Evaluation of Linear and Non-Linear Models to Describe Temperature-Dependent Development of Scopula subpunctaria (Lepidoptera: Geometridae) and Its Stage Transition Models

Abstract

Scopula subpunctaria (Herrich-Schaeffer), is a significant insect pest affecting tea plantations in China; however, its thermal developmental characteristics remain inadequately understood. This study examined the immature developmental stages of S. subpunctaria under eight constant temperature regimes (13, 16, 19, 22, 25, 28, 31, and 33 °C) in controlled laboratory conditions. Results indicated an inverse relationship between temperature and the total duration of the immature stages (egg to pupa), with developmental time decreasing from 105.8 days at 13 °C to 29.3 days at 31 °C. Specifically, the developmental durations for eggs, larvae, and pupae ranged from 5.4 to 20.3 days, 15.4 to 52.3 days, and 8.1 to 33.3 days, respectively, in 13 °C to 31 °C temperature range. Using an ordinary linear model, the estimated lower developmental threshold temperatures were 8.61 °C for eggs, 8.40 °C for larvae, 9.39 °C for pupae, and 8.85 °C for the total immature stage, with corresponding thermal constants of 114.94, 302.11, 149.93, and 558.99 degree-days (DD), respectively. Comparative analysis of eleven nonlinear models revealed substantial variation in estimates of lower and upper temperature thresholds, while estimates of optimal temperatures showed minor differences. Based on statistical criteria and biological relevance, the Briere-2 model was selected to characterize egg development, the Lactin-1 model for larval development, and the Briere-1 model for pupal and total immature stages. Stage transition models for eggs, larvae, pupae, and the total immature period were constructed using a two-parameter Weibull function integrated with the respective nonlinear models. This study provides foundational insights into the thermal developmental characteristics of S. subpunctaria and offers predictive tools for forecasting stage-specific emergence in tea plantations.

1. Introduction

Scopula subpunctaria (Herrich-Schaeffer) (Lepidoptera: Geometridae) is one of the prevalent insect pests affecting tea trees. It distributes across most tea-producing provinces in China, including Jilin, Jiangsu, Zhejiang, Shandong, Hunan, and Hubei [1,2,3]. Its high leaf consumption capacity, rapid reproduction and wide distribution render significant pest potential. The larvae tend to perch on the leaf margins, gnawing semi-permeable reticular spots on the edges of young leaves. In the later stage, the larvae often chew the leaves into large, smooth “C”-shaped notches. During large-scale outbreaks, they frequently consume all the leaves of tea plants, leaving only the main veins, which significantly impacts tea production and quality [3].

Insects are poikilothermic organisms that have adapted to specific temperature ranges. Temperature is the most crucial abiotic factor affecting insect development [4,5,6,7,8], survival [9], reproduction [10,11], –growth parameter [12], and distribution [13], Species-specific thermal characteristics, such as lower and upper temperature thresholds or the optimum temperature, can define the boundaries of arthropod biological activities [14,15]. As the temperature rises from the lower temperature threshold to the upper temperature threshold, the developmental rate increases [15]. The relationship between temperature and developmental rate is non-linear near lower and upper temperature thresholds but approximately linear at moderate temperatures [16]. Numerous linear and non-linear developmental rate functions or models have been proposed to depict this relationship [8,16,17,18,19,20,21,22,23,24,25,26]. The simple linear function enables the estimation of two crucial parameters: the lower temperature threshold (LT) and thermal constant (K). Nonlinear models offer a more precise description of the developmental rate across a broader temperature range [16,18,19,22,23].

Stage transition (emergence) models of insects invariably integrate a temperature-dependent rate model with a distribution model to forecast stage-specific transitions [7,10,13,27,28]. As crucial components of population dynamic models, stage transition models play a vital role in predicting stage-specific emergence peak times, outbreaks, the invasive range of some species, which are instrumental in formulating effective management programs [4,7,13,14,27,28].

Several studies have addressed the partial biology of S. subpunctaria, such as outbreak reason and control methods [29], scanning electron microscopic observation on ultrastructure of mouthparts, antennae and sex glands [1,30], adult emergence and mating behaviors [2], the effects of nutrition supplementation on longevity and fecundity [31], the temperature-dependent oviposition model [3]. the antennal transcriptome and olfaction-related genes [32,33], predation of natural enemy [34]. The relationship between developmental rate and temperature for S. subpunctaria has not been reported in detail. A more accurate prediction of the developmental rate across a broader temperature range is needed. Moreover, the information on immature stage transition models of S. subpunctaria is lacking. Thus, the objectives of this study are as follows:

(1)

To determine the influence of temperature on the development of the immature stages of S. subpunctaria.

(2)

To compare two linear temperature-dependent model and estimate lower temperature threshold and thermal constant for each stage.

(3)

To evaluate eleven nonlinear models and select the most suitable one to depict the relationship between the developmental rate and temperature for each stage.

(4)

To establish stage transition models for the immature stages of S. subpunctaria.

2. Materials and Methods

2.1. Insect Rearing

A colony of S. subpunctaria was established in Insect Ecology laboratory at Xinyang Agriculture and Forestry University, which was originally collected from one tea plantations in Xiaomiao Village, Shisanliqiao County, Shihe District, Xinyang City, Henan Province, China (32°01′35″ N, 113°59′21″ E) in March 2019. About 30 pairs of S. subpunctaria adults were collected from tea plantations. These adults were then placed in a breeding cage (50 × 50 × 45 cm) and provided with cotton moistened with 10% sugar solution. The cotton moistened with the 10% sugar solution was replaced daily. Six to eight fresh tea Camellia sinensis (L.) shoots (about 20 cm long) were inserted into a conical flask (300 mL) filled with tap water, which were placed into the breeding cage to enable S. subpunctaria adults to lay eggs [2]. Every day, the freshly laid eggs on the tea shoots were collected and transferred to another breeding cage. Once the eggs hatched into larvae, fresh tea leaves were provided as food and replaced promptly until the larvae developed into pupae. The newly developed pupae (<24 h) were individually placed in 300 mL conical flasks, which were sealed with a piece of fabric mesh. After emerging from the pupae, the adults were reared using the same methods as described above to produce the next generation. The colony was kept in an insect rearing room maintained at a temperature of 25 ± 1 °C, 14:10 (L:D) h, and relative humidity of 65 ± 5%. The insect population was used for experiments after two generations.

2.2. Temperature-Dependent Development

Temperature-dependent development of eggs, larvae, and pupae were investigated at eight designated temperatures (13, 16, 19, 22, 25, 28, 31, and 33 °C) in growth chambers (RTOP-310 Y, Zhejiang Top Cloud-agri Technology Co. Ltd., Hangzhou, China). The photoperiod in all chambers was set at 14:10 (L:D) h and the relative humidity was maintained as 75 ± 5%.

Newly laid eggs were collected from the breeding cage every 24 h These eggs were then placed in the designated temperature chambers to observe the development of each immature stage. A total of 300 eggs in each temperature chamber were monitored and examined daily. A binocular stereomicroscope (Olympus SZ51, Olympus Corporation, Tokyo, Japan) was used to check for their development into larvae and pupae. Each egg was placed individually into a Petri dish (9 cm in diameter and 2 cm in height). After hatching into larvae, fresh tea leaves were provided as food. The fresh tea leaves were replaced daily until pupation. Once pupation occurred at each temperature, the pupae were individually placed into a conical flask (300 mL), which was sealed with a piece of fabric mesh. The development of eggs, larvae, and pupae at each temperature was recorded daily until the emergence of adults or death. Due to the high mortality rate of eggs at 33 °C, no continuous development under this temperature was monitored.

2.3. Developmental Rate Model

2.3.1. Determining the Lower Temperature Threshold and Thermal Constant

The developmental rates (1/d) of immature stages of S. subpunctaria at the examined constant temperatures were employed to fit temperature-driven linear and nonlinear functions. In accordance with the law of “total effective temperatures” (Equation (1)), two linear models, namely the ordinary linear model (Equation (2)) and the Ikemoto linear model (Equation (3)) [35], were utilized to estimate the respective parameters:

$$D\left(T - LT\right)=K·\mathsf{\text{(1)}}$$

(1)

$$1/D=-LT/K+T/K$$

(2)

$$D·T=K+LT·D$$

(3)

where D is the developmental duration (d), T is temperature (°C), LT is lower temperature threshold (°C), and K is thermal constant (DD).

The lower temperature threshold (°C) and thermal constant (DD) for immature stages of S. subpunctaria were estimated by ordinary and Ikemoto models. The data points within the intermediate temperature range (16–28 °C) were utilized for this estimation.

2.3.2. Evaluation of Nonlinear Model

Eleven nonlinear developmental rate models were chosen to depict the relationship between temperature and the developmental rates of the immature stages of S. subpunctaria. The equations and their parameters showed as follows

Logan-3 nonlinear model [21]:

$$r(T)=\psi \left[{\displaystyle \frac{{\left(T-T_b\right)}^2}{{\left(T-T_b\right)}^2+D^2}}-e^{-{\displaystyle \frac{\left(T_{max}-(T-T_b)\right)}{∆T}}}\right]$$

(4)

where r(T) is the mean developmental rate at temperature T (°C) (same in all nonlinear models); ψ is development rate at the base temperature (T_b), T_max is lethal maximum temperature threshold, ΔT is width of high temperature boundary area, T_b is arbitrary base temperature (not a lower developmental threshold); D is empirical constant.

Logan-6 nonlinear model [14,18]:

$$r\left(T\right)= \psi \left[e^{\rho T} - e^{\left(\rho T_{max} - \left({\displaystyle \frac{T_{max} - T}{∆T}}\right) \right)}\right]$$

(5)

where ψ is the maximum developmental rate, ρ is a constant defining the rate at optimal temperature), T_max is the lethal maximum temperature, and ΔT is the temperature range over which physiological breakdown becomes the overriding influence.

Logan-10 nonlinear model [14,18]:

$$r\left(T\right)=\mathrm{\alpha }\left[{\displaystyle \frac{1}{\left(1+k{e}^{ - \rho ·T}\right)}}{e}^{ - {\displaystyle \frac{\left(T_{max} - T\right)}{∆T}}}\right]$$

(6)

where α and k are empirical constants, and ρ, T_max, and ΔT are same as in Logan-6 model.

Lactin-1 nonlinear model [22]:

$$r\left(T\right)= e^{\rho T} - e^{\left(\rho T_{max} - \left({\displaystyle \frac{T_{max} - T}{∆T}}\right) \right)}$$

(7)

where ρ, T_max, and ΔT are same as in Logan-6 model.

Lactin-2 nonlinear model [22]:

$$r\left(T\right)= e^{\rho T} - e^{\left(\rho T_{max} - \left({\displaystyle \frac{T_{max} - T}{∆T}}\right) \right)}+ \lambda$$

(8)

where ρ, T_max, and ΔT are same as in Logan-6 model, and λ allows the curve to intersect the abscissa at suboptimal temperatures.

Briere-1 nonlinear model [23]:

$$r\left(T\right)=aT\left(T - T_{min}\right)\sqrt{\left(T_{max} - T\right)}$$

(9)

where a is an empirical constant, T_min is the low temperature developmental threshold, and T_max is the lethal temperature threshold.

Briere-2 nonlinear model [23]:

$$r\left(T\right)=aT\left(T - T_{min}\right){\left(T_{max} - T\right)}^{1/d}$$

(10)

where a, T_min, and T_max are as in Birere-1, and d is an empirical constant.

3rd-order polynomial nonlinear model [20,36]:

$$r\left(T\right)=a{T}^3+b{T}^2+cT+d$$

(11)

where a, b, c, and d are empirical constants.

Simplified beta type nonlinear model [24]:

$$r\left(T\right)=\rho \left(\alpha - {\displaystyle \frac{T}{10}}\right){\left({\displaystyle \frac{T}{10}}\right)}^{\beta }$$

(12)

where ρ, α, and β are empirical constants.

Inverse second-order polynomial nonlinear model [25]:

$$r\left(T\right)={\displaystyle \frac{a}{1+bT+c{T}^2}}$$

(13)

where a, b, and c are empirical constants.

Biophysical nonlinear model [21]:

$$r\left(T\right)={\displaystyle \frac{RHO25\mathrm{*}\frac{T}{298.15}\mathrm{*}exp\left\{\frac{HA}{R}\left(\frac{1}{298.15} - \frac{1}{T}\right)\right\}}{1+exp\left\{\frac{HL}{R}\left(\frac{1}{TL} - \frac{1}{T}\right)\right\}+exp\left\{\frac{HH}{R}\left(\frac{1}{TH}-\frac{1}{T}\right)\right\}}}$$

(14)

where T is Kelvin temperature (°C + 273.15), R is the universal gas constant (1.987 cal deg⁻¹ mol⁻¹), RHO25 the development rate at 25 °C assuming no enzyme inactivation, HA the enthalpy of activation of the reaction that is catalyzed by a rate-controlling enzyme (cal mol⁻¹), TL the Kelvin temperature at which the rate-controlling enzyme is half active and half low-temperature inactive, TH the Kelvin temperature at which the rate-controlling enzyme is half active and half high-temperature inactive, HL the change in enthalpy associated with low temperature inactivation of the enzyme, and HH the change in enthalpy associated with high-temperature inactivation of the enzyme.

To select the most suitable model for describing the temperature-dependent developmental rate of eggs, larvae, pupae, and the entire immature stage, the following comparison criteria were taken into account:

a: The model ought to depict the data precisely. The adjusted coefficient of determination (Adj. r²) and the residual sum of squares (RSS) offer complementary insights regarding the goodness of fit and the utility for predicting observations [14]. Additionally, the Akaike’s information criteria [37] and Bayesian-Schwarz information criteria [38] were employed to assess the goodness of fit of nonlinear models [39].

The Adj. r², AIC, and BIC are defined as follows,

$$Adj. r^2={\displaystyle \frac{RSS/\left\{n-\left(\theta +1\right)\right\}}{SS/\left(n-1\right)}}$$

(15)

$$AIC=n\left[ln\left(RSS\right)\right]-\left[n-2\left(\theta +1\right)\right]-nln\left(n\right)$$

(16)

$$BIC=n\left[ln\left(RSS\right)\right]+\left(\theta +1\right)ln\left(n\right)-nln\left(n\right)$$

(17)

where RSS is the residual sum of squares, n is the number of observations, θ is the number of parameters, SS is the total sum of squares.

A higher Adj. r² value implies a better explanatory power of the model. Lower AIC and BIC values signify the preferred model, which has the fewest parameters and a better fit. The nonlinear models were ranked according to the AIC values, while BIC values were utilized for confirmatory purposes.

b: Whether the estimated parameters of the models possess biological significance. For the process of development, the crucial biological parameters required are the low temperature threshold, optimal temperature, and high temperature threshold [8,23,27,36,40].

2.4. Stage Distribution Model

The variations in the developmental durations for eggs, larvae, pupae, and total immature stages were simulated using a two-parameter Weibull function [7,27,41].

$$f\left(Px\right)=1-\mathrm{e}\mathrm{x}\mathrm{p}(-{(Px/a)}^b)$$

(18)

where f(P_x) is the cumulative proportion of stage emergence at physiological age P_x, a is the scale parameter, and b is the parameter of curve shape.

Prior to fitting the data to the Weibull function, cumulative frequency distributions of developmental durations were created by summing the frequencies across consecutive ages for each insect stage in days. These cumulative probability distributions were normalized to a maximum value of 1 by dividing the frequency at each age by the total frequency. Ultimately, standardized cumulative distribution curves were generated for various temperatures, and the age scale in days for each stage was converted into physiological age (P_x) using the rate summation method [18,27]:

$$Px={\int }_0^{n}r\left(T_i\right)\approx {\sum }_{i=0}^{n}r\left(T_i\right)$$

(19)

where P_x is the physiological age at nth day, r(T_i) is the developmental rate at temperature T at time i (day). The cumulative probability distribution of developmental durations for each stage was modeled using physiological age as the independent variable. Parameter estimation for the two-parameter Weibull model was conducted utilizing software TableCurve 2D v4 [42].

2.5. Stage Transition Model

The stage transition model was integrated with both the nonlinear developmental rate model and the distribution model corresponding to each immature stage. The simulation of stage transitions was performed utilizing the computational techniques proposed by Choi and Kim [43]. Physiological ages were determined by aggregating developmental rates across varying temperature conditions, which subsequently served as inputs for the two-parameter Weibull distribution model. The cumulative proportion of the cohort transitioning between stages was estimated using the Weibull function for each stage at a specified physiological age. The proportion of the cohort transitioning within the physiological age interval between i and i + Δi was calculated by subtracting the cumulative proportion at physiological age i from that at physiological age i + Δi.

2.6. Statistical Analyses

The influence of temperature on the developmental durations of different immature stages was assessed using one-way analysis of variance (ANOVA), and mean comparisons were conducted employing the Tukey test [44]. Parameter estimation for all functions was performed utilizing the Table Curve 2D v4 software [42].

3. Results

3.1. Temperature-Dependent Development

Notably, successful development from egg to adult occurred within the temperature range of 13 to 31 °C, whereas larval progression was inhibited at 33 °C (

Table 1. Number of insects completing the developmental stages and developmental durations (days, mean ± SE) of each stage of Scopula subpunctaria at different constant temperatures.

Temperature (°C)	Number of Insects Completing the Developmental Stages				Developmental Duration (Days, Mean ± SE)
	Egg	Larva	Pupa	Total Immature	Egg	Larva	Pupa	Total Immature
13	255	185	162	162	20.3 ± 0.09a	52.3 ± 0.28a	33.3 ± 0.30a	105.8 ± 0.32a
16	207	166	152	152	18.2 ± 0.11b	42.2 ± 0.31b	24.9 ± 0.39b	85.5 ± 0.44b
19	260	200	182	182	9.4 ± 0.07c	28.2 ± 0.21c	14.8 ± 0.17c	52.3 ± 0.28c
22	229	172	156	156	8.6 ± 0.12d	21.5 ± 0.17d	11.4 ± 0.10d	41.2 ± 0.20d
25	279	228	210	210	7.0 ± 0.04e	17.9 ± 0.10e	9.8 ± 0.07e	34.6 ± 0.11e
28	227	139	112	112	6.0 ± 0.04f	15.9 ± 0.14f	8.1 ± 0.08f	29.7 ± 0.20f
31	209	58	38	38	5.4 ± 0.09g	15.4 ± 0.16f	8.5 ± 0.15f	29.3 ± 0.21f
33	135	- *	-	-	9.6 ± 0.22	-	-	-

Means in the same column followed by the same letters are not significantly different (p < 0.05,). * No observed values.

). Temperature exerted a significant influence on the developmental durations of all immature stages (ANOVA: Egg: F = 5176.97, df = 6, 1659, p < 0.0001; Larva: F = 3315.11, df = 6, 1192, p < 0.0001; Pupa: F = 1838.04, df = 6, 1005, p < 0.0001; Table 1). Total immature development durations decreased significantly as temperature increased, from 105.8 ± 0.32 d at 13 °C to 29.3 ± 0.21 d at 31 °C (F = 10941.0, df = 6, 1005, p < 0.0001; Table 1).

3.2. Lower Temperature Threshold and Thermal Constant

Both the ordinary linear and Ikemoto linear models demonstrated strong linear correlations between developmental rates and temperatures ranging from 16 °C to 28 °C (Figure 1;

Table 2. Two linear temperature-driven rate models of immature stages of Scopula subpunctaria at constant temperatures and estimations of lower temperature threshold (LT) and thermal constant (K).

Stage	Method	Linear Regression			LT (°C)	K (DD)
		Equation	r2	p
Egg	Ordinary	1/D = − 0.0749 + 0.0087 T	0.955	0.0010	8.61	114.94
	Ikemoto	DT = 97.923 + 10.409 D	0.965	0.0028	10.41	97.92
Larva	Ordinary	1/D = − 0.0278 + 0.0033 T	0.991	0.0004	8.40	302.11
	Ikemoto	DT = 286.989 + 9.072 D	0.991	0.0004	9.07	286.99
Pupa	Ordinary	1/D = − 0.0627 + 0.0067 T	0.988	0.0006	9.39	149.93
	Ikemoto	DT = 139.231 + 10.233 D	0.989	0.0005	10.23	139.23
Total immature	Ordinary	1/D = − 0.0158 + 0.0018 T	0.989	0.0005	8.85	558.99
	Ikemoto	DT = 518.977 + 9.741 D	0.988	0.0006	9.74	518.97

). While the ordinary linear model yielded lower estimates of developmental thresholds and higher thermal constants compared to the Ikemoto model, both models effectively described the temperature-dependent development of S. subpunctaria. The lower temperature thresholds for the egg, larva, pupa, and total immature stages of S. subpunctaria, as estimated using the ordinary linear model, were determined to be 8.61, 8.40, 9.39, and 8.85 °C, respectively. Corresponding thermal constants were calculated as 114.94, 302.11, 149.93, and 558.99 degree-days (DD), respectively (Table 2).

3.3. Evaluation of Nonlinear Model

Drawing upon previously published studies, eleven nonlinear models were selected to characterize the developmental rates across the various life stages of S. subpunctaria. According to established evaluation criteria, the Briere-2, Logan-3, Logan-6, Lactin-1, and Lactin-2 models were identified as most appropriate for describing egg development. For larvae, the 3rd-order polynomial, Logan-10, Lactin-1, Lactin-2, and Simplified beta type models were deemed most suitable. The pupal stage was best represented by the Simplified beta type, Briere-2, Briere-1, 3rd-order polynomial, and Lactin-1 models, while the total immature stage was optimally described by the 3rd-order polynomial, Simplified beta type, Briere-1, Biophysical, and Briere-2 models (

Table 3. Statistical evaluation of 11 nonlinear models describing the relationship between temperature and developmental rate of immature stages of Scopula subpunctaria.

Model	θ *	Egg				Larva				Pupa				Total Immature
		Adj. r2	AIC	BIC	Rank	Adj. r2	AIC	BIC	Rank	Adj. r2	AIC	BIC	Rank	Adj. r2	AIC	BIC	Rank
Logan-3	5	0.922	−75.98	−67.50	2	0.976	−90.55	−83.87	10	0.959	−77.29	−70.61	7	0.963	−96.21	−89.53	9
Logan-6	4	0.913	−73.83	−65.43	3	0.985	−91.09	−84.36	8	0.967	−75.83	−69.10	9	0.975	−96.17	−89.44	10
Logan-10	5	0.790	−68.02	−59.54	7	0.985	−93.72	−87.05	2	0.943	−74.97	−68.29	11	0.966	−96.69	−90.02	8
Lactin-1	3	0.896	−72.11	−63.79	4	0.990	−93.13	−86.34	3	0.978	−77.84	−71.06	5	0.983	−97.99	−91.21	6
Lactin-2	4	0.853	−69.63	−61.23	5	0.988	−92.75	−86.02	4	0.974	−77.51	−70.78	6	0.979	−97.36	−90.63	7
Briere-1	3	0.835	−68.43	−60.11	6	0.988	−91.78	−85.00	7	0.980	−78.54	−71.75	3	0.985	−98.64	−91.85	3
Briere-2	4	0.945	−77.51	−69.11	1	0.987	−91.84	−85.11	6	0.978	−78.67	−71.94	2	0.981	−98.08	−91.35	5
3rd-order polynomial	4	0.755	−65.57	−57.17	10	0.996	−100.31	−93.58	1	0.976	−78.27	−71.54	4	0.986	−100.26	−93.53	1
Simplified beta type	3	0.805	−67.07	−58.75	8	0.989	−92.14	−85.36	5	0.982	−79.22	−72.43	1	0.986	−99.12	−92.34	2
Inverse second-order polynomial	3	0.766	−65.64	−57.32	9	0.987	−90.94	−84.15	9	0.967	−75.09	−68.31	10	0.977	−95.71	−88.93	11
Biophysical	4 **					0.920	−84.14	−77.41	11	0.941	−76.75	−70.02	8	0.964	−98.37	−91.64	4

Adj. r²: adjusted coefficient of determination, AIC: Akaike information criterion, BIC: Bayesian-Schwarz information criterion. * The number of parameters in the model. ** Wagner et al. (1984) [16] identified the four parameter-reduced model, as the most appropriate.

).

These nonlinear models in Figure 1 yielded estimates of optimal temperatures (T_opt) with minor variation. Specifically, for eggs, T_opt varied from 27.95 °C (Inverse second-order polynomial) to 30.97 °C (Logan-3). For larvae, T_opt ranged from 29.41 °C (Inverse second-order polynomial) to 31.00 °C (Briere-1). For pupae, T_opt varied from 28.79 °C (Inverse second-order polynomial) to 30.51 °C (Logan-3). For total immature stage, T_opt ranged from 29.33 °C (Inverse second-order polynomial) to 30.84 °C (Simplified beta type).

Parameter estimates and optimal temperatures derived from six selected models are presented in

Table 4. Estimated parameter values of selected nonlinear models for immature stages of Scopula subpunctaria.

Model	Parameter	Egg	Larva	Pupa	Total Immature
Logan-10	α	0.369583912	0.18297454	0.28257244	0.12540107
	k	15012000	232.52114988	311.7746717	170.983033
	ρ	0.585155887	0.20000218	0.2221569	0.19952189
	Tmax	31.61119936	21.5246653	22.3878743	12.5555097
	ΔT	2.010732873	14.6180631	14.6837764	18.5193126
	Topt	29.21	30.52	29.52	30.74
Lactin-1	ρ	0.170162484	0.15097219	0.16439678	0.15169148
	Tmax	34.84809776	36.2640752	35.2401609	36.3094117
	ΔT	5.869139753	6.61874375	6.07655787	6.58979692
	Topt	28.98	29.64	29.16	29.72
Lactin-2	ρ	0.160603639	0.11831839	0.11118654	0.10218944
	Tmax	35.11256752	38.5338873	38.4834116	40.1557488
	ΔT	6.215585126	8.43263416	8.93582277	9.76431074
	λ	−0.00544366	−0.0141233	−0.0502311	−0.0144183
	Topt	28.89	30.09	29.52	30.38
Briere-1	a	0.000120594	0.0000294493	0.0000705029	0.000016594
	Tmin	8.19549661	4.25813298	7.08951066	5.27901386
	Tmax	34.3700706	38.1968281	36.2294195	37.7622876
	Topt	28.45	31.00	29.79	30.79
Briere-2	a	0.00016442	0.0000595	0.00013764	0.000033227
	Tmin	−1.586829	1.02724264	4.24356967	2.41313463
	Tmax	33.0129531	33.1589323	32.1854061	32.9974716
	d	7.41364712	4.73023501	5.020426	4.67706889
	Topt	30.88	30.04	29.48	29.93
Simplified beta type	ρ	0.00578986	0.002769	0.00453869	0.00138198
	α	3.63180641	4.30782715	3.93422199	4.20528039
	β	3.46366565	2.62869997	3.04926494	2.75205603
	Topt	28.18	31.00	29.63	30.84

. Considering Akaike Information Criterion (AIC) rankings, the number and biological interpretability of parameters, the Briere-2 model was identified as the most suitable for describing temperature-dependent developmental rates of S. subpunctaria eggs. Lactin-1 was preferred for larvae, while Briere-1 was optimal for both pupae and total immature stages. Based on these selected models, the optimal temperatures for developmental rates were estimated as 30.88 °C for eggs, 29.64 °C for larvae, 29.79 °C for pupae, and 30.79 °C for the total immature stage (Table 4).

3.4. Stage Distribution Model

The relationship between the cumulative proportion of developmental completion and physiological age for eggs, larvae, pupae, and total immature stage was effectively modeled using a two-parameter Weibull function (Egg: F = 966.28; df = 1, 69; p < 0.0001; Larva: F = 3664.01; df = 1, 86; p < 0.0001; Pupa: F = 2652.01; df = 1, 83; p < 0.0001; Total immature: F = 3891.84; df = 1, 106; p < 0.0001; Figure 2;

Table 5. Estimated parameter values (mean ± SE) of distribution models for immature stages of Scopula subpunctaria.

Stage	Parameter (Mean ± SE)		r2
	a	b
Egg	0.9886 ± 0.00751	9.5506 ± 0.89202	0.934
Larva	1.0128 ± 0.00237	12.2890 ± 0.45325	0.977
Pupa	1.0132 ± 0.00458	8.3567 ± 0.40481	0.970
Total immature	1.0088 ± 0.00143	21.3284 ± 0.82102	0.974

). The distribution patterns for all stages were similar (Figure 2). The parameter a, representing the expected physiological age at 50% developmental completion, was estimated as 0.9886, 1.0128, 1.0132, and 1.0088 for eggs, larvae, pupae, and the total immature stage, respectively (Table 5). Notably, the parameter b was lower in the pupal stage compared to the egg and larval stages, indicating greater variability in developmental durations during the pupal stage.

The two-parameter Weibull function $f\left(Px\right)=1-\mathrm{e}\mathrm{x}\mathrm{p}(-{(Px/a)}^b)$ was applied.

3.5. Stage Transition Model

The emergence rate of individuals within a cohort, as a function of cohort age (days) and temperature (°C), was simulated by integrating a nonlinear developmental rate model with stage distribution model, constituting a stage transition model for each developmental stage (Figure 3). Overall, the emergence rate curves across all stages exhibited a similar pattern. At lower temperatures, emergence rates corresponded to prolonged developmental durations accompanied by greater variability. As temperature increased, the curves narrowed and variability decreased, until temperatures exceeded approximately 32.0 °C, beyond which the curves broadened again and displayed a delayed peak emergence time (Figure 3).

4. Discussion

This study investigated the temperature-dependent developmental durations of immature stages of S. subpunctaria under constant temperature conditions. Temperature exerted a significant influence on the developmental durations across all immature stages. Notably, S. subpunctaria demonstrated sensitivity to high temperatures, as survival through the immature stages was not observed at 33 °C.

The ordinary linear model, initially introduced by Ludwig [17], requires minimal data and is computationally straightforward. It provides approximate estimates of the lower temperature threshold (LT) and thermal constant (K) that are comparable to those derived from more complex models [16,36]. Consequently, this model has been widely employed to characterize insect developmental rates within mid-temperature ranges [7,8,14,15,25,26,35,36,43,45]. However, the ordinary linear model has limitations: it is applicable only within an intermediate temperature range (15–30 °C), which results in an overestimation of the LT due to extrapolation of the linear segment into a region where the relationship is inherently nonlinear [15]. Furthermore, this model does not estimate the optimal temperature or the upper temperature threshold. Ikemoto and Takai [35] proposed an alternative linear regression model to estimate LT and K, which tends to yield higher LT values and lower K values compared to the ordinary linear model.

To address the limitations of linear models, various nonlinear models have been developed to more accurately describe developmental rates across a broader temperature spectrum. Several of these models have been successfully applied to estimate both lower and upper temperature thresholds for development [8,15,18,19,23,26,27,40]. Model selection typically prioritizes statistical criteria such as goodness of fit; however, additional considerations include balancing model fit with structural complexity, the reliability of estimated temperature thresholds, and the biological interpretability of parameters [8,40].

In the present study, nonlinear models were selected based on the adjusted coefficient of determination (Adj. r²), Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC), all of which are independent of parameter count and provide robust measures of model fit [37,39]. Moreover, the capacity of a model to estimate critical temperature points (i.e., T_min, T_opt, and T_max) and the biological relevance of its parameters were deemed essential criteria for model selection.

To evaluate the eleven nonlinear models, the top five models ranked by AIC were selected for further analysis, focusing on the biological interpretation of parameters and the estimation of critical temperature thresholds (i.e., T_min, T_opt, and T_max). Among these top five nonlinear models, the Briere-2 model was identified as the most suitable for characterizing egg development due to its relatively low parameter count and superior AIC ranking. For larval development, although the 3rd-order polynomial model demonstrated a favorable ranking, it failed to estimate low and high temperature thresholds. In contrast, the Lactin-1 model, which possessed fewer parameters and a higher Adj. r² value compared to the Logan-10 model, was ultimately chosen as the optimal nonlinear model for modeling larval stage transitions. Regarding pupal development, both the 3rd-order polynomial and Simplified Beta-type models were unable to estimate key temperature points. The Briere-1 model, exhibiting fewer parameters and a higher Adj. r² value relative to the Lactin-1 and Briere-1 models, was therefore selected to describe pupal development. Applying a similar rationale, the Briere-1 model was also deemed the best nonlinear model for representing the developmental rate of the total immature stage. Consequently, the Briere-2, Lactin-1, and Briere-1 models were applied to their respective stage transition models.

In this study, the cumulative distributions of egg, larval, pupal, and total immature stages were effectively modeled using a two-parameter Weibull function, with physiological age serving as the independent variable. Notably, larval developmental periods were longer and exhibited greater variability compared to those of eggs and pupae. The Weibull function has been widely employed to describe the cumulative proportion of developmental completion in insects [7,13,27,43,46,47]. Integrating the distribution model with the nonlinear developmental rate models enables the prediction of stage-specific emergence rates under varying temperature conditions in the field. This investigation represents the first effort to develop stage transition models for S. subpunctaria, which are essential components for constructing a comprehensive population model for this species, analogous to models developed for other arthropods [46,47].

5. Conclusions

Overall, this study provides foundational insights into the thermal biology of S. subpunctaria across a range of constant temperatures, simulates developmental rates using both linear and nonlinear models, and establishes stage transition models. The findings are expected to facilitate accurate predictions of S. subpunctaria population development timing in tea plantations and support the formulation of integrated pest management strategies. Furthermore, the stage transition models developed herein constitute a critical component for constructing a comprehensive population model that will elucidate the seasonal fluctuations and population dynamics of S. subpunctaria in the field.