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Journal of Thermal Biology 36 (2011) 403–408

Contents lists available at ScienceDirect

Journal of Thermal Biology
journal homepage: www.elsevier.com/locate/jtherbio

A model for the time–temperature–mortality relationship in the
chill-susceptible beetle, Alphitobius diaperinus, exposed to fluctuating
thermal regimes
H. Colinet a,b,n, L. Lalouette b, D. Renault b
a
b

Earth and Life Institute ELI, Biodiversity Research Centre BDIV, Catholic University of Louvain, Croix du Sud 4-5, B-1348 Louvain-la-Neuve, Belgium
Universite´ de Rennes 1, UMR CNRS 6553 Ecobio, 263 Avenue du Ge´ne´ral Leclerc CS 74205, 35042 Rennes Cedex, France

a r t i c l e i n f o

abstract

Article history:
Received 27 May 2011
Accepted 15 July 2011
Available online 22 July 2011

Exposing insects to a fluctuating thermal regime (FTR) compared with constant low temperature (CLT)
significantly reduces cold-induced mortality. The beneficial effects of FTR result from physiological
repair during warming intervals. The duration and the temperature experienced during the recovery
period are supposed to strongly impact the resulting cold survival; however, disentangling the effects of
both recovery variables had not been broadly investigated. In this study, we investigate cold tolerance
(lethal time, Lt50) of the polyphagous beetle Alphitobius diaperinus. We examined adult survival under
various CLTs (0, 5, 10 and 15 1C), and under 20 different FTR conditions, where the 0 1C exposure
alternated with various recovery temperatures (Rt) (5, 10, 15 and 20 1C) combined with various
recovery durations (Rds) (0.5, 1, 2, 3 and 4 h). Under CLTs, Lt50 increased with temperature until no
mortality occurred above the upper limit of cold injury zone (ULCIZ). Under FTRs, Lt50 increased with
both Rt and Rd. The magnitude of the survival gain was clearly boosted when Rt was above the ULCIZ
(at 20 1C). Based on a data matrix of lethal times with multiple Rt Rd combinations, a predictive
model showed that cold survival increased exponentially with Rt and Rd. This model was subsequently
validated with additional survival tests. We suggest that increasing recovery durations associated with
optimal recovery temperatures eventually leads to a progressive chilling compensation.
& 2011 Elsevier Ltd. All rights reserved.

Keywords:
Temperature
Thermal fluctuations
Cold
Insect
Survival
Recovery

1. Introduction
Many arthropods, especially those from temperate and cold
regions, may face periods of potentially harmful low temperatures during development or adult life (Denlinger and Lee, 2010).
Environmental conditions are generally not static and diurnal
thermal variations can have profound impacts on the life history
traits of insects (MacMillan et al., 2005; Marshall and Sinclair,
2009; Fischer et al., 2011; Renault, 2011). Several studies demonstrated that exposing insects to fluctuating thermal regimes
(FTRs) (i.e., cold-exposure interrupted by periodic short warming
pulses) could significantly reduce adult mortality (Coulson and
Bale, 1996; Nedvˇed et al., 1998; Hancˇ and Nedvˇed, 1999; Renault
et al., 2004; Colinet et al., 2006; Koˇsta´l et al., 2007; Colinet and
Hance, 2010; Renault, 2011; Terblanche et al., 2010). FTRs
promote cold survival of insects despite the sudden thermal
variations experienced by individuals. As the total duration at
n
Corresponding author at: Earth and Life Institute ELI, Biodiversity Research
Centre BDIV, Catholic University of Louvain, Croix du Sud 4-5, B-1348 Louvain-laNeuve, Belgium.
E-mail address: herve.colinet@uclouvain.be (H. Colinet).

0306-4565/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jtherbio.2011.07.004

low temperature (i.e. overall dose of cold) is shorter under FTRs
than under constant low temperature (CLT) (Hancˇ and Nedvˇed,
1999; Renault et al., 2004; Colinet et al., 2006), the beneficial
effects of warming pulses results, at least in part, from a reduced
amount of accumulated injuries under FTRs. However, this only
partially explains the benefits of FTRs as the cold survival remains
much higher under FTRs than CLTs, when subtracting the time
spent at a high temperature (e.g. Hancˇ and Nedvˇed, 1999; Renault
et al., 2004); the beneficial effects of warming pulses also provide
a periodic opportunity to physiologically repair chilling injuries,
which otherwise accumulate under CLTs (e.g. Colinet et al., 2007a,
2007b; Koˇsta´l et al., 2007; Lalouette et al., 2007).
Survival of insects exposed to low temperature depends on
combination of factors, such as the duration and temperature of
exposure (e.g. Leather et al., 1993; Sømme, 1996; Bale, 2002;
Chown and Terblanche, 2006). Cold-induced mortality is generally determined by exposing insects to CLTs (Bale, 2002; Sømme,
1996). However, chill-susceptible species may exploit daily bouts
at warm temperature to recover from chilling injuries.
Several predictive models based on a degree-time concept
have been proposed to describe the time–temperature–mortality
relationship under CLTs (e.g. Turnock et al., 1983; Powell, 2003;

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H. Colinet et al. / Journal of Thermal Biology 36 (2011) 403–408

Jing et al., 2005). However, these may not be useful for predicting
mortality under controlled or natural FTR conditions. The magnitude of benefits resulting from FTRs likely depends on factors
such as the recovery duration (Rd) and the recovery temperature
(Rt); however, these have been poorly documented. Furthermore,
those studies that have looked at these two variables have
typically done so independently (e.g. Nedvˇed et al., 1998; Hancˇ
and Nedvˇed, 1999; Jing et al., 2005; Stotter and Terblanche, 2009).
In order to model the cold survival under FTRs, we performed a
comprehensive assessment of the recovery effects based on
several combinations between Rt and Rd variables and developed
a data matrix of survival within these ranges of Rt and Rd.
There may be significant species-specificity in the way the
recovery processes are achieved in insects (Hancˇ and Nedvˇed,
1999). For instance, improvement of cold survival under FTRs
generally occurs within a large range of temperatures (from 5 to
30 1C), but in flesh flies, beneficial effects of warming pulses only
occur at 15 1C (Chen and Denlinger, 1992). Therefore, to understand the recovery processes, Rt and Rd must be broadly investigated for each biological model. The polyphagous beetle
A. diaperinus (Coleoptera: Tenebrionidae), commonly known as
the lesser mealworm, was chosen as a study model. Because of its
invasive status, the thermal biology of this tropical species has
been studied extensively in recent years (e.g. Salin et al., 1998,
2000; Renault et al., 2002, 2003), and ample data have shown that
FTRs promote the cold survival of A. diaperinus (e.g. Renault et al.,
2004; Renault, 2011; Lalouette et al., 2007).
Turnock et al. (1983) proposed a simple conceptual framework
to describe different temperature zones within the thermobiological span of invertebrates. In the cold injury zone, the survival
and development are negatively affected by low temperature (in a
cumulative fashion), and no physiological repair is possible.
Within the neutral zone, the temperature is not detrimental but
there is no physiological repair. Finally, within the active zone,
previous chilling injury can be repaired. Insects exposed to FTRs
might thus have opportunity for physiological repair only when
recovery temperature is optimal. We therefore assumed that the
upper limit of cold injury zone (ULCIZ; Nedvˇed et al., 1998), that
have been estimated to be around 15 1C in adult A. diaperinus
(Renault et al., 2004), might be situated between the cold injury
and active zone as described by Turnock et al. (1983).
In this study we tested the following assumptions (i) if Rt
remains below the ULCIZ, the beneficial effects of FTR would only
result from a reduced daily cold dose (i.e. degree-time concept),
(ii) when Rt is above the ULCIZ threshold, the beneficial effects of
FTR would result from a reduced amount of accumulated chilling
injuries (i.e. reduced cold dose) combined with the additional
opportunities for physiological repair. We also assumed that cold
survival will increase with increasing Rd. Beyond a certain level of
Rd, chilling injuries are expected to be fully recovered (i.e.
complete compensation) and survival is expected to be similar
to untreated controls, especially when Rd is combined with
optimal Rt.

2.2. Survival experiments
Groups of 10 beetles were placed into Petri dishes and were
assigned to one of the following experimental conditions:
(i) constant low temperatures (CLTs) (0, 5, 10, 15 and 20 1C);
(ii) fluctuating thermal regimes (FTRs) (Table 1), where the cold
exposure at 0 1C was interrupted by a daily transfer to various
recovery temperatures (Rt from 5 to 20 1C) and for various
recovery durations (Rd from 0.5 to 4 h) (see Table 1). Programmed
thermo-regulated incubators (Model SANYO MIR-153) were used
for all the assays. Temperatures were checked using automatic
recorders (Hobos data logger, model U12-012, Onset Computer
Corporation, accuracy 70.35 1C). The actual temperature changed across the different temperatures at a rate of about
0.7–0.8 1C/min. All beetles were deprived from water and food
during the cold assays. For each experimental condition, a Petri
dish containing 10 beetles was removed for the cold-incubator
and transferred to the laboratory conditions (25 1C; Light/Dark:
16/8 h) every day, in order to assess survival of beetles with water
supply. This procedure was repeated daily for all treatments until
mortality reached 100%. Survival was scored as the proportion of
insects that exhibited limb movements and could stand on their
legs. Renault (2011) has recently addressed a methodological
issue concerning the assessment of cold survival, which may vary
with the observation period after the stress. We thus performed a
series of preliminary tests in order to establish the appropriate
post-cold stress interval for an accurate assessment of the
survival. Groups of 10 beetles were exposed to 0 1C alternating
with 20 1C for various Rd (from 0.5 to 4 h) (see Table 1). Petri
dishes were removed at daily intervals and beetles were assessed
for survival one, two and seven days after they were removed
from the experimental conditions. In all the assays, survival,

Table 1
Experimental conditions used to determine lethal times of adult A. diaperinus. The
beetles were exposed at a range of constant low temperature (0, 5, 10, 15 and
20 1C) or fluctuating thermal regimes. Under fluctuating thermal conditions, the
0 1C exposure was interrupted daily by warming pulses at a range of recovery
temperatures (0, 5, 10, 15 and 20 1C) and for various recovery durations (0, 0.5, 1,
2, 3 and 4 h).
Recovery temperature
(1C)

Daily recovery duration
(h)

Daily duration at 0 1C
(h)

0

0

24

5

0
0.5
1
2
3
4

24
23.5
23
22
21
20

10

0
0.5
1
2
3
4

24
23.5
23
22
21
20

15

0
0.5
1
2
3
4

24
23.5
23
22
21
20

20

0
0.5
1
2
3
4

24
23.5
23
22
21
20

2. Material and methods
2.1. Rearing and acclimation conditions
Adult A. diaperinus were collected from poultry house litter at
Treffendel (Brittany, France, 2100 000 W, 48120 2400 N) in October 2010.
The insects were then reared under controlled conditions at 25 1C
(Light/Dark: 16/8 h) and supplied with water and commercial dry
dog food ad libitum. All the insects used for this study were of
maximum two months of age.

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H. Colinet et al. / Journal of Thermal Biology 36 (2011) 403–408

expressed as the lethal time for 50% of the population (Lt50), was
similar when assessed two and seven days after the end of the
cold stress (Fig. 1). However, in two distinct experimental conditions (for Rt ¼1 and 3 h), lethal times (Lt50) were significantly
lower when assessed on day one compared to day two or seven
(Fig. 1). Some individuals were moribund (or did not move any
appendage) on day one, but were still moving on day two or
seven, which resulted in an underestimation of Lt50 when
assessed on day one. We thus decided to examine insects’ survival
two days after the cold stress in all subsequent assays.
2.3. Statistical analysis
Survival data, expressed as lethal time for 50% and 90% of the
population (Lt50 and Lt90, respectively) were obtained from probit
analysis, as previously described (Renault et al., 2004), using
MINITAB Statistical Software Release 13 (MINITAB Inc., State
College, Pennsylvania). The Lt50 and Lt90 values generated from
each experimental condition were regressed against Rd (explanatory variable) for each Rt condition (fixed variable). F-tests
were used to compare linear versus quadratic models and to
determine which equation best fit the regression. Additionally,
F-tests were used to test the null hypothesis that slopes were
identical (i.e. parallel lines). These analyses were conducted on
Prism V 5.01 (GraphPad software, Inc., San Diego, California). To
model the relationship between lethal times (Lt50 or Lt90) and the
two recovery variables (Rt and Rd) simultaneously, an iterative
least squares estimation procedure was used based on the
following equation: Lt50 or Lt90 ¼Y0 exp(k Rd Rt). Convergence of
the iterative procedure provided parameter estimates, which

Fig. 1. Lethal time (Lt50 7 SE) of adult A. diaperinus exposed to FTR, where the 0 1C
exposure alternated with 20 1C for various durations (0.5, 1, 2, 3 and 4 h). Survival
was evaluated after one (white bar), two (gray bar) and seven days (black bars)
following the end of cold stress. Moribund insects were considered as dead. The
symbol (*) indicates when Lt50 values were significantly different, based on
fiducial limits overlap (a ¼ 0.05).

405

were then used to draw tri-dimensional plots in STATISTICA
V.5.5 software (Statsoft, France).
2.4. Additional tests: model validation
We performed a series of additional tests to assess the
agreement between model-predicted and observed lethal times
(Lt50 or Lt90) under a range of FTRs. Lethal times were determined
in beetles exposed to 0 1C alternating with some Rt and Rd not
previously tested: Rt¼ 7 1C with Rd¼2 h, Rt ¼12 1C with Rd ¼1 h,
Rt¼12 1C with Rd¼2 h, and Rt¼12 1C with Rd¼4 h. Lethal times
were determined using the method described above. To evaluate
model predictions we used the method of Pineiro et al. (2008).
That is, predicted values were regressed against observed ones
and a F-test was used to compare the slope against a 1:1 line.

3. Results
3.1. Effect of recovery duration and temperature
Lt50 under CLTs increased with the temperature of exposure.
Lt50 was 108.29 76.88 h at 0 1C, 302.3279.15 h at 5 1C and
554.64714.64 h at 10 1C (P o0.05 based on fiducial limits). It
was not possible to determine the Lt50 of beetles exposed to 15
and 20 1C as no mortality occurred after 6 weeks of experiment.
The assays were stopped after 6 weeks, as longer exposure to such
experimental conditions would encounter mortality unrelated to
thermal conditions (i.e. starvation and/or desiccation).
Under fluctuating thermal conditions, the lethal times increased
significantly with the Rd (Fig. 2A,B). Note that the value of the assay
0–15 1C with 3 h of recovery is lacking due to a technical problem
that occurred during the assay. Model comparisons indicated that a
linear equation best fitted the relationships (Po0.05). Parameter
estimates of all the regressions are provided in Table 2. All the
regression slopes significantly differed from zero (Table 2) and also
differed among treatments (Po0.05). The slope of each regression
represents the gain of survival (in hours) with increasing Rd (in
hours) and for various temperatures during the recovery (Rt). The
magnitude of this gain varied according to Rt (the temperature
during the recovery). The slope was rather shallow (15.24 h h 1 of
recovery) when Rt was 5 1C, then it increased regularly with every
5 1C step of Rt. The slope then increased markedly when Rt reached
20 1C (85.94 h h 1 of recovery) (Table 2).
3.2. Modeling cold survival in relation to recovery variables
To model the recovery time–temperature–mortality, an iterative least squares estimation procedure was performed on the
following equation: Lt50 or Lt90 ¼Y0 exp(k Rd Rt). When run with a

Fig. 2. Positive relationship between the lethal time of adult A. diaperinus and the recovery duration, at a range of recovery temperatures (0, 5, 10, 15 and 20 1C). Refer to
Table 1 for description of the experimental conditions. A: Lt50 (7 SE) and B: Lt90 (7 SE).

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H. Colinet et al. / Journal of Thermal Biology 36 (2011) 403–408

Table 2
Results of the regressions describing the relationship between lethal times and recovery duration (as explanatory variable), at various recovery temperatures (Rt) (as fixed
variable). Parameter estimates and deviation from zero F-tests.
Dependent variable

Explanatory variable

Fixed variable Rt (1C)

Slope 7 SE

Y-intercept

r2

F

P

Lt50 (h)

Rd (from 0 to 4 h)

5
10
15
20

15.24 7 1.20
33.66 7 3.88
54.03 7 2.24
85.94 7 6.97

105.5
103.5
110.1
106.4

0.975
0.949
0.994
0.974

160
75.28
581
151.6

o0.001
o0.001
o0.001
o0.001

Lt90 (h)

Rd (from 0 to 4 h)

5
10
15
20

24.57 7 6.36
42.80 7 7.77
66.49 7 6.86
113.9 7 9.70

150.2
163.7
173.3
158.3

0.788
0.883
0.96
0.971

14.89
30.33
93.77
137.8

0.018
0.005
0.002
o0.001

Fig. 3. Three-dimensional plot describing the relationship between survival times Lt50 (A) and Lt90 (B), recovery temperature (Rt) and recovery duration (Rd) in coldexposed A. diaperinus. Surface fitting derived from iterative least squares estimation procedure based on Lt¼ Y0 exp(k Rd Rt) equation with parameters fitted to
experimental data.

Table 3
Model-predicted (expected) versus observed lethal times (Lt50 and Lt90) for a
range of recovery temperature (Rt) and recovery duration (Rd) combinations.
Rt
(1C)

Rd
(h)

Lt50
Lt50 7 SE
expected (h) observed (h)

7
12
12
12

2
1
2
4

149.82
144.62
178.78
261.61

137.01 7 3.48
167.02 7 12.69
177.457 3.11
273.237 9.68

Lt90
Lt90
expected (h) observed 7SE
(h)
217.65
210.55
256.90
382.48

195.797 7.12
256.967 37.73
223.917 6.32
322.177 14.45

least square loss of function, 96.91% of the variance was explained
(R2 ¼0.984) for the Lt50 model (Fig. 3A), and 95.74% (R2 ¼0.978)
for Lt90 (Fig. 3B). The values of the parameters at the convergence
were Y0 ¼116.99 or 172.55 and k¼0.0176 or 0.0165 for the
models explaining Lt50 and Lt90, respectively. These parameters
were used to draw the tri-dimensional surface plots (Fig. 3A,B).
3.3. Model validation
The observed lethal times resulting from the additional tests
are summarized in Table 3. To evaluate the model prediction, the
observed values were regressed against the model-predicted
ones, and F-tests were used to compare the slope against a 1:1
line. The slopes were equal to 1.07870.209 and 0.630 70.186 for
Lt50 and Lt90, respectively. For both models, the slopes did not
differ significantly from a 1:1 line (F ¼0.138, P¼0.728 and
F¼ 3.948, P¼0.117 for Lt50 and Lt90, respectively), providing
evidence of the excellent agreement between model-predicted
and observed lethal times under the various FTRs tested.

4. Discussion
The distinction between dead and living individuals presents a
recurring problem when assessing insect survival under controlled conditions (Baust and Rojas, 1985). When subjected to
cold stress, insects may accumulate extensive chilling injuries
requiring, in some cases, a long recovery period before movement
is regained. Thus, it is crucial to determine the appropriate period
for an accurate estimate of survival. In this study, we found that
assessing survival after only 24 h underestimated the actual lethal
time, which corroborates the notion that the methodological
procedure has to be carefully chosen to avoid biased results
(Renault, 2011).
Survival of insects exposed to sub-optimal temperature
depends on both the duration and the temperature of exposure
(i.e. the cold dose) (e.g. Leather et al., 1993; Sømme, 1996; Bale,
2002). When the temperature decreases and/or the duration of
exposure increases, insects may suffer from cumulative physiological damages that lead to a high level of mortality. In the
present study, lethal times increased with temperature under
CLTs until no mortality could be determined in beetles exposed to
15 and 20 1C for up to 6 weeks. Renault et al. (2004) estimated
that temperature becomes injurious below 15 1C (ULCIZ) in adult
A. diaperinus, which most likely explains why mortality did not
occur at 15 and 20 1C.
Under fluctuating thermal regimes, the duration of survival
significantly increased with both the recovery duration and the
recovery temperature. The positive effect of FTRs appears as a
general phenomenon, promoting cold survival in most of the
species tested to date (e.g. Coulson and Bale, 1996; Nedvˇed et al.,
1998; Hancˇ and Nedvˇed, 1999; Renault et al., 2004; Jing et al.,

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H. Colinet et al. / Journal of Thermal Biology 36 (2011) 403–408

2005; Colinet et al., 2006; Koˇsta´l et al., 2007; Colinet and Hance,
2010). Under FTRs, the cold exposure alternates with short pulses
at optimal temperatures, allowing a physiological repair of the
accumulated chilling injuries. The mechanisms underlying the
positive impact of FTRs involve, (1) the re-establishment of ion
gradient homeostasis (Koˇsta´l et al., 2007), (2) the anabolism or
catabolism of compatible solutes (Wang et al., 2006; Lalouette
et al., 2007), (3) the consumption of amino acids (Colinet et al.,
2007a; Lalouette et al., 2007), (4) the expression of heat shock
proteins (Hsps) at the transcript level (Wang et al., 2006;
Tollarova´-Borovanska´ et al., 2009; Colinet et al., 2010) and at
the protein level (Colinet et al., 2007b), (5) the activation of
antioxidant system to counter reactive oxygen species production
(Lalouette et al., 2011), and (6) the upregulation of various
proteins that play key roles in proteolysis, cytoskeleton and
energy metabolism (Colinet et al., 2007b).
FTRs promote cold survival of insects despite the sudden
thermal variations experienced by individuals. Terblanche et al.
(2010) recently showed that different levels of diel thermal
variations (temperature variability) experienced during some
acclimation treatments resulted in different (or even opposite)
plastic responses, with the large thermal variations being not
necessarily as beneficial as the shallow thermal variations. In the
context of the present study, thermal variations were applied not
during acclimation but during the stress period itself. Interestingly, the largest thermal variation (highest recovery temperature) was associated with the greatest cold survival. The rapid
thermal variations between stress conditions (0 1C) and recovery
temperature (e.g. 20 1C) did not induce mortality but, on the other
hand, improved the survival, suggesting that the potential detrimental effects of sudden thermal variations are much lower than
the beneficial effects on recovery (Colinet et al., 2006).
Disentangling the effects of the recovery duration from those
of the recovery temperature has not been investigated in insects.
Both variables had a strong impact on the cold survival of adult
A. diaperinus. The beneficial effects arising from these two factors
likely share some similarities. The duration of recovery and the
recovery temperature determine the daily cold dose, consequently affecting the cold survival. However, there might be some
specificity in the way these two variables act on cold survival. The
recovery temperature likely determines whether a physiological
repair is possible, e.g. above some specific thermal thresholds (i.e.
at permissive temperatures). The recovery duration determines to
what extent the physiological repair is achieved and to what
extent the homeostasis restored.
Under fluctuating thermal conditions, survival increased regularly
with the duration of recovery. The longer the recovery duration the
longer the beetles survived. These data provide evidence for a
beneficial effect of FTRs at every recovery duration tested, even when
it was as short as 0.5 h. Jing et al. (2005) also found that such short
recovery periods were sufficient to induce a small but detectable
beneficial effect on the survival of the migratory locust eggs exposed
to 10 1C. The magnitude of the survival gain with increasing
recovery duration (i.e. slope) significantly varied according to the
temperature applied during the recovery period. The duration of
survival was boosted when the recovery temperature was set above
the ULCIZ (e.g. at 20 1C). This suggests that at recovery temperatures
below the ULCIZ, the extension of the duration of survival only
resulted from the reduced cold dose. On the contrary, above the
ULCIZ, the beneficial effects of FTRs resulted from a reduced cold dose
associated with additional benefits derived from the opportunity for
physiological repair. Similarly, Hancˇ and Nedvˇed (1999) noted in the
chill-sensitive collembolan, Orchesella cincta, a weak effect when the
recovery temperature was low, followed by a quasi-exponential
increase with increasing recovery temperature, before decreasing
when temperature became supra-optimal (heat stress).

407

During the thermal fluctuation, a high metabolic activity has
been observed in adult A. diaperinus because of the high energy
demand for physiological repair mechanisms (Lalouette et al.,
2011). The longer the recovery period the more chilling injuries
can potentially be repaired. Survival was clearly enhanced when
both the recovery period and temperature were the highest (i.e.
4 h and 20 1C). Exposure to cold stress deeply perturbs the
physiological and cellular homeostasis, and return to the homeostatic equilibrium occurs over several hours/days (Overgaard
et al., 2007; Colinet, in press; Lalouette et al., 2011). It has
previously been shown that adult A. diaperinus exposed to 0 1C
alternating with 20 1C for 12 h on a daily basis resulted in no coldrelated mortality (Lalouette et al., 2007), suggesting a complete
compensation of cold effects when extended recovery duration
were applied. Although in the present study we did not assess
recovery durations longer than 4 h, we can assume that increasing the recovery duration in association with an optimal recovery
temperature would result in a progressive chilling compensation
until no cold-related mortality could be detected. In fact, when
the three-dimensional plot of the lethal time, the recovery
temperature and duration were smoothed with a surface algorithm, an exponential trend became clear. This exponential
dependence corroborates the view that increasing recovery durations associated with optimal recovery temperatures would
eventually lead to a progressive chilling compensation, expressed
as an asymptotical increase of lethal time (i.e. no detectable
mortality).
We performed a series of additional tests in order to assess the
validity of our model. An excellent agreement between theoretical and observed values was obtained. This provides support to
the validity of the model describing the interrelationships among
the survival and the two recovery factors. The degree of accumulated chilling injuries increases in a complex multiplicative
manner with cooling temperatures and duration (Nedvˇed et al.,
1998; Jing et al., 2005). It appears that time- and temperaturedependent recovery process also follows this trend. The models
predicting insect cold survival under natural conditions should
consider including the effects of warming pulses. As the ability to
recover and the speed by which insects are able to recover from
chilling injuries highly depend on the basal level of cold tolerance
of each species (Bale, 1993; Hancˇ and Nedvˇed, 1999; Colinet and
Hance, 2010); these species-specific effects should also be taken
into account in further ecological studies.

Acknowledgments
This study was supported by the Fonds de la Recherche
Scientifique – FNRS in Belgium. This paper is number BRC128 of
the Biodiversity Research Centre. We thank B.N. Philip for helpful
comments on an earlier version of the manuscript.

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