Comparison of Thermal Performance 
Equations in Describing Temperature-
Dependent Developmental Rates of Insects: 
(I) Empirical Models
Authors: Pei-Jian Shi, Gadi V.P. Reddy, Lei Chen, & 
Feng Ge
This is a pre-copyedited, author-produced PDF of an article accepted for publication in Annals of 
the Entomological Society of America following peer review. The version of record [insert 
complete citation information here] is available online at: https://dx.doi.org/10.1093/aesa/sav121.
Shi, Pei-Jian , Gadi V. P. Reddy, Lei Chen, and Feng Ge. "Comparison of Thermal Performance 
Equations in Describing Temperature-Dependent Developmental Rates of Insects: (I) Empirical 
Models." Annals of the Entomological Society of America 109, no. 2 (December 2015): 211-215. 
DOI: 10.1093/aesa/sav121.
Made available through Montana State University’s ScholarWorks 
scholarworks.montana.edu 
Comparison of Thermal Performance Equations in
Describing Temperature-Dependent Developmental
Rates of Insects: (I) Empirical Models
Pei-Jian Shi,1 Gadi V. P. Reddy,2,3 Lei Chen,4 and Feng Ge5
1Co-Innovation Centre for Sustainable Forestry in Southern China, Bamboo Research Institute, Nanjing Forestry University,
Nanjing 210037, China (peijianshi@gmail.com), 2Montana State University, Western Triangle Ag Research Center, 9546 Old Shelby
Rd., P. O. Box 656, Conrad, MT 59425 (reddy@montana.edu), 3Corresponding author, e-mail: reddy@montana.edu, 4Graduate
School of Environmental Science, Hokkaido University, N19W8, Sapporo 060-0819, Japan (lei-chen1029@eis.hokudai.ac.jp), and
5State Key Laboratory of Integrated Management of Pest Insects and Rodents, Institute of Zoology, Chinese Academy of
Sciences, Beijing 100101, China (gef@ioz.ac.cn)
Abstract
Temperature greatly affects the developmental duration of insects at their different stages, and many mathe-
matical models exist for describing their temperature-dependent developmental rates. It is important to choose
a suitable model to predict outbreaks of pest insects under climate change. However, previous comparisons
among these models were usually based on a single species. In the present study, we compared the six nonlin-
ear models (the Brie´re-1, Brie´re-2, Lactin, Performance-2, beta, and Ratkowsky models) based on the goodness
of fit and the trade-off between the model’s goodness of fit and structural complexity, using 10 temperature-
dependent developmental rate datasets on insects to make the conclusions general. We found that the square
root model (i.e., the Ratkowsky model) fitted all datasets well, and the curve shape produced by this model also
approximates the curve shape of thermodynamically based mathematical models. The square root model was
originally derived to be applicable to the growth rates of bacteria, and until now it has been generally ignored in
entomology. We were mainly concerned with the predicted results obtained by using this model on observa-
tions of temperature-dependent developmental rates. We found that the square root model described well the
pooled developmental rates in the low-, mid-, and high-temperature ranges, and we believe that it merits wider
use in entomology.
Key words: goodness of fit, nonlinear, pest outbreak, square root model, structural complexity
Evidence has demonstrated that temperature has a significant influ-
ence on the developmental rate of insects (Uvarov 1931, Campbell
et al. 1974). Under global climate change, pest outbreaks exhibit
more complex dynamics and become more difficult to be predicted.
To predict the developmental duration of important agricultural
and forest pest insects exposed to different temperatures, many
mathematical models have been proposed (e.g., Logan et al. 1976;
Sharpe and DeMichele 1977; Schoolfield et al. 1981; Wang et al.
1982; Lactin et al. 1995; Brie`re et al. 1999; Ikemoto 2008; Shi et al.
2011; Re´gnie`re et al. 2012; Wang et al. 2013). These models have
also been tried in the study of the effect of temperature on the
growth rate of ectotherms (e.g., insects, mites, spiders, crabs,
shrimps, fishes). Yin et al. (1995) proposed a beta function to reflect
the effect of temperature on the developmental rate of crops.
Ratkowsky et al. (1983) proposed a square root model to predict
the effect of temperature on the growth rate of bacterial cultures.
However, the latter two models have received little attention in
entomological investigations. The previous studies in entomological
research frequently compared the Logan, Lactin, and Brie´re models
using a single species. From different studies, the conclusion as to
which was the best model was usually different. In addition, the ma-
jority of previous studies used small sample sizes ranging from five
to eight temperatures, which made the estimates of the parameters
of the nonlinear models less precise.
In the present study, we chose several nonlinear models and data
of temperature-dependent developmental rate with large sample
sizes to examine the models’ trade-off between goodness of fit and
structural complexity. We attempted to find a general model that
could well describe the majority of datasets on temperature-
dependent developmental rates of insects. This work is also moti-
vated by the studies of ontogenetic growth equation for animals and
plants. A beta sigmoid growth equation obtained from the integral
of the beta function has showed better fitting flexibility than some
traditional growth models such as the exponential, Gompertz,
Logistic, and von Bertalanffy models (Yin et al. 2003, Voorend et al.
2014, Shi et al. 2016). Thus, it is valuable to further compare the ex-
isting temperature-dependent developmental or growth models to
find a better model than the beta function. Then we can obtain more
new candidate ontogenetic growth models by using the integral
form for the developmental or growth equation.
Materials and Methods
Data
Ten datasets of temperature-dependent developmental rates of in-
sects were chosen (among which Sample 2 is a predatory mite, not
an insect species). Table 1 lists the studied species, developmental
stages, sample sizes, and data sources. We also compared the fitted
results by using the mean and raw data of developmental rates of fe-
male fruit flies (Drosophila buzzatii Patterson & Wheeler; i.e.,
Sample 10).
Models
In the present study, we used a total of six nonlinear models that all
have the “conceptual” lower and upper developmental thresholds
(T1 and T2) at which developmental rates equal zero.
(i) Brie´re-1 equation (Brie`re et al. 1999):
r ¼ aTðT  T1Þ T2  Tð Þ1=2 (1)
Here, r represents developmental rate at temperature T; a is a con-
stant; T1 and T2 represent “conceptual” lower and upper develop-
mental thresholds at which developmental rates equal zero.
(ii) Brie´re-2 equation (Brie`re et al. 1999):
r ¼ aTðT  T1Þ T2  Tð Þ1=b (2)
Here, the parameters are the same as those in the Brie´re-1 equation
except an additional parameter b, which makes the curve fitting
more flexible. Then the Brie´re-1 equation is substantially a special
case of the Brie´re-2 equation when b¼2.
(iii) Lactin equation (Lactin et al. 1995):
r ¼ kþ e q T  e q TuðTuTÞ=d (3)
Here, k, q, Tu, and d are constants. And Tu is defined as the upper
“lethal” temperature, which is higher than the “conceptual” upper
developmental threshold (Lactin et al. 1995). In the Lactin equation,
there are no T1 and T2 parameters. However, the curve produced by
the Lactin equation has two intersections with the X-axis (i.e.,
Temperature-axis). We can obtain T1 and T2 by a numerical
method.
(iv) Performance-2 equation (Shi et al. 2011, Wang et al. 2013):
r ¼ c T  T1ð Þ 1 eK TT2ð Þ
 
(4)
Here, c and K are constants. This equation supersedes the
Performance-1 equation (Wang et al. 2013), which has more param-
eters and produces a very similar curve shape to the Performance-2
equation.
(v) Beta equation (Yin et al. 1995, 2003; Shi et al. 2016):
r ¼ rm T2  T
T2  Tm
 
T  T1
Tm  T1
 TmT1
T2Tm
(5)
Here, rm represents the maximal developmental rate at
temperature Tm. Note: all parameters in the beta equation have clear
meanings.
(vi) Ratkowsky equation (Ratkowsky et al. 1983):
ffiffi
r
p ¼ c T  T1ð Þ 1 eK TT2ð Þ
 
(6)
This equation looks like the Performance-2 equation except there is
a square root of r in the leaf-hand side of equation. Thus, this equa-
tion is often called the square root equation.
Model Evaluation
In order to compare the goodness of fit and the trade-off between
model’s goodness of fit and structural complexity among these six
nonlinear models, we used the residual sum of squares (RSS), coeffi-
cient of determination (R2), adjusted coefficient of determination
(R2adj), root mean square error (RMSE), and corrected Akaike infor-
mation criterion (AICc).
RSS ¼
Xn
i¼1 ri  r^ ið Þ
2 (7)
Here, n represents the sample size; ri represents the i-th observation
of developmental rate, and ri with a hat represents the predicted de-
velopmental rate.
R2 ¼ 1
Xn
i¼1 ri  r^ ið Þ
2
Xn
i¼1 ri  rið Þ
2
(8)
Table 1. Ten datasets of temperature-dependent developmental rates of insects
Sample Species Order: Family Developmental stage Sample size Source
1 Helicoverpa armigera Hu¨bner Lepidoptera: Noctuidae Pupa 23 Wu et al. (2009)
2 Kampimodromus aberrans Oudemans Acari: Phytoseiidae Egg þ larva þ 1st–2nd nymphs 9 Broufas et al. (2007)
3 Toxorhyynchites brecipalpis Theobald Diptera: Culicidae Egg 19 Trpis (1972)
4 Bactrocera dorsalis Hendel Diptera: Tephritidae Egg 19 Messenger and Flitters (1958)
5 Aedes aegypti L. Diptera: Culicidae Larva 15 Gilpin and McClelland (1979)
6 Bemisia tabaci (B-biotype) Gennadius Hemiptera: Aleyrodidae From egg to adult 7 Xiang et al. (2007)
7 Lipaphis erysimi Kaltenbach Hemiptera: Aphididae From egg to adult 12 Liu and Meng (2000)
8 Myzus persicae Sulzer Hemiptera: Aphididae From egg to adult 11 Liu and Meng (1999)
9 Epilachna varivestis Mulsant Coleoptera: Coccinellidae Larva 8 Shirai and Yara (2001)
10 Drosophila buzzatii Patterson & Wheeler Diptera: Drosophilidae From egg to adult 9–1,204a de Jong (2010)
a9 is the number of mean data of developmental rates, and 1,204 is the number of raw data of developmental rates.
Here, ri with a dash represents the mean of the observed develop-
mental rates. And the adjusted coefficient of determination is as
follows:
R2adj ¼ 1
n 1
n p 1 R
2
 
(9)
Here, p represents the number of parameters in a nonlinear model
including one for the error term. We also used the RMSE because
the magnitude of the RMSE can be considered to be the “average de-
viation” between the observed data and the predicted values.
RMSE ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
RSS=ðn  p þ 1Þ
p
(10)
AICc ¼ 2Lþ 2pn= n p 1ð Þ (11)
where
L ¼  n
2
ln
RSS
n
 
(12)
We used RSS and R2 as the indicators to evaluate the model’s good-
ness of fit, and used R2adj, RMSE, and AICc as the indicators to eval-
uate the trade-off between a model’s goodness of fit and structural
complexity.
The “nlinfit” function in Matlab 6.5 was used to carry out the
nonlinear fitting. For robust estimation, “nlinfit” uses an iterative
reweighted least squares algorithm. At every iteration, the robust
weights are recalculated based on each observation’s residual from
the previous iteration. These weights downweight outliers, so that
their influence on the fit is decreased. Iterations continue until the
weights converge. We provided the Matlab procedures for carrying
out the nonlinear fitting for a given dataset of temperature-
dependent developmental rates (see Supp Material 1 [online only]
for details).
Results
Supp Table 1 (see Supp Material 2 [online only] for details) shows
the comparative results. From the viewpoint of a trade-off between
the model’s goodness of fit and structural complexity (based on the
calculated R2adj or RMSE or AICc), the Brie´re-2 equation was better
than the Brie´re-1 equation in five cases with larger sample size
(n>10), but the latter performed better for the datasets with smaller
sample size (n<10). However, the Brie´re-2 equation had a better
goodness of fit than the Brie´re-1 equation for all datasets investi-
gated. Because of an additional parameter, b, the Brie´re-2 equation
is more flexible in producing a curve shape. From Sample 10 with a
small sample size of mean data, we found that the Brie´re-1 equation
obtained a better trade-off between the model’s goodness of fit and
structural complexity than the Brie´re-2 equation; however, it is
worse than the Brie´re-2 equation when using the raw data. Except
for Samples 4 and 7, the Ratkowsky equation had a better goodness
of fit than the Brie´re-2 equation for all other samples. It was also
better than the Brie´re-2 equation based on the calculated R2adj,
RMSE, and AICc. Although the beta equation fitted the data of the
observed developmental rates well, it was demonstrated to be no
better than the Ratkowsky equation from the point of view of good-
ness of fit or from the trade-off between model’s goodness of fit and
structural complexity. We also found that the curve shape produced
by the beta equation was more similar to that produced by the
Ratkowsky equation, relative to the others. For the remaining two
equations (i.e., the Lactin and Performance-2 equations), the good-
ness of fit or the trade-off between model’s goodness of fit and
structural complexity was worse than the Brie´re-2 equation or the
Ratkowsky equation. The data showed that in the mid-temperature
range, a perfect linear relationship between temperature and devel-
opmental rate did not exist. However, the Lactin and Performance-2
equations had assumed a very approximate linear relationship be-
tween them. Thus, they could not perform better than the others for
most datasets.
The lower and upper developmental thresholds are a matter of in-
terest to some investigators, because they want to know the predicted
critical temperatures where insects stop developing. Traditionally, the
linear fitting was used to calculate the lower developmental threshold
by using the developmental rate data in the mid-temperature range.
The intersection between the straight line and the X-axis is reckoned
as the lower developmental threshold. For estimating the upper devel-
opmental threshold, the Logan, Brie´re, Lactin, and Performance equa-
tions are usually used. From the model’s goodness of fit or the trade-
off between model’s goodness of fit and structural complexity, the
Lactin equation was better than the Performance-2 equation for each
species investigated. The Brie´re-1 and Brie´re-2 equations provided an
approximate estimate on the “conceptual” upper developmental
threshold, but their estimates on the “conceptual” lower developmen-
tal threshold were lower than the estimates by using the Lactin and
Performance-2 equations. The beta and Ratkowsky equations
described the effect of temperature on the developmental rates of in-
sects in the mid-temperature range better than the Lactin and
Performance-2 equations.
Supp Figs. 1–11 (see Supp Material 3 [online only] for details) vi-
sually exhibit the fitted results using these six nonlinear equations to
10 datasets. There are nine insect species and one mite species (i.e.,
Sample 2). Supp Figs. 10 and 11 (online only) described the same in-
sect species (D. buzzatii), but with different sample sizes. Supp Fig.
10 (online only) used the mean data, whereas Supp Fig. 11 (online
only) used the raw data. The open circles represent the observations
of developmental rates, and the curves represent the predicted devel-
opmental rates based on different nonlinear models. For every spe-
cies, the aforementioned six nonlinear models were used to fit the
data of temperature-dependent developmental rates, respectively.
Discussion
As an empirical method, the predicted lower developmental thresh-
old by using the linear equation in the mid-temperature range usu-
ally approximates its actual lower terminating temperature for
development. However, the Brie´re-1, Brie´re-2, beta, and Ratkowsky
equations only provide “conceptual” lower and upper developmen-
tal thresholds, and the estimates of these two extreme temperatures
by these nonlinear equations do not accord with the actual observed
threshold temperatures. Although the Lactin and Performance-2
equations can provide the approximate estimates to the actual devel-
opmental thresholds, these two equations cannot reflect an approxi-
mate exponential growth relationship between temperature and
developmental rate in the low-temperature range (Uvarov 1931,
Campbell et al. 1974). If the data of developmental rates at low tem-
peratures are lacking, the Lactin and Performance-2 can fit the data
well (see Sample 6 in Supp Fig. 6 [online only]). However, these two
equations cannot overall reflect the nonlinear relationship between
temperature and pooled developmental rates in the low- and mid-
temperature ranges. We believe that these two nonlinear equations
can do well on the condition that the data in the low-temperature
range are lacking. Thus, we do not approve using the Lactin and
Performance-2 equations to fit the data including the developmental
rates in the low-temperature range. The Brie´re-1 and Brie´re-2
equations also applied to many samples, but the latter was shown to
be better than the former actually as a special case of the latter. We
consider that the additional parameter (i.e., in the Brie´re-2 equation)
is justified, making data fitting more flexible. The Brie´re-2 equation
was almost applicable to all samples investigated. For Sample 9, the
last datum was intentionally excluded to check the predicting abili-
ties of different models. We found that the Brie´re-2, Lactin, and
Performance-2 equations provided an impractical prediction for the
developmental rate at the excluded temperature, whereas the beta
and Ratkowsky equations provided acceptable predictions at this
excluded temperature. In this case, we should approve the use of the
Ratkowsky equation rather than the Brie´re-2 equation. In fact, the
curve shape produced by the Ratkowsky equation approximates
that produced by the SSI or ROR models based on thermodynamic
theories (Ikemoto 2008, Ikemoto et al. 2013, Ratkowsky et al.
2005, Corkrey et al. 2012). These two thermodynamic equations
also predicted a slightly concave curve in the high-temperature
range. In practice, it is impossible to observe reliable developmental
rates at temperatures near the upper developmental threshold be-
cause of high mortality at these temperatures. Thus, it is difficult
to determine which model is most reasonable in predicting
developmental rates near the actual upper developmental threshold.
However, we found that the degree of curvature of the Ratkowsky
equation near the upper developmental threshold was very small
when developmental rates in the high-temperature range exhibited a
sharp drop with increased high temperatures (see Supp Figs. 1 and 4
[online only]). We also showed an example of growth rate of
Flavobacterium hibernum McCammon, a lactose-utilizing bacte-
rium from a freshwater Antarctic lake (Fig. 1; McCammon et al.
1988). In fact, the Ratkowsky equation has been demonstrated to
apply to many other datasets of growth rates of bacteria
(Ratkowsky et al. 2005), as that model was originally developed to
describe the temperature dependence of bacterial growth.
Schoolfield et al. (1981) consider that the temperature-dependent
rate models are applicable to both developmental rate and growth
rate of poikilotherms. The Logan model (Logan et al. 1976) is al-
ways used to describe the effect of temperature on the developmen-
tal rate of insects, and is also used for the growth rate of insects
calculated from life table methods. Thus, it is feasible to introduce
the Ratkowsky equation to the investigation of temperature-depen-
dent developmental rate of insects, considering its good performance
in fitting data.
In all the models, temperatures such as T1 and T2 should just be
looked upon as mathematical symbols and not predictors of mini-
mum or maximum temperatures of a range at which development
can occur. For the square root model (i.e., the Ratkowsky model),
Ratkowsky et al. have never maintained that these temperatures are
anything more than notional (or notational) temperatures, and there
is no organism on Planet Earth (bacterial, archaeal, eucaryotic) that
we are aware of which can grow at temperatures approaching T1
from above. That is, usually the true minimum observable tempera-
ture for growth is several degrees (C or K) above the notional mini-
mum temperature. At the upper end of the temperature scale, the
difference between the notional maximum temperature T2 and the
observable maximum temperature Tmax may be much smaller than
at the lower end (perhaps as small as 1C), so it is very difficult to
get reliable data at this upper threshold temperature because of high
mortality rates. Thus, we advocate that when using any of the six
models, there should be no thoughts that these equations are doing
anything more than possibly providing a good fit to existing data on
temperature dependence of growth rate. The models simply do not
incorporate a prediction of the range of temperature dependence.
Supplementary Data
Supplementary data are available at Annals of the Entomological Society of
America online.
Acknowledgments
We are deeply grateful to Prof. David Ratkowsky (University of Tasmania,
Australia) for providing invaluable comments on the early version of this
study, and we also thank Prof. Gerdien de Jong (Utrecht University, The
Netherlands) and Dr. Takaya Ikemoto (Teikyo University School of
Medicine, Japan) for providing the partial data used in the present study. P.-J.
Shi was supported by the National Natural Science Foundation for Young
Scholars of China (31400348), the Priority Academic Program Development
of Jiangsu Higher Education Institutions, and the Startup Foundation of
Nanjing Forestry University (GXL038). F. Ge was supported by the National
Natural Science Foundation of China (31221091) and the R&D Special
Foundation for Public Welfare Industry (Agriculture 201303019-05).
Fig. 1. Effect of temperature on the growth rate of F. hibernum. (A)
Temperature-dependent growth rate. The small open circles represent the
observed growth rates (1/min), and the curve represents the predicted growth
rates at different temperatures. (B) Square root of growth rate.
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