^{ 1 }Department of Mathematics and Physics, Agricultural University, Plovdiv, Bulgaria

^{ 2 }Department of Viticulture, Agricultural University, Plovdiv, Bulgaria

^{ 1 }

^{ 2 }

^{ 1 }

^{ * }corresponding author: letchov@au-plovdiv.bg

The process related to the changes in dimensions and mass of grape berry passes through two growth phases separated by a lag phase, and can be described by a double sigmoid curve. The onset of the growth phases and their duration are important factors for understanding the growth processes in grape berries. A new method for their quantitative determination was developed in the present study. In this method, the phase transition dates correspond to the times at which the rate of change of the curvature of the logistic (sigmoid) curve reaches an extreme value. The method was tested on three seedless grape varieties, Sultanina, Ruby Seedless and Rusalka 3, and the changes in grape berry dimensions and mass were tracked from anthesis to harvest. For each of the varieties, a double logistic model of change in berry length, width and mass from anthesis to harvest was developed and the metrics of growth - beginning, stabilisation and end of growth - for each of the two phases were determined. It was found that the metrics in mass and berry dimensions do not match and shift relative to each other over time. A comparison of growth metrics with phenological metrics, such as anthesis, veraison and ripening, showed that phenophases cannot be used as a time scale to record the acceleration of growth processes, as they shifted in time with growth metrics. An exception was veraison, which coincided with the beginning of the accelerated growth of grapes during the second growth phase, following the lag phase. The time scale presented in the current research is a new tool for monitoring growth processes and could help clarify the links between visible changes in the grape berries and the ongoing processes within them. The developed method can also be used for the analysis of various growth processes that follow the logistic law.

The process of grape berry development has been intensively studied to improve the quantity and quality of production. Grape berry growth is a dynamic process, which includes a complex sequence of molecular genetics and biochemical changes (

The current study was mostly inspired by Coombe's research (

The accurate identification of the onset and duration of growth phases is important for producers and processors, as this can affect the quality and management of the grape crop. Tracking the growth of berries is usually done by observing phenological events such as budburst, anthesis, veraison and ripening. In the present study a double logistic model was applied to the timing of events in grape berry growth in terms of berry mass (M) and the berry dimensions length (L) and width (W). To describe growth, two simple sigmoids are considered in a single model, explaining two growth cycles. The sigmoid model is similar to

A method based on the double logistic model has been developed for timing the main events in berry growth: beginning, stabilisation and end of growth for each of the two growth phases. The growth metrics are defined following the methodology of

The data were collected in 2010 from the vineyards in the experimental fields of the Agricultural University of Plovdiv in Bulgaria (42.05N, 24.77E; 300 m asl), located at the foot of the Rhodope Mountains. Table grape cultivars with stenospermocarpic seedlessness were studied: Sultanina, Ruby Seedless and Rusalka 3 - part of the Ampelographic Collection of the Department of Viticulture at the University. The vines are grown on a Moser training system with a pruning load of 8 to 10 spurs with two winter buds and one cane with 11 winter buds. The vines are grafted onto a

In the vineyards, 30 vines of each variety were selected to study grape growth. During the year, every seven days from the emergence of flower buttons to harvest, five bunches of each studied cultivar were selected. Each cluster was conditionally divided into three longitudinal sectors (^{©}), and the fresh mass with an accuracy of 0.001g using analytical scales (Bosch^{©} SAE 200 and Bosch^{©}-waagen seit 1852).

In parallel with the measurements described above, systematic phenological observations were carried out each week to record the following phenological growth stages: budding (BBCH code: 07 and 08), first leaf unfolded and spread away from shoot (code: 11), inflorescence emerge (code: 53), flowering (code: 61, 65, 69), ripening of berries (code: 81, 83, 85), berries ripe for harvest (code: 89)/ technological maturity (brix/acid ratio ≥ 25).

Grape berry growth consists of two successive cycles, each with distinctive characteristics (_{1} and g_{2}, which describe the first and second growth stages:

(1) G(t) = g_{1}(t; _{1}
_{2}(t; _{2}
_{1} and _{2} are the vectors of the parameters

During each phase, the growth parameters (berry fresh mass M, width W and length L) follow a sigmoid pattern, and therefore their change can be described by the logistic function (

(2)

+

where

when (g(t)/g_{f}) << 1), _{f} is the value of the growth function at the end of the growth phase;

(3)

The growth function parameters (^{2}) between measured data and model estimates. “Lmfit” high-level interface to non-linear optimisation and curve fitting problems for Python was used to fit the logistics model (Equation 2) to the measured data (

).

After fitting the logistic model to the measured M, L and W, the curvature change rate (CCR) method was applied to identify the moments of onset and end of active growth, as well as the point of time when the growth processes became stable. In this method, the moments of acceleration and stabilisation of growth processes are defined as being when the CCR of the growth curve reaches extreme values (_{0}), stabilisation of growth (t_{inf}), and end of growth (t_{f}). The metrics t_{0} and t_{f} are the moments at which the CCR of the logistic growth curve and the acceleration of growth reach maximum values; t_{inf} is the inflection point of the logistic curve at which time the CCR has a minimum value (Figure 1) and the growth proceeds at maximum rate. The metrics t_{0}, t_{f} and t_{inf} are defined for every growth phase (Figure 2). Here, we determine the extremes of CCR explicitly by applying an approximation to the curvature of a plane curve (see Supplementary data - Appendix 1). In this approximation, the curvature of the logistics curve can be calculated as the second derivative of the logistics function (Equation 2). Then CCR is defined as the first derivative of the curvature of the curve. The extremes of the CCR of the logistics curve give the growth metrics as:

(4a)

(4b)

(4c)

The vertical lines indicate transition dates: beginning of growth (t_{0}), stabilisation of growth (t_{inf}), and end of growth (t_{f}). DOY = day of year.

The vertical lines indicate transition dates: beginning of growth (t_{0}), stabilisation of growth (t_{inf}), and end of growth (t_{f}). DOY = day of year.

The equations (4a, 4b, 4c) give the main transition points of grape berry growth as functions of the berry dimensions (or mass) g_{0} at the beginning of a phase, the maximum of berry dimensions (or mass) g_{f}, and the specific growth rate

DOY = day of year.

DOY = day of year.

Very often the logistic function is used in another form (

The inflorescence began (BBCH code 53) around 20 April (Table 1). The beginning of anthesis was at the end of May and the beginning of June. It was earliest fo Sultanina (25 May) and it was 10 days late for Rusalka 3 (4 June). Full anthesis for all three varieties was in early June. The longest anthesis was for Ruby Seedless (14 days), while for Sultanina and Rusalka 3 it was about half as long (7 to 9 days).

titre du tableau
Varieties
Inflorescence
Anthesis
Veraison
Ripening
beginning
beginning
full
end
beginning
full
end
Sultanina
23 April
25 May
30 May
2 June
20 July
24 July
28 July
10 September
(113)
(145)
(150)
(153)
(201)
(205)
(209)
(253)
Ruby Seedless
April 20
May 31
June 5
June 13
27 July
3 August
7 August
29 September
(110)
(151)
(156)
(164)
(208)
(215)
(219)
(272)
Rusalka 3
19 April
4 June
8 June
10 June
25 July
28 July
1st August
26 August
(109)
(155)
(159)
(161)
(206)
(209)
(213)
(241)

DOY in the brackets

Veraison began at the end of July (between 20 and 27 July) and continued 12 days for Ruby Seedless, 9 days for Sultanina and 8 days for Rusalka 3. It is noteworthy that for Rusalka 3, anthesis and veraison took place most rapidly, lasting for about a week, while for Ruby Seedless, these processes lasted about two weeks. It should be noted that the anthesis-veraison period was about twice as long as the veraison–ripening period for the Rusalka 3 variety.

The mass and linear dimensions (length and width) of the berries in all three grape varieties, Sultanina, Ruby Seedless and Rusalka 3, followed the double logistic growth pattern throughout the growing season (Figure 3), with the coefficient of determination R^{2} being between 0.944 and 0.998. The fitting parameters of the logistic models g_{1} (first phase) and g_{2} (second phase) are presented in Table 2. During each of the two phases, grape berry length, width and mass followed the logistic model with very high coefficients of determination values and low SSE and RMSE. The logistic curve had a better fit for measured sizes and mass during the first growth phase; this is due to the larger differences between the measured and model values of grape size and weight at the end of growth when they reached maturity. One of the possible reasons for this is that the berries begin to dry after reaching maturity, which leads to a reduction in their size and mass.

Vertical lines = growth metrics during the first (green) and second (red) development phase. Sultanina, Ruby seedless and Rusalka 3 grape varieties. DOY = day of year. The coefficient of determination R^{2} refers to the entire growth period.

Vertical lines = growth metrics during the first (green) and second (red) development phase. Sultanina, Ruby seedless and Rusalka 3 grape varieties. DOY = day of year. The coefficient of determination R^{2} refers to the entire growth period.

The parameters _{1} and g_{2} provided valuable information about the growth processes during the first and second growth phases of the grape berries. For both processes, parameter _{1} ≈ 0, as the accumulated biomass (or dimensions) during anthesis, was not taken into account in the present study. For the second phase, _{2} = b_{1f} + d_{1} ≈ _{1}, that is, _{2} represents the available biomass (or dimensions reached) at the end of the first growth phase. As can be seen in Tables 2 and 3, the estimates for M, L and W of grape berries given by the double logistic model are very close to the measured values. For both Sultanina and Rusalka 3 varieties, 39 % of biomass accumulated during the first growth phase, while for Ruby seedless variety, 65 % of the total biomass accumulated. For all three varieties, most of the increase in the linear dimensions L and W occurred during the first growth phase: 63 % of the final length L and width W of the berries for Sultanina, 64.4 % of L and 75 % of W for Ruby Seedless, and 76 % of L and 69 % of W for Rusalka 3.

titre du tableau
Sultanina
Parameter
M
_{1}
M
_{2}
L
_{1}
L
_{2}
W
_{1}
W
_{2}
3.96E-07
1.92E-08
1.90E-06
1.68E-08
1.94E-05
3.73E-06
1.270
2.023
11.29
3.97
10.27
4.58
0.002
1.258
2.62
13.86
1.68
11.85
0.207
0.170
0.243
0.180
0.202
0.132
SSE
0.008
0.656
0.330
4.801
0.669
7.697
R
^{2}
0.999
0.969
1.000
0.948
0.999
0.934
RMSE
0.019
0.173
0.122
0.467
0.174
0.592
Ruby seedless
Parameter
M
_{1}
M
_{2}
L
_{1}
L
_{2}
W
_{1}
W
_{2}
1.34E-05
1.04E-08
1.63E-05
9.42E-09
4.20E-03
2.85E-06
2.276
1.179
11.680
2.489
12.570
2.521
0.018
2.316
4.151
15.640
1.810
14.380
0.149
0.152
0.202
0.166
0.114
0.113
SSE
0.160
0.286
4.241
1.426
0.987
0.479
R
^{2}
0.995
0.981
0.996
0.964
0.986
0.992
RMSE
0.085
0.085
0.439
0.255
0.714
0.110
Rusalka 3
Parameter
M
_{1}
M
_{2}
L
_{1}
L
_{2}
W
_{1}
W
_{2}
3.67E-05
9.36E-09
1.02E-05
9.88E-09
1.19E-03
7.77E-09
1.983
3.151
14.06
1.33
12.95
4.11
0.003
1.986
3.11
17.23
1.70
14.77
0.151
0.152
0.227
0.151
0.143
0.154
SSE
0.039
1.977
1.623
1.330
4.035
4.860
R
^{2}
0.998
0.959
0.999
0.974
0.995
0.942
RMSE
0.041
0.300
0.272
0.246
0.428
0.470

Grape varieties are Sultanina, Ruby and Rusalka 3. Indices 1 and 2 refer to the first and second growth phases. SSE = sum of squares due to error, R^{2} = coefficient of determination, RMSE = Root mean squared error

titre du tableau
Measured
Sultanina
Ruby seedless
Rusalka 3
Mass (M), g
3.3975 ± 0.7332
3.5278 ± 0.3235
5.4493 ± 0.3214
Length (L), mm
17.70 ± 0.75
18.50 ± 0.74
20.49 ± 0.98
Width (W), mm
16.52 ± 1.37
16.91 ± 0.39
19.52 ± 0.57

Sultanina, Ruby seedless and Rusalka 3 grape varieties. Data are mean ± standard deviation (SD).

At the beginning of the first phase, for all three grape varieties, the specific rate of biomass accumulation µ_{1М }varied between 0.15 and 0.21 per day, the specific rate of elongation µ_{1L} ϵ [0.20, 0.24] per day, and the specific rate of widening of the berries µ_{1W} ϵ [0.11, 0.20] per day. At the beginning of the second growth phase, µ_{2М} varied between 0.15 and 0.17 per day, µ_{2L} ϵ [0.15, 0.18] per day, and µ_{2W} ϵ [0.11, 0.15] per day. For Sultanina, μ_{1M} was 18 % greater than μ_{2M}, μ_{1L} was 26 % greater than μ_{2L}, and μ_{1W} was 18 % greater than μ_{2W}. For the Ruby Seedless and Rusalka 3 varieties, the specific growth rates at the beginning of the two phases were almost the same in mass and width: μ_{1M} ≈ μ_{2M} and μ_{1W} ≈ μ_{2W}. The specific rate of lengthening of the berries at the beginning of the first phase was higher than that at the beginning of the second phase for all three varieties; _{1L} > μ_{2L}, with μ_{1L} being greater than μ_{2L }by 18 % for the Ruby Seedless variety, 26 % for Sultanina and 34 % for Rusalka 3.

Table 4 gives the growth metrics during the first and second phases of grape berry development: t_{0} (beginning of active growth), t_{inf} (moment at which growth stabilises) and t_{f }(end of active growth), defined by the CCR method.

_{0} (beginning of active growth)

The increase in linear dimensions, L and W, of the berries began one to two weeks earlier than the start of active biomass production, so that by the end of anthesis there was a slight decrease in the density of the berries. For Sultanina, the onset of the active increase in linear dimensions began 12 days after anthesis had ended (in early June), and the active accumulation of biomass began about three weeks after anthesis. For Ruby Seedless and Rusalka 3, the increase in linear dimensions accelerated at the end of anthesis, while biomass accumulation was delayed by about a week.

_{inf}

The changes in grape mass, length and width stabilised earlier for the Sultanina and Rusalka 3 varieties, with t_{inf} ≈ 178 DOY (26 June), while for the Ruby Seedless variety, growth was delayed by about a week. The stabilisation of the biomass accumulation rate was delayed by about a week compared to the stabilisation of berry growth in width and by about 11 days in length. It should be noted that the stabilisation of growth occurred in the middle of the anthesis-veraison period;

titre du tableau
Sultanina
Metric
M
_{1}
M
_{2}
L
_{1}
L
_{2}
W
_{1}
W
_{2}
t
_{0}
172
206
166
205
165
200
t
_{inf}
183
220
175
218
176
218
t
_{f}
194
233
185
231
187
235
Δ
22
27
19
26
22
35
length of lag phase
12
20
13
Ruby seedless
Metric
M
_{1}
M
_{2}
L
_{1}
L
_{2}
W
_{1}
W
_{2}
t
_{0}
176
218
167
214
161
212
t
_{inf}
192
233
178
228
181
232
t
_{f}
207
248
189
242
202
252
Δ
31
30
22
28
41
40
length of lag phase
11
25
10
Rusalka 3
Metric
M
_{1}
M
_{2}
L
_{1}
L
_{2}
W
_{1}
W
_{2}
t
_{0}
168
225
163
220
160
227
t
_{inf}
183
241
173
235
176
242
t
_{f}
199
256
183
251
192
257
Δ
31
31
20
31
32
30
length of lag phase
26
37
35

Sultanina, Ruby Seedless and Rusalka 3 varieties Indices 1 and 2 refer to the first and second growth phases.

_{f} (end of active growth)

At the end of the first phase, the growth processes rapidly slowed down and growth entered the lag phase. The accumulation of biomass in the first phase subsided about two weeks earlier than the elongation of the berries, and one week earlier than the widening;

_{1} = (t_{f} – t_{0}) (duration of first phase)

For all three varieties studied, the anthesis-veraison phase lasted one and a half months (45 days). The period of active growth Δ_{1} was shorter than the duration of the anthesis-veraison period; this is because active growth did not begin immediately after the end of anthesis and it ended before veraison. The Sultanina variety had the shortest growth period, which lasted from 19 to 22 days (L, W and M);

The duration of the lag phase (defined by growth metrics as the period between the end of the first growth phase t_{1f} and the beginning of the second growth phase t_{20}) is presented in Table 3.

For all three varieties, the duration of the lag phase for berry mass Δ_{M} and width Δ_{W} was shorter than that for berry length. For Sultanina and Ruby Seedless, the deceleration of berry growth in terms of mass and width lasted for less than two weeks (10 to 13 days), while the deceleration in the lengthening of the berries lasted for 3 weeks (20 to 25 days). For the Rusalka 3 variety, Δ_{L} and Δ_{W} lasted for approximately 35 days, while Δ_{M} lasted for 26 days. The slowing down of the growth processes was greater with respect to L and W compared to that of berry mass M. During the lag phase, berry mass increased almost twice as much as growth in L and W, which led to some compaction of the berries. During the first phase, accumulated mass was 24 % for Rusalka 3, approximately 22 % for Sultanina, and 14 % for Ruby seedless. At the end of the first phase, Sultanina and Ruby seedless had increased in length by 9 % and Rusalka 3 by 11 %. The growth in width W during the lag phase was smallest for the Ruby Seedless variety with an increase in berry width during the first phase of 11 %, while for Rusalka 3 and Sultanina it was 12 and 14 % respectively.

_{0} (beginning of active growth)

The onset of active development of Sultanina and Ruby Seedless berries during the second phase began during veraison, with active growth beginning earliest in terms of W, followed by L and then M. For Sultanina, accelerated growth in M and L began during full veraison, while W development accelerated at the beginning of veraison. For Ruby seedless, W and L growth began during full veraison, while accelerated biomass production began at the end of veraison. For the Rusalka 3 variety, the growth processes accelerated after veraison. Berry elongation started one week after veraison, while widening and biomass accumulation began two weeks after veraison. In the second phase, the accelerated growth of Sultanina berries began the earliest at the end of July, followed by Ruby Seedless in early August and Rusalka 3 in late August. For Ruby Seedless, accelerated biomass accumulation began about a week later than the accelerated development of the linear dimensions of the berries. Acceleration was delayed by almost a week for Rusalka 3 (5 days) and for Sultanina (6 days) in terms of length and width respectively.

_{inf} (moment

During the second phase, the growth processes of the Sultanina variety stabilised the earliest at the beginning of August (218 to 220 DOY), while those of the Ruby Seedless variety stabilised in the middle of August. For the varieties Sultanina and Ruby Seedless, growth stabilisation occurred before the middle of the veraison-maturity period, while for Rusalka 3 it was delayed by 10 to 20 days and was reached at the end of August, shortly before this variety reached technological maturity.

_{f} (end of active growth)

The end of the second phase (t_{2f}) - marked by accelerated attenuation of growth processes due mainly to the mechanisms of cell size enlargement - occurred the earliest for the Sultanina variety at the end of August, for Ruby Seedless at the beginning of September, and for Rusalka 3 in the first half of September.

_{2} = (t_{f} – t_{0}) (duration of the second phase)

The duration of biomass accumulation in the second phase was about one month (30 days), while the growth period in length was the shortest for the Sultanina variety (25 days); for Ruby Seedless it lasted 28 days and for Rusalka 3 it lasted 30 days. In the second phase, the widening period varied, being the shortest for the Rusalka 3 variety (30 days), while Sultanina lasted 35 days and Ruby Seedless 41 days. For the Sultanina variety, the period of active development of the berries began during veraison and ended three weeks before the onset of technological maturity. Because the veraison- ripening period lasted 44 days, it was about 1.5 to 1.7 times longer than the active growth duration Δ_{2}. In Ruby seedless, the growth processes began immediately after the end of veraison and ended three weeks before technological maturity. The veraison-maturity period lasted 53 days, or about twice as long as the M and L and 1.3 times longer than the W growth period. For the Rusalka 3 variety, the growth processes in the second phase began one week for L and two weeks for M and W after the end of veraison, and ended two weeks after reaching maturity. The duration of the second phase was 30 days and was longer than the period veraison-technological maturity (23 days). This anomaly is most probably due to the fact that harvest was carried out before reaching technological maturity.

For all three varieties, the duration of the first phase Δ_{1} was less than or equal to the duration of the second phase Δ_{2}. For Sultanina, Δ_{1} = 0.74Δ_{2}, and for the other two varieties Δ_{1} ≈ 0.9Δ_{2}. The duration of the two phases was almost the same in terms of biomass accumulation and widening; however, the berry lengthening process was longer in the second phase and Δ_{1L} ≈ 0.74Δ_{2L}.

Defining growth metrics allows us to determine the average growth rate during a given phase. During the first phase, the process of biomass accumulation in the berries proceeded at an average rate of 66 mg/day for the three varieties, the process being the fastest in Rusalka 3 at 74 mg/day and slowest in Sultanina at 54 mg/day; meanwhile, the rate was 65 mg/day in Ruby Seedless (Figure 4). Biomass accumulation rates during the second phase of Sultanina and Ruby Seedless were higher: 75 and 104 mg/day respectively. For the Rusalka 3 variety, the rate of biomass growth during the second phase was about half that of the first phase (only 39 mg/day).

The average rates of increase in linear dimensions in the first phase were significantly higher (3 to 6 times) than the growth rates during the second phase. The average lengthening rates were 0.60 mm/day for the first phase and 0.12 mm/day for the second growth phase. The average growth rates in terms of width were 0.11 mm/day in the first phase and 0.39 mm/day in the second phase. Growth processes during the lag phase slowed down, but did not stop, as the rate of biomass accumulation in the berries varied from 0.017 g/day for Rusalka 3 to 0.024 g/day for Sultanina and Ruby Seedless. The slowing down in growth in terms of length and width during the lag phase was significant compared to the first phase. For the Sultanina and Ruby Seedless varieties, the lengthening was about 0.06 mm/day^{and the width increased by 0.09 mm/day. For the Rusalka 3 variety, the growth rates of L and W during the lag phase were the same (0.04 mm/day). The rates of accumulation of biomass during the first phase was 2.4 (Sultanina), 2.8 (Ruby Seedless) and 4.5 (Rusalka 3) times higher than those during the lag phase. The processes of lengthening and widening of the berries during the first phase were about 10 times and 5 to 7 times faster respectively. After the lag phase, there was a 2-fold (Rusalka 3), 3-fold (Sultanina) and 4.5-fold (Ruby Seedless) increase in biomass accumulation rate, while the lengthening was almost twice as fast, and the growth rate in terms of W was 1.5 times faster.}

Over the years, several logistic growth models have been proposed to describe the growth of individual plants, such as the logistic and Gompertz models (

In addition to describing growth well and in detail, the double logistic model allows new metrics to be introduced for tracking the growth of grapes. The developed method - based on the rate of curvature change of the logistic curve - allows the onset of important events in grape berry growth to be explicitly defined. The events in traditional phenology register the visible changes in the berry, while the growth metrics are related to the kinematics of the ongoing processes in the berry, rate and acceleration of the growth processes. Over the years, berry growth has been monitored by examining changes to either one of the dimensions of the berries or berry mass. Naturally, the question arises as to which extensive property (_{0}, t_{inf}, t_{f} shift depending on which physical property of the berries is used (M, L, or W), and the differences between the growth metrics - depending on the property against which they are determined - are greater in the first phase than in the second phase. The onset of the first L and W growth phase for the Sultanina variety began two weeks after the end of anthesis, and the acceleration of M growth began three weeks after anthesis. For Ruby Seedless, the acceleration of L and W growth began at the end of anthesis, while the acceleration of M was delayed by two weeks. In Rusalka 3, the beginning of L and W growth began at the end of anthesis, and M was delayed by about a week. The beginning of the second phase for the Sultanina and Ruby Seedless began during full veraison. For the Rusalka 3 variety, the second phase of L growth began one week after the end of veraison, while active M and W growth was delayed by about two weeks. For Sultanina and Ruby Seedless varieties, berry growth ended 3 to 4 weeks before technological maturity. For the Rusalka 3 variety, growth continued for 2 to 3 weeks after reaching technological maturity. All these differences in the timing of growth metrics were most likely due to the fact that the reasons for these changes differed depending on the physical property that was changed. The shift in time of the growth metrics in relation to phenological events raises the question of interdependence between the visible changes in the berries and the kinematics of the growth processes taking place in them.

When comparing the two types of metrics, phenological and growth metrics, it should be borne in mind that phenological observations were made weekly, which affected the accuracy of these estimates. Therefore, in future studies on the relationship between the two types of metrics, phenological observations and measurements of grape berry mass and dimensions should be performed more frequently, and a more detailed phenological scale should be used (

The value of the specific growth rate µ is an important parameter related to the length Δ of a given phase. The explicit determination of growth metrics (see Equations 4a,b,c) shows that the length of the growth period (t_{f} – t_{0}) depends on the growth rate μ (Equation 4d), provided there is no external influence during growth: _{M}, the length μ_{L} and the width _{W} of the berries at the beginning of the first phase μ_{1} were greater than or almost equal to those of the second phase _{2} for all three varieties. Therefore, the length of growth in the first phase was less than or equal to the length of growth in the second phase.

In general, the duration of growth in a given phase also depends on external factors, such as precipitation and temperature. Therefore, taking into account the relationship between the length of phase Δ and

Grapevine phenology observations and measurements of grape berry dimensions and mass of three seedless varieties of