A weight‐based phenology model for immature stages of the red‐headed cockchafer, Adoryphorus coulonii (Burmeister) (Coleoptera: Scarabaeidae: Dynastinae), a pest of pastures in south‐eastern Australia

A new method of modelling insect phenology which can be used when stadia occupancy times are unobserved is described. the method was motivated by a study of larvae of the red‐headed cockchafer which were reared and weighed regularly for a range of constant temperatures using largely field‐collected first, second and third‐instar larvae. These larvae had undergone an unknown but significant proportion of their development in the field. the method first models development rate as a function of temperature using relative growth rates in mean larval weight. Predicted mean larval weight was then used as a physiological time scale in an ordinal regression model of the proportion of the population in each of first to third‐instar stage. an algorithm to predict proportions in the field of each stage up to and including the pupal stage given a known starting date for the population and daily maximum and minimum soil temperatures is described and tested using data from two field sites. Growth rates under laboratory conditions were slower than those in the field probably due to the handling required for weighing. As a result, an adjustment to the rate parameter and a 10d̀C lower development threshold were required for the growth and phenology models to predict trajectories for mean larval weight and proportion of second‐instar larvae which were consistent with observations from the field sites.


INTRODUCTION
The red-headed cockchafer, Adoryphorus coulonii (Burmeister), is a serious pest of improved pastures in southeastern Australia where the soil-dwelling larvae destroy the roots of grasses and clovers (McQuillan & Ireson 1987).To minimise economic losses, prediction of the phenology of the immature stages of A.cou/onii is important.The phenology of the red-headed cockchafer is largely temperature-driven in a buffered environment due to its predominantly subterranean life cycle and lack of exposure to daylength as a synchronising cue.The life cycle takes 2 years to complete with oviposition regionally synchronous in mid to late spring.There are three larval instars prior to pupation.Newly eclosed adults undergo a 6-month subterranean diapause and there may be two non-interbreeding populations in the same area of ground with life cycles out of phase by 12 months (Hardy 1981).
Insect phenology models attempt to predict the proportion of the population in each development stage at a given time from the start of oviposition.These models are *Author to whom correspondence should be addressed.
typically calibrated using stage-frequency data collected in the field (see Manly 1989 for a review of models estimated from these types of data).Phenology can be modelled by determining a physiological time scale (e.g.day-degrees) using constant temperature studies and then using this scale and field-collected stage-frequency data to: (i) estimate a stage progression model based on a statistical model of stage duration times (Kempton 1979;Munholland & Kalbfleisch 1991;Munholland & Dennis 1992); or (ii) estimate a statistical model of the proportion of the population in each stage using ordinal regression models or continuation ratio models (Stedinger et al. 1985;Dennis et al. 1986;Dennis & Kemp 1988;Candy 1991).Other methods have been used to model phenology (e.g.Wagner et ul. 1984a,b), but the method described here is more similar to those described above.
A new method was used to model the phenology of A. coulonii.This approach is based on a temperaturedependent growth model for cohort mean larval weight (i.e.mean of weights of all instars present assuming a single population produced from synchronous oviposition) and a model which predicts the proportion of the population in each stage given mean larval weight.The reasons why this approach was adopted were: (i) the lack of stage frequency data covering all larval instars; (ii) the unknown level of development of field-collected larvae used in the constant temperature experiments; and (iii) the small proportion of individuals that successfully coniplcted development of the stadium at which they were collected for some instars and temperature regimens.A phcnology model based on weight growth does not appear to have been used previously.Van Straalen (1985) used a tempcrdture-dependent growth model for body length to predict the progression of cohorts of Collembola through length classes.Moulting frequency was included in the calculation of the physiological time scale, but particular pre-imaginal instars were not identified in the model.The difference here is that, unlike Collembola, immature red-headed cockchafers fall into easily observable, discrete developmental stages which are of intrinsic interest.So a method of predicting proportions in each stage was required.

Growth models for cohort mean larval weight
The instantaneous relative growth rate may be modelled a s a function of temperature alone whcre t is time from hatch or moult depending on instar, W is mean larval weight at time t , T is temperature, and 0 is a vcctor of unknown parameters.
T i m of the start of growth, to, can be taken as zero without loss of generality.Mean initial hatch weight is given by WO.Integrating ( I ) with respect to time, t , gives the increment for a period between times tI and r2 as {In(&) -In(m)}(t.-~~) -' = ./ ( T , I I ) (4 I t is clear that this model of growth rates, using a general function of temperature, can be fitted to the empirical relative growth rates, y. given by the left-hand side of (2) and calculated using periodic measurements of larval weight.Final weight, W,, for the period, t2-rl given initial mean weight, WI, is Note that for this form of model, growth rate depends only oi1.flT.B) and the period length t2-tl.The age of the insect at time 1 2 , given by 1 2 -r ~) is not rcquired; that is, to need not be known (see below).A more flexible growth rate function is the von Bertalanfl'y model (Ratkowsky 1990) and Garcia (1994) gives the state-space form of this model which does not require to to be known.The statespace model is where u is the asymptote, h is the rate parameter, rind c is a shape parameter.In our application, at least one of these parameters is a function of temperature.
Under field situations, temperature varies wil:h time, 1, so equation ( 1) becomes ( 4) Integrating (4) between times tI and t 2 .t 2 > / I , gives Although any arbitrary starting date tI less than t 2 (given WI is known) could be used in ( 5 ) , in application of our phenology algorithm, a time of hatch and inil.ialmean weight, WO, will be assumed.A similar algebraic solution to the von Bertalanffy growth model for Ihctuating temperatures is more difficult.T o simplify the problem i t will be assumed that only the rate parameter, h, is ;I function of temperature: h = ,f{ T ( I ) , H } .Model (3) then becomes Two forms of the functionf( T , 0 ) were considcred for each of relative growth rate, y, and the von Bertalanffy rate parameter, h.First, the linear day-degree model given by where the lower developmental threshold tempcrature I\ given by 01 while 0 2 IS the rate of development (Howe 1967).Second, a model proposed by Ratkowsky (1990) which takes into account both lower, 01, and upper, 0 2 , threshold temperatures for development IS

Ordinal regression model using mean larval weight
A model analogous to the class of ordinal regression models fitted to stage frequency data (Candy 1901) whcrc time, t , measured either as calendar days or day-degrees is used to predict the mean of the distribution of unobserved development level.The proportion of thc population of' physiological age, or development lcvel X( 14,).whcre mean larval weight is w, below the first cut-point parameter.(1. are first instar, those with a value between X I and the second cut-point parameter, uz. are allocated to the second instar.and those above u.7 are third instar.The variable X ( W ) is analogous to LIT([) ofStedinger ct d. (1985).Thus, time is replaced by mean weight here which is itself ;in integral over temperature and time.Weight is highly positively correlated with physiological age for a given quality ofdiet.For the data from the constant temperature study, the diet was uniform and not limited in supply.
If nii is the number of larvae of instarj and weight class i with mid-class value for weight class i given by wi, Ni is the total number of larvae in the sample for weight class i, mii is the cumulative number over instars I to j (i.e.mi, = 1 nib), and the cumulative density function (dfl of X ( w ) for a given mean weight IY, is given by G(.u), then the ordinal regression model gives the expected value of mi, as where , !I and wi are the regression parameter and predictor variable, respectively, and the mean of X is given by -pwi for weight class i.The number in the final class is obtained by difference.If E(m,,) = Nipi,, and 8 . .The properties of this class of models and the estimation procedure used here have been described by Candy (1991).
In constant temperature studies of the cabbage butterfly, Pieris rapue, Gilbert (1988) found mean pupal weight decreased with increasing temperature with a rectangular hyperbola adequately describing the relationship.To test if any similar effect occurred in the constant temperature study, Pearson residuals (McCullagh & Nelder 1989) were calculated after the fit of model (8) and assessed for any trend with temperature.

Constant temperature study
A total of 22 batches collected as one of instars I, 11 or 111 was reared under constant temperature regimens ranging from 7°C to 31°C.Larvae were housed in single rearing cells to avoid competition and fed liberally on fresh carrot slices.Table 1 gives details of these batches including the number of larvae which developed through to at least the next instar (or pre-pupa stage for the case of field collected third-instar larvae) and the mean weight at the date they first entered the incubator.All larvae in a batch were weighed at approximately weekly intervals for the higher temperature regimens and 2-weekly for the remainder.Relative growth rates were calculated for mean larval weight from consecutive observation times.Since the identity of individual larvae was not maintained from measurement to measurement, only growth in the mean of the individual weights for each instar could be calculated.Growth periods in which more than one individual moulted (or entered the non-feeding stage in the case of third-instar larvae) were excluded.To maximise the number of growth periods available for modelling, it was found necessary to combine first and second-instar larvae so that moults of first-instar larvae could be ignored.
Empirical relative growth rates, y, based on equation (2).were calculated for first and second instars combined, for each batch and period as:

CI
where Wl is the mean weight at the end of the period, & is the mean weight at the start of the period and cl is the length of the growth period in days.The factor 100 is used to stiindardise the relative growth rates to a period o f 100 days at the regimen temperature, and avoid small vnlues if a daily rate is used.For the von Bertalanffy model.equation (3) was fitted to the observed mean weights at the end of each increment period, Wr, with ii constant value for each of the N and c parameters and a separate estimate o f the rate parameter, h. for each batch.
Given the restrictions on periods used for model fitting described above, ;I total of 35 pairs of values of ( W2, Wl) were available for fitting growth rate models for the combined lirst and second instar, and 69 pairs for the third instar.
For the fit of the ordinal regression model, the number of larvae in each combination of instar and weight class were pooled across all 22 batches.Due to the large range in weight from freshly hatched first t o fully developed third instnrs.unequnl weight classes were used.More important was the division of the weight scale into classes in the range where first instar progress to second instar, and the range where second progress to third instar.A division of these ranges into suficient classes to detect this change was required.Outside these critical ranges divisions were broad.As well.numbers in each instar and weight class for each batch were determined to test any efl-ect of temperature on the progression of larvae through the weight and instar cliisses.A total o f 2 I79 larvae were classified by weight and instar for [he constant temperature rearings.

Field data
Two adjacent sheep-grazing properties, Rockville and Invcrull near Oatlands (42"18'S, 147"21'E), in the Midl a n d s region of Tasmania were selected for field sampling.

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pasture on ii clay-loam duplex soil over a clay subsoil, while the Inverell site was botanically similar, but the soil was a deeper sandy loam.Both sites had experienced outbreaks of larvae during the previous 5 years.
Field sampling was carried out using two methods.First, a single block of soil was taken at each sampling occasion using a spade to a depth of approximately 20 cm within a fixed 25m by 2Sm plot.Populations were sam-pled at regular intervals (approximately weekly or fortnightly) at both sites for the period 8 March 1990 to 6 May 1992 and individual larvae from 21 random subsample of at least 50 were weighed and classified to instar.In an unrelated experiment, from 26 March 1991 at Rockville and 12 April 1991 at Inverell, a second sam- pling method was used where 100 core samples, each 9 cm in diameter and lO--l2cm deep, were taken every 2.5m o n an adjacent 25 ni x 25 m grid.All larvae in the core samples were classified to instar and then replaced after counting at approximately monthly intervals.Larvae in this sample were not weighed.
The start of a new generation was assumed t o be 31 October 1990 for both sites based on the first appenrancc of eggs at both sites at the next spade sampling occasion on 15 November 1990.This allows a reasonably accunitc (i.e.* 2 weeks) estimate of the date of the start of the population.Note that the eggs are extremely dillicult to find because of their small size so this stage was excluded from the stage frequency data.The population progressed through each instar until some pupae were collected in mid-March at lnverell and early April at Rockville in 1992.Only sampling from IS November 1990 will be considered for the following.The individu..il Inrval weights were tabulated into the same weight clusscs used in the constant temperature studies for each instar for each of the two sites; 4514 and 51 16 larvae were classified for Inverell and Rockville, respectively.
Due to the late start of the core sampling, the stages collected consisted almost entirely of second and thirdinstar larvae and no first-instar larvae were present in the core samples after April.Absence of the egg and lirstinstar from this stage frequency data set meant that it could not be used to calibrate a phenology model.The total number in each stage over all 100 core samples was obtained for each sampling occasion at each sitc.These data were used to compare the observed and predicted number of second-instar larvae expressed as a pwportion of the total population of second and third-instar Iiirvae ( i s .first-instar larvae were excluded) with predictions obtained using the algorithm described below.
Daily soil minimum and maximum temperature n t ii depth of 5cm were recorded at the Inverell site during I March 1989 to 31 March 1992 which covered most ofthc field sampling dates.Missing values in these soil teniperature data were estimated using a regression of daily soil minimum and maximum on corresponding daily minimum and maximum air temperatures from ii nearby meteorological station (Candy 1993).Non-linear interpolation between temperature extrema using cubic splines was carried out to obtain temperature at 15-min intervals (Candy 1990(Candy , 1993)).

Predicting Adoryphorus coulonii field phenology
The models described above were implemented in a n algorithm which predicts the phenology ant1 cohort ( 5 ) When the proportion of third-instar larvae obtained in (3) reached 0.95, mean larval weight was incremented using model ( 6) and parameter estimates in Table 2.The mean weight and date at the time at which the proportion first exceeded 0.95 was recorded to allow the mean weight to be incremented.Although incrementing mean larval weight when all larvae have entered the third instar is not necessary to progress the cohort to pupal stage (this is already carried out in step ( 4)), it is useful to compare predicted mean larval weight with mean third-instar larval weight.mean weight of A .coulonii under fluctuating temperatures in the field.Predictions obtained were then compared to the time of appearance of each larval instar and pupal stages observed in the spade samples and the proportion of second and third-instar larvae that are second iristar observed in the core samples.Also, mean larval weight predicted from the growth models was compared to mean weight of each larval instar obtained from the field sampling.The date of initiation of the new generation was taken as the 31 October 1990 for both Inverell and Rockville sites and the method of progressing the population through development stages is described in terms of a cohort of 100 individuals.The method of prediction of field phenology is given by the following steps.
( I ) Oviposition is assumed to have occurred simultaneously for all 100 individuals at 12 h on 31 October 1990.Development time (or incubation period) from oviposition to hatch is accumulated in terms of day-degrees with a 12°C lower threshold until 285 day-degrees have been accumulated (Candy 1993).All 100 eggs are assumed to hatch on this day.
( 2 ) The mean weight of the 100 first-instar larvae at hatch is assumed to be 8.0mg.Mean larval weight (i.e.mean of all instars) is accumulated using model (7) (with a given value of the lower threshold, 0 , ) incorporated in equation (3, soil temperature at 15-inin intervals, and the initial mean hatch weight.Using this procedure, the mean weight of all larvae present was predicted at each sanipling date. (3) Using the ordinal regression model ( 8) with logistic link function, logarithm of mean larval weight obtained in step (2), and parameter estimates obtained from the constant temperature regimens, the proportion of the cohort in each instar was predicted.These proportions were multiplied by 100 to give the predicted number in each instar.
(4) Since (3) cannot be used to progress third-instar larvae to the pupal stage, second-instar larvae progressing to third instar in the sampling period (i.e. between consecutive sampling dates) were maintained as a separate subcohort with the date of the end of the period recorded for this sub-cohort.Day-degrees with a 12°C lower threshold were accumulated from this date and all third-instar larvae in the sub-cohort progressed to the pupal stage when 800 day-degrees had accumulated (Candy 1993).

R E S U L T S Initial development
The initial weight of a freshly hatched first instar based on the two laboratory batches averaged 8ing (range = 5.8-10.6mg).The mean initial weight of the other firstinstar batches were all above 27mg (Table I), so that these larvae had probably already undergone some developnient when they were collected.Some of the difference between laboratory and field first-instar larvae could be due to soil and food i n the gut, which is a significant component of larval weight for each instar up until the non-feeding pre-pupal stage (McQuillan & Webb 1994).The mean weight of freshly moulted second-instar larvae at the first measurement of the laboratory reared batch was 56.3 nig, while the other field-collected second-instar batches had mean initial weights greater than l00mg (Table I ) .The average weight of third-instar larvae at the measurement period when they were first observed in batches reared from field-collected second-instar larvae was 343.4 mg.The average weight of the second-instar larvae at these batch's previous measurement was 242.6 nig.The larvae empty their gut just prior to moult, but replenish it within a day, so larvae with empty guts are unlikely to be observed.Clearly, a significant proportion of development had already occurred for most of the field-collected batches of third-instar larvae (Table I).

Growth of first and second-instar larvae
The parameters of the von Bertalanffy model (3) could not be estimated successfully unless the asymptote and shape parameters n and c were fixed at preset values of  With these preset parameter values a separate estimate of the h parameter was obtained for each temperature.In terms of residual sum of squares, pooled across temperature regimens, for mean larval weight, Wz, of combined first and second instars, the von Bertalanffy model gave an improvement of 3% compared to weight predicted using mean relative growth.For third-instar larvae this figure was 6%.
Weighted non-linear least squares was used to fit the models (6) and ( 7) to each of the mean value of y , -V, and the estimated shape parameter, 6, for each temperature for combined first and second instars.The statistical weights were the number of periods used at each temperature.The shape of the relationship between mean of relative growth, y, and temperature was very similar to that for h, and temperature.For brevity, only the results for the models of relative growth, y , will be given, as very similar results were obtained when the von Bertalanffy rate parameter, h, was modelled.The linear model (6) was not appropriate given that a peak in growth rate, measured as either y or b occurred between 15 and 25°C.
However, the values or lower, O i , and upper, 02, threshold temperatures for model (7) could not be estimated successfully.Figure 1 shows the fit of model ( 7) to these values of ,jj, with standard errors shown, for combined first and second instars for a range of lower threshold temperatures.Given the high degree of variability in growth rates within temperature regimens, reflected in standard errors in Fig.I and the low growth rates for the 2 .5 2 a, c 1 .5 0 12°C compared with the 7°C regimen batches, there is a large degree of uncertainty about the value of the threshold parameters O1 and 02.In the validation of these models, described later, the value of the lower threshold is more critical in predicting field phenology than the upper threshold.As a result, the upper threshold was fixed at 35°C and combined with this, values of 5, 7. 10, and 12°C were used for the lower threshold, 01.The remaining two parameters for model (7) were estimated by weighted non-linear least squares.In the case o f 0, set less than 12"C, the data for the 12°C regimen batches were excluded from the fit.Figure 1 shows the fitted model for each value of 0 , .When the 12°C regimen data values were included, the fitted model underestimated the growth rates considerably for the 20°C regimen which provides the most reliable mean rate (Fig. I).Table2 gives results for the fit of model ( 7).From the fitted growth lcurves for model ( 7) the main difference between the curves is in predicted growth rates between 5 and 15°C (Fig. I).I n the validation of this model, described later, these rliflrerences were important when predicting field phenology.

Growth of final-instar larvae, prepupal and pupal development
The approach of using a temperature-dependent growth model for mean larval weight and the ordinal regression model to predict proportion in each stage given mean weight cannot be used to progress third-instar larvae to the prepupal and then pupal stages.This is btxause the third instar loses approximately 30% of its weight by evacuating its gut contents when it enters the non-feeding stage before progressing to the prepupal s~agt:.The approach here assumes a monotonically increasing mean larval weight.Progressing third instars to prelpupal and pupal stages uses a different approach described later.However, to predict growth in mean third-instar weight up until entry to the non-feeding period of this stadium.the linear day-degree model (6) was fitted to mean relative growth rate, y, and b for this instar i n the same way as that described for combined first and second instars.There was no evidence of non-linearity in the relationship with temperature but the variability in relative growth rates was of a similar order to that seen for the combined first and second instars.Again the relationship between 7 and temperature was very similar to that of i m and temperature.The results of the fit are given in Table2 for relative growth.

Ordinal regressions
For the ordinal regression model (8), the alternative l i n k functions described earlier combined with each of II', and In(wl) were tried.The best model was obtained using In(w,) with a logit link function.Parameter estimates and fit statistics are given for data from each of the two field sites and the constant temperature rearings in Tablc 3. Figure2 shows, for the constant temperature data, the fitted trajectory of proportion in each instar against log transformed mean weight along with the corresponding observed proportions plotted against class mid-points transformed to the logarithmic scale.Figure 3 compares the fitted models for each data set.Using the parameter estimates obtained from the constant temperature data, the residual deviance for each of the two field sites showed a significant lack of fit ( P < 0.01) with the lack of fit greatest for Inverell.The fit of the model with common parameters for both field sites compared to the fit of the model with site-specific parameters was used to test the difference between sites.This difference was significant ( P < 0.01) (Table3).The simplest way to compare these models is to determine the weight at which half the population is first instar and half second instar or later and, similarly, the weight when half the population is second instar or earlier and half third instar.These mean (or equivalently median) weights can be calculated as exp(-ccl//l) and exp{-(al + ccz)//3} respectively, noting that C C ~ is parameterised as a difference (Candy 1991).These mean moult weights are given in Table 3.I t is clear Fitted proportion in larval instar.Constant temfrom these values that the mean weights at which moulting occurs are very similar for larvae at the Rockville site and the larvae in the constant temperature experiments, but are considerably less for larvae at the Inverell site.The implications of these results for predicting field phenology are discussed later.
For the constant temperature data, Pearson residuals were calculated from observed and predicted proportions in each weight class within each batch.These residuals and their batch means were each graphed against temperature (graphs not shown).In both cases, there was no obvious residual effect of temperature.

Adoryphorus coulonii field phenology
Initially, step (2) of the field phenology algorithm used the parameter estimates obtained from the constant temperature studies.Figure 4a shows the observed mean weight of each larval instar at the Inverell site along with predictions of mean larval weight (i.e.mean over all instars) obtained in step (2) using model ( 7) (with each lower threshold temperature of 5, 7, 10 and 12°C) and equation ( 5). Figure 4b shows mean weekly maximum and minimum soil temperatures a t 5 c m depth for the field sanipling period.The results were similar a t the Iiockville site for which the model predictions are the same as those for Inverell.It is clear from Fig. 4a that the model under-estimates niean larval weight, the trajectory of which should lie between observed mean weights for second and third instars for most of the period from hatch which is predicted to have occurred on the I January 1991 (i.e. the first instar was found in field sampling for only a short period).The predicted mean weight trajectory should lie closer to the mean for second instar when this stage predominates and closer to that li)r the third instar when that stage predominates.As ii result, the predicted proportion of second-instar larvae from step (3) did not decline significantly ovtr the sampling period whereas for the core sample data, nearly all I 4 5 second-instar larvae had moulted by the end of this period.The results using the von Bertalanffy model combined with model ( 7) fitted to the estimate rate parameter, h, were very similar to those shown for the simpler relative growth rate model.
To test the effect of changing the weights at which moulting is predicted to occur, the ordinal regression model parameters obtained from the fit to the Inverell field data (Table 3) were used.This change had little effect on mean larval weight projections since it only changes the time when the growth model for third-instar larvae is used.In this case, the improvement in similarity to the field sampling in the appearance of larval instars is only slight, occurring only 1-2 weeks earlier.
To provide a model which more adequately reflected mean weight growth and phenology in the field, the field data were used to adjust the weight growth model.Unfortunately, this could only be achieved in a rudimentary manner using trial-and-error manipulation of two parameters.The reasons a more rigorous approach could not be used are: (i) the field predictions are based on the simulation algorithm given above which combined a number of models with many parameters; and (ii) the field data were not adequate for direct calibration.
The approach adopted involved manipulating starting date for the cohort and the f14 parameter in model ( 7).Significant improvement in predictions could not be achieved by changing starting date which was subsequently maintained at 31 October 1990.However, predicted trajectories for mean larval weight (Figs 4c.d) and Proportion of second instar larvae observed for Rockville (6) and Inverell ( + ) with predicted proportion obtained from the phenology algorithm using the modified model ( 7) with 5 ( -) , 7 (--), 10 (----), and 12°C (--) lower threshold temperatures.
proportion of second-instar larvae (Fig. 5) which were sensible given field data were obtained by increasing the mean relative growth rate for first and second-instars by multiplying O4 in model (7) by factors of 100/70, 100/65, 100/60, and lO0/55 for values of lower threshold temperature of 5, 7. 10, and 12°C.respectively.This is equivalent to reducing the period defined for relative growth of 100 days to the same relative growth rate for a period of 70, 65, 60 and 55 days, respectively.Using the ordinal regression parameters for the Inverell site required a scaling factor of 100/63 rather than 100/60 for the 10°C threshold in order to give predicted proportions of second-instar larvae very similar to those in Fig. 5. Figure 6 shows the estimated population density for each site obtained from the core samples.

DISC U SSlO N
In constant temperature studies, the difficulty of relying on time taken to complete development of a life stage to calibrate a development rate model is that individuals must be reared from start to finish of the stage with no information obtained on development rate for the period in-between.For insects such as A .coulonii with long-lived larval stages the time required for rearing may make study of development times difficult.Here, the problems with rearing immature stages of A .ctoulonii in the laboratory meant that the data on development time alone were inadequate and, for most batches, misleading due to prior development of field-collected specimens.This was expressed in the inability to obtain reasonable parameter estimates for the linear day-degree model for each instar and lnverell (----).Vertical bars represent SE. using empirical development rates based on stadia occupancy times.To overcome these problems we have introduced a new method of modelling phenology using growth rate of mean larval weight as a measure of development rate.In this approach, any growth period provides data for modelling development rate; therefore, making total reliance on completion of development unnecessary.As with development time, growth rate of larval weight will depend on quality and quantity of diet as well a s temperature.We have also shown how the laboratory calibrated larval weight growth model can be adjusted using data from destructive field sampling.A number of limitations of the data available for modelling i n this study became apparent and suggestions on how these can be avoided in future studies of this nature are included in the discussion below.
A greater number of temperature regimens used in laboratory rearing would have been useful to confirm the shape of.and improve estimation for, the temperature response curves for weight growth of combined first and second-instar larvae and third-instar larvae.The limited number of temperatures used resulted in model (7) being over-parameterised, but this in itself is not of great concern as the model is used in our phenology algorithm purely as a method of non-linear interpolation between temperatures used in the laboratory to the full range of predicted soil temperatures in the field.Of possibly greater concern is the uncertainty about the mean growth rates seen in Fig. 1 due to their large standard errors.The variability in growth rates and the anomalous result for the 12°C regimen probably reflect the sensitivity of A .c~oukviii to laboratory rearing including regular handling to measure larval weight.In future studies of this nature, the amount of handling could be reduced by selecting, at each occasion, a random subsample of larvae for weighing .
Due to these difficulties, validation of the growth and phenology models using field data was all the more crucial.For mean larval weight growth, choosing a range of values for the lower threshold parameter, O,, allowed the 'best' value of those tested to be selected based on the beliaviour of predictions compared to field data.From Figs 4c, 4d, and 5, it is clear that the 5 and 7°C lower threshold values allow development to continue through winter so that continued growth in mean larval weight using these values in step (3) resulted in the predicted proportion of second instar declining too early compared to observed values (Fig. 5 ) and growth trajectories for mean larval weight which appear to increase unrealistically during winter (Fig. 4c,d).From Fig. 5 it appears that the 10°C threshold gives the best prediction of proportion second instars.Obtaining a more precise value of the upper threshold parameter is relatively unimportant as its influence on the shape of the temperature response curve occurs in the temperature range which is outside the range of ambient soil temperatures.In contrast, the adjustment of relative growth rate for laboratory reared first and second instar, via the scaling of O4 using the Factors given earlier for each temperature threshold, was required to more accurately reflect field results (Fig. 4a.c.d).Although the adjustment of H4 for each value of the lower ,Lhreshold parameter could not be carried out rigorously, the high degree of uncertainty in estimates of 04, reflected in the standard error of its estimate (Table 2), suggests that 0, is also an appropriate parameter to adjust.The adjustments used are within the statistical limits of the constant temperature data, since scaling up predicted rates with these factors gave maximum relative growth rates which were within the upper standard error limit for the 15°C regimen (i.e. from 1.5 to 2.5, see Fig. I).In comparison, the maximum relative growth rate for the field sampling for second instars was 4.2 for the period 25 January 199 I to X February 1991 at Inverell and for first instars was 4.5 for the period 8 January 1991 to 17 January 1991 at Rockville.However, it should be remembered that i n the case of field sampling these relative growth rates incorporate both between-and within-cohort variation.A component of the difference in growth rate and development time between field and constant temperature rearing5 could be due to the acceleration of development rates induced by fluctuating temperatures (Hagstrum & Hagstruin 1970).
The slight improvement in fit of the von Bertalanlry growth model did not noticeably improve predictions for the field data.However, if the models described here are used in other studies, the von Bertalantry model should be considered, particularly if the calibration and validation data sets are less noisy than those used here.
The results of the field sampling and model predictions (using the above adjustments) at the Rockville site are very similar to those at Inverell.Timing of appe:arance of each instar observed in the spade sampling and the proportion of second-instar larvae in the population observed from core sampling are very similar at both sites.However, one clear ditrerence between the sites is that 211 Rockville, mean third-instar weight increased through summer 1991/1992 while, at Inverell, mean weight changed little over this period.This difference could be due t o the higher population density at the lnverell site (Pig.h ) , possibly combined with the poorer soil conditions at this site.The soil at lnverell is drier over summer due to its shallower depth.There are large fluctuations in the mean weight of each instar at both sites and this a n be attributed largely to the lack of replication of the s p d c sampling.Dates when first observations of each imniaturt' stage were made in the field sampling at these two sitcs correspond well to the description of the life cycle o f A .coulonii given by Hardy (1981).
Some other features of the field data worth noting are: (i) the observed dip in the proportion of second instar in autumn (Fig. 5 ) which occurred at both sites making sampling variation an unlikely cause; (ii) the persistence of second instar at around 20% in late spring when model predictions decline to around 4% (Fig. 5); and (iii) the increase in population density around May seen in Fig. 6 generalise the models presented here to include these factors and possibly explain the growth differences observed between the two sites.Alternatively, in similar studies, it may be more efficient to incorporate soil type and population density enects in both the growth and phenology models using data from laboratory rearing using a range of soil types and densities per rearing cell.
In summary, the best model of mean larval weight growth rate was found to be model (7) for mean relative growth rate with a 10°C lower threshold combined with scaling O4 by assuming the model applies to a period of 60 days for Rockville and 63 days for Inverell, rather than 100 days.We believe that despite the limitations imposed by the data, a useful model of the phenology of A .coulonii and, more generally, a potentially useful modelling method for insect phenology have been developed.with the greater increase observed at Inverell.These features of the data could be the result of two separate populations developing slightly out of phase with each other, but with each developing synchronously across the two sites.Although oviposition is generally synchronous, there may be a number of episodes of mass oviposition, which explains the increase in population density, particularly at Inverell, and the phenological observations (i) and (ii).Multiple episodes of oviposition giving rise to cohorts of different ages mixing in unknown proportions in the sampled population can seriously limit our ability to estimate and/or validate phenology models.If one episode initiates a large majority of the new population or the episodes are close together in time the negative effect on the accuracy of the phenology algorithm will be reduced.Despite this problem, the phenology model presented here should be useful in predicting field phenology of the red-headed cockchafer if the timing of oviposition episodes can be estimated with reasonable accuracy.If multiple cohorts which are of similar size and/or are sufficiently out of phase to seriously affect the accuracy of predictions, then estimates of oviposition dates and relative densities for the different cohorts are required to combine stage proportions across cohorts.This assumes there is no difference in mortality or growth rate between cohorts.
One of the assumptions employed in applying the ordinal regression model (8) in the field is that the same relationship between stage proportions and mean larval weight holds as that estimated from the constant temperature data.This assumption was reasonable for the Rockville site, but this was not so for the Inverell site.The lower mean weight at moults at the Inverell site could, like the stagnation of growth rates for third-instar larvae, be related to the high population density and/or poor quality diet at this site.Apart from our inability to incorporate such effects here, the lack of any obvious trend in Pearson residuals with temperature for model (8) gives some confidence in applying the phenology model without regard for differences in temperature profiles between sites and years.
By April 1992, predicted mean larval weight was, in general, lower than observed mean weight of third-instar larvae at Rockville (Fig. 4d), but higher than that observed at Inverell (Fig. 4c).Combining the ordinal regression model parameters estimated from the Inverell field data with a slightly slower growth rate for Inverell as compared to Rockville (i.e.100/63 compared to 100/60 scaling factor applied to 0,) for the selected 10°C lower threshold gave almost identical predictions for the proportion of the second instar to those for the Rockville simulations.The trajectory for mean larval weight for Inverell in this case (graph not shown) reached 383.4mg by April 1992 which is, in general, lower than the observed mean weight for the third instar at this time (Fig. 4c).Without further data from sites covering a range of population densities and soil types, it is not possible to + Pwi, then C -' ( p ) = 8 defines the link function in generalized linear model terminology (Nelder & Wedderburn 1972; McCullagh & Nelder 1989).Commonly used statistical distributions for X and their corresponding link functions are the normal/probit [G-I(p) = @ -' ( p ) where @(.) is the normal cd' , logistic/logit [ G -] ( p ) = In{p( 1 -p)-I}], and extreme value/complementary log-log [C-l ( p ) = In{ -In( 1 -p ) } ] .

Fig. 2 .
Fig.2.Observed and fitted proportion in larval instar (x, first; 8, second; + , third) versus the logarithm of the mean larval weight for constant temperature rearings.

Table I
Summary of constant temperature studies of larval Adoryphorus coulonii development "Number progressing to prepupal stage for third instar.'Known hatch dates From observed eggs.'Known development time from laboratory hatched first instars.

Table 2
Results of fit of larval weight growth models

Table 3
Results of fit of log-logistic ordinal regression model (8) for larval weight