j,p2) in which each shoot was characterized by the estimated value of its flowering threshold aj,p and its parameter of variance relative to flowering j,p2, with a model (aj,p, p2) in which the shoots of each tree had a common parameter of variance p2 relative to flowering. The test was highly significant (2lL = 1142, d.f. = 71, 2 = 202·5, P = 0·00). The hypothesis of the same p2 for all shoots of a tree was therefore rejected, and a fortiori the hypothesis of the same parameter of variance relative to flowering s2 for all shoots of the tree population. The j,p2 values were highly variable among trees (j,p coefficient of variation on trees 3 and 6: 72·2 and 74·2 %, respectively). The j,p2 traduced, in part, the variability in development time accumulation among the flower buds of a flowering shoot. Two sources of variation could be put forward. First, slight significant positive correlations were found between j,p and the number of flowers of the shoot (r = 0·37, n = 89, P < 0·001 for tree 3, and r = 0·52, n = 93, P < 0·001 for tree 6). This suggested a differential distribution of assimilates among flower buds when they are numerous on a shoot, leading to variable rates of development, and a larger j,p on the shoot. Secondly, variable j,p2 among shoots of a tree could be related to differences in the effective temperature of their flower buds, linked to the shoot position in the canopy (inside/outside, cardinal orientation).
The flowering thresholds ap' of the trees at the 520 m site estimated by the tree model followed a normal distribution with mean t and variance ßt2 (Kolmogorov–Smirnov’s test, n = 10, ks = 0·2129, P = 0·68).
|
DISCUSSION |
|---|
|
TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
|---|
Model structure
A stochastic model simulating the time course of flowering of an asynchronically flowering population of trees has been presented. It is based on a synthetic representation of the flowering variability. Flowering at a particular level (tree or population) is the aggregation of flowering of the lower-level components (flowering unit or tree, respectively). The basic component is the flowering unit, whose flower buds are supposed to follow the same stochastic process of development. The flowering unit must be determined for each species. The model considers the flowering variability both between and within trees. Within-tree variability is related to the time course of flowering at the flowering unit level and to differences in flowering time between flowering units. Between-tree variability is related to different tree flowering thresholds due to genetic and environmental factors. The inputs are a development time scale and the number of flower buds per flowering unit or per tree, obtained by observation or as plant growth model outputs. The aggregative structure of the model allows different levels of detail in the outputs, from the flowering unit to the population time course of flowering.
The general form of the model [eqn (3)] has a large number of parameters specific to the shoots and trees upon which they are estimated and is not relevant for use on other trees. Three hypotheses on the distribution of the parameter values were proposed to simplify the model and derive the ‘shoot’ model, which requires only four parameters quantifying the components of the flowering variability. These hypotheses have biological meaning and their pertinence should be tested for each species. The influence of environmental and genetic factors on flowering phenology tends to be under-estimated by the hypotheses as the individual tree has no influence on the distribution of the differences dj,p and on the parameter of variance relative to flowering at the shoot level. Each tree is characterized by its own flowering threshold and by the number of flower buds on its shoots. Nevertheless, the simulated time course of flowering of n trees is not the superposition at different times of n flowering of the same tree. The model randomly samples, in an estimated distribution, a flowering threshold for each flowering shoot whose weight is its number of flower buds. It then reproduces the differences in flowering phenology among trees and among shoots within a tree, in the limits of the variability of the data used to estimate the model parameters. These data must therefore be representative of the existing variability, in particular with respect to the influence of genetic and environmental factors, in order to use the model on other populations.
The simulations performed with different levels of flowering synchronization between and within trees show that the main flowering patterns are well represented by the ‘shoot’ model (Fig. 1). For example, Gentry (1974) found five distinct flowering patterns among the American Bignoniaceae, mainly related to different strategies with regard to pollinators (see also Bawa, 1983; Ims, 1990). They correspond to various levels of synchronization within and between trees, and can be well simulated by the model: Fig. 1B corresponds to the ‘cornucopia’ flowering pattern and Fig. 1D to the ‘big band’ flowering pattern (Gentry, 1974).
Two simpler models were derived from the ‘shoot’ model: the ‘tree’ model that neglects within-tree flowering asynchrony, and the ‘population’ model that also neglects the between-tree flowering asynchrony (e.g. Agostini et al., 1999) and considers that all the flower buds of the population follow the same stochastic process of development (i.e. no genetic or environmental influence).
Model adaptability
The model can be adapted to the biological characteristics of a particular species or environment, and to the objectives and needs of a particular study. The hypotheses made to derive the ‘shoot’ model may not be suited for some species. It is then possible to use different distributions for the differences dj,p and the tree flowering thresholds in eqns (4) and (5), or to fix a tree flowering sequence in the case of a strong environmental (Primack, 1980) or genetic (Primack, 1980; Pors and Werner, 1989; Carthew, 1993; O’Brien and Calder, 1993; Mitchell-Olds, 1996; Hof et al., 1999) influence on flowering phenology. Likewise, a particular distribution of the parameter of variance relative to flowering at the shoot level can be used. The analytic formulation of the ‘shoot’ model would then be more complex.
The model can be used to simulate the time course of flowering for a predictive or descriptive purpose, but the accuracy of the simulation is not the same. The most accurate way to reproduce the observed time course of flowering is to estimate the parameters at the tree and/or flowering unit level and to fix their value in the simulation process. The variability is then only induced by the stochastic process of flower bud development. If the objective is to predict the time course of flowering, then simulations are run with the flowering thresholds randomly sampled in distributions whose parameters are also estimated. Simulations are less accurate, but the model is of more general use, provided that the environment and the population whose time course of flowering is simulated are included in the domain of validity of the model defined by the data upon which parameters were fitted. Our study on P. cattleianum showed that overall flowering duration and mid-bloom date are well estimated, even if parameters are fitted in another place. Moreover, the range of the most likely simulated data can be estimated from a large number of simulations (Figs 2–4).
A limitation of the predictive ability of the model may arise from a strong lack of balance in the number of flower buds of the components of a flowering level, as illustrated by the model validation results for P. cattleianum at the 200 and 890 m sites. Heavily flowering trees may influence the population time course of flowering, particularly if their flowering thresholds are similar, or if they are early or late flowering. To estimate these flowering thresholds and to fix them in the simulation process is one way to overcome this problem. The general model [eqn (3)] and its different levels of simplification can therefore respond to the different objectives and needs of a study, with different levels of simulation accuracy in consequence. The cost of higher accuracy is more precise observations, more complex parameter estimation and simulation limited to the flowering upon which parameters are fitted.
The flowering model can be connected to classical budburst models which provide the development time scale (e.g. Chuine et al., 1999). The critical development threshold used in the budburst models corresponds to the shoot or the tree flowering threshold of our model where it then follows a specified distribution whose parameters are fitted from the data.
The model calculates the probability that a flower bud bursts between ti – 1 and ti. If individual flowers last 1 d, the model simulates directly the population of open flowers each day. But if individual flowers last nf days, the model simulates the onset of flowering for each flower. The population of flowers open at ti is then calculated by adding the number of flowers that opened on the nf – 1 days preceding ti (and which are still open) to the number of flowers that open between ti – 1 and ti. Similar calculations can be made for particular events during the flowering period, such as the period of pollen release or stigma receptivity (e.g. O’Brien and Calder, 1993), or the period in which flowers are susceptible to pests or pathogens.
The flowering model has been built for and tested on tree species. However, it is applicable to any flowering species, including annual and herbaceous species. Requisites are the identification of flowering units, as defined in the model construction, and the determination of a development time scale.
Application of the model to Psidium cattleianum
Flowering in P. cattleianum shows both between- and within-tree asynchrony, the latter being more pronounced than the former. The ‘shoot’ model is then the most relevant to describe the time course of flowering, as indicated by its high likelihood, whereas the ‘population’ model has the lowest likelihood. Therefore, taking asynchrony into account in a flowering model improves the accuracy of the time course of flowering fit for species affected by this phenomenon.
Data show that flowering is not a smooth process at the tree or the population level (Figs 2–4). The irregularities are due to the simultaneous flowering of several shoots. The aggregative structure of the ‘shoot’ and the ‘tree’ models, and the integration of flowering variability, allow simulation of such irregularities (Fig. 1B and C), whereas the population model does not as it considers the flower bud population as a whole and therefore smoothes the flowering process.
The model parameters and ßs estimated using the 520 m data set, embrace the genetic and environmental variation of flowering phenology. Although further studies are needed to confirm this result, the assumption of a similar normal distribution of the differences dj,p on all the trees is acceptable, indicating that the shoot flowering threshold distribution around the tree flowering threshold is not affected by the tree, and also that the different flowering shoots of a tree are comparable with respect to flowering. The tree flowering thresholds follow a normal distribution at the 520 m site. This distribution also fits the distribution of the tree flowering thresholds estimated at the sites at 200 and 890 m (n = 8, ks = 0·3045, P = 0·37; and n = 10, ks = 0·1873, P = 0·81, respectively). The genetic and environmental variability in tree flowering thresholds included in the normal distribution estimated at 520 m therefore appears representative of this variability at other sites with different genetic and environmental conditions.
The hypothesis of a unique parameter of variance relative to flowering at the shoot level is not verified. This parameter is highly variable between the flowering shoots of a tree, suggesting the effect of physiological or micro-environmental factors. Moreover, the distributions of this parameter value are significantly different among individuals, indicating a tree effect, probably related to genetic and/or environmental factors. However, our data cannot be used to test this. Further work is needed to specify the factors affecting the parameter variability within and between trees. From a practical point of view, we retain this hypothesis for strawberry guava as the ‘shoot’ model gives a satisfactory fit of the data.
Model applications
The model applications are wide and cover fields of research where simulation of the time course of flowering is useful: flowering phenology (e.g. Primack, 1980; Augspurger, 1983; Carthew, 1993); interactions between floral biology and reproductive ecology [effective mating population; gene exchange (e.g. Fripp et al., 1987); outcrossing rates (e.g. Murawski and Hamrick, 1991, 1992); foraging behaviour of pollinators (e.g. Bawa, 1983); seed and fruit set (e.g. Carthew, 1993)]; distribution of airborne pollen release; integrated management of flower pests and disease; and integration in a production model for crops whose pollination is a key step in seed and fruit set (Lescourret et al., 1999).
The model is also a tool to quantify, at the individual or the population level, the within-tree variability and the tree flowering thresholds. These parameters are useful as selection criteria (Hof et al., 1999), to study variability in a population, or to estimate the contribution of genetic and environmental factors to flowering variability, given a relevant experimental design.
|
LITERATURE CITED |
|---|
|
TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
|---|
-
Agostini D, Habib R, Chadoeuf J. 1999. A stochastic approach for a model of flowering in kiwifruit ‘Hayward’. Journal of Horticultural Science and Biotechnology 74: 30–38.
Andersen TB. 1991. A model to predict the beginning of the pollen season. Grana 30: 269–275.
Augspurger CK. 1983. Phenology, flowering synchrony, and fruit set of six neotropical shrubs. Biotropica 15: 257–267.
Bawa KS. 1983. Patterns of flowering in tropical plants. In: Jones CE, Little RJ, eds. Handbook of experimental pollination. New York: Scientific and Academic Editions, 394–410.
Belmonte J, Roure JM. 1991. Characteristics of the aeropollen dynamics at several localities in Spain. Grana 30: 364–372.
Boudry P, Wieber R, Saumitou-Laprade P, Pillen K, Van Dijk H, Jung C. 1994. Identification of RLFP markers closely linked to the bolting gene B and their significance for the study of the annual habit in beets (Beta vulgaris L.). Theoretical and Applied Genetics 88: 852–858.[CrossRef]
Boyer WD. 1973. Air temperature, heat sums, and pollen shedding phenology of longleaf pine. Ecology 54: 420–426.[CrossRef]
Cannell MGR, Smith RI. 1983. Thermal time, chill days and prediction of budburst in Picea sitchensis. Journal of Applied Ecology 20: 951–963.[CrossRef]
Carthew SM. 1993. Patterns of flowering and fruit production in a natural population of Banksia spinulosa. Australian Journal of Botany 41: 465–480.[CrossRef]
Chuine I, Cour P, Rousseau DD. 1998. Fitting models predicting dates of flowering of temperate-zone trees using simulated annealing. Plant, Cell and Environment 21: 455–466.[CrossRef]
Chuine I, Cour P, Rousseau DD. 1999. Selecting models to predict the timing of flowering of temperate trees: implications for tree phenology modelling. Plant, Cell and Environment 22: 1–13.
Citadin I, Raseira MCB, Herter FG, Baptista da Silva J. 2001. Heat requirement for blooming and leafing in peach. HortScience 36: 305–307.
Dacünha-Castelle D, Duflo M. 1982. Probabilités statistiques. I. Problèmes à temps fixe. Paris: Masson.
Dennis B, Kemp KP, Beckwith RC. 1986. Stochastic model of insect phenology: estimation and testing. Environmental Entomology 15: 540–546.
Dodd JC, Estrada A, Jeger MJ. 1992. Epidemiology of Colletotrichum gloeosporioides in the tropics. In: Bailey JA, Jeger MJ, eds. Colletotrichum biology, pathology and control. Wallingford: CAB International.
Frenguelli G, Bricchi E, Romano B, Mincigrucci G, Spieksma FTM. 1989. A predictive study on the beginning of the pollen season for Gramineae and Olea europea L. Aerobiologia 5: 64–70.[CrossRef]
Fripp YJ, Griffin AR, Moran GF. 1987. Variation in allele frequencies in the outcross pollen pool of Eucalyptus regnans F. Muell. throughout a flowering season. Heredity 59: 161–171.
Gentry AH. 1974. Flowering phenology and diversity in tropical Bignoniaceae. Biotropica 6: 64–68.[CrossRef]
Hänninen H. 1987. Effects of temperature on dormancy release in woody plants: implications of prevailing models. Silva Fennica 21: 279–299.
Hof L, Keiser LCP, Elberse IAM, Dolstra O. 1999. A model describing the flowering of single plants, and the heritability of flowering traits of Dimorphotheca pluvialis. Euphytica 110: 35–44.[CrossRef]
Ickovic MR, Boussioud-Corbières F, Sutra JP, Thibaudon M. 1989. Hay fever symptoms compared to atmospheric pollen counts and floral phenology within Paris suburban area in 1987 and 1988. Aerobiologia 5: 30–36.[CrossRef]
Ims RA. 1990. The ecology and evolution of reproductive synchrony. Tree 5: 135–140.
Janick J, Moore JN. 1975. Advances in fruit breeding. West Lafayette: Purdue University Press.
Kramer K. 1994. Selecting a model to predict the onset of growth of Fagus sylvatica. Journal of Applied Ecology 31: 172–181.[CrossRef]
Lehman EL. 1983. Theory of point estimation. New York: Wiley and Sons.
Lescourret F, Blecher N, Habib R, Chadoeuf J, Agostini D, Pailly O, Vaissière B, Poggi I. 1999. Development of a simulation model for studying kiwi fruit orchard management. Agricultural Systems 59: 215–239.[CrossRef]
Mathsoft. 1999. S-Plus 2000 guide to statistics, volume 1. Seattle: Mathsoft.
Medan D, Bartoloni N. 1998. Fecundity effects of dichogamy in an asynchronically flowering population: a genetic model. Annals of Botany 81: 373–383.
Mitchell-Olds T. 1996. Genetic constraints on life-history evolution: quantitative-trait loci influencing growth and flowering in Arabidopsis thaliana. Evolution 50: 140–145.[CrossRef][Web of Science]
Mosseler A, Papadopol CS. 1989. Seasonal isolation as a reproductive barrier among sympatric Salix species. Canadian Journal of Botany 67: 2563–2570.
Murawski DA, Hamrick JL. 1991. The effect of the density of flowering individuals on the mating systems of nine tropical tree species. Heredity 67: 167–174.[Web of Science]
Murawski DA, Hamrick JL. 1992. Mating system and phenology of Ceiba pentandra (Bombacaceae) in Central Panama. Heredity 83: 401–404.
Murfet IC. 1977. Environmental interaction and the genetics of flowering. Annual Review of Plant Physiology 28: 253–278.[Web of Science]
Murray MB, Cannell MGR, Smith RI. 1989. Date of budburst of fifteen tree species in Britain following climatic warming. Journal of Applied Ecology 26: 693–700.[CrossRef]
Nienstaedt H. 1974. Genetic variation in some phenological characteristics of forest trees. In: Lieth H, ed. Phenology and seasonality modelling. New York: Springer Verlag, 389–400.
Normand F, Habib R. 2001. Phenology of strawberry guava (Psidium cattleianum) in Réunion Island. Journal of Horticultural Science and Biotechnology 76: 541–545.
O’Brien SP, Calder DM. 1993. Reproductive biology and floral phenologies of the sympatric species Leptospermum myrsinoides and L. continentale (Myrtaceae). Australian Journal of Botany 41: 527–539.[CrossRef]
Osawa A, Shoemaker CA, Stedinger JR. 1983. A stochastic model of balsam fir bud phenology utilizing maximum likelihood parameter estimation. Forest Science 29: 478–490.
Pors B, Werner PA. 1989. Individual flowering time in a goldenrod (Solidago canadensis): field experiment shows genotype more important than environment. American Journal of Botany 76: 1681–1688.[CrossRef]
Primack RB. 1980. Variation in the phenology of natural populations of montane shrubs in New Zealand. Journal of Ecology 68: 849–862.[CrossRef]
Sprent P. 1989. Applied nonparametric statistical methods. London: Chapman & Hall.
CiteULike 
Connotea 
Del.icio.us What's this?
This article has been cited by other articles:
|
|
 |
|
  B. G. Forde Is it good noise? The role of developmental instability in the shaping of a root system J. Exp. Bot., October 1, 2009; 60(14): 3989 - 4002. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||