Frequency Distribution of Citrus Rust Mite (AcariEriophyidae) Damage on Fruit in Hamlin Orange Trees
Y YANG J C ALLEN J L KNAPPl ANDP A STANSLy2
Department of Entomology and Nematology University of Florida Gainesville FL 32611
EnvironEntomol24(5)1018-1023(1995)ABSTRACT Frequency distribution of the citrus rust mite Phlfllocoptruta oleivora (Ash-mead) damage on Hamlin orange Citrus sinensis fruit was studied from 24 August to 13October 1993 in Lake Alfred FL The study plot consisted of 4-yr-old Hamlin orange treeswith a north-south row OIientation Fruit on the north quadrant of the tree were found tohave the highest mean surface damage followed by the east south and west quadrants Thefrequency distribution of fruit surface damage changed with mean damage levels Then themean damage was low most of the fruit had no rust mite damage With increasing mean fruitdamage the proportion of fruit without damage decreased and the proportion of fruit withhigher damage correspondingly increased The resulting frequency distribution changed froman exponential decay curve to a more or less symmetrical unimodal curve with the peakshifting toward higher damage classes as mean fruit surface damage was increased The fre-quency dishibution was fitted to a 2-variable logistic distribution function of mean fruit surfacedamage and damage class using maximum likelihood estimation method Fruit without rustmite damage was considered a discrete point at zero and its relative frequency was determinedas the height of the cumulative logistic at zero The model approximated the actual data wellat low mean fruit surface damage but gave a poor fit at high mean values
KEY WORDS Phyllocoptruta oleivora frequency distribution yield loss model
EXTENSIVEFEEDING BY the citrus rust mite Phyl-locoptruta oleivora (Ashmead) causes fruit surfacediscoloration (russet) (Albrigo and McCoy 1974McCoy and Albrigo 1975) and reports indicatethat heavy surface russet reduces growth and in-creases drop of the damaged fruit (Allen 19781979 Yang et a1 1994) Reduced fruit grade andgrowth and increased fruit drop directly affect cit-rus crop yield Mite damage is not equally distrib-uted over all the fruit in a grove (Hall et a1 1991)and furtllermore only a high percentage of surfacedamage shows obvious effects on fruit growth anddrop (Allen 1978 1979) It is therefore importantto know the fractions of fruit in a grove that fallinto different damage categories (the frequencydistribution) This would then permit us to calcu-late average losses over the distribution from re-duced fruit grade reduced growth and increaseddrop (Allen 1978 Allen et al 1994) Allen andStamper (1979) reported tllat the relative frequen-cy distribution of mite damage on Valencia andPineapple orange Citrus sinensis and on Dun-can grapefruit Citrus paradisi can be describedwith a modified beta distribution with the mean
ICitms Research and Education Center Lake Alfred FL33850-2299
as its only parameter Because zero is the lowerlimit of the beta distribution proportion of fruitwithout damage (at zero) cannot be estimated Inthis article we seek to develop a simpler c1osed-form cumulative distribution function that avoidsthe somewhat awkward beta function in integralform Our purpose was 2-fold 1st to determine thefrequency distribution of percentage of damage onHamlin orange fruit and 2nd to express the dis-tribution in terms of the mean percentage of sur-face damage with a simple mathematical formulawhich could be used to construct loss models fromrust mite damage
Materials and Methods
This study was conducted at a commercial citrusgrove in Lake Alfred FL from 24 August to 13October 1993 when late-season fruit surface dam-age occurred The grove consisted of 4-yr-oldHamlin orange Citms sinensis trees on Swinglerootstock The trees were =2 111 high The studyplot consisted of an area of =2 ha with 8 rows oftrees running from north to south each row con-sisting of =35 trees The sampling area was locatedat the center of the study plot Ten trees weretagged at each of the central 6 rows before anyvisible mite damage occurred Ten fmit at 05-15
0046-225X951018-1023$02000copy 1995EntomologicalSocietyof America
Octolwr 1995 YANGET ALFREQUENCYDISTRIBUTIONOF MITE DAMAGEON FRUIT 1019
41tilIIIEIIItICIII41
50
40
30
20
10
o230
bull Northo Eastbull Southltgt West
240 250 260 270Julian day (1993)
280
Table 1 Estimates for parameters nand b of cquation2 with different levels of mcan fruit surface damage
Iil 1 OhselVed distlibution of damaged fmit ontrees (Lake Alfred FL 1993)
m above ground were selected randomly fromeach of the 4 quadrants (south east north andWtst) of a tagged tree for a total of 40 fruit pertree Sampling was made once every 1-2 wk for atotal of 6 sample dates The total number of fruitfor each sample date was 40 X 10 X 6 (2400) The2400 randomly selected fruit were assumed to berepresentative of all fmit in the grove at the timeof sampling
The percentage of fruit surface area damagedby the rust mite was recorded for each fruit Thiswas determined by visually estimating the per-ctntage pf russeted area on one side of each fruitand then turning the fruit 1800 and repeating theestimation on the other side Damage estimationwas based on the portion of the fruit surface thatwas completely discolored with a resolution of 5The region under direct solar exposure was usuallyavoided by the citrus rust mite and was consideredas undamaged A comparative study by IGA in-dicated that average variation in damage estima-tion for thc samc person and among different peo-ple was 5-10 The study plot was under regularmanagement during the study vithout any pesti-cide application
Fruit were grouped into a zero class and 10intervals of percentage of surface damage that is0 1-10 11-20 91-100 Frequency datawere fitted to a logistic distribution function usingthe maximum likelihood estimation procedure byDennis et al (1986)
Results
Fruit surface damage in the study grove oc-curred late in August because of a slow buildup incitrus rust mite population Mean fruit surfacedamage was about 06 on 24 August but in-creased quickly during September By 13 Octoberthe mean fruit surface damage reached 34 (Fig1 Table 1)
Damaged fruit were not equally distributedamong the 4 quadrants of a tree (Fig 1) In theearly stage of mite damage when the mean fruit
surface damage was low fruit on the east quadrantof the tree had the highest mean surface damagefollowed by the north quadrant But in the latestage of mite damage when the mean fruit surfacedamage was increased fruit on the north quadrantof the tree had the highest mean surface damagefollowed by the east quadrant The west quadrantalways had the lowest mean surface damage Bythe time of the last observation (13 October 1993)mean damage for the north east south and westquadrants were 42 39 30 and 24 respectivelyBecause fruit surface damage is directly related tototal mite population supported by the fruit thenorth side of the tree should have the highest mitepopulation followed by the east south and westPublished research showed that citrus rust miteswere usually unevenly distributed in trees (Yothersand Mason 1930 Albrigo and McCoy 1974 Allenand McCoy 1979 and Hubbard 1885)
Frequency Distribution of Damaged FruitThe frequency distribution of fruit surface damagechanged with mean damage (Fig 2) When themean damage was low most of the fruit had norust mite damage With the increase of mean fruitdamage the proportion of fruit without damagedecreased and the proportion of fruit with higherdamage correspondingly increased The resultingfrequency distribution changed from an exponen-tial decay curve to a more or less symmetricalllni-modal curve with the peak shifting toward higherdamage classes as mean fnlit surface damage wasincreased (Fig 2)
Allen and Stamper (1979) tested beta gammaand normal distributions for their frequency dis-tribution data on Valencia and Pineapple orangesand grapefruit they found beta distribution gavethe best fit to their data These 3 theoretical dis-tributions have no closed forms for their cumula-tive density functions Furthermore beta and gam-ma distributions are discontinuous at point zeroThis means proportion of fruit without damage(that is zero damage) cannot be determined usingthe fitted beta or gamma distribution But fruitwithout damage is of primary interest as far asmanagement is concerned
These considerations led us to look for alterna-tive distribution functions Based on the trend of
1020
2500
s 2000bull--0 1500bull(1)c 1000EJ 500Z
o
ENVIRONMENTAL ENTOMOLOCY
o 0 0 0 0 0 0 0 0 0 0~ N M v ~ w ~ ~ ~ 0~
Damage class ()
Vol 24 no 5
13-0ct
27-Sep
14-Sep
Date07-Sep
31-Aug
24-Aug
Fig 2 Obselved frequency distribution of mite damage over time 011 fruit (Lake Alfred FL 1993)
Using equation 2 and the maximum likelihoodestimation procedure by Dennis et al (1986) pa-rameters a and h were obtained for each of the 6sets of data (Table 1) They were found to changewith mean fruit surface damage (IL) We therefore
1 1F(x) = A [ (x _ a)] (2)
1 + exp ---b-
where F(x) = the proportion of fruit with a per-centage damage of less than or equal to x and
1A = F(lOO) = ------
1 + exp( _100b- a)
Equation 2 should be interpreted as follows theproportion of fruit without damage is the cumu-lative density up to zero that is F(O) the propor-tion of fruit between damage (XI X2) is F(X2) -F(XI) where Xl gt 0 X2 gt 0 and X2 gt Xl Equation2 reaches a value of 1 at the upper limit of x =100 This makes its probability density function in-tegrate to 1 between (-00 100) The densityfunction is
the frequency data (Fig 2) we used the logisticdistribution to fit the frequency data using themaximum likelihood estimation procedure byDennis et aI (1986) The logistic distIibution func-tion is
1F(x) = [ (x _ a)] (1)
1+ exp ---b-
where a b = parameters The mean of the logisticdistribution is a the variance of the logistic distri-bution is
2~b23
(Patel et a1 1976) We found that the frequencydistribution over damage classes (0 100) in ourstudy could be described using equation 1 Thisapproach assumed that the frequency at zero dam-age class was the height of the cumulative logisticdistribution at zero
Although the logistic distribution (equation 1)describes a continuous distribution from negativeinfinity to positive infinity our damage classes arelimited to the range of (0 100) Because the fre-quency at zero damage class was assumed theheight of the cumulative logistic distribution atzero we were dealing with a range of (-00 100)in the logistic distribution The truncated logisticdistribution is
Tnble 2 Pnrumetr timntc for Intion 4 wId 5 by 2 different method
1021
Mthod Equation Pardmeter Parameter Parameter
MLE-NLlN 4 ao = -110444 al = 12588 a2 = -2140175 ho = 116649 h1 = -42538 h2 = 020180
MLEd 6 ao = -118936 al = 12954 a2 = -160044ho = 123524 hI = -55660 h2 = 0]99]
R2
0996809472093631gtNAC
x2936698g1JNAC8441892h
NN
Iit Iuations 4 and 5 separately using the maximum likelihood estimates for II and b from each of the 6 sets of dataIgt Rsults for the combined 6 spts of data Not applicabled InsNt 1uations 4 and 5 into equation 6 before data fitting using maximum likelihood estimation method
assumed that parameters a and b were functionsof mean fmit surface damage (p) that is a(p) andb(p) The following function was found to give agood fit to parameter a in relation to mean fmitsurface damage (p)
a(p) = ao + alP + a2exp(-P) (4)where 0o a a2 = parameters The following func-tion was found to give a good fit to parameter b(p)
h(p) = ho + blexp( -h2p) (5)
where b() hIgt h2 = parameters The final form oftil( cumulative frequency distribution was the fol-lowing 2-variable logistic function of mean fmitsurface damage (p) and damage class (x)
1 1F(x p) = - ------ (6)
A 1 + exp[- x ~(P)]where a(p) and b(p) are functions of p as definedin equations 4 and 5 The corresponding probabil-ity density function is
1 [x - a(p)]f(x p) = 2X bW ex
p[ - b(P)] 2 (7)
A x - a(p)1 + exp - h(p)
In equations 6 and 7
1
A = F(lOO) = [ I100 - a(p)1 + exp - h(p)
45 120
30 115Q
III 15 bullbull~41 110Gi
0 EE 0 Parameter 8 ~lUbull bull Parameter b 105 lUlU middot15 QQ
-30 1000 10 20 30 40
mu = Mean damage ()
Fig 3 Helationshipbetween parameters a and h inthe logisticequation (equation2) and mean fruit surfacedamage
We used 2 methods to estimate parameters inequations 4 and 5 In the 1st method parametersa and b in equation 2 were estimated using datafor each sample date using the maximum likeli-hood estimation procedure by Dennis et al (1986)Parameters in equations 4 and 5 were then esti-mated using estimates for parameters a and b fromeach of the 6 sets of data using SAS-NLIN pro-cedure (SAS Institute 1985) Parameter estimatesare shown in Table 2 This is a 2-step method Ina better approach we replaced a(p) and b(p) inequation 6 with equations 4 and 5 and then usedthe maximum likelihood estimation procedure byDennis et aI (1986) for simultaneous estimation ofall 6 parameters (ao a] a2 b() bl b2) based on theoriginal 6 sets of data The results are shown inTable 2 Based on the chi-square values the si-multaneous estimates gave a better fit than the2-step method Parameter estimates from thisI-step method were then used in the model Thuswe have
a(p) = -118936 + 12954p (8)
- 160044 exp(- p)
b(p) = 123524 - 55660 exp(-O1991p) (9)
The relationships of parameters a(p) and b(p) tomean fmit surface damage (p) are shown in Fig3 The predicted frequency distribution is shownin Fig 4 The probability density function can beobtained by replacing parameters a(p) and b(Jl) inequation 7 with equations 8 and 9 The predictedprobability density function is shown in Fig 5where probability for 0 damage class is not shownThe probability for 0 damage class can be calcu-lated from equation 6 by setting damage x at O
Discussion
Properties of the CwnuIative Frequency Dis-tribution Function The logistic distribution func-tion (equation 1) has been used to model insectphenology (Dennis et al 1986 Kemp et al 1986Dennis and Kemp 1988) as a stochastic processHere we used the truncated logistic function(equation 6) for describing the frequency distri-bution of mst mite damage on citms fmit Asshown in Fig 3 parameter a(Jl) exhibits a sharpincrease when the mean fmit surface damage islow and a slower linear increase with further in-
1022 ENVIRONMENTAL ENTOMOLOGY Vol 24 no 5
2500=~ 2000bull-- 15000bullQ) 1000cE~ 500Z
o o 0 0 0 0 0 0 0~ N M ~ ~ ~ ~
Damage class ()
oco o 0C1gt 0
13-0ct
Date
Fig 4 Predicted frequency distribution of mite damage over time on fmit (Lake Alfred FL 1993)
mainly caused by large deviations in a few datapoints These deviations might be caused by ran-dom errors Although the chi-square statistic mightreject the hypothesis of the logistic the fit is adf-quate for practical pU1)oses Vhen the mean dam-age was between 0 and 25 model predictionswere at an accuracy of gt75 as compared withthe observed data Model predictions were poor athigher mean damage values
Application of the Cmnulative DistributionFWlction The cumulative frequency distributionfunction (equation 6) will enable us to determinethe proportion of fmit that falls into a specificdamage class if the mean fmit surface damage islmown For example the proportion of fmit thatfalls between damage Xl and X2 is F(x2 IL) - F(x]IL) In commercial citms production it is oftennecessary to determine the proportion of fmit thatcan go to the fresh fruit market If fruit with morethan x percentage of surface damage is rejectedfrom the fresh fruit market then the proportion offruit that can go to the fresh fruit market (thepackout) would simply be
crease in mean damage (equation 8 Fig 3) pa-rameter b(J1) also exhibits a sharp increase but ap-proaches a constant value with further increasingdamage (equation 9 Fig 3) Because a(f1) is themean of the untnmcated logistic distributionwhereas b(J1) is positively related to standard de-viation of the untnmcated logistic distribution thisindicates that as the peak of the density functionshifts towards higher damage there is little changein the variance after the data mean exceeds =20This is similar to shifting a density curve to a high-er mean without changing the shape (variance)High chi-square values (Tables 1 and 2) were
008
iiicC
~004cce~
100 1 1F(x IL) = - ------
A [ x - a(IL)]1+ exp - b(lL)
(10)
Fig 5 Predicted probability density function of mitedamage on fruit where probability at zero damage classis not shown (Lake Alfred FL 1993)
For example if we assume the packout level is x= 5 and the mean fruit damage in a Hamlinorange grove is IL = 10 then the proportion of
October 1995 YANG ET AL FHEQUENCY DISTHIBUTION OF MITE DAMAGE ON FHUlT 1023
fmit that can go for the fresh fmit market can becalculated from equation 10
F(5 10) = [ 1 ] = 05773A 5 - a(10)
1 + exp - b(lO)
where a(10) and b(lO) can be calculated fromequations 8 and 9 respectively This means that5773 fmit have a damage of 510 and theycan go for the fresh fmit market and 4227(100-5773) fmit have a damage gt10 and theycan only go for the processed fruit market
Another intended application of the establishedequation is to determine yield loss from rust mitedamage Rust mite damage reduces growth and in-creases drop of damaged fruit (Allen 1978 1979Allen et aI 1994 Yang et al 1994) But these ef-fects are not uniformly distributed over damageclasses with larger effects on heavily damagedfruit It is therefore necessary to integrate theseeffects over all damage classes based on the fre-qu(ncy distribution of damaged fruit Mathemati-cal models describing the relationships betweenfruit growth and drop and fruit surface damagehave been developed (Allen 1978 1979 Yang etal 1994) Allen et al (1994) established differentialequations to estimate volume loss from reducedfmit growth and drop These differential equationscombine the frequency distribution model withgrowth and drop models The frequency distribu-tion model (equation 6) developed in this studycould also be used in a similar way This modelshould be further improved and tested using fielddata before it could be applied in rust mite man-agement practices
Acknowledgments
We thank James 1 Jones and Harvey L Cromroy(Univfrsity of Florida) for reviewing the manuscript Wealso thank James H Matis (Texas AampM University) andHamon C Littell (University of Florida) for their help indata analysis We thank Coca-Cola for allowing us to con-duct this study in their citrus grove and for their kindnessand help We are also grateful to Dow-ElanCo and Hel-ena Cl1lmical for their financial assistance Published asFlorida Agricultural Experiment Station Journal SeriesNo H-04022
References Cited
Albrigo L G and C W McCoy 1974 Characteristicinjury by citrus rust mite to orange leaves and fruitProc Fla State Home Soc 87 48-55
Allen J C 1978 The effect of citrus rust mite dama~eon citrus fruit drop J Econ Entomol 71 746-50
1979 The effect of citrus rust mite damage on citrusfruit growth J Econ Entomol 72 195-201
Allen J C and C W McCoy 1979 The thermalenvironment of the citrus rust mite Agric Meteorol20 411-425
Allen J C and J H Stamper 1979 Frequency dis-tribution of citrus rust mite damage on citrus fruit JEcon Entomol 72 327-330
Allen J C Y Yang J L Knal)ll and P A Stansly1994 The citrus rust mite story a modelling ap-proach to a fruit-mite-pathogen system pp 619-639In D Hosen F Bennett and J Capinera [eds] lestmanagement in the subtropics biological control-aFlorida perspective Intercept Andover UK
Dennis B and W P Kern) 1988 Further statisticalinference methods for a stochastic model of insectphenology Environ Entomol 17 887-893
Dennis B W P Kemp and R C Beckwith 1986Stochastic model of insect phenology estimation andtesting Environ Entomol 15 540-546
Hall D G C C Childers and J E Eger 1991Estimating citrus rust mite (Acari Eriophyidae) levelson fruit in individual citrus trees Environ Entomol20 382-390
Hubbard H G 1885 Insects affecting the orangeUS Dep Agric Div Entomol Spec Hep 1885
Kemp W P B Dennis and R C Beckwith 1986Stochastic phenology model for the western budworm(Lepidoptera Tortricidae) Environ Entol1lo1 15547-554
McCoy C W and L G Alhrigo 1975 Feeding in-jury to the orange caused by the rust mite Phyll()c()p-tnlta oleiv()ra (Prostigmata Eriophyoidea) Ann En-tomo1 Soc Am 68 289-297
Patel J K C H Kapadia and D B Owen 1976Handbook of statistical distributions Marcel DekkerNew York
SAS Institute 1985 SAS users guide statistics ver-sion 5 ed SAS Institute Cary NC
Yang Y J C Allen J L Knapp and P A Stansly1994 Citms rust mite (Acari Eriophyidae) damageeffects on Hamlin orange fruit growth and drop En-viron Entomol 23 244-247
Yothers W W and A C Mason 1930 The citrusrust mite and its control US Dep Agric Bull 176
Received for publication 4 August 1994 accepted 17Apri1199S
Octolwr 1995 YANGET ALFREQUENCYDISTRIBUTIONOF MITE DAMAGEON FRUIT 1019
41tilIIIEIIItICIII41
50
40
30
20
10
o230
bull Northo Eastbull Southltgt West
240 250 260 270Julian day (1993)
280
Table 1 Estimates for parameters nand b of cquation2 with different levels of mcan fruit surface damage
Iil 1 OhselVed distlibution of damaged fmit ontrees (Lake Alfred FL 1993)
m above ground were selected randomly fromeach of the 4 quadrants (south east north andWtst) of a tagged tree for a total of 40 fruit pertree Sampling was made once every 1-2 wk for atotal of 6 sample dates The total number of fruitfor each sample date was 40 X 10 X 6 (2400) The2400 randomly selected fruit were assumed to berepresentative of all fmit in the grove at the timeof sampling
The percentage of fruit surface area damagedby the rust mite was recorded for each fruit Thiswas determined by visually estimating the per-ctntage pf russeted area on one side of each fruitand then turning the fruit 1800 and repeating theestimation on the other side Damage estimationwas based on the portion of the fruit surface thatwas completely discolored with a resolution of 5The region under direct solar exposure was usuallyavoided by the citrus rust mite and was consideredas undamaged A comparative study by IGA in-dicated that average variation in damage estima-tion for thc samc person and among different peo-ple was 5-10 The study plot was under regularmanagement during the study vithout any pesti-cide application
Fruit were grouped into a zero class and 10intervals of percentage of surface damage that is0 1-10 11-20 91-100 Frequency datawere fitted to a logistic distribution function usingthe maximum likelihood estimation procedure byDennis et al (1986)
Results
Fruit surface damage in the study grove oc-curred late in August because of a slow buildup incitrus rust mite population Mean fruit surfacedamage was about 06 on 24 August but in-creased quickly during September By 13 Octoberthe mean fruit surface damage reached 34 (Fig1 Table 1)
Damaged fruit were not equally distributedamong the 4 quadrants of a tree (Fig 1) In theearly stage of mite damage when the mean fruit
surface damage was low fruit on the east quadrantof the tree had the highest mean surface damagefollowed by the north quadrant But in the latestage of mite damage when the mean fruit surfacedamage was increased fruit on the north quadrantof the tree had the highest mean surface damagefollowed by the east quadrant The west quadrantalways had the lowest mean surface damage Bythe time of the last observation (13 October 1993)mean damage for the north east south and westquadrants were 42 39 30 and 24 respectivelyBecause fruit surface damage is directly related tototal mite population supported by the fruit thenorth side of the tree should have the highest mitepopulation followed by the east south and westPublished research showed that citrus rust miteswere usually unevenly distributed in trees (Yothersand Mason 1930 Albrigo and McCoy 1974 Allenand McCoy 1979 and Hubbard 1885)
Frequency Distribution of Damaged FruitThe frequency distribution of fruit surface damagechanged with mean damage (Fig 2) When themean damage was low most of the fruit had norust mite damage With the increase of mean fruitdamage the proportion of fruit without damagedecreased and the proportion of fruit with higherdamage correspondingly increased The resultingfrequency distribution changed from an exponen-tial decay curve to a more or less symmetricalllni-modal curve with the peak shifting toward higherdamage classes as mean fnlit surface damage wasincreased (Fig 2)
Allen and Stamper (1979) tested beta gammaand normal distributions for their frequency dis-tribution data on Valencia and Pineapple orangesand grapefruit they found beta distribution gavethe best fit to their data These 3 theoretical dis-tributions have no closed forms for their cumula-tive density functions Furthermore beta and gam-ma distributions are discontinuous at point zeroThis means proportion of fruit without damage(that is zero damage) cannot be determined usingthe fitted beta or gamma distribution But fruitwithout damage is of primary interest as far asmanagement is concerned
These considerations led us to look for alterna-tive distribution functions Based on the trend of
1020
2500
s 2000bull--0 1500bull(1)c 1000EJ 500Z
o
ENVIRONMENTAL ENTOMOLOCY
o 0 0 0 0 0 0 0 0 0 0~ N M v ~ w ~ ~ ~ 0~
Damage class ()
Vol 24 no 5
13-0ct
27-Sep
14-Sep
Date07-Sep
31-Aug
24-Aug
Fig 2 Obselved frequency distribution of mite damage over time 011 fruit (Lake Alfred FL 1993)
Using equation 2 and the maximum likelihoodestimation procedure by Dennis et al (1986) pa-rameters a and h were obtained for each of the 6sets of data (Table 1) They were found to changewith mean fruit surface damage (IL) We therefore
1 1F(x) = A [ (x _ a)] (2)
1 + exp ---b-
where F(x) = the proportion of fruit with a per-centage damage of less than or equal to x and
1A = F(lOO) = ------
1 + exp( _100b- a)
Equation 2 should be interpreted as follows theproportion of fruit without damage is the cumu-lative density up to zero that is F(O) the propor-tion of fruit between damage (XI X2) is F(X2) -F(XI) where Xl gt 0 X2 gt 0 and X2 gt Xl Equation2 reaches a value of 1 at the upper limit of x =100 This makes its probability density function in-tegrate to 1 between (-00 100) The densityfunction is
the frequency data (Fig 2) we used the logisticdistribution to fit the frequency data using themaximum likelihood estimation procedure byDennis et aI (1986) The logistic distIibution func-tion is
1F(x) = [ (x _ a)] (1)
1+ exp ---b-
where a b = parameters The mean of the logisticdistribution is a the variance of the logistic distri-bution is
2~b23
(Patel et a1 1976) We found that the frequencydistribution over damage classes (0 100) in ourstudy could be described using equation 1 Thisapproach assumed that the frequency at zero dam-age class was the height of the cumulative logisticdistribution at zero
Although the logistic distribution (equation 1)describes a continuous distribution from negativeinfinity to positive infinity our damage classes arelimited to the range of (0 100) Because the fre-quency at zero damage class was assumed theheight of the cumulative logistic distribution atzero we were dealing with a range of (-00 100)in the logistic distribution The truncated logisticdistribution is
Tnble 2 Pnrumetr timntc for Intion 4 wId 5 by 2 different method
1021
Mthod Equation Pardmeter Parameter Parameter
MLE-NLlN 4 ao = -110444 al = 12588 a2 = -2140175 ho = 116649 h1 = -42538 h2 = 020180
MLEd 6 ao = -118936 al = 12954 a2 = -160044ho = 123524 hI = -55660 h2 = 0]99]
R2
0996809472093631gtNAC
x2936698g1JNAC8441892h
NN
Iit Iuations 4 and 5 separately using the maximum likelihood estimates for II and b from each of the 6 sets of dataIgt Rsults for the combined 6 spts of data Not applicabled InsNt 1uations 4 and 5 into equation 6 before data fitting using maximum likelihood estimation method
assumed that parameters a and b were functionsof mean fmit surface damage (p) that is a(p) andb(p) The following function was found to give agood fit to parameter a in relation to mean fmitsurface damage (p)
a(p) = ao + alP + a2exp(-P) (4)where 0o a a2 = parameters The following func-tion was found to give a good fit to parameter b(p)
h(p) = ho + blexp( -h2p) (5)
where b() hIgt h2 = parameters The final form oftil( cumulative frequency distribution was the fol-lowing 2-variable logistic function of mean fmitsurface damage (p) and damage class (x)
1 1F(x p) = - ------ (6)
A 1 + exp[- x ~(P)]where a(p) and b(p) are functions of p as definedin equations 4 and 5 The corresponding probabil-ity density function is
1 [x - a(p)]f(x p) = 2X bW ex
p[ - b(P)] 2 (7)
A x - a(p)1 + exp - h(p)
In equations 6 and 7
1
A = F(lOO) = [ I100 - a(p)1 + exp - h(p)
45 120
30 115Q
III 15 bullbull~41 110Gi
0 EE 0 Parameter 8 ~lUbull bull Parameter b 105 lUlU middot15 QQ
-30 1000 10 20 30 40
mu = Mean damage ()
Fig 3 Helationshipbetween parameters a and h inthe logisticequation (equation2) and mean fruit surfacedamage
We used 2 methods to estimate parameters inequations 4 and 5 In the 1st method parametersa and b in equation 2 were estimated using datafor each sample date using the maximum likeli-hood estimation procedure by Dennis et al (1986)Parameters in equations 4 and 5 were then esti-mated using estimates for parameters a and b fromeach of the 6 sets of data using SAS-NLIN pro-cedure (SAS Institute 1985) Parameter estimatesare shown in Table 2 This is a 2-step method Ina better approach we replaced a(p) and b(p) inequation 6 with equations 4 and 5 and then usedthe maximum likelihood estimation procedure byDennis et aI (1986) for simultaneous estimation ofall 6 parameters (ao a] a2 b() bl b2) based on theoriginal 6 sets of data The results are shown inTable 2 Based on the chi-square values the si-multaneous estimates gave a better fit than the2-step method Parameter estimates from thisI-step method were then used in the model Thuswe have
a(p) = -118936 + 12954p (8)
- 160044 exp(- p)
b(p) = 123524 - 55660 exp(-O1991p) (9)
The relationships of parameters a(p) and b(p) tomean fmit surface damage (p) are shown in Fig3 The predicted frequency distribution is shownin Fig 4 The probability density function can beobtained by replacing parameters a(p) and b(Jl) inequation 7 with equations 8 and 9 The predictedprobability density function is shown in Fig 5where probability for 0 damage class is not shownThe probability for 0 damage class can be calcu-lated from equation 6 by setting damage x at O
Discussion
Properties of the CwnuIative Frequency Dis-tribution Function The logistic distribution func-tion (equation 1) has been used to model insectphenology (Dennis et al 1986 Kemp et al 1986Dennis and Kemp 1988) as a stochastic processHere we used the truncated logistic function(equation 6) for describing the frequency distri-bution of mst mite damage on citms fmit Asshown in Fig 3 parameter a(Jl) exhibits a sharpincrease when the mean fmit surface damage islow and a slower linear increase with further in-
1022 ENVIRONMENTAL ENTOMOLOGY Vol 24 no 5
2500=~ 2000bull-- 15000bullQ) 1000cE~ 500Z
o o 0 0 0 0 0 0 0~ N M ~ ~ ~ ~
Damage class ()
oco o 0C1gt 0
13-0ct
Date
Fig 4 Predicted frequency distribution of mite damage over time on fmit (Lake Alfred FL 1993)
mainly caused by large deviations in a few datapoints These deviations might be caused by ran-dom errors Although the chi-square statistic mightreject the hypothesis of the logistic the fit is adf-quate for practical pU1)oses Vhen the mean dam-age was between 0 and 25 model predictionswere at an accuracy of gt75 as compared withthe observed data Model predictions were poor athigher mean damage values
Application of the Cmnulative DistributionFWlction The cumulative frequency distributionfunction (equation 6) will enable us to determinethe proportion of fmit that falls into a specificdamage class if the mean fmit surface damage islmown For example the proportion of fmit thatfalls between damage Xl and X2 is F(x2 IL) - F(x]IL) In commercial citms production it is oftennecessary to determine the proportion of fmit thatcan go to the fresh fruit market If fruit with morethan x percentage of surface damage is rejectedfrom the fresh fruit market then the proportion offruit that can go to the fresh fruit market (thepackout) would simply be
crease in mean damage (equation 8 Fig 3) pa-rameter b(J1) also exhibits a sharp increase but ap-proaches a constant value with further increasingdamage (equation 9 Fig 3) Because a(f1) is themean of the untnmcated logistic distributionwhereas b(J1) is positively related to standard de-viation of the untnmcated logistic distribution thisindicates that as the peak of the density functionshifts towards higher damage there is little changein the variance after the data mean exceeds =20This is similar to shifting a density curve to a high-er mean without changing the shape (variance)High chi-square values (Tables 1 and 2) were
008
iiicC
~004cce~
100 1 1F(x IL) = - ------
A [ x - a(IL)]1+ exp - b(lL)
(10)
Fig 5 Predicted probability density function of mitedamage on fruit where probability at zero damage classis not shown (Lake Alfred FL 1993)
For example if we assume the packout level is x= 5 and the mean fruit damage in a Hamlinorange grove is IL = 10 then the proportion of
October 1995 YANG ET AL FHEQUENCY DISTHIBUTION OF MITE DAMAGE ON FHUlT 1023
fmit that can go for the fresh fmit market can becalculated from equation 10
F(5 10) = [ 1 ] = 05773A 5 - a(10)
1 + exp - b(lO)
where a(10) and b(lO) can be calculated fromequations 8 and 9 respectively This means that5773 fmit have a damage of 510 and theycan go for the fresh fmit market and 4227(100-5773) fmit have a damage gt10 and theycan only go for the processed fruit market
Another intended application of the establishedequation is to determine yield loss from rust mitedamage Rust mite damage reduces growth and in-creases drop of damaged fruit (Allen 1978 1979Allen et aI 1994 Yang et al 1994) But these ef-fects are not uniformly distributed over damageclasses with larger effects on heavily damagedfruit It is therefore necessary to integrate theseeffects over all damage classes based on the fre-qu(ncy distribution of damaged fruit Mathemati-cal models describing the relationships betweenfruit growth and drop and fruit surface damagehave been developed (Allen 1978 1979 Yang etal 1994) Allen et al (1994) established differentialequations to estimate volume loss from reducedfmit growth and drop These differential equationscombine the frequency distribution model withgrowth and drop models The frequency distribu-tion model (equation 6) developed in this studycould also be used in a similar way This modelshould be further improved and tested using fielddata before it could be applied in rust mite man-agement practices
Acknowledgments
We thank James 1 Jones and Harvey L Cromroy(Univfrsity of Florida) for reviewing the manuscript Wealso thank James H Matis (Texas AampM University) andHamon C Littell (University of Florida) for their help indata analysis We thank Coca-Cola for allowing us to con-duct this study in their citrus grove and for their kindnessand help We are also grateful to Dow-ElanCo and Hel-ena Cl1lmical for their financial assistance Published asFlorida Agricultural Experiment Station Journal SeriesNo H-04022
References Cited
Albrigo L G and C W McCoy 1974 Characteristicinjury by citrus rust mite to orange leaves and fruitProc Fla State Home Soc 87 48-55
Allen J C 1978 The effect of citrus rust mite dama~eon citrus fruit drop J Econ Entomol 71 746-50
1979 The effect of citrus rust mite damage on citrusfruit growth J Econ Entomol 72 195-201
Allen J C and C W McCoy 1979 The thermalenvironment of the citrus rust mite Agric Meteorol20 411-425
Allen J C and J H Stamper 1979 Frequency dis-tribution of citrus rust mite damage on citrus fruit JEcon Entomol 72 327-330
Allen J C Y Yang J L Knal)ll and P A Stansly1994 The citrus rust mite story a modelling ap-proach to a fruit-mite-pathogen system pp 619-639In D Hosen F Bennett and J Capinera [eds] lestmanagement in the subtropics biological control-aFlorida perspective Intercept Andover UK
Dennis B and W P Kern) 1988 Further statisticalinference methods for a stochastic model of insectphenology Environ Entomol 17 887-893
Dennis B W P Kemp and R C Beckwith 1986Stochastic model of insect phenology estimation andtesting Environ Entomol 15 540-546
Hall D G C C Childers and J E Eger 1991Estimating citrus rust mite (Acari Eriophyidae) levelson fruit in individual citrus trees Environ Entomol20 382-390
Hubbard H G 1885 Insects affecting the orangeUS Dep Agric Div Entomol Spec Hep 1885
Kemp W P B Dennis and R C Beckwith 1986Stochastic phenology model for the western budworm(Lepidoptera Tortricidae) Environ Entol1lo1 15547-554
McCoy C W and L G Alhrigo 1975 Feeding in-jury to the orange caused by the rust mite Phyll()c()p-tnlta oleiv()ra (Prostigmata Eriophyoidea) Ann En-tomo1 Soc Am 68 289-297
Patel J K C H Kapadia and D B Owen 1976Handbook of statistical distributions Marcel DekkerNew York
SAS Institute 1985 SAS users guide statistics ver-sion 5 ed SAS Institute Cary NC
Yang Y J C Allen J L Knapp and P A Stansly1994 Citms rust mite (Acari Eriophyidae) damageeffects on Hamlin orange fruit growth and drop En-viron Entomol 23 244-247
Yothers W W and A C Mason 1930 The citrusrust mite and its control US Dep Agric Bull 176
Received for publication 4 August 1994 accepted 17Apri1199S
1020
2500
s 2000bull--0 1500bull(1)c 1000EJ 500Z
o
ENVIRONMENTAL ENTOMOLOCY
o 0 0 0 0 0 0 0 0 0 0~ N M v ~ w ~ ~ ~ 0~
Damage class ()
Vol 24 no 5
13-0ct
27-Sep
14-Sep
Date07-Sep
31-Aug
24-Aug
Fig 2 Obselved frequency distribution of mite damage over time 011 fruit (Lake Alfred FL 1993)
Using equation 2 and the maximum likelihoodestimation procedure by Dennis et al (1986) pa-rameters a and h were obtained for each of the 6sets of data (Table 1) They were found to changewith mean fruit surface damage (IL) We therefore
1 1F(x) = A [ (x _ a)] (2)
1 + exp ---b-
where F(x) = the proportion of fruit with a per-centage damage of less than or equal to x and
1A = F(lOO) = ------
1 + exp( _100b- a)
Equation 2 should be interpreted as follows theproportion of fruit without damage is the cumu-lative density up to zero that is F(O) the propor-tion of fruit between damage (XI X2) is F(X2) -F(XI) where Xl gt 0 X2 gt 0 and X2 gt Xl Equation2 reaches a value of 1 at the upper limit of x =100 This makes its probability density function in-tegrate to 1 between (-00 100) The densityfunction is
the frequency data (Fig 2) we used the logisticdistribution to fit the frequency data using themaximum likelihood estimation procedure byDennis et aI (1986) The logistic distIibution func-tion is
1F(x) = [ (x _ a)] (1)
1+ exp ---b-
where a b = parameters The mean of the logisticdistribution is a the variance of the logistic distri-bution is
2~b23
(Patel et a1 1976) We found that the frequencydistribution over damage classes (0 100) in ourstudy could be described using equation 1 Thisapproach assumed that the frequency at zero dam-age class was the height of the cumulative logisticdistribution at zero
Although the logistic distribution (equation 1)describes a continuous distribution from negativeinfinity to positive infinity our damage classes arelimited to the range of (0 100) Because the fre-quency at zero damage class was assumed theheight of the cumulative logistic distribution atzero we were dealing with a range of (-00 100)in the logistic distribution The truncated logisticdistribution is
Tnble 2 Pnrumetr timntc for Intion 4 wId 5 by 2 different method
1021
Mthod Equation Pardmeter Parameter Parameter
MLE-NLlN 4 ao = -110444 al = 12588 a2 = -2140175 ho = 116649 h1 = -42538 h2 = 020180
MLEd 6 ao = -118936 al = 12954 a2 = -160044ho = 123524 hI = -55660 h2 = 0]99]
R2
0996809472093631gtNAC
x2936698g1JNAC8441892h
NN
Iit Iuations 4 and 5 separately using the maximum likelihood estimates for II and b from each of the 6 sets of dataIgt Rsults for the combined 6 spts of data Not applicabled InsNt 1uations 4 and 5 into equation 6 before data fitting using maximum likelihood estimation method
assumed that parameters a and b were functionsof mean fmit surface damage (p) that is a(p) andb(p) The following function was found to give agood fit to parameter a in relation to mean fmitsurface damage (p)
a(p) = ao + alP + a2exp(-P) (4)where 0o a a2 = parameters The following func-tion was found to give a good fit to parameter b(p)
h(p) = ho + blexp( -h2p) (5)
where b() hIgt h2 = parameters The final form oftil( cumulative frequency distribution was the fol-lowing 2-variable logistic function of mean fmitsurface damage (p) and damage class (x)
1 1F(x p) = - ------ (6)
A 1 + exp[- x ~(P)]where a(p) and b(p) are functions of p as definedin equations 4 and 5 The corresponding probabil-ity density function is
1 [x - a(p)]f(x p) = 2X bW ex
p[ - b(P)] 2 (7)
A x - a(p)1 + exp - h(p)
In equations 6 and 7
1
A = F(lOO) = [ I100 - a(p)1 + exp - h(p)
45 120
30 115Q
III 15 bullbull~41 110Gi
0 EE 0 Parameter 8 ~lUbull bull Parameter b 105 lUlU middot15 QQ
-30 1000 10 20 30 40
mu = Mean damage ()
Fig 3 Helationshipbetween parameters a and h inthe logisticequation (equation2) and mean fruit surfacedamage
We used 2 methods to estimate parameters inequations 4 and 5 In the 1st method parametersa and b in equation 2 were estimated using datafor each sample date using the maximum likeli-hood estimation procedure by Dennis et al (1986)Parameters in equations 4 and 5 were then esti-mated using estimates for parameters a and b fromeach of the 6 sets of data using SAS-NLIN pro-cedure (SAS Institute 1985) Parameter estimatesare shown in Table 2 This is a 2-step method Ina better approach we replaced a(p) and b(p) inequation 6 with equations 4 and 5 and then usedthe maximum likelihood estimation procedure byDennis et aI (1986) for simultaneous estimation ofall 6 parameters (ao a] a2 b() bl b2) based on theoriginal 6 sets of data The results are shown inTable 2 Based on the chi-square values the si-multaneous estimates gave a better fit than the2-step method Parameter estimates from thisI-step method were then used in the model Thuswe have
a(p) = -118936 + 12954p (8)
- 160044 exp(- p)
b(p) = 123524 - 55660 exp(-O1991p) (9)
The relationships of parameters a(p) and b(p) tomean fmit surface damage (p) are shown in Fig3 The predicted frequency distribution is shownin Fig 4 The probability density function can beobtained by replacing parameters a(p) and b(Jl) inequation 7 with equations 8 and 9 The predictedprobability density function is shown in Fig 5where probability for 0 damage class is not shownThe probability for 0 damage class can be calcu-lated from equation 6 by setting damage x at O
Discussion
Properties of the CwnuIative Frequency Dis-tribution Function The logistic distribution func-tion (equation 1) has been used to model insectphenology (Dennis et al 1986 Kemp et al 1986Dennis and Kemp 1988) as a stochastic processHere we used the truncated logistic function(equation 6) for describing the frequency distri-bution of mst mite damage on citms fmit Asshown in Fig 3 parameter a(Jl) exhibits a sharpincrease when the mean fmit surface damage islow and a slower linear increase with further in-
1022 ENVIRONMENTAL ENTOMOLOGY Vol 24 no 5
2500=~ 2000bull-- 15000bullQ) 1000cE~ 500Z
o o 0 0 0 0 0 0 0~ N M ~ ~ ~ ~
Damage class ()
oco o 0C1gt 0
13-0ct
Date
Fig 4 Predicted frequency distribution of mite damage over time on fmit (Lake Alfred FL 1993)
mainly caused by large deviations in a few datapoints These deviations might be caused by ran-dom errors Although the chi-square statistic mightreject the hypothesis of the logistic the fit is adf-quate for practical pU1)oses Vhen the mean dam-age was between 0 and 25 model predictionswere at an accuracy of gt75 as compared withthe observed data Model predictions were poor athigher mean damage values
Application of the Cmnulative DistributionFWlction The cumulative frequency distributionfunction (equation 6) will enable us to determinethe proportion of fmit that falls into a specificdamage class if the mean fmit surface damage islmown For example the proportion of fmit thatfalls between damage Xl and X2 is F(x2 IL) - F(x]IL) In commercial citms production it is oftennecessary to determine the proportion of fmit thatcan go to the fresh fruit market If fruit with morethan x percentage of surface damage is rejectedfrom the fresh fruit market then the proportion offruit that can go to the fresh fruit market (thepackout) would simply be
crease in mean damage (equation 8 Fig 3) pa-rameter b(J1) also exhibits a sharp increase but ap-proaches a constant value with further increasingdamage (equation 9 Fig 3) Because a(f1) is themean of the untnmcated logistic distributionwhereas b(J1) is positively related to standard de-viation of the untnmcated logistic distribution thisindicates that as the peak of the density functionshifts towards higher damage there is little changein the variance after the data mean exceeds =20This is similar to shifting a density curve to a high-er mean without changing the shape (variance)High chi-square values (Tables 1 and 2) were
008
iiicC
~004cce~
100 1 1F(x IL) = - ------
A [ x - a(IL)]1+ exp - b(lL)
(10)
Fig 5 Predicted probability density function of mitedamage on fruit where probability at zero damage classis not shown (Lake Alfred FL 1993)
For example if we assume the packout level is x= 5 and the mean fruit damage in a Hamlinorange grove is IL = 10 then the proportion of
October 1995 YANG ET AL FHEQUENCY DISTHIBUTION OF MITE DAMAGE ON FHUlT 1023
fmit that can go for the fresh fmit market can becalculated from equation 10
F(5 10) = [ 1 ] = 05773A 5 - a(10)
1 + exp - b(lO)
where a(10) and b(lO) can be calculated fromequations 8 and 9 respectively This means that5773 fmit have a damage of 510 and theycan go for the fresh fmit market and 4227(100-5773) fmit have a damage gt10 and theycan only go for the processed fruit market
Another intended application of the establishedequation is to determine yield loss from rust mitedamage Rust mite damage reduces growth and in-creases drop of damaged fruit (Allen 1978 1979Allen et aI 1994 Yang et al 1994) But these ef-fects are not uniformly distributed over damageclasses with larger effects on heavily damagedfruit It is therefore necessary to integrate theseeffects over all damage classes based on the fre-qu(ncy distribution of damaged fruit Mathemati-cal models describing the relationships betweenfruit growth and drop and fruit surface damagehave been developed (Allen 1978 1979 Yang etal 1994) Allen et al (1994) established differentialequations to estimate volume loss from reducedfmit growth and drop These differential equationscombine the frequency distribution model withgrowth and drop models The frequency distribu-tion model (equation 6) developed in this studycould also be used in a similar way This modelshould be further improved and tested using fielddata before it could be applied in rust mite man-agement practices
Acknowledgments
We thank James 1 Jones and Harvey L Cromroy(Univfrsity of Florida) for reviewing the manuscript Wealso thank James H Matis (Texas AampM University) andHamon C Littell (University of Florida) for their help indata analysis We thank Coca-Cola for allowing us to con-duct this study in their citrus grove and for their kindnessand help We are also grateful to Dow-ElanCo and Hel-ena Cl1lmical for their financial assistance Published asFlorida Agricultural Experiment Station Journal SeriesNo H-04022
References Cited
Albrigo L G and C W McCoy 1974 Characteristicinjury by citrus rust mite to orange leaves and fruitProc Fla State Home Soc 87 48-55
Allen J C 1978 The effect of citrus rust mite dama~eon citrus fruit drop J Econ Entomol 71 746-50
1979 The effect of citrus rust mite damage on citrusfruit growth J Econ Entomol 72 195-201
Allen J C and C W McCoy 1979 The thermalenvironment of the citrus rust mite Agric Meteorol20 411-425
Allen J C and J H Stamper 1979 Frequency dis-tribution of citrus rust mite damage on citrus fruit JEcon Entomol 72 327-330
Allen J C Y Yang J L Knal)ll and P A Stansly1994 The citrus rust mite story a modelling ap-proach to a fruit-mite-pathogen system pp 619-639In D Hosen F Bennett and J Capinera [eds] lestmanagement in the subtropics biological control-aFlorida perspective Intercept Andover UK
Dennis B and W P Kern) 1988 Further statisticalinference methods for a stochastic model of insectphenology Environ Entomol 17 887-893
Dennis B W P Kemp and R C Beckwith 1986Stochastic model of insect phenology estimation andtesting Environ Entomol 15 540-546
Hall D G C C Childers and J E Eger 1991Estimating citrus rust mite (Acari Eriophyidae) levelson fruit in individual citrus trees Environ Entomol20 382-390
Hubbard H G 1885 Insects affecting the orangeUS Dep Agric Div Entomol Spec Hep 1885
Kemp W P B Dennis and R C Beckwith 1986Stochastic phenology model for the western budworm(Lepidoptera Tortricidae) Environ Entol1lo1 15547-554
McCoy C W and L G Alhrigo 1975 Feeding in-jury to the orange caused by the rust mite Phyll()c()p-tnlta oleiv()ra (Prostigmata Eriophyoidea) Ann En-tomo1 Soc Am 68 289-297
Patel J K C H Kapadia and D B Owen 1976Handbook of statistical distributions Marcel DekkerNew York
SAS Institute 1985 SAS users guide statistics ver-sion 5 ed SAS Institute Cary NC
Yang Y J C Allen J L Knapp and P A Stansly1994 Citms rust mite (Acari Eriophyidae) damageeffects on Hamlin orange fruit growth and drop En-viron Entomol 23 244-247
Yothers W W and A C Mason 1930 The citrusrust mite and its control US Dep Agric Bull 176
Received for publication 4 August 1994 accepted 17Apri1199S
Tnble 2 Pnrumetr timntc for Intion 4 wId 5 by 2 different method
1021
Mthod Equation Pardmeter Parameter Parameter
MLE-NLlN 4 ao = -110444 al = 12588 a2 = -2140175 ho = 116649 h1 = -42538 h2 = 020180
MLEd 6 ao = -118936 al = 12954 a2 = -160044ho = 123524 hI = -55660 h2 = 0]99]
R2
0996809472093631gtNAC
x2936698g1JNAC8441892h
NN
Iit Iuations 4 and 5 separately using the maximum likelihood estimates for II and b from each of the 6 sets of dataIgt Rsults for the combined 6 spts of data Not applicabled InsNt 1uations 4 and 5 into equation 6 before data fitting using maximum likelihood estimation method
assumed that parameters a and b were functionsof mean fmit surface damage (p) that is a(p) andb(p) The following function was found to give agood fit to parameter a in relation to mean fmitsurface damage (p)
a(p) = ao + alP + a2exp(-P) (4)where 0o a a2 = parameters The following func-tion was found to give a good fit to parameter b(p)
h(p) = ho + blexp( -h2p) (5)
where b() hIgt h2 = parameters The final form oftil( cumulative frequency distribution was the fol-lowing 2-variable logistic function of mean fmitsurface damage (p) and damage class (x)
1 1F(x p) = - ------ (6)
A 1 + exp[- x ~(P)]where a(p) and b(p) are functions of p as definedin equations 4 and 5 The corresponding probabil-ity density function is
1 [x - a(p)]f(x p) = 2X bW ex
p[ - b(P)] 2 (7)
A x - a(p)1 + exp - h(p)
In equations 6 and 7
1
A = F(lOO) = [ I100 - a(p)1 + exp - h(p)
45 120
30 115Q
III 15 bullbull~41 110Gi
0 EE 0 Parameter 8 ~lUbull bull Parameter b 105 lUlU middot15 QQ
-30 1000 10 20 30 40
mu = Mean damage ()
Fig 3 Helationshipbetween parameters a and h inthe logisticequation (equation2) and mean fruit surfacedamage
We used 2 methods to estimate parameters inequations 4 and 5 In the 1st method parametersa and b in equation 2 were estimated using datafor each sample date using the maximum likeli-hood estimation procedure by Dennis et al (1986)Parameters in equations 4 and 5 were then esti-mated using estimates for parameters a and b fromeach of the 6 sets of data using SAS-NLIN pro-cedure (SAS Institute 1985) Parameter estimatesare shown in Table 2 This is a 2-step method Ina better approach we replaced a(p) and b(p) inequation 6 with equations 4 and 5 and then usedthe maximum likelihood estimation procedure byDennis et aI (1986) for simultaneous estimation ofall 6 parameters (ao a] a2 b() bl b2) based on theoriginal 6 sets of data The results are shown inTable 2 Based on the chi-square values the si-multaneous estimates gave a better fit than the2-step method Parameter estimates from thisI-step method were then used in the model Thuswe have
a(p) = -118936 + 12954p (8)
- 160044 exp(- p)
b(p) = 123524 - 55660 exp(-O1991p) (9)
The relationships of parameters a(p) and b(p) tomean fmit surface damage (p) are shown in Fig3 The predicted frequency distribution is shownin Fig 4 The probability density function can beobtained by replacing parameters a(p) and b(Jl) inequation 7 with equations 8 and 9 The predictedprobability density function is shown in Fig 5where probability for 0 damage class is not shownThe probability for 0 damage class can be calcu-lated from equation 6 by setting damage x at O
Discussion
Properties of the CwnuIative Frequency Dis-tribution Function The logistic distribution func-tion (equation 1) has been used to model insectphenology (Dennis et al 1986 Kemp et al 1986Dennis and Kemp 1988) as a stochastic processHere we used the truncated logistic function(equation 6) for describing the frequency distri-bution of mst mite damage on citms fmit Asshown in Fig 3 parameter a(Jl) exhibits a sharpincrease when the mean fmit surface damage islow and a slower linear increase with further in-
1022 ENVIRONMENTAL ENTOMOLOGY Vol 24 no 5
2500=~ 2000bull-- 15000bullQ) 1000cE~ 500Z
o o 0 0 0 0 0 0 0~ N M ~ ~ ~ ~
Damage class ()
oco o 0C1gt 0
13-0ct
Date
Fig 4 Predicted frequency distribution of mite damage over time on fmit (Lake Alfred FL 1993)
mainly caused by large deviations in a few datapoints These deviations might be caused by ran-dom errors Although the chi-square statistic mightreject the hypothesis of the logistic the fit is adf-quate for practical pU1)oses Vhen the mean dam-age was between 0 and 25 model predictionswere at an accuracy of gt75 as compared withthe observed data Model predictions were poor athigher mean damage values
Application of the Cmnulative DistributionFWlction The cumulative frequency distributionfunction (equation 6) will enable us to determinethe proportion of fmit that falls into a specificdamage class if the mean fmit surface damage islmown For example the proportion of fmit thatfalls between damage Xl and X2 is F(x2 IL) - F(x]IL) In commercial citms production it is oftennecessary to determine the proportion of fmit thatcan go to the fresh fruit market If fruit with morethan x percentage of surface damage is rejectedfrom the fresh fruit market then the proportion offruit that can go to the fresh fruit market (thepackout) would simply be
crease in mean damage (equation 8 Fig 3) pa-rameter b(J1) also exhibits a sharp increase but ap-proaches a constant value with further increasingdamage (equation 9 Fig 3) Because a(f1) is themean of the untnmcated logistic distributionwhereas b(J1) is positively related to standard de-viation of the untnmcated logistic distribution thisindicates that as the peak of the density functionshifts towards higher damage there is little changein the variance after the data mean exceeds =20This is similar to shifting a density curve to a high-er mean without changing the shape (variance)High chi-square values (Tables 1 and 2) were
008
iiicC
~004cce~
100 1 1F(x IL) = - ------
A [ x - a(IL)]1+ exp - b(lL)
(10)
Fig 5 Predicted probability density function of mitedamage on fruit where probability at zero damage classis not shown (Lake Alfred FL 1993)
For example if we assume the packout level is x= 5 and the mean fruit damage in a Hamlinorange grove is IL = 10 then the proportion of
October 1995 YANG ET AL FHEQUENCY DISTHIBUTION OF MITE DAMAGE ON FHUlT 1023
fmit that can go for the fresh fmit market can becalculated from equation 10
F(5 10) = [ 1 ] = 05773A 5 - a(10)
1 + exp - b(lO)
where a(10) and b(lO) can be calculated fromequations 8 and 9 respectively This means that5773 fmit have a damage of 510 and theycan go for the fresh fmit market and 4227(100-5773) fmit have a damage gt10 and theycan only go for the processed fruit market
Another intended application of the establishedequation is to determine yield loss from rust mitedamage Rust mite damage reduces growth and in-creases drop of damaged fruit (Allen 1978 1979Allen et aI 1994 Yang et al 1994) But these ef-fects are not uniformly distributed over damageclasses with larger effects on heavily damagedfruit It is therefore necessary to integrate theseeffects over all damage classes based on the fre-qu(ncy distribution of damaged fruit Mathemati-cal models describing the relationships betweenfruit growth and drop and fruit surface damagehave been developed (Allen 1978 1979 Yang etal 1994) Allen et al (1994) established differentialequations to estimate volume loss from reducedfmit growth and drop These differential equationscombine the frequency distribution model withgrowth and drop models The frequency distribu-tion model (equation 6) developed in this studycould also be used in a similar way This modelshould be further improved and tested using fielddata before it could be applied in rust mite man-agement practices
Acknowledgments
We thank James 1 Jones and Harvey L Cromroy(Univfrsity of Florida) for reviewing the manuscript Wealso thank James H Matis (Texas AampM University) andHamon C Littell (University of Florida) for their help indata analysis We thank Coca-Cola for allowing us to con-duct this study in their citrus grove and for their kindnessand help We are also grateful to Dow-ElanCo and Hel-ena Cl1lmical for their financial assistance Published asFlorida Agricultural Experiment Station Journal SeriesNo H-04022
References Cited
Albrigo L G and C W McCoy 1974 Characteristicinjury by citrus rust mite to orange leaves and fruitProc Fla State Home Soc 87 48-55
Allen J C 1978 The effect of citrus rust mite dama~eon citrus fruit drop J Econ Entomol 71 746-50
1979 The effect of citrus rust mite damage on citrusfruit growth J Econ Entomol 72 195-201
Allen J C and C W McCoy 1979 The thermalenvironment of the citrus rust mite Agric Meteorol20 411-425
Allen J C and J H Stamper 1979 Frequency dis-tribution of citrus rust mite damage on citrus fruit JEcon Entomol 72 327-330
Allen J C Y Yang J L Knal)ll and P A Stansly1994 The citrus rust mite story a modelling ap-proach to a fruit-mite-pathogen system pp 619-639In D Hosen F Bennett and J Capinera [eds] lestmanagement in the subtropics biological control-aFlorida perspective Intercept Andover UK
Dennis B and W P Kern) 1988 Further statisticalinference methods for a stochastic model of insectphenology Environ Entomol 17 887-893
Dennis B W P Kemp and R C Beckwith 1986Stochastic model of insect phenology estimation andtesting Environ Entomol 15 540-546
Hall D G C C Childers and J E Eger 1991Estimating citrus rust mite (Acari Eriophyidae) levelson fruit in individual citrus trees Environ Entomol20 382-390
Hubbard H G 1885 Insects affecting the orangeUS Dep Agric Div Entomol Spec Hep 1885
Kemp W P B Dennis and R C Beckwith 1986Stochastic phenology model for the western budworm(Lepidoptera Tortricidae) Environ Entol1lo1 15547-554
McCoy C W and L G Alhrigo 1975 Feeding in-jury to the orange caused by the rust mite Phyll()c()p-tnlta oleiv()ra (Prostigmata Eriophyoidea) Ann En-tomo1 Soc Am 68 289-297
Patel J K C H Kapadia and D B Owen 1976Handbook of statistical distributions Marcel DekkerNew York
SAS Institute 1985 SAS users guide statistics ver-sion 5 ed SAS Institute Cary NC
Yang Y J C Allen J L Knapp and P A Stansly1994 Citms rust mite (Acari Eriophyidae) damageeffects on Hamlin orange fruit growth and drop En-viron Entomol 23 244-247
Yothers W W and A C Mason 1930 The citrusrust mite and its control US Dep Agric Bull 176
Received for publication 4 August 1994 accepted 17Apri1199S
1022 ENVIRONMENTAL ENTOMOLOGY Vol 24 no 5
2500=~ 2000bull-- 15000bullQ) 1000cE~ 500Z
o o 0 0 0 0 0 0 0~ N M ~ ~ ~ ~
Damage class ()
oco o 0C1gt 0
13-0ct
Date
Fig 4 Predicted frequency distribution of mite damage over time on fmit (Lake Alfred FL 1993)
mainly caused by large deviations in a few datapoints These deviations might be caused by ran-dom errors Although the chi-square statistic mightreject the hypothesis of the logistic the fit is adf-quate for practical pU1)oses Vhen the mean dam-age was between 0 and 25 model predictionswere at an accuracy of gt75 as compared withthe observed data Model predictions were poor athigher mean damage values
Application of the Cmnulative DistributionFWlction The cumulative frequency distributionfunction (equation 6) will enable us to determinethe proportion of fmit that falls into a specificdamage class if the mean fmit surface damage islmown For example the proportion of fmit thatfalls between damage Xl and X2 is F(x2 IL) - F(x]IL) In commercial citms production it is oftennecessary to determine the proportion of fmit thatcan go to the fresh fruit market If fruit with morethan x percentage of surface damage is rejectedfrom the fresh fruit market then the proportion offruit that can go to the fresh fruit market (thepackout) would simply be
crease in mean damage (equation 8 Fig 3) pa-rameter b(J1) also exhibits a sharp increase but ap-proaches a constant value with further increasingdamage (equation 9 Fig 3) Because a(f1) is themean of the untnmcated logistic distributionwhereas b(J1) is positively related to standard de-viation of the untnmcated logistic distribution thisindicates that as the peak of the density functionshifts towards higher damage there is little changein the variance after the data mean exceeds =20This is similar to shifting a density curve to a high-er mean without changing the shape (variance)High chi-square values (Tables 1 and 2) were
008
iiicC
~004cce~
100 1 1F(x IL) = - ------
A [ x - a(IL)]1+ exp - b(lL)
(10)
Fig 5 Predicted probability density function of mitedamage on fruit where probability at zero damage classis not shown (Lake Alfred FL 1993)
For example if we assume the packout level is x= 5 and the mean fruit damage in a Hamlinorange grove is IL = 10 then the proportion of
October 1995 YANG ET AL FHEQUENCY DISTHIBUTION OF MITE DAMAGE ON FHUlT 1023
fmit that can go for the fresh fmit market can becalculated from equation 10
F(5 10) = [ 1 ] = 05773A 5 - a(10)
1 + exp - b(lO)
where a(10) and b(lO) can be calculated fromequations 8 and 9 respectively This means that5773 fmit have a damage of 510 and theycan go for the fresh fmit market and 4227(100-5773) fmit have a damage gt10 and theycan only go for the processed fruit market
Another intended application of the establishedequation is to determine yield loss from rust mitedamage Rust mite damage reduces growth and in-creases drop of damaged fruit (Allen 1978 1979Allen et aI 1994 Yang et al 1994) But these ef-fects are not uniformly distributed over damageclasses with larger effects on heavily damagedfruit It is therefore necessary to integrate theseeffects over all damage classes based on the fre-qu(ncy distribution of damaged fruit Mathemati-cal models describing the relationships betweenfruit growth and drop and fruit surface damagehave been developed (Allen 1978 1979 Yang etal 1994) Allen et al (1994) established differentialequations to estimate volume loss from reducedfmit growth and drop These differential equationscombine the frequency distribution model withgrowth and drop models The frequency distribu-tion model (equation 6) developed in this studycould also be used in a similar way This modelshould be further improved and tested using fielddata before it could be applied in rust mite man-agement practices
Acknowledgments
We thank James 1 Jones and Harvey L Cromroy(Univfrsity of Florida) for reviewing the manuscript Wealso thank James H Matis (Texas AampM University) andHamon C Littell (University of Florida) for their help indata analysis We thank Coca-Cola for allowing us to con-duct this study in their citrus grove and for their kindnessand help We are also grateful to Dow-ElanCo and Hel-ena Cl1lmical for their financial assistance Published asFlorida Agricultural Experiment Station Journal SeriesNo H-04022
References Cited
Albrigo L G and C W McCoy 1974 Characteristicinjury by citrus rust mite to orange leaves and fruitProc Fla State Home Soc 87 48-55
Allen J C 1978 The effect of citrus rust mite dama~eon citrus fruit drop J Econ Entomol 71 746-50
1979 The effect of citrus rust mite damage on citrusfruit growth J Econ Entomol 72 195-201
Allen J C and C W McCoy 1979 The thermalenvironment of the citrus rust mite Agric Meteorol20 411-425
Allen J C and J H Stamper 1979 Frequency dis-tribution of citrus rust mite damage on citrus fruit JEcon Entomol 72 327-330
Allen J C Y Yang J L Knal)ll and P A Stansly1994 The citrus rust mite story a modelling ap-proach to a fruit-mite-pathogen system pp 619-639In D Hosen F Bennett and J Capinera [eds] lestmanagement in the subtropics biological control-aFlorida perspective Intercept Andover UK
Dennis B and W P Kern) 1988 Further statisticalinference methods for a stochastic model of insectphenology Environ Entomol 17 887-893
Dennis B W P Kemp and R C Beckwith 1986Stochastic model of insect phenology estimation andtesting Environ Entomol 15 540-546
Hall D G C C Childers and J E Eger 1991Estimating citrus rust mite (Acari Eriophyidae) levelson fruit in individual citrus trees Environ Entomol20 382-390
Hubbard H G 1885 Insects affecting the orangeUS Dep Agric Div Entomol Spec Hep 1885
Kemp W P B Dennis and R C Beckwith 1986Stochastic phenology model for the western budworm(Lepidoptera Tortricidae) Environ Entol1lo1 15547-554
McCoy C W and L G Alhrigo 1975 Feeding in-jury to the orange caused by the rust mite Phyll()c()p-tnlta oleiv()ra (Prostigmata Eriophyoidea) Ann En-tomo1 Soc Am 68 289-297
Patel J K C H Kapadia and D B Owen 1976Handbook of statistical distributions Marcel DekkerNew York
SAS Institute 1985 SAS users guide statistics ver-sion 5 ed SAS Institute Cary NC
Yang Y J C Allen J L Knapp and P A Stansly1994 Citms rust mite (Acari Eriophyidae) damageeffects on Hamlin orange fruit growth and drop En-viron Entomol 23 244-247
Yothers W W and A C Mason 1930 The citrusrust mite and its control US Dep Agric Bull 176
Received for publication 4 August 1994 accepted 17Apri1199S
October 1995 YANG ET AL FHEQUENCY DISTHIBUTION OF MITE DAMAGE ON FHUlT 1023
fmit that can go for the fresh fmit market can becalculated from equation 10
F(5 10) = [ 1 ] = 05773A 5 - a(10)
1 + exp - b(lO)
where a(10) and b(lO) can be calculated fromequations 8 and 9 respectively This means that5773 fmit have a damage of 510 and theycan go for the fresh fmit market and 4227(100-5773) fmit have a damage gt10 and theycan only go for the processed fruit market
Another intended application of the establishedequation is to determine yield loss from rust mitedamage Rust mite damage reduces growth and in-creases drop of damaged fruit (Allen 1978 1979Allen et aI 1994 Yang et al 1994) But these ef-fects are not uniformly distributed over damageclasses with larger effects on heavily damagedfruit It is therefore necessary to integrate theseeffects over all damage classes based on the fre-qu(ncy distribution of damaged fruit Mathemati-cal models describing the relationships betweenfruit growth and drop and fruit surface damagehave been developed (Allen 1978 1979 Yang etal 1994) Allen et al (1994) established differentialequations to estimate volume loss from reducedfmit growth and drop These differential equationscombine the frequency distribution model withgrowth and drop models The frequency distribu-tion model (equation 6) developed in this studycould also be used in a similar way This modelshould be further improved and tested using fielddata before it could be applied in rust mite man-agement practices
Acknowledgments
We thank James 1 Jones and Harvey L Cromroy(Univfrsity of Florida) for reviewing the manuscript Wealso thank James H Matis (Texas AampM University) andHamon C Littell (University of Florida) for their help indata analysis We thank Coca-Cola for allowing us to con-duct this study in their citrus grove and for their kindnessand help We are also grateful to Dow-ElanCo and Hel-ena Cl1lmical for their financial assistance Published asFlorida Agricultural Experiment Station Journal SeriesNo H-04022
References Cited
Albrigo L G and C W McCoy 1974 Characteristicinjury by citrus rust mite to orange leaves and fruitProc Fla State Home Soc 87 48-55
Allen J C 1978 The effect of citrus rust mite dama~eon citrus fruit drop J Econ Entomol 71 746-50
1979 The effect of citrus rust mite damage on citrusfruit growth J Econ Entomol 72 195-201
Allen J C and C W McCoy 1979 The thermalenvironment of the citrus rust mite Agric Meteorol20 411-425
Allen J C and J H Stamper 1979 Frequency dis-tribution of citrus rust mite damage on citrus fruit JEcon Entomol 72 327-330
Allen J C Y Yang J L Knal)ll and P A Stansly1994 The citrus rust mite story a modelling ap-proach to a fruit-mite-pathogen system pp 619-639In D Hosen F Bennett and J Capinera [eds] lestmanagement in the subtropics biological control-aFlorida perspective Intercept Andover UK
Dennis B and W P Kern) 1988 Further statisticalinference methods for a stochastic model of insectphenology Environ Entomol 17 887-893
Dennis B W P Kemp and R C Beckwith 1986Stochastic model of insect phenology estimation andtesting Environ Entomol 15 540-546
Hall D G C C Childers and J E Eger 1991Estimating citrus rust mite (Acari Eriophyidae) levelson fruit in individual citrus trees Environ Entomol20 382-390
Hubbard H G 1885 Insects affecting the orangeUS Dep Agric Div Entomol Spec Hep 1885
Kemp W P B Dennis and R C Beckwith 1986Stochastic phenology model for the western budworm(Lepidoptera Tortricidae) Environ Entol1lo1 15547-554
McCoy C W and L G Alhrigo 1975 Feeding in-jury to the orange caused by the rust mite Phyll()c()p-tnlta oleiv()ra (Prostigmata Eriophyoidea) Ann En-tomo1 Soc Am 68 289-297
Patel J K C H Kapadia and D B Owen 1976Handbook of statistical distributions Marcel DekkerNew York
SAS Institute 1985 SAS users guide statistics ver-sion 5 ed SAS Institute Cary NC
Yang Y J C Allen J L Knapp and P A Stansly1994 Citms rust mite (Acari Eriophyidae) damageeffects on Hamlin orange fruit growth and drop En-viron Entomol 23 244-247
Yothers W W and A C Mason 1930 The citrusrust mite and its control US Dep Agric Bull 176
Received for publication 4 August 1994 accepted 17Apri1199S
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