Rev Colomb Cienc Pecu 2016; 29:199-209

Revista Colombiana de Ciencias Pecuarias

Original articles

199

Comparison of non-linear models to describe growth of Iranian Guilan sheep¤

Comparación de modelos no lineales para describir el crecimiento de ovejas iraní Guilan

Comparação de modelos não lineares para descrever o crescimento de ovinos iraniano Guilan

Navid Ghavi Hossein-Zadeh1*, PhD; Mohammad Golshani2, MSc.

1Department of Animal Science, Faculty of Agricultural Sciences, University of Guilan, Rasht, Iran.

2Agricultural Organization of Guilan Province, Rasht, Iran.

(Received: September15, 2015; accepted: April 14, 2016)

doi: 10.17533/udea.rccp.v29n3a05

¤ Tocitethisarticle:GhaviHossein-ZadehN,GolshaniM.Comparisonofnon-linearmodelstodescribegrowthofIranianGuilansheep.RevColombCiencPecu2016;29:199-209.

* Correspondingauthor:NavidGhaviHossein-Zadeh.DepartmentofAnimalScience,FacultyofAgriculturalSciences,UniversityofGuilan,Rasht,41635-1314,Iran.Tel:+981333690274.Fax:+981333690281.E-mail:nhosseinzadeh@guilan.ac.ir

Summary

Background: non-linearmathematicalmodels,empiricallydevelopedbyplottingbodyweightagainstage,areusedtodescribethegrowthcurveindifferentanimals.Objective:todescribethegrowthpatterninGuilansheepusingnon-linearmodels.Methods:sixnon-linearmathematicalequations(Brody,Negativeexponential,Logistic,Gompertz,vonBertalanffyandRichards)wereusedtodescribethegrowthcurvesinGuilansheep.TheAgriculturalOrganizationofGuilanprovince(Rasht, Iran)provided thedatasetusedinthisstudy.Thedatasetincluded42,257weightrecordsoflambsfrombirthto240daysofageduringyears1994to2014.Eachmodelwasseparatelyfittedtobodyweightrecordsofalllambs,malesandfemales,usingtheNLINandMODELproceduresofSAS.Themodelsweretestedforgoodnessoffitusingadjustedcoefficientofdetermination,rootmeanssquareerror(RMSE),Durbin-Watsonstatistic,akaike’sinformationcriterion(AIC)andbayesianinformationcriterion(BIC).Results:theRichardsmodelprovidedthebestfittothegrowthcurveinfemalesandalllambs,withthelowestRMSE,AIC,andBICvaluescomparedtotheothermodels.TheBrodymodelprovidedthebestfitofgrowthinmalelambsduetothelowervaluesofAICandBICcomparedtotheothermodels.Thenegativeexponentialmodelprovidedtheworstfitofgrowthformales,femalesandalllambs.Conclusion:theevaluationofthegrowthequationsusedinthisstudyindicatesthepotentialofnon-linearfunctionsforfittingbodyweightrecordsofGuilansheep.

Keywords: body weight, fat-tailed sheep, growth function, growth pattern.

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Resumen

Antecedentes: los modelosmatemáticosnolinealesdesarrolladosempíricamentemediantelacomparacióndelpesocorporalcontralaedadsehanusadoparadescribirlacurvadecrecimientoendiferentesanimales.Objetivo: describirelpatróndecrecimientoenovejasGuilanusandomodelosnolineales.Métodos: seis ecuacionesmatemáticas no lineales (Brody,Exponencial negativo,Logístico,Gompertz, vonBertalanffyyRichards) seutilizaronparadescribir lascurvasdecrecimientoenovejasGuilan.ElconjuntodedatosutilizadosenesteestudioseobtuvodelaOrganizaciónparalaAgriculturadelaprovinciaGuilan(Rasht,Irán),ycomprendió42,257registrosdepesodecorderos,recogidosdesdeelnacimientohastalos240díasdeedaddurantelosaños1994a2014.Cadamodelofueajustadoporseparadoalosregistrosdepesocorporalde todos los corderos,machosy hembras, utilizando los procedimientosNLINyMODELdeSAS.Losmodelosfueronprobadosparalabondaddelajustemedianteelcoeficientededeterminaciónajustado,laraízcuadradadelerrorcuadráticomedio(RMSE),elestadísticodeDurbin-Watson,elcriteriodeinformacióndeakaike(AIC)yelcriteriodeinformaciónbayesiano(BIC).Resultados: el modelodeRichardsproporcionóelmejorajustedelacurvadecrecimientoenlashembrasytodosloscorderosdebidoalosvaloresmásbajosdeRMSE,AICyBICconrespectoalosotrosmodelos.ElmodeloBrodyproporcionóelmejorajustedelacurvadecrecimientodeloscorderosmachosdebidoalosvaloresmásbajosdeAICyBICconrespectoalosotrosmodelos.ElmodeloExponencialnegativoporporcionóelpeorajustedelacurvadecrecimientoparamachos,hembrasytodosloscorderos.Conclusión: laevaluacióndediferentesecuacionesdecrecimientoutilizadasenesteestudioindicaelpotencialdelasfuncionesnolinealesparaelajustedelosregistrosdepesocorporalenovejasderazaGuilan.

Palabras clave: función de crecimiento, ovejas de cola gorda, patrón de crecimiento, peso corporal.

Resumo

Antecedentes: os modelosmatemáticosnãolinearesdesenvolvidosempiricamenteatravésdarepresentaçãográficadopesocorporalcomaidadetêmsidoadequadosparadescreveracurvadecrescimentoemdiferentesanimais. Objetivo:oobjetivodesteestudofoidescreveropadrãodecrescimentoemovinosGuilanusandomodelos não lineares.Métodos: seis equaçõesmatemáticas não lineares (Brody,Negativo exponencial,Logístico,Gompertz,vonBertalanffyeRichards)foramutilizadasparadescreverascurvasdecrescimentoemovelhasGuilan.OconjuntodedadosutilizadonesteestudofoiobtidoapartirdaOrganizaçãoAgrícoladaprovínciadeGuilan(Rasht,Irã)ecomposta42.257registrosdepesodecordeirosqueforamrecolhidosdesdeonascimentoaté240diasdeidade,durante1994a2014.Cadamodelofoiajustadoseparadamenteparaosregistrosdepesocorporaldetodososcordeiros,machosefêmeas,utilizandoosprocedimentosNLINemodeloemSAS.Osmodelosforamtestadosparaaqualidadedoajustepormeiodocoeficientededeterminaçãoajustado,araizquadradadeerroquadráticomédio(RMSE),oestatísticoDurbin-Watsonestatística,ocritériodeinformaçãodeAkaike(AIC)ecritériodeinformaçãoBayesiano(BIC).Resultados: o modelodeRichardsprovidenciaomelhorajustedacurvadecrescimentonasfêmeasetodososcordeiros,devidoaosmenoresvaloresdeRMSE,AICeBICdoqueoutrosmodelos.OmodeloBrodydaomelhorajustedacurvadecrescimentoemcordeirosdevidoaosmenoresvaloresdeAICeBICdoqueoutrosmodelos.Omodeloexponencialnegativoofereceopiorajustedecurvadecrescimentoparamachos,fêmeasetodososcordeiros.Conclusão: avaliaçãodiferentesequaçõesdecrescimentoutilizadosnesteestudoindicaramopotencialdasfunçõesnãolinearesparaamontagemderegistosdepesocorporaldeGuilan.

Palavra chave: função crescimento, ovinos de cauda gorda, padrão de crescimento, peso corporal.

Introduction

Guilan sheep is a fat-tailed breed in Iran,numberingsome400,000animalsinthenorthofthecountry,anddistributedinthenorthernandwesternpartsofGuilanProvinceinthemountainsbetweenAssalem,Khalkhal,Oshkourat,andDeilaman.ThisbreedcanalsobefoundinsomeareasoftheGuilan-Zanjanborder.Meanadultliveweightinthisbreedis

35Kg(77lbs)forramsand31Kg(67lbs)forewes.Thecoatisyellowish-whitetopurewhite,withbrownpatchesonthehead,face,andthebottomofthelegs.Thisbreedisvaluedmainlyduetoitsabilitytoliveinmountainousareaswithrain-fedfoothillsandfoothillsteppeswith1300mm(51 inches) annual rainfall.Youngewesare randomlyexposed to the ramsforthefirsttimeatapproximately1.5yearsofage.Ewesarekeptintheflockupto7yearsofage.Ewesare

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supplemented,dependingupon requirements, for afewdaysafterlambing.Ramsarekeptuntilamaleoffspring is available for replacement.During thebreedingseason,single-sirepensareusedallocating20-25ewesperram.Lambsremainwiththeirdamuntilweaning.Lambsareear-taggedandweightedimmediately after lambing.During the sucklingperiod,lambsarefeddryalfalfaafterthreeweeksofage.Lambsareweanedatapproximately90daysofage.Animalsarekeptonnaturalpastureduringspring,summerandautumn.Sinceenvironmentalconditionsareadverse,theanimalsarekeptindoorsduringthewintermonths.Theflockismainlykeptonpastureandfedcerealpasture,withsupplementalfeed,includingalfalfaandwheatstraw,providedespeciallyaroundthematingseason(Eteqadiet al.,2014).

Growth is defined as increase in liveweight ordimension against age. Changes in live weightor dimension for a period of time are explainedbygrowthcurves(Keskinet al.,2010).Analysisofanimalgrowthperformancethroughthelifespanishelpful to establish appropriate feeding strategiesand thebestageforslaughtering.Studies focusingongrowthcurveshaveincreasedinrecentyearsdueto thedevelopmentofnewcomputationalmethodsfor faster andmore accurate analyses aswell astheavailabilityofnewmodels(Souzaet al.,2013).Slowgrowth rates resulting in lowmarketweighthasbeenidentifiedtobeoneofthefactorslimitingprofitabilityinproductionsystem(Nooret al.,2001;Abegazet al.,2010).Growthrateisrelatedtorateofmaturingandmatureweightandtheselattertraitshavebeensuggestedtoberelatedwithotherlifetimeproductivityparametersinsheep(Bedieret al.,1992;Abegazet al.,2010).

Non-linearmathematicalmodels, empiricallydevelopedbyplottingbodyweightagainstage,aresuitable to describe the growth curve in differentanimal groups (Malhado et al., 2009). The useofmathematical growthmodels provides a goodway of condensing the information into a fewparameterswith biologicalmeaning, to facilitateboththeinterpretationandtheunderstandingofthephenomenon(Fitzhugh,1976;Malhadoet al.,2009).Growthcurveparametersprovidepotentiallyusefulcriteriaformodifyingtherelationshipbetweenbodyweight and age through selection (Kachman and

Gianola,1984)andanoptimumgrowthcurvecanbeobtainedby selection fordesiredvaluesof growthcurveparameters(BathaeiandLeroy,1998).Growthcurves provide several applications to animalproduction,suchas(1)evaluationoftheresponsetotreatmentsovertime;(2)analysisoftheinteractionbetweensubpopulations(ortreatments)andtime;and(3)identificationofheavieranimalsatyoungerageswithinapopulation(BathaeiandLeroy,1996;Freitas,2005;Malhado et al.,2009).

No previous studies have been conducted ongrowth curve characteristics of theGuilan sheep.Therefore,theobjectiveofthisstudywastodescribethegrowthpatterninGuilansheepusingnon-linearmodels.For thispurpose,sixmathematicalmodels(Brody,Negative exponential,Logistic,Gompertz,vonBertalanffy,andRichards)currentlyusedtofitthegrowthpatternofanimalswereexaminedtoevaluatetheir efficiency in describing the growth curve ofGuilansheep.

Materials and methods

ThedatasetusedinthisstudywasobtainedfromtheAgriculturalOrganization ofGuilan Province(Rasht,Iran)andcomprised42,257weightrecords,whichwerecollectedon18,972lambsfrombirthto240daysofagefromyears1994to2014.Thedatawerescreenedseveraltimesanddefectiveandoutofrangerecordsweredeleted.Aftereditingtheinitialdata set, 41,894bodyweight records (madeupof19,879formalesand22,015forfemalelambs)wereusedforstatisticalanalysis.

Non-linear growth functions used to describethegrowthcurvesofGuilansheeparepresentedinTable1.TheBrody,Negativeexponential,Logistic,Gompertz,VonBertalanffyandRichards functionswerefittothedatatomodeltherelationshipbetweenweightandage.Eachmodelwasfittedseparatelytobodyweightrecordsofalllambs,maleandfemalelambsusing theNLINandMODELprocedures inSAS(SASInstituteInc.,Cary,NC,USA;2002)andtheparameterswereestimated.TheNLINprocedureproduces least squares orweighted least squaresestimatesof theparametersof anon-linearmodel.Foreachnon-linearmodeltobeanalyzed,themodel

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(usingasingledependentvariable)andthenamesandstartingvaluesoftheparameterstobeestimatedmustbespecified.Whennon-linearfunctionswerefitted,theGauss-Newtonmethodwasusedastheiterationmethod.TobeginthisprocesstheNLINprocedurefirstexaminesthestartingvaluespecificationsoftheparameters. If a grid of values is specified,NLINprocedureevaluates the residual sumof squaresateachcombinationofparametervaluestodeterminethe set of parameter values producing the lowestresidualsumofsquares.Theseparametervaluesareusedfortheinitialstepoftheiteration.TheMODELprocedureanalyzesmodelsinwhichtherelationshipsamongthevariablescompriseasystemofoneormorenonlinear equations. Primary uses of theMODELprocedureareestimation,simulation,andforecastingof nonlinear simultaneous equationmodels (GhaviHossein-Zadeh,2014).

Table 1. Functional forms of equations used to describe the growth curve of Guilan sheep.

Equation Functional form

Brody

Negative exponential

Logistic

Gompertz

von Bertalanffy

Richards

y = body weight at age t (day); a = asymptotic weight, which is interpreted as mature weight; b = is an integration constant related to initial animal weight. The value of b is defined by the initial values for y and t; k = is the maturation rate, which is interpreted as weight change in relation to mature weight to indicate how fast the animal approaches adult weight; m = is the parameter that gives shape to the curve by indicating where the inflection point occurs.

Themodels were tested for goodness of fit(qualityofprediction)usingadjustedcoefficientofdetermination( 2adjR ),residualstandarddeviationorrootmeans square error (RMSE),Durbin-Watsonstatistic(DW),Akaike’sinformationcriterion(AIC)andBayesianinformationcriterion(BIC).

2adjR wascalculatedusingthefollowingformula:

( )( ) ( )

2 211 1adjn

R Rn p

−= − − −

Where:

2R =isthemultiplecoefficientofdetermination

( 2 1 RSSRTSS

= − ).

TSS=isthetotalsumofsquares.RSS=istheresidualsumofsquares.n=isthenumberofobservations(datapoints).p=isthenumberofparametersintheequation.

The 2R value is an indicatormeasuring theproportionoftotalvariationaboutthemeanofthetraitexplainedbythegrowthcurvemodel.Thecoefficientofdeterminationliesalwaysbetween0and1,andthefitofamodelissatisfactoryif 2R isclosetounity.

RMSEisasortofgeneralizedstandarddeviationandwascalculatedasfollows:

1RSSRMSE

n p=

− −Where.

RSS=istheresidualsumofsquares.n=isthenumberofobservations(datapoints).p=isthenumberofparametersintheequation.

RMSE is one of themost important criteria tocomparethesuitabilityofgrowthmodels.Therefore,thebestmodelistheonewiththelowestRMSE.

DW was used to detect the presence ofautocorrelationintheresidualsfromtheregressionanalysis. In fact, the presence of autocorrelatedresiduals suggests that the function may beinappropriateforthedata.TheDurbin-Watsonstatisticrangesinvaluefrom0to4.Avalueneartwoindicatesnoautocorrelation;avaluetoward0indicatespositiveautocorrelation;avaluetoward4indicatesnegativeautocorrelation (GhaviHossein-Zadeh, 2014).DWwascalculatedusingthefollowingformula:

1 ctaybe−

=+

ktbey ae−−=

( )1 mkty a be −−= −

( )1 kty a be−= −( )kty a ae−= −

( )31 kty a be−= −

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( )212

1

nt ttn

tt

e eDW

e−

=

−= ∑

∑Where:

te =istheresidualattimet.1te − =istheresidualattimet-1.

AICwascalculatedasusingthefollowingequation(BurnhamandAnderson,2002):

( )ln 2AIC n RSS p= × +

AIC is a good statistic to comparemodels ofdifferentcomplexitybecauseitadjuststheRSSforthe number of parameters in themodel.A smallernumericalvalueofAIC indicates abetterfitwhencomparingmodels.

BICcombinesmaximumlikelihood(datafitting)andchoiceofmodelbypenalizingthe(log)maximumlikelihoodwithatermrelatedtomodelcomplexity,asfollows:

( )ln lnRSSBIC n p nn

= +

AsmallernumericalvalueofBICindicatesabetterfitwhencomparingmodels.

Results

Estimated parameters of non-linear growthfunctions forGuilan sheep are presented inTable2.Also,goodnessoffitstatisticsforthesixgrowthmodelsfittedtobodyweightrecordsarepresentedinTable3. 2adjR valueshadlittledifferencesamongthemodelsforalllambs,malesandfemales,butBrodymodelandRichardsequationprovidedthegreatest

2adjR values for all lambs,males and females, and

Negative exponential provided the lowest valuesof 2adjR formales,femalesandalllambs(Table3).Also,NegativeexponentialfunctionhadthelowestvaluesofDWformales,femalesandalllambs,butotherfunctionshadlittledifferencesinthisregard.Ingeneral,DWvalueswererelativelylowfordifferentmodelsandtheNegativeexponentialmodelprovided

the lowestvalue formales, females andall lambs.TheRichardsequationprovidedthelowestvaluesofAICandBICforfemalesandalllambs,buttheBrodymodelprovidedthelowestvaluesofAICandBICformalelambs.RichardsandBrodyequationsprovidedthelowestvaluesofRMSEforfemalesandalllambs.Also,Richards,BrodyandvonBertalanffyequationsprovidedthelowestvaluesofRMSEformalelambs.But,NegativeexponentialmodelprovidedthegreatestvaluesofRMSE,AICandBICforalllambs,malesandfemalelambs.Therefore,theBrodymodelwasselectedasthebestmodelforfittingthegrowthcurveinmales,andtheRichardsmodelprovidedthebestfit of growth curve in females and all lambs.TheNegativeexponentialmodelprovidedtheworstfitofgrowthcurveformales,femalesandalllambs.Theparameteraisconsideredasanestimateofasymptoticweight.ForthedatasetstudiedheretherewerelowasymptoticweightestimatesforBrodyandRichardsequationsasthebestfittedgrowthmodels(Table2).

Thekparameter,whichrepresentsmaturationrate,isanother important trait tobeconsidered,since itindicates thegrowthspeed toreach theasymptoticweight. In this study, females generally showedhigher values for this parameter thanmales.Thenegativecorrelationof-0.99to-0.98betweena and k parametersobtained,wasbasedonthebestmodelformales,femalesandalllambs.Observedbodyweightsofanimals from1 to240daysofagearedepictedinFigure1.Asshown,therewasasigmoidaltrendforbodyweightsalongwith increase inage.Also,predicted bodyweights (Kg) as a function of age(days)obtainedwithdifferentgrowthmodelsforalllambs,malesandfemales,arepresentedinFigure2.Theestimatedgrowthcurvesweretypicallysigmoid.

Discussion

Onceanappropriatemodelof thegrowthcurveisselected,selectionemphasiscanthenbedirectedexclusively to the level of the growth curve. It isimportanttodevelopanoptimalstrategytoachieveadesiredgrowthshapethroughchangingthegrowthmodelparameters.Inpreviousstudies,abroadrangeofgrowthmodelshavebeenselected,dependingonhowaccuratelytheyfitthedata.Tariqet al.(2013)selectedtheMorgan-Mercer-Flodinmodelasthebest

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Item Parameter Model

Brody Negative exponential

Logistic Gompertz von Bertalanffy Richards

All lambs a 27.42 (0.19)

24.60 (0.12) 21.68 (0.045)

23.13 (0.07) 24.03 (0.09) 31.68 (1.85)

b 0.89 (0.001)

- 5.69 (0.05) 2.01 (0.008) 0.50 (0.001) 0.96 (0.01)

k 0.0068 (0.00009)

0.0096 (0.0001)

0.0251 (0.0001)

0.0152 (0.00009) 0.0122 (0.00009) 0.0041 (0.0008)

m - - - - - -0.73 (0.06)

Male lambs

a 29.07 (0.30)

25.80 (0.19) 22.46 (0.07)

24.08 (0.11) 25.10 (0.14) 30.73 (1.84)

b 0.90 (0.002)

- 5.81 (0.07) 2.03 (0.01) 0.50 (0.002) 0.93 (0.03)

k 0.0064 (0.0001)

0.0091 (0.0001)

0.0248 (0.0002)

0.0149 (0.0001) 0.0119 (0.0001) 0.0052 (0.0010)

m - - - - - -0.87 (0.10)

Female lambs

a 26.21 (0.23)

23.69 (0.15) 21.04 (0.06)

22.38 (0.09) 23.20 (0.11) 33.07 (3.71)

b 0.89 (0.002)

- 5.61 (0.07) 2.00 (0.01) 0.49 (0.002) 0.98 (0.01)

k 0.0071 (0.0001)

0.0099 (0.0001)

0.0254 (0.0002)

0.0155 (0.0001) 0.0125 (0.0001) 0.0032 (0.0010)

m - - - - - -0.65 (0.07)

Table 2. Parameter estimates for the different growth models in Guilan sheep (standard errors are in parentheses).

fitted equation for growth curve inMengali sheepbreedofBalochistan.Similar to the results of thisstudy,Gbangbocheet al.(2008)selectedtheBrodymodelasthebestfunctiontofitthegrowthcurveoftheAfricanDwarfsheep.Ontheotherhand,Freitas(2005) reported thatLogistic,VonBertalanffy andBrody functionsweremore versatile tofit growthcurvesinsheep.IntheBergamascasheepinBrazil(McManus et al., 2003), among thefitted growthmodels(Brody,Richards,andLogistic),theLogisticmodel showed the goodness of fit. Lambe et al. (2006)selectedtheRichardsandGompertzmodelsfortheiraccuracyoffitamongfourcompetingmodels(Gompertz,Logistic,RichardsandtheExponentialmodel).TheGompertzandvonBertalanffymodelsshowed the best fit inMorkaraman andAwassilambs (Topal et al., 2004); Gompertzmodel inSuffolk sheep (Lewiset al., 2002).Malhadoet al. (2009) reported that bothGompertz andLogistic

functionspresented thebestfit of growth curve inDorpersheepcrossedwiththelocalBrazilianbreedsMoradaNova,RaboLargo,andSantaInês.BathaeiandLeroy (1996), evaluatinggrowth inMehrabanIranianfat-tailedsheep,selectedtheBrodyfunctionbecauseof simplicity of interpretation and ease ofestimation.Sarmentoet al.(2006)observedthattheGompertz function presented the best adjustmentwhen compared to the othermodels in studies ofgrowthcurvesofSantaInêssheepherdsinthestateofParaíba,Brazil.Akbaset al. (1999) studied thefittingperformanceofBrody,Negativeexponential,Gompertz, Logistic and vonBertalanffymodelstodataonweight-ageofKivircikandDaglicmalelambsandfoundthattheBrodymodelwasthebestequationfordescribingthegrowthoflambs.Similarto thecurrent results,Goliomytiset al. (2006)andGhaviHossein-Zadeh(2015)reportedtheexcellentfitof theRichards function to theweight–agedata

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Table 3. Goodness of fit for different growth curves in Guilan sheep.

Item Statistics ModelBrody Negative

exponentialLogistic Gompertz von

BertalanffyRichards

All lambs 0.9530 0.9378 0.9520 0.9527 0.9528 0.9530

DW 0.70 0.53 0.71 0.71 0.71 0.70

RMSE 3.16 3.64 3.20 3.17 3.17 3.16

AIC 542406 554126 543311 542664 542536 542393

BIC 96559 108270 97463 96817 96688 96554

Male lambs 0.9532 0.9372 0.9522 0.9529 0.9530 0.9532

DW 0.71 0.53 0.72 0.71 0.71 0.71

RMSE 3.20 3.70 3.23 3.21 3.20 3.20

AIC 242981 248794 243383 243084 243029 242982

BIC 46254 52059 46656 46357 46302 46263

Female lambs 0.9540 0.9395 0.9529 0.9537 0.9538 0.9540

DW 0.70 0.54 0.71 0.71 0.71 0.70

RMSE 3.09 3.55 3.13 3.10 3.10 3.09

AIC 269871 275901 270378 270025 269951 269856

BIC 49756 55779 50264 49910 49837 49750

2adjR : Adjusted coefficient of determination; RMSE: Root means square error; DW: Durbin–Watson; AIC: Akaike information criteria; BIC: Bayesian Information Criteria.

2adjR

2adjR

2adjR

ofKaragouniko andShall sheep, respectively.Ontheotherhand,daSilvaet al.(2012)observedthatRichardsmodelwasproblematicduringtheprocessofconvergenceinSantaInêssheep.

AccordingtothelowDWvaluesobtainedinthecurrent study fromfitting thenon-linearmodelsofthe growth curve, itwas concluded that therewaspositive autocorrelation between residuals for allmodels.TheNegativeexponentialmodelprovidedthemostpositiveautocorrelationamongdifferentmodels.It seems that residual autocorrelation presented noproblem, or only a very slight one in the currentstudy.Positiveautocorrelationisaserialcorrelationinwhichapositiveerrorforoneobservationincreasesthechancesofapositiveerrorforanotherobservation(GhaviHossein-Zadeh,2014).Thelowerestimatesofaparameter,obtainedfromthebestfittedmodelsinthisstudy,mayindicatethatanimalsarelighterasadultsandmaybeconsideredfast-growing,asthesesheeprequirelesstimetoreachmaturitycomparedtootherbreeds(daSilvaet al.,2012).

The definition of an optimum adult weightis controversial as it depends on the species,breed, selectionmethod,management system andenvironmental conditions (Malhado et al., 2009).Itwas reported that average adult liveweight inGuilansheepis35Kg(77lbs)forramsand31Kg(67lbs)forewes(Eteqadiet al.,2014).Thesevaluesweregenerallyconsistentwith theestimatesof theaparameterobtainedfromthebestfittedmodelsinthisstudy.

Contrary to the current results, da Silva et al. (2012) reported high asymptoticweight estimatesinSantaInêssheep,buttheiranimalswerebetween120and774daysofage.Lôboet al.(2006)evaluatedthegrowthcurveofSantaInêssheepfrombirthto550daysofageandobtainedgreatera value thanthe current study.Also,McManuset al. (2003) inthestudyofgrowthcurvefrombirthto2,000daysofage inBergamáciasheepobtainedagreaterestimateofaparametercomparedwith this study.Generally,consistentwiththecurrentresults,Malhadoet al.(2008)

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a b

c

Figure 1. Observed body weights for all lambs (a), males (b), and female lambs (c).

studiedthegrowthcurveofTexel×SantaInêscrossedsheepfrombirthupto365daysofageandobtaineda valuesrangingfrom29.14to32.16Kg.Also,Malhadoet al.(2009)reportedavaluesof29.35to32.41inthefitofdifferentgrowthfunctionsfortheMoradaNova,RaboLargo,andSantaInêsBraziliansheepbreeds.TheseBrazilianbreedswereweightedfrombirthupto

210daysofageinthestudybyMalhadoet al.(2009).Toral(2008)warnedthatinferencesaboutestimatedparametersmayoftenbesubjecttoerrorsbecausethedatasetsthatareusedtofitnon-lineargrowthfunctionsdonotincludeanimalgrowthuntilmaturity.Similarly,Garneroet al.(2005)statedthatwhentheavailabledata cover only growth before reachingmaturity,

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Figure 2. Predicted body weights (Kg) as a function of age (days) obtained with different growth models for all lambs (a), male (b), and female (c) lambs.

a b

c

thesedatacanproduceerroneousa and kestimates.Animalswithhighk valuesshowprecociousmaturityinrelationtothosewithlowerk valuesandsimilarinitialweight.Thegreatervaluesofkforfemalelambsinthisstudyindicateshighermaturityrates(i.e.,theyreachedmatureweightearlier).

Because of the narrow deviation range in theweight at birth, the variation among the k valuesbecomes a reliable predictor of the growth rate(Malhado et al., 2009).Therefore, animalswithhigher k values reach asymptoticweight earlier.Bathaei andLeroy (1996) estimatedmuch higher

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k valuesforbodyweightthanthecurrentstudy,formalesandfemalesoftheMehrabanIranianfattailedsheep,whenapplyingtheBrodygrowthmodelovera48-monthperiod.Thisdifferencecouldbeattributedtothetimeunit(month)andtimeperiod(fouryears)theyusedcomparedwiththepresentstudy,andthedifferent growth function used (Goliomytis et al., 2006).Goonewardeneet al. (1981),Rogerset al. (1987)andPerottoet al.(1992)obtaineddifferentk valueswhenapplyingdifferentgrowthfunctionstothesameweight-agedata,indicatingthatthemodelofchoiceaffectsparameterestimates.

Themost important biological correlation for agrowth curve is betweena and k parameters.Thenegativecorrelationbetweentheseparametersinthecurrentstudyimpliesthattheearliestanimalsarelesslikelytoexhibithighadultweight;i.e.,animalsthathave higher adultweights generally present lowergrowthratesthananimalswithaloweradultweight(daSilvaet al.,2012).Also,similartothesigmoidalpatternsofgrowthcurvesinthisstudywereobtainedbyotherstudies(Goliomytiset al.,2006;Malhadoet al.,2009;daSilvaet al.,2012).

The sixnon-linear functions investigated in thepresentstudywereadequateindescribingthegrowthpatterninGuilansheep.However,theBrodymodelprovidedthebestfitofgrowthcurveinmales,andtheRichardsmodel for females and all lambsdueto the lower values ofRMSE,AIC, andBIC andgreater values of 2adjR than the othermodels.Theresultsofthisstudyshowthatcurrentgrowthcurvemodelswould be very helpful in genetic breedingprograms, flock nutritional management, anddecisionmaking on culling of animalswith slowgrowth rate. In this condition, selection could bemadeforanimalswithfastergrowthrateandorthatare early-maturing after observing their individualgrowth curve.After selecting a desiredmodel ofthegrowthcurveforthisbreedofsheep,itisworthnotingtodevelopappropriategeneticselectionandmanagement strategiesalongwith theprovisionofgoodenvironmentalconditionsintheflocktoachieveasuitableshapeofgrowthcurveviamodifyingthemodelparameters.

Acknowledgements

The corresponding author would like toacknowledgetheUniversityofGuilanforfinancialsupportofthisresearch(ProjectNo.989).

Conflicts of interest

The authors declare they have no conflicts ofinterestwith regard to thework presented in thisreport.

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