ESTIMATE OF Alabama argillacea ( HÜBNER ) ( LEPIDOPTERA : NOCTUIDAE ) DEVELOPMENT WITH NONLINEAR MODELS

The objective of this work was to evaluate which nonlinear model [Davidson (1942, 1944), Stinner et al. (1974), Sharpe & DeMichele (1977), and Lactin et al. (1995)] best describes the relationship between developmental rates of the different instars and stages of Alabama argillacea (Hübner) (Lepidoptera: Noctuidae), and temperature. A. argillacea larvae were fed with cotton leaves (Gossypium hirsutum L., race latifolium Hutch., cultivar CNPA 7H) at constant temperatures of 20, 23, 25, 28, 30, 33, and 35oC; relative humidity of 60 ± 10%; and photoperiod of 14:10 L:D. Low R2 values obtained with Davidson (0.0001 to 0.1179) and Stinner et al. (0.0099 to 0.8296) models indicated a poor fit of their data for A. argillacea. However, high R2 values of Sharpe & DeMichele (0.9677 to 0.9997) and Lactin et al. (0.9684 to 0.9997) models indicated a better fit for estimating A. argillacea development.


INTRODUCTION
Alabama argillacea (Hübner) (Lepidoptera: Noctuidae) is a native species of Central and South America found in most areas where cotton is cultivated, from South Canada to northern Argentina (Carvalho, 1981).This species occurs during the whole cultivation period of cotton plants, with increasing populations as the cycle progresses.This insect defoliates cotton plants from the lower to the upper parts, with most damage occurring during the last three instars (Bellettini et al., 1999).A. argillacea is the main cotton defoliator pest, greatly impacting cotton plant productivity in Argentina, Brazil, Colombia, Mexico, Nicaragua, Paraguay, Peru, and the USA (Almestar et al., 1977;Falcon & Daryl, 1977;Cies, 1978;Alvarez & Sanchez, 1982;Nyffeler et al., 1987;Michel, 1994;Ramalho, 1994;Lobos, 1999).A. argillacea is a lesser pest in South-Central Brazil but, except for the State of Bahia, in the Northeast this pest can cause damage during the whole cotton crop cycle (Gravena & Cunha, 1991).
The introduction of another cotton pest in Brazil, Anthonomus grandis Boheman (Coleoptera: Curculionidae), complicated pest control in this crop, resulting in a significant reduction in the area cultivated (Ramalho et al., 1989).New areas for agriculture in Brazil such as the savannah regions in the States of Mato Grosso do Sul and Goiás have added the importance of A. argillacea as a cotton crop pest.Besides, the appearance of populations resistant to insecticides has been increasing problems with pest control in this crop.
The relationship between insect developmental rate and temperature represents an important ecological variable for modelling population dynamics of these organisms.Linear models were the first developed for insects (Howe, 1967) but the lack of linearity of insect developmental rate at low and high temperatures suggests that such models are inadequate to describe this parameter for these organisms.Since early 1980 this problem has led to increasing interest in developing nonlinear phenological models in integrated pest management programs (Wagner et al., 1984).
Nonlinear models (Logan et al., 1976) have been elaborated for several insect species in certain circumstances.Davidson (1942Davidson ( , 1944) ) described the insect developmental rate as a function of temperature using logistic equations.Stinner et al. (1974) described the temperature effect on developmental rate as a modified sigmoid equation that results in a symmetrical curve at higher temperatures.Sharpe & DeMichele (1977) formulated a complex biophysical model, later modified by Schoolfield et al. (1981), which describes a nonlinear response of developmental rate of insects exposed to low and high temperatures as well as a linear response at intermediate temperatures (Wagner et al., 1984).Lactin et al. (1995) modified the nonlinear model of Logan et al. (1976) by eliminating the parameter ψ and introducing parameter intercepts λ, which allowed estimation of the developmental threshold.
Because linear models are not very precise sources of information on developmental rate inhibition at extreme temperatures, the purpose of this research was to determine which nonlinear model (Davidson, 1942(Davidson, , 1944;;Stinner et al., 1974;Sharpe & DeMichele, 1977;Lactin et al., 1995) better describes the temperature effect on the developmental rate of A. argillacea.

MATERIAL AND METHODS
This research was developed at the Biological Control Unit (UCB)/Embrapa Algodão, in Campina Grande, State of Paraíba, Brazil.Specimens of A. argillacea were collected in Touros, State of Rio Grande do Norte, and maintained in BOD at constant temperatures of 20, 23, 25, 28, 30, 33, and 35 o C, relative humidity of 60 ± 10%, and photoperiod of 14:10 L:D.Larvae of this species were fed with cotton plant leaves (Gossypium hirsutum L., race latifolium Hutch., cultivar CNPA 7H) while adults received a solution of honey (20%) and distilled water (80%).
Newly emerged adults were used to form 15 pairs of A. argillacea, with five of them per PVC cage (Medeiros, 1997).First egg clutches of A. argillacea were placed in Petri dishes (9.0 x 1.5 cm) with a moist cotton ball, and observed daily to determine their incubation period and viability at the same temperatures and humidity conditions used for A. argillacea pairs.
The fifty-first instar of A. argillacea larvae were individualized in plastic cups (250 ml) for each treatment.A cylindrical plastic tube (2.5 cm) with distilled water was inserted in a circular hole in the cover of each plastic cup.One fresh cotton leaf was removed from apical parts of plants and used daily as food for A. argillacea larvae.The dorsal surface of these larvae was marked with dye (Day-Glo Colour Corp) to facilitate observation of ecdyses.Everyday, results were recorded and cotton leaves substituted.Mean developmental rate of egg, instars, prepupae, and pupae of A. argillacea at different temperatures was estimated with the formula: where r(T) is the mean developmental rate at temperature T ( o C); di, individual observations of development period in days; and n, number of observations.This method is recommended by Logan et al. (1976) to account for linearity in the transformation of development period to developmental rate.
Developmental rate is the reciprocal of development period in days and represented by values from 0 to 1.These rates are used in development models where data are added daily.Organism development is completed when the sum of their daily developmental rate reaches value 1 (Curry & Feldman, 1987).Therefore, the integral of the developmental rate function along time (as in the models of Davidson, 1942Davidson, , 1944;;Stinner et al., 1974;Sharpe & DeMichele, 1977;Lactin et al., 1995) can be used to simulate the development of an organism submitted to different temperatures.For this reason, descriptive nonlinear procedures have been used to analyze relationships between developmental rate of A. argillacea and temperature as: 1) logistic equation of Davidson (1942Davidson ( , 1944)): where r(T) is the mean developmental rate at temperature T ( o C); a, value which defines the place of the regression line in relation to the x axis; b, slope of the curve line; k, constant defining the upper limit of the sigmoid line; Ti, temperature in the environmental chamber.
The parameters a, b, and k were estimated with the regression nonlinear model of Marquardt using PROC NLIN (Sas Institute Inc., 2000).This method is used to determine the minimum square of the parameters estimated with this model.
The parameters c, k 1 , and k 2 were estimated with Marquardt's method using PROC NLIN (Sas Institute Inc., 2000).
3) biophysical model of Sharpe & DeMichele (1977), modified by Schoolfield et al. (1981) where r(T) is the mean developmental rate at temperature T ( o K); R, universal gas constant (1.987 cal degree -1 mole -1 ); RHO 25 , developmental rate at 25 o C (298.15 o K), assuming no enzyme inactivation; H A , enthalpy of activation of the reaction catalyzed by a rate-controlling enzyme; T L , Kelvin temperature at which the rate-controlling enzyme is half active and half low-temperature inactive; H L , change in the enthalpy associated with low temperature inactivation of the enzyme; T H , Kelvin temperature at which the rate-controlling enzyme is half active and half hightemperature inactive; and H H , change in the enthalpy associated with high-temperature inactivation of the enzyme.
Parameters RHO 25 , H A , T H , and H H were estimated with Marquardt's method using PROC NLIN (Sas Institute Inc., 2000), with the procedure adopted by Wagner et al. (1984).
The numerator of the Sharpe & DeMichele (1977) equation explains the dependence of developmental rate with temperature in the absence of inactivation at low or high temperatures, while the first and the second exponential equations in the denominator explain respectively the inhibition at low and high temperatures (Wagner et al., 1984).These authors developed a method to determine if a model with six, four, or two parameters adjusts to the data.This method tests the nonlinearity of data for extreme temperatures (low and high), that would indicate inhibition at those temperatures.The model is constituted by six parameters and is better adjusted to data if both extreme temperatures have a significant effect on inhibition.The parameters T H and H H assume constant values of 1,000 and 100,000,000, respectively, when high temperatures have no significant effect on inhibition.If low temperatures have no significant effect on inhibition, the parameters T L and H L receive constant values of 100 and -100,000,000, respectively.Therefore, the model with four parameters will be better adjusted to data in both cases.When both low and high temperatures have no effect on inhibition, the model with two parameters is better adjusted to data; and the four parameters T H , H H , T L , and H L will have constant values of 1,000; 100,000,000; 100; and -100,000,000, respectively.
4) The model of Lactin et al. (1995)  where r(T) is the mean developmental rate at temperature T ( o C); T L , lethal temperature ( o C); ρ, rate of increase at optimal temperature; ∆ T , difference between the lethal and optimal temperature of development; and λ, the parameter that makes the curve intercept the x-axis, allowing the estimation of a developmental threshold.
The parameters T L , ρ, ∆ T , and λ were estimated with the method of Marquardt using PROC NLIN (Sas Institute Inc., 2000).
Determination coefficient (R 2 ) of nonlinear models cannot be calculated with linear models [(R 2 = 1 -(SQR/SST)], because most of the nonlinear models show non-identifiable intercepts.In this case, the Sas uses the sum of the noncorrelated squares instead of the sum of total squares (Freund & Littell, 1986).The R 2 of these models were calculated as R 2 = 1 -(S 2 y /S 2 td ), where S 2 y is the variance of the residues of the model and S 2 td is the variance of observed means of developmental rate.
The R 2 values for the logistic model of Davidson (1942Davidson ( , 1944) ) from 0.0001 to 0.1179 (Table 2) and the sigmoid model of Stinner et al. (1974) from 0.0099 to 0.8296 (Table 3) were low, which suggests that neither are appropriate for describing data obtained with A. argillacea.On the other hand, high values of R 2 for the biophysical model of Sharpe & DeMichele (1977) from 0.9677 to 0.9997 (Table 4), and Lactin et al. (1995) from 0.9685 to 0.9997 (Table 5) showed better adjustment to A. argillacea data.
Inhibition of development of A. argillacea due to temperature occurs at 35°C, while this was not significant at 20°C.The version of the Sharpe & DeMichele (1977) model had values of 100 and -100,000,000 for T L and H L , respectively, because inhibition of development of A. argillacea was significant at higher temperatures.Larvae of A. argillacea showed higher tolerance to high temperatures which is represented by a high value of H H with the model of Sharpe & DeMichele (1977) (Table 4) and a low value of ∆ T with the model of Lactin et al. (1995) (Table 5).
The value of the parameter T H of the Sharpe & DeMichele (1977) model is the temperature ( o K) at which the enzyme that controls developmental rate of insects is partially inhibited.Value of T H for A. argillacea was 306.3°K (Table 4), therefore this species presents thermal stress at 33.3°C.This indicates that the estimate of maximum thermal action by the model of Sharpe & DeMichele (1977) was realistic.
The parameter T L of the model of Lactin et al. (1995) represents the temperature ( o C), at which the insect does not survive for a long period.The estimated T L values for A. argillacea were similar to those observed (Table 5), because transformation of these values to current temperatures (T L + 20) shows that the lethal one for A. argillacea is 56.83°C.The λ values calculated with the model of Lactin et al. (1995) were < 0 (Table 5), indicating that this model can be used to calculate minimum temperature for each instar and developmental stage of A. argillacea.T L = lethal temperature (°C).∆ T = difference between development at lethal and optimal temperatures.λ = parameter that makes the curve intercept the x-axis, allowing development threshold estimation.Sharpe & DeMichele (1977) and Lactin et al. (1995) models appropriately described relationships between developmental rate and temperature for A. argillacea (Fig. 1).

DISCUSSION
The logistic equation of Davidson (1942Davidson ( , 1944) ) and the sigmoid model of Stinner et al. (1974) do not appropriately describe relationships between the developmental rate of different stages of A. argillacea and temperature.
Although these models have been used to describe the relationship between the developmental rate and temperature of insect species, they present the following problems: (1) the model of Stinner et al. (1974) assumes symmetrical shape at both sides of optimal temperature and for this reason does not appropriately describe insect development at high temperatures; and (2) the model of Davidson (1942Davidson ( , 1944) ) offers low descriptive precision at both ends of the relationship curves between developmental rate and temperature (Wagner et al., 1984).Harari et al. (1998) pointed out that the Davidson (1942Davidson ( , 1944) ) model was not adequate in the case of development of Maladera matrida Argaman (Coleoptera: Scarabaeidae) because it estimated longer developmental rate at higher rather than at optimal temperatures.
The biophysical model of Sharpe & DeMichele (1977) describes a nonlinear response between developmental rate at low and high temperatures, as well as a linear response at intermediate temperatures.For this reason, Wagner et al. (1984) and Fan et al. (1992) consider that this nonlinear model better describes the effect of constant temperatures on insect development.The model was applied and evaluated by Gould & Elkinton (1990), Orr & Obrycki (1990), Fan et al. (1992), Morales-Ramos & Cate (1993), Judd &McBrien (1994), andHarari et al. (1998) and was considered appropriate for determining developmental rate of organisms studied.Lactin et al. (1995), modified the nonlinear model of Logan et al. (1976) by eliminating the parameter Ψ and introducing the parameter λ (intercept), which allowed estimation of development threshold for insects.The resulting model was applied and evaluated by Briere & Pracros (1998) and is appropriate for describing the relationship between developmental rate of different stages of Lobesia botrana Dennis & Schiffermüller (Lepidoptera: Tortricidae) and temperature.Models of Sharpe & DeMichele (1977) and Lactin et al. (1995) were more precise for describing the relationship between developmental rate of different stages of A. argillacea and temperature (Fig. 1) because both described an asymmetrical shape around high temperatures (Fig. 1).Briere & Pracros (1998) stated that the relationship between developmental rate and temperatures in insects is nonlinear and presents an asymmetrical shape composed of three sections: the first is represented at low temperatures where the increase in developmental rate is nonlinear from development zero; the second section, where the developmental rate becomes proportional to temperature increase; and the third, which starts with the optimal and finishes with lethal temperature.
Our results suggest that Sharpe & DeMichele (1977) and Lactin et al. (1995) models are more precise for describing the relationships between developmental rate of different instars and stages of A. argillacea and temperature.These results can therefore be used to forecast the occurrence of different stages and instars of A. argillacea in cotton crops, and enable greater precision in choosing the best periods for controling this pest.
resulted from modifications in the nonlinear model ofLogan et al. (1976):
r(T) is the mean developmental rate at temperature T (°C), di, individual observations of development period in days and n, number of observations. :