CAPÍTULO V: LA ACTUACIÓN PROBATORIA – PRUEBA DEL PAGO EN EL
6. Discusión de resultados
Each leaf, other than ‘O’ leaves, on each of the 54 control shoots was grown to the harvest date using the initial leaf area as estimated by (4.9) from the initial, pre- treatment (or establishment) measurements of leaf length and width and accumulated day-degrees using the following steps:
Predict leaf area at harvest (steps C3 to C7) At establishment
Exclude old ('O') leaves (step C1)
Determine largest individual leaf area (step C2) Select shoot i Select leaf j Sum leaf areas for shoot Predict area of leaf pair at harvest ( steps C4 to C7) Sum leaf areas for shoot Predict total area of new season's leaves at harvest for shoot i (Step C9 or T3)
Grow leaves and total shoot leaf area using steps C1 to C7
(step T1)
Using batch size, proportion surviving,
and ELAL predict loss of leaf area
(step T2)
Assume no recruited leaves due to
disbudding so set additional leaf
area to zero (step T3a) Subtract from control leaf area
(step T3b)
Recruit new leaf pair
k (Step C8)
Figure 4.12 Flowchart of a process/simulation model of the impact of browsing by C. bimaculata larvae on E. nitens shoot growth. ELAL is the 'effective leaf area loss per larva'.
j=j+1
k=k+1 i=i+1
Growth without larval feeding (control shoots)
Growth with larval feeding (treatment shoots)
C2) Determine the largest area for an individual leaf for the shoot, Lmax, which will be assumed to be for a fully expanded leaf (see Section 3.2.2).
C3) For each leaf estimate the time of leaf initiation x6 in terms of day-degrees from the 1st August with a 6oC lower threshold temperature where
{
}
[
]
(
)
x6 Te6 c le L c b 1 1 1 = + − − − − − , exp ln ln / max ,le is the leaf area at initial measurement, predicted using length and width measurements and (4.9), Te,6 (=DD[6]) is the day-degrees from 1st of August to the establishment date, and parameters b and c are given by b= −2 0364. and
c=2.2774. This estimate of x6 is based on (3.5) (Section 3.3.1) but note that there the estimates of b and c in Table 3.4 are for the optimum threshold of
3oC . The 6oC threshold was used here so that time of leaf initiation was compatible with the time scale used to predict maximum FELA [see (3.1) in
Section 3.2.2]. The estimates of b and c used here were also obtained by the
RC/LMM estimation procedure (Section 3.2.3).
C4) Scale x6 using a separate ‘shrinkage’ parameter for each of the early and late treatments but common to all shoots at the particular time. The estimation of these parameters will be describe below but they operate by shrinking x6
towards the ‘mean’ value of x6 based on the Weibull ‘distribution’ described by (3.1) to give x *6 x6 x6 x6 x6 * ( ) = +λ − where ) 1 ( 1 0 6 − α + Γ θ + = X x , 0 0 =
X , estimates of θ and α are given in Table 3.2, and λ is the shrinkage parameter. An alternative was to replace x in the above by the time, also 6 calculated in DD[6] units, to peak proportion of maximum FELA (see Fig. 3.6)
which corresponds to the mode of the Weibull distribution. This approach gave considerably poorer predictions, when combined with the rest of the algorithm, than using the mean as was found in Section 3.2.3.
C5) Predict the fully expanded leaf area (FELA), L, for each leaf using (3.1) with x6
*
replacing x.
C6) Predict day-degrees from 1st August to leaf initiation using a 3oC threshold as
{
}
[
]
(
)
x3 =Te,3 +exp c−1ln −ln 1−le /L −c b−1
where Te,3 is the day-degrees with 3oC threshold, DD[3], from 1st of August to the establishment date and parameter estimates for b and c given in Table 3.4
as b= −3 0003. and c=2.3045.
C7) Using (3.4) predict individual leaf area at the harvest date, lh, as
{
}
[
]
lh L Th x b
c c
= 1−exp −( ,3 − 3) / ′
where ln
( )
b′ =−c−1b. If the harvest leaf area was predicted to be less than that at establishment, lh <le, then lh was set equal to le.Leaves recruited during the period between initial measurement and harvest
C8) Recruit and grow new leaves. The number of recruited leaves per shoot was calculated as the difference between final number of leaves harvested and the number of leaves measured on the control shoots for the particular tree at the initial measurement. This number was averaged over the two control shoots for each tree, divided by two, and finally rounded to the nearest integer to give an estimate of the number of recruited leave pairs, n. This reflects the way in
which new leaves are recruited from the naked bud in pairs (Jacobs, 1955, p.21). This estimate of recruitment of new leaf pairs is subject to measurement error since recruitment is inferred from the difference in the final number of leaves compared to the initial number of leaves on each control shoot. However, there were some control shoots for which this difference was negative but generally these negative values were small with -1 being a
common value and the largest negative value being -7. These can be explained as small leaves that were naturally shed during the above period. For the caged control such small, shed leaves would have desiccated and so be
indistinguishable from the general frass in the bottom of the cage control. Also there could have been unintentional losses of leaves at harvest. These negative values were reset to zero. If shed leaves were mostly newly recruited then the above estimate of recruitment can be considered the net gain in new leaves so
that the contribution these leaves make to the predicted leaf area at harvest should be a realistic value. Note that at harvest, leaf area due to newly recruited leaves cannot be determined separately to the leaf area of leaves existing at initial measurement. The number of day-degrees with threshold 3o
C between establishment and harvest, Th,3 −Te,3, was divided by n to give the average
interval between recruitment of new leaf pairs. This process was repeated for the 6oC threshold. Therefore the time of initiation of each simulated leaf pair, in terms of day-degrees from 1st August for each of thresholds 3 and 6oC was known so that steps (C4) to (C7) could then be repeated for these new leaves.
Predicted total leaf area for the shoot at harvest
C9) The predicted total leaf area for the shoot at harvest (excluding ‘O’ leaves) could then be calculated as the sum of ‘grown on’ leaf areas for leaves existing at establishment plus twice the sum of ‘grown on’ leaf areas for leaf pairs recruited between establishment and harvest. This predicted total leaf area was then compared with the actual leaf area at harvest.
Calibration of the simulation algorithm for control shoot growth
To predict growth in leaf area for control shoot leaves the above simulation algorithm was calibrated by estimating the ‘shrinkage’ parameter λ in step (C4). It is clear from Section 3.2.2 that to predict the growth of leaves accurately, good estimates of
FELA are required. Unfortunately, the observed values of FELA were predicted with
poor precision as seen in Fig. 3.6. In addition, the specification of the starting date for day-degree accumulation as input to (3.1) is somewhat arbitrary but reasonable since it occurs within the winter period when growth is at a minimum. However, by using a day-degree scale with lower development threshold and starting day-degree
accumulations from sometime in the winter dormancy period allows a greater generality in the use of (3.1) than simply using Julian days for the time scale. Even so, with temperatures exceeding the 6oC threshold for some of this period then accumulating day-degrees from different dates in this winter period will give
different values for DD[6]. The underlying problem is the unknown time of onset of
the physiological processes which initiate leaf development at both the Esperance sites in 1985 and the Gould’s Block site in 1992.
In an attempt to overcome this problem (3.1) was calibrated to the Gould’s Block experiment by estimating a ‘shrinkage’ parameter λ for the day-degree time scale. For this calibration and to simplify the calculation of day-degree accumulations between the various dates in the above algorithm, X0 in step (C4) was set to zero so that all day-degree accumulations start from 1st August.
The calibration of the growth prediction algorithm for control shoots used a profile log-likelihood for the unknown λin step (C4). This log-likelihood was calculated assuming the sum of squares, over the control shoots, of the residuals for relative growth rate (RG) for given λ, RSS(λ), is distributed as a scaled chi square statistic with single degree of freedom, σ2χ12 where the scale parameter σ2 is the variance of the residuals This approach was used at the individual leaf-level in Section 3.2 to adjust the leaf expansion model. Since the initial leaf area (LAI) for the shoot used in
to calculate RG is the observed LAI then
{
}
{
}
[
]
RSS( )λ =
∑
C S, ln LAF −ln LAFɵ ( )λ 2.
Using a grid of values for λ the profile likelihood was calculated separately for each of the early and late treatments by calculating RSS( ) /λ σ2 using the 36 control shoots in the early treatment and then separately for the 18 control shoots at the late treatment. The value of λ which gives the smallest value of 2
/ ) (λ σ
RSS is the
maximum profile likelihood estimate (MPLE), λˆ. The estimate ofσ2 is
RSS(λɵE) /35 for the early treatment, where λɵE is the MLPE estimate, and similarly for by RSS(λˆL)/17 for the late treatment. For the early treatment the MLPE estimate was λɵE =0 19. while for the late treatment as λɵL =0 28. . These estimates are similar to that obtained in Section 3.2 of 0.2. Figure 4.13 shows the profile log-likelihood and approximate 95% support intervals based on a 2
1
χ .
To compare predictions to actual total shoot leaf areas, measured leaf area at harvest
0
0.1
0.2
0.3
0.4
0
4
8
12
16
20
Shrinkage factor
-2
x
p
ro
fil
e
lo
g-
lik
el
ih
oo
d
Figure 4.13 Profile log-likelihood for shrinkage factor for early (solid line) and late (hashed line) treatments with approximate 95% confidence intervals shown with the offset for clarity and to reflect the smaller sample size for the late treatment.