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Evaluation of presence-absence sampling plans for the diamondback moth (Plutella xylostella) (Lepidoptera: Plutellidae)

Jianhua Mo1, Greg Baker2 and Mike Keller3

1Yanco Agriculture Institute, NSW Agriculture, Yanco, NSW 2703, Australia
2
Entomology Unit, SARDI, GPO Box 397, SA 5001, Australia
3
Plant and Pest Science, Waite Campus, The University of Adelaide SA 5005, Australia.
Corresponding author: jianhua.mo@agric.nsw.gov.au

Abstract

Two sets of sequential presence-absence sampling plans for the management of diamondback moth (DBM) were developed and evaluated, one for the classification of levels of the proportions of plants infested with larvae and the other for the classification of levels of larval density. The action thresholds investigated were 0.15, 0.25, 0.35 and 0.45 for proportion-based sampling plans and 0.2, 0.4, 0.6 and 0.8 larvae/plant for density-based sampling plans. Under the proportion-based sampling plans, the expected correct decision rates were ≥95% for 86-87% of all possible population levels and the expected average sample size was ≤50 plants for 73-87% of all possible population levels. Re-sampling analyses showed average sample sizes of <40 plants in reaching the ≥95% accuracy. For density-based sampling plans, an empirical proportion-density model was first established. The resulting model was highly significant (P<0.001) and explained 97% of the total variation in the independent variable. Satisfactory performance (≥95% accuracy at ≤50 plants sample size) of the density-based sampling plans can be expected when the true population density does not lie in the vicinity of the action threshold. In conclusion, the sequential binomial sampling plans presented here can be used effectively in the monitoring of DBM populations for decision making.

Introduction

Diamondback moth (DBM), Plutella xylostella (L.) (Lepidoptera: Plutellidae), is a key pest of the important vegetable group that includes cabbage, broccoli, cauliflower, collards, rapeseed, mustard and Chinese cabbage. DBM is notorious for its rapid development of resistance to insecticides. In recent years, concerns over the resistance problem, human health and environment have forced the Brassica vegetable industry in many countries to implement some form of integrated pest management (IPM) and insecticide resistance management (IRM) (Talekar & Shelton 1993). However, grower adoption of IPM/IRM strategies has been slow. In Australia, most growers still spray their crops largely on a calendar basis. To encourage more growers to adopt threshold-based spray programs, which is the cornerstone of IPM and IRM, easy-to-use and time-efficient sampling plans are needed, as well as extension effort to convey the benefits of IPM and IRM.

One easy-to-use sampling method is presence-absence sampling. Recording for each sampling unit only the presence/absence of the target pest, presence-absence sampling provides an attractive alternative to enumerative sampling, in which the number of pests in each sampling unit has to be recorded. The advantage is obvious for pest species forming aggregation clusters or those that are easily overlooked because of small size or cryptic behaviour of the pests and when the density of the pest population is high (Jones 1994). Presence-absence sampling is also the logical choice for IPM programs using proportion-based action thresholds, which are commonly used in horticulture crops. Sequential presence-absence sampling, or presence-absence sampling implemented under a sequential rule, provides the additional attractiveness of being potentially time-efficient as it enables decisions regarding pest population levels being made at minimal sample sizes.

Although DBM does not form clusters, its earlier instars are quite small (<5mm) and easily overlooked on their leafy host plants. The larvae also tend to feed in hidden locations and wriggle away or drop down when disturbed. As a result, accurate recording of the number of larvae on a plant can be very difficult under field conditions. This paper investigates the efficiency and reliability of presence-absence sampling in the sequential classifications of DBM population levels under Wald’s (1947) sequential probability ratio test (SPRT). Sequential sampling plans for both proportion-based and density-based action thresholds were developed and evaluated. The action thresholds used were based on those practised by Brassica vegetable growers in Australia. Practical applications of the sampling plans are discussed.

Methods

Data description

Sampling data from four host crops, Brussels sprouts, cabbage, cauliflower and broccoli, collected in three states, South Australia (SA), Victoria (Vic) and Queensland (Qld), were used in this study. Sample sizes ranged from 35 to 300 plants. Data sets with a sample size of less than 100 plants were used for setting up the sampling plans and the rest for validating the sampling plans.

Sequential sampling plans

Under SPRT, a population is classified as below or above a prescribed action threshold (the AT) according to the positions of sample points relative to two parallel stop lines, the lower stop line and the upper stop line. The position of a sample point (n, Tn) is determined by the total number of plants sampled (n) and the number of infested plants found (Tn). The two stop lines are determined by the value of the AT and the distribution of the number of infested plants in the target population. In presence-absence sampling, the subject of interest is the proportion of infested plants in the target population. Since the real proportion of infested plants in a population at any given time is a fixed value, the number of infested plants found during a random sampling process observes binomial distribution, regardless of the patchiness of the distribution of the target organism causing the infestation. Stop lines under binomial distribution are calculated according to Fowler and Lynch (1987):

where p0 and p1 are the nominal proportions of infested plants around the AT (p0 <the AT <p1), α the error rate for recommending control when in fact the infested proportion is below the AT, β the error rate for recommending no control when in fact the infested proportion is above the AT. If Tn <Tlower, the population level is considered below the AT. If Tn >Tupper, the population level is considered below the AT. If Tlower ≤Tn ≤Tupper, the population level relative to the AT cannot be determined and more plants need to be sampled.

In this study, α was set to 0.1 and β to 0.05. The lower β value was chosen to guard against the error of recommending no-control decision when control is needed. Based on personal survey of local Brassica growers, four ATs each were investigated for the classification of the proportions of infested plants (0.15, 0.25, 0.35 and 0.45) and larval density (0.2, 0.4, 0.6 and 0.8 larvae/plant). For the classification of the proportions of infested plants, p0 and p1 were set to AT-0.05 and AT+0.05, respectively. For the classification of larval density, p0 and p1 were set to AT-0.1 and AT+0.1, respectively.

Conversion between proportions of infested plants and densities

Classification of the mean density with presence-absence sampling requires the conversion of density-based ATs into proportion-based ATs. This was done with the inverse of the empirical equation of Gerrard and Chiang (1970):

where p is the proportion of plants infested with larvae, m is the larval density, γ and δ are parameters to be estimated. This equation was used because of its independence from underlying distributions.

Evaluation

Performances of the sampling plans were evaluated with the operational characteristics (OC) and the average sample number (ASN) curves. The OC curve is a plot of the probability of “no intervention” (or no spray) versus the true population level (larval density or proportions of plants with larvae in this study). For each sampling plan, the range of population levels for which OC≤0.05 or OC≥0.95 (OC95) was determined. The OC95 ranges correspond to population levels for which the expected rate of correct classification is ≥95%. The ASN curve is a plot of the average sample size over the population level. As growers do not normally sample more than 50 plants in their monitoring of DBM populations, ranges of population levels for which ASN≤50 plants (ASN50) were determined for each sampling plan. Calculations of the OC and ASN values were done with the algorithms of Nyrop and Binns (1992).

Validation

Validation of the sampling plans was performed by simulated re-sampling of 20 independent data sets. The sample sizes of these data sets ranged from 100 plants to 600 plants. The proportion of infested plants was 0.02-0.56 and the larval density was 0.02-1.46 larvae/plant. Individual plants within a data set were randomly selected with replacement. The initial sample size was set to ten plants and the increment to one plant. Sampling was terminated when a decision could be made with regard to the population level relative to the AT. For each AT, 1000 simulations were run, at the end of which the percentages of simulation runs which correctly classified the population level relative to the AT (Correct%) and the average sample size (ASNsim) were calculated.

Results

Classification of proportions of infested plants

Sampling plans for the four proportional ATs are given in Table 1. The OC curves under these sampling plans were parallel curves centred on their respective ATs (Figure 1). Depending on the action thresholds, the OC95 ranges were found at a minimum of 0.07 proportion units below the AT (proportion of plants with larvae <AT + 0.07) and a minimum of 0.05~0.06 proportion units above the AT (proportion of plants with larvae <AT + 0.05~0.06). These OC95 ranges represented 87∼88% of all possible levels of the proportions of infested plants (0-1).

Table 1. Sequential stop lines at 4 proportion-based action thresholds and 4 density-based action thresholds

Action Threshold

Upper stop line (Tupper)

Lower stop line (Tlower)

15% plants infested (0.15)

2.7762 + 0.1452 n

-3.5643 + 0.1452 n

25% plants infested (0.25)

4.1768 + 0.2477 n

-5.3625 + 0.2477 n

35% plants infested (0.35)

5.0953 + 0.3489 n

-6.5418 + 0.3489 n

45% plants infested (0.45)

5.5524 + 0.4497 n

-7.1285 + 0.4497 n

0.2 larvae/plant

2.0580 + 0.1441 n

-2.6422 + 0.1441 n

0.4 larvae/plant

4.0978 + 0.2709 n

-5.2610 + 0.2709 n

0.6 larvae/plant

5.8037 + 0.3716 n

-7.4512 + 0.3716 n

0.8 larvae/plant

7.2786 + 0.4556 n

-9.3448 + 0.4556 n

As the AT increased from 0.15 to 0.45, peak ASN values increased from 81 plants to 160 plants (Figure 1). The ASN50 ranges were found at a minimum of 0.08~0.15 proportion units below the AT and a minimum of 0.05~0.12 proportion units above the AT (Table 2). The larger separation distances of the two ranges were associated with the higher ATs. Overall, these ASN50 ranges represented 73-87% of all possible population levels, with higher representation shown by sampling plans for the lower ATs (Table 2).

Figure 1. Operating characteristic (OC, =probability of not intervening) and average sample number (ASN) functions for the classification of proportions of plants infested with DBM larvae at 4 action thresholds (15%, 25%, 35%, and 45% infested plants) using sequential presence-absence sampling.

Table 2. Ranges of population levels (prop. of infested plants or larval density) over which the expected correct classification rate was at least 95% (OC95) (corresponding to OC≤0.95 or OC≤0.05) and those over which the expected average sample size is ≤50 plants (ASN50). The matching maximal ASN value or maximal error rate for each range was given for cross-references. Numbers in brackets are percentages of the widths of the specified ranges over the width of the entire population level range (0-1)

 

Action threshold

OC95 range

Maximal ASN

ASN50 range

Maximal error rate

Proportion of

0.15

p<0.08 or p>0.20 (88%)

52

p<0.07 or p>0.20 (87%)

0.05

infested plants (p)

0.25

p<0.18 or p>0.31 (87%)

73

p<0.14 or p>0.33 (81%)

0.01

 

0.35

p<0.28 or p>0.41 (87%)

88

p<0.21 or p>0.46 (75%)

0.00

 

0.45

p<0.38 or p>0.51 (87%)

94

p<0.30 or p>0.57 (73%)

0.00

Number of

0.2

m<0.05 or m>0.31

28

Any m values

1

larvae/plant (m)

0.4

m<0.18 or m>0.58

50

m<0.20 or m>0.58

0.07

 

0.6

m<0.31 or m>0.87

70

m<0.28 or m>1.00

0.01

 

0.8

m<0.42 or m>1.16

85

m<0.33 or m>1.40

0.01

Results of re-sampling analyses of these sampling plans were shown in Table 3. For AT=0.15, the percentage of correct decisions was 100% for all data sets in which the proportions of infested plants differed from the AT by at least 0.05 proportion units. For AT=0.25, 0.35, and 0.45, the percentages of correct decisions were slightly lower (95-99%). The average numbers of plants sampled to reach these decisions were 17, 36, 30, and 35 under the sampling plans for the AT=0.15, 0.25, 0.35, and 0.45, respectively. Over 70% of the data tested within these population ranges had an average sample size of less than 50 plants, irrespective of the ATs. As expected, the rates of correct decisions were much lower and the average sample numbers were much higher for data sets in which the proportions of infested plants were very close to the ATs ( | p - AT | <0.05) (Table 3).

Table 3. Percentages of simulation runs that correctly classified the proportion of infested plants in each of 20 independent data sets relative to each of the four proportional action thresholds (0.15, 0.25, 0.35 and 0.45) (Correct%) and the corresponding average sample sizes (ASNsim). One thousand simulation runs were performed for each data set. Within each data set, individual plants were randomly sampled with replacement. Initial sample size was set to 10 and the sample increment to 1. N=number of plants in the data set, p=proportion of plants with larvae in the data set.

     

Correct%

 

ASNsim

Data

N

p

0.15

0.25

0.35

0.45

 

0.15

0.25

0.35

0.45

S-cabb

600

0.17

83.3

98.4

100.0

100.0

 

71

70

39

26

PCB98

420

0.55

100.0

100.0

100.0

100.0

 

11

15

28

58

PCB99

360

0.63

100.0

100.0

100.0

100.0

 

10

13

20

34

Bshrt11

100

0.73

100.0

100.0

100.0

100.0

 

10

11

14

21

Bshrt12

100

0.71

100.0

100.0

100.0

100.0

 

10

11

15

22

Bshrt29

100

0.2

100.0

100.0

100.0

100.0

 

29

24

20

17

CabIHD1

100

0.34

100.0

99.5

56.2

99.5

 

18

49

163

65

CabIHD2

100

0.63

100.0

100.0

100.0

100.0

 

10

12

20

33

Bshrt2-68

100

0.26

100.0

70.7

98.6

100.0

 

27

122

73

38

Bshrt5-32

100

0.16

73.7

99.0

100.0

100.0

 

82

63

35

25

Bshrt3-10

100

0.12

79.3

100.0

100.0

100.0

 

87

43

30

22

Bshrt26-50

100

0.12

75.9

100.0

100.0

100.0

 

84

43

29

22

Bshrt37-63

100

0.17

82.5

98.7

100.0

100.0

 

78

68

37

26

Bshrt61-69

100

0.13

67.3

99.8

100.0

100.0

 

92

47

31

23

Bshrt44-48

100

0.71

100.0

100.0

100.0

100.0

 

10

11

15

23

Bshrt18-40

100

0.38

100.0

100.0

88.1

95.8

 

15

35

121

98

Bshrt13-56

100

0.35

100.0

99.4

56.1

99.0

 

16

43

158

75

Bshrt19-21

100

0.4

100.0

100.0

100.0

100.0

 

34

27

22

18

Bshrt27-34

100

0.17

80.7

98.6

100.0

100.0

 

75

68

38

26

Bshrt46-57

100

0.19

92.0

94.8

100.0

100.0

 

59

91

43

28

Relationships between p and m

The 108 data sets used in the construction of p-m relationship can be grouped into 5 groups according to the crop variety and the states from which the data were collected: cabbage, cauliflower and broccoli in Victoria, cabbage in Queensland and Brussels sprouts in South Australia. Linear regressions of ln[-ln(1-p)] over ln(m) were performed for each of the 5 groups of data sets (Figure 2). These regression lines did not differ significantly in either the slopes (F=0.9129, df=4, 98, P>0.05) or the intercepts (F=2.2437, df=4, 102, P>0.05). Hence a common regression line was fitted to represent the relationship for all data sets. The resulting regression line was highly significant (F=3104.6903, df=1, 106, P<0.001), explaining 97% of total variation in the dependent variable (Figure 2).

Figure 2. Plot of ln[-ln(1-p)] on ln(m) from 5 groups of data sets and the common regression line, where m is the larval density and p the proportion of plants with larvae present. The two parallel lines on either side of the common regression line show the 95% confidence interval of the regression.

Classification of larval density

Sampling plans for the four density-based ATs are given in Table 1. The OC curves of these sampling plans (Figure 3) were not as steep as the OC curves of the sampling plans for the classification of the proportions of infested plants (Figure 2). As a result, the OC95 ranges were located further away from the ATs (Table 2). To have a ≥0.95 probability of making a no-spray decision, the larval density had to be lower than the AT by at least 0.15 larvae/plant at AT=0.2 larvae/plant and by at least 0.38 larvae/plant at AT=0.8 larvae/plant. Conversely, to have a ≥0.95 probability of making a spray decision, the larval density had to be higher than the AT by at least 0.11 larvae/plant at AT=0.2 larvae/plant and by at least 0.36 larvae/plant at AT=0.8 larvae/plant.

The peak ASN values increased from 41 plants at the AT=0.2 larvae/plant to 160 plants at the AT=0.8 larvae/plant (Figure 3). The ASN50 range at AT=0.2 larvae/plant covered the entire range of possible population levels. For the other three ATs, the ASN50 ranges were found 0.2~0.47 larvae/plant below the AT and 0.18~0.6 larvae/plant above the AT (Table 2). The higher the AT, the further away the ASN50 range was from the AT.

Figure 3. Operating characteristic (OC =probability of not intervening) and average sample number (ASN) functions for the classification of DBM larval density at 4 action thresholds (0.2, 0.4, 0.6, and 0.8 larvae/plant) using sequential presence-absence sampling.

Results of the re-sampling analyses showed that the percentage of correct decisions was >95% for all data sets in which the larval density differed from the AT by at least 0.08 larvae/plant when AT=0.2 larvae/plant and by at least 0.16 larvae/plant when AT=0.6 larvae/plant. (Table 4). At the AT=0.8 larvae/plant, the percentages of correct classifications were consistently high (>99%). The average sample sizes taken to reach these decisions were 18 plants at AT=0.2 larvae/plant, 35 plants at AT=0.4 larvae/plant, 31 plants at AT=0.6 larvae/plant, and 45 plants at AT=0.8 larvae/plant (Table 4). As expected, simulations for those data sets with larval density close to the AT yielded less accurate results and used larger sample sizes.

Table 4. Percentages of simulation runs that correctly classified the larval density in each of 20 independent data sets relative to each of the four proportional action thresholds (0.2, 0.4, 0.6 and 0.8 larvae/plant) (Correct%) and the corresponding average sample sizes (ASNsim). One thousand simulations were run for each data set. Within each data set, individual plants were randomly sampled with replacement. Initial sample size was set to 10 and the sample increment to 1. N=number of plants in the data set, m=number of larvae per plant in the data set.

     

Correct%

 

ASNsim

Data

N

m

0.2

0.4

0.6

0.8

 

0.2

0.4

0.6

0.8

S-cabb

600

0.26

80.6

99.1

100.0

100.0

 

45

55

39

34

PCB98

420

1.18

100.0

100.0

100.0

99.9

 

10

16

34

79

PCB99

360

0.92

100.0

100.0

100.0

100.0

 

10

13

24

44

Bshrt11

100

1.49

100.0

100.0

100.0

100.0

 

10

11

17

27

Bshrt12

100

1.31

100.0

100.0

100.0

100.0

 

10

11

18

29

Bshrt29

100

0.02

100.0

100.0

100.0

100.0

 

22

22

22

22

CabIHD1

100

0.53

99.8

98.2

82.0

100.0

 

15

58

166

82

CabIHD2

100

1.28

100.0

100.0

100.0

100.0

 

10

13

24

44

Bshrt2-68

100

0.28

99.0

98.0

99.7

100.0

 

22

65

54

48

Bshrt5-32

100

0.18

32.5

99.6

100.0

100.0

 

48

49

36

32

Bshrt3-10

100

0.12

69.4

100.0

100.0

100.0

 

52

35

31

28

Bshrt26-50

100

0.12

70.2

100.0

100.0

100.0

 

52

36

30

28

Bshrt37-63

100

0.2

75.0

99.5

100.0

100.0

 

46

53

38

33

Bshrt61-69

100

0.2

61.8

100.0

100.0

100.0

 

52

38

32

29

Bshrt44-48

100

1.46

100.0

100.0

100.0

100.0

 

10

11

18

30

Bshrt18-40

100

0.52

100.0

99.8

32.0

99.3

 

13

39

197

123

Bshrt13-56

100

0.44

100.0

86.0

74.0

99.9

 

14

54

183

92

Bshrt19-21

100

0.04

99.7

100.0

100.0

100.0

 

26

24

23

23

Bshrt27-34

100

0.22

76.0

99.6

100.0

100.0

 

47

52

38

34

Bshrt46-57

100

0.24

87.0

97.8

100.0

100.0

 

37

63

42

36

Discussion

There have been no published reports of binomial sampling for DBM. A sequential sampling plan based on enumerative data was developed for DBM monitoring in Malaysia (Hing & Sivapragasam 1997). Application of this sampling plan elsewhere is limited because of its use of relatively high action thresholds (7-14 larvae/plant) and involvement of a distribution-specific parameter (k of the negative-normal distribution).

Two sets of sequential binomial sampling plans were developed in this study, one targeting the proportion of plants infested with DBM larvae and the other larval density, with the objective of providing simple and efficient monitoring methods for DBM populations. The results showed that all four proportion-based sampling plans performed well in classifying the proportions of infested plants relative to the action thresholds. The expected rate of correct classification was ≥95% for most possible population levels (87-88%). This was confirmed in the simulated re-sampling of 20 independent data sets, which showed a 100% correct classification rate for all data sets in which the proportion of infested plants was not in the immediate vicinity of the action threshold (separated by >0.05 proportion units). Although the expected sample size to achieve the ≥95% accuracy was up to 94 plants, a sample size of 50 plants was sufficient when the true population level differed from the action thresholds by a minimum of 0.07-0.15 proportion units. In fact, simulated re-sampling of actual data showed average sample sizes of <40 plants in reaching ≥95% accuracy.

Classification of larval densities relative to action thresholds using sequential binomial sampling plans involves the additional step of converting the density-based action thresholds into proportion-based action thresholds. This inevitably introduces some errors to the sampling plans due to the uncertainty of any proportion-density models (Schaalje et al. 1991). As a result, the expected performance of the density-based sampling plans was not as satisfactory as the proportion-based sampling plans. However, satisfactory performance (≥95% accuracy at ≤50 plants sample size) can be expected from the density-based sampling plans if the true population density is some distances away from the action threshold (>0.38 larvae/plant below the action threshold and >0.57 larvae/plant above the action threshold). The required separation distances were much smaller for the lower action thresholds. This was again confirmed in the re-sampling analyses of 20 independent data sets. A correct decision rate of >95% was achieved at an average samples sizes of ≤ 45 plants for all data sets in which the true population density differed from the action thresholds by a minimum of 0.16 larvae/plant. Presence-absence sampling is a special form of binomial sampling with a cut-off point of 1. Many studies have shown that the precision of classifying population densities with sequential binomial sampling can be increased by using higher cut-off points (Binns 1990, Boeve & Weiss 1997, Naranjo et al. 1996, Nyrop & Binns 1991). Other cut-off points were not investigated in this study as the action thresholds practised by most Brassica vegetable growers in Australia are mostly below one larva/plant.

As in any sequential sampling plan, when the true population level was close to the action threshold, the probability of making correct decisions were relatively low (<0.95) and the average sample sizes were relatively high (up to 160 plants). To avoid excessive sample sizes with no promise of significant improvement on sampling precisions, it is suggested that the maximum sample size be limited to 50 plants. If no decision is reached by that sample size, growers are advised to make their own decisions according to other relevant information such as historical pest level data, crop development stage and pheromone trapping data. Termination of a sequential sampling plan in this way introduces errors (Fowler & Lynch 1987). However, the likelihood of taking such an action is not high, as seen from results of this study.

Selection between the two sets of sampling plans depends on grower's primary concern about DBM infestation and crop development stage. If the primary concern is the proportion of plants infested with DBM larvae, or the crop is in a development stage sensitive to DBM damage and one larva is sufficient to cause significant damage to the host plant, then the use of proportion-based sampling plans would be the better choice. If the primary concern is the abundance of DBM larvae in the crop, or the crop is in a development stage not particularly sensitive to DBM damage and the total damage to a host plant is the accumulative work of all larvae, then density-based sampling plans may be considered.

The proportion-based sequential presence-absence sampling plans developed here can also be used in multi-pests situations. In these situations, infestations by individual pest species can be combined and treated as the infestation by a single pest species. The total number of plants infested with any one of the pest species still follows the binomial distribution. Since the development of a proportion-based sequential presence-absence sampling plan is affected only by the nominal action thresholds and allowed error rates and not by any species-specific parameters, the same sampling plans can be used. However, there are two things that need to be considered when applying the sampling plans to multi-pests situations. Firstly, the action threshold may need to be adjusted. Theoretically, the new action threshold can be calculated as 1 - Π [1-at(i)], where Π denotes the product and at(i) is the action threshold of species i. In practice, if the action thresholds for individual pest species are somewhat arbitrary in the first place, it is just as acceptable for the grower to nominate the new action threshold based on his/her tolerance level to the combined infestation of the pest species concerned. Secondly, it only makes sense to combine those pest species for which the management actions are similar. Ideally the pest species to be combined should also have similar feeding habits and growers have similar tolerance levels to them, such as lepidopteran pests in Brassica crops.

Acknowledgements

We wish to thank Ms N. Endersby of the Department of Natural Resources & Environment of Victoria and Mr. J. Duff of the Department of Primary Industries of Queensland for providing some of the data. This work was part of a Horticulture Australia Limited-funded project on the management of DBM.

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