Theoretical Population Biology 83 (2013) 82–94
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Theoretical Population Biology
journal homepage: www.elsevier.com/locate/tpb
Stage and age structured Aedes vexans and Culex pipiens
(Diptera: Culicidae) climate-dependent matrix population model
Željka Lončarić a , Branimir K. Hackenberger b,∗
a
BioQuant, Našička 4, 31000 Osijek, Croatia
b
Department of Biology, Josip Juraj Strossmayer University in Osijek, Trg Ljudevita Gaja 6, 31000 Osijek, Croatia
article
info
Article history:
Received 15 July 2011
Available online 24 August 2012
Keywords:
Mosquitoes
Variable carrying capacity
Climate dependency
Population model
Transient dynamics
abstract
Aedes vexans and Culex pipiens mosquitoes are potential vectors of many arbovirial diseases. Due to
the ongoing climate changes and reappearance of some zoonoses that were considered eradicated,
there is a growing concern about potential disease outbreaks. Therefore, the prediction of increased
adult population abundances becomes an essential tool for the appropriate implementation of mosquito
control strategies. In order to describe the population dynamics of A. vexans and C. pipiens mosquitoes
in temperate climate regions, a 3-year period (2008–2010) climate-dependent model was constructed.
The models represent a combination of mathematical modeling and computer simulations, and include
temperature, rainfall, photoperiods, and the flooding dynamics of A. vexans breeding sites. Both models
are structured according to the developmental stages, and by individuals’ ‘‘age’’ (i.e., time spent in
each developmental stage), as we wanted to enable a time delay between the appearances of different
developmental stages of mosquitoes. The time delay length is temperature dependent, with temperature
being the most important factor influencing morphogenesis rates in immatures and gonotrophic cycle
durations in adult mosquitoes. To determine which developmental stages are the most sensitive and are
those at which control measures should be aimed, transient elasticities were calculated. The analysis
showed that both mosquito species reacted to perturbation of the same matrix elements; however, in
the C. pipiens model, the stage with greatest proportional sensitivity (i.e., elasticity) during most of the
three-year reproduction season contained adults, while in the A. vexans model it contained larvae. The
models were validated by comparing 7-day model outputs with data on human bait collection (HBC)
obtained from the Public Health Institute of Osijek-Baranja, with both model outputs showing valid
compatibility with field data over the three-year period. The proposed models can easily be modified
to describe population dynamics of other mosquito species in different geographical areas, as well as for
assessing the efficiency and optimization of different mosquito control strategies.
© 2012 Elsevier Inc. All rights reserved.
1. Introduction
During the past few years, the geographical distribution of
arthropod-borne zoonoses has expanded considerably (Chevalier
et al., 2004). Climate changes can cause the emergence and reemergence of vector-borne diseases by changing their geographical distribution as well as their dynamics (Confalonieri et al., 2007).
Several studies predict that diseases such as malaria, dengue and
West Nile virus will have increased transmission intensity and that
their spatial distribution will expand in correlation with climate
changes (Hales et al., 2002; Martens et al., 1995; Ogden et al., 2008;
Hongoh et al., 2012). In August 2007, the first indigenous transmission of chikungunya in Europe was reported from a rural area
∗
Corresponding author.
E-mail addresses: [email protected], [email protected]
(B. K. Hackenberger).
0040-5809/$ – see front matter © 2012 Elsevier Inc. All rights reserved.
doi:10.1016/j.tpb.2012.08.002
in Emilia-Romagna, Italy (Townson and Nathan, 2008). Considering the ongoing climatological changes and reappearance of some
zoonoses that are considered to be eradicated (Kallio-Kokko et al.,
2005; Akritidis et al., 2010; Polley, 2005), there is a growing concern about potential disease outbreaks. Therefore, in order to predict increased mosquito abundances and implement appropriate
control strategies it is very important to know which environmental parameters govern the population dynamics of mosquitoes.
In particular, temperature is one of the most important environmental factors influencing insect physiology and behavior
(Ratte, 1985), and mosquitoes like all poikilotherms are highly dependent on the ambient temperature for successful development
(Ahumada et al., 2004). As all mosquitoes have aquatic larval and
pupal stages and thus require water for breeding and development, heavy rainfall was correlated with increased mosquito abundances and in some areas with subsequent disease outbreaks by
several authors (Hu et al., 2006; Kelly-Hope et al., 2002; Lindsay
et al., 1993). Foreseeable annual changes in environmental factors
Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94
such as photoperiod and temperature are used by mosquitoes as
cues to initiate or cease their activity, because seasonal activity
patterns change with latitude and differ among mosquito species
(Knight et al., 2003). Diapause as an adaptation for hibernation
is common in mosquitoes from northern and temperate latitudes
(Vinogradova, 2007), so, in the temperate belt, a seasonal cycle of
Aedes and Culex species involves its development and reproduction
during the spring–summer period and a reproductive diapause
during the autumn–winter time (Mori and Wada, 1978; Thomson
et al., 1982; Vinogradova, 2007).
There are several ways to predict peaks in mosquito populations
and risks of disease outbreaks. The most common one is by
correlating mosquito abundance data with various environmental
variables such as temperature, precipitation, rainfall, and tidal
events (Reisen et al., 2008; Yang et al., 2009), or with climatological
and hydrological model outputs (Thomson et al., 2006; Shaman
et al., 2002). All these models have a relatively good correlation
with mosquito abundance data and are able to predict an increase
in mosquito populations or an increase in disease outbreak risks;
however, these models can be constructed only for areas in which
systematic monitoring with available large quantities of data about
mosquito abundance has been carried out.
Over the last two decades different types of model have
been developed to describe the population dynamics of various
mosquito species (Focks et al., 1993; Fouque and Baumgartner,
1996; Shone et al., 2006; Shaman et al., 2006; Erickson et al., 2010),
two of them being matrix models (Schaeffer et al., 2008; Ahumada
et al., 2004). Age-structured matrix models were developed
independently by Bernardelli (1941), Lewis (1942), and Leslie
(1945, 1948). These models classified organisms into discrete
age classes and incorporated age-specific vital rates such as
survival probability and fecundity of each age class from one age
class to the next. Lefkovitch (1965) developed a stage-structured
model for species whose development is better described using
characteristics such as organism size, weight, and physiological,
morphological, or developmental state. In our work, stage and
age structured models were combined for modeling the dynamics
of the two most common mosquito species in the city of Osijek
′ ′′
′
′′
(45°33 4 N, 18°41 38 E, Croatia), A. vexans and C. pipiens. C.
pipiens mosquitoes are cosmopolitan species that can be found
in almost all known urban and suburban temperate and tropical
regions. They are known to be very potent disease vectors for
several diseases such as the West Nile and St. Louis encephalitis
viruses, avian malaria, and filarial worms (Bogh et al., 1998; Reisen
et al., 1992; Turell et al., 2002). A. vexans vector competence is
usually considered as negligible, but several studies have shown
that this mosquito species could be involved in vector-disease
transmissions (Yildirim et al., 2011; Goddard et al., 2002a,b; Molaei
and Andreadis, 2006).
The goal of this research design was to construct a stage
and age structured model that could enable time delay in the
presence of different developmental stages of C. pipiens and A.
vexans mosquitoes. This research design also implemented a
mechanistic model construction approach using biological and
ecological characteristics of mosquitoes with factors that influence
and limit their growth and development in temperate geographical
areas. The models were constructed based on available literature
and monitoring data on C. pipiens and A. vexans mosquitoes and
were used for a simulation of mosquito population dynamics in the
city of Osijek during 2008–2010. Model outputs were compared
with data on human bait collection (HBC), and general correlation
existed in both models. To determine which developmental stages
are the most sensitive and are those at which control measures
should be aimed, transient elasticities were calculated for each
mosquito species.
83
Fig. 1. Environmental parameters used in the model construction. Mean daily
temperature (°C) and number of rainy days in Osijek during the three-year period
(2008–2010). In the model the photoperiod at latitude 45 °N was used. Danube’s
water levels (cm) from measurement station Batina in the three-year period
(2008–2010).
2. Methods
A discrete-time, stage and age structured matrix model, with
one-day projection interval, was constructed in order to simulate
the population dynamics of C. pipiens and A. vexans mosquitoes.
The model is constructed for a three-year period (2008–2010),
and only females are modeled. In this section, the mosquito lifecycle and the specific biological characteristics of the modeled
species are described. Then, three-dimensional projection matrix
construction, specific influences that are modeled for each
species, and the general structure of the models is described.
Environmental parameters used in the model construction are
shown in Fig. 1.
2.1. Study area
The city of Osijek is located in the continental part of Croatia,
and, according to Koppen’s classification, it belongs to the climate
type Cfwbx. This is a moderately warm rainy climate with
no dry periods and with precipitation uniformly distributed
throughout the year. The mean temperature of the coldest month
(January) usually does not drop below −0.4 °C, while the mean
temperature of the warmest month (July) usually does not exceed
21.4 °C. Minimum temperatures during the winter can sometimes
be below −25 °C, and, during the summer, the maximum
temperatures can sometimes exceed 40 °C. Annual precipitation
amounts to 900 mm, with a maximum in June, and a secondary
maximum in September, while the annual relative air humidity
ranges between 77% and 92%. However, during the past ten years,
strong and sudden temperature and rain variations have become
quite usual. Located 10 km northeast of the city of Osijek is a
marshland area and Kopački Rit Nature Park. The basic ecological
feature of Kopački Rit is given by its flooding dynamics; the
landscape of the whole region depends on the flood intensity.
The parts of the marshland change size, form, and function
depending on the quantity of risen water originating mostly
from the Danube, and to a smaller extent from the River Drava.
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Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94
This area represents an ideal breeding site for different species
of floodwater mosquitoes, among which A. vexans is the most
abundant, being 75.59% of mosquito fauna. C. pipiens mosquitoes
are also continuously present in the city area throughout the year
(Merdić et al., 2010).
2.2. Mosquito biology
Common characteristics: Mosquitoes are holometabolous insects,
which means that they undergo complete metamorphosis. Their
life cycle consists of four different developmental stages: embryo
(egg), larva, pupa, and imago or adult. A few days after oviposition,
larvae hatch from the eggs and start their development in the
water. Several days later, depending on environmental conditions
and food abundance, larvae turn into pupae which do not
feed anymore. After approximately two days, adult mosquitoes
emerge from the pupae. Shortly after emergence, females are
fertilized, with mating occurring only once in their lifetime.
The female usually passes through several gonotrophic cycles,
whose number depends on numerous environmental factors. Each
gonotrophic cycle consists of host seeking, blood feeding, and
oviposition. Temperature is one of the most important abiotic
factors affecting the complete mosquito life cycle—development,
growth, and survival of immature mosquitoes (Clements, 1992),
and blood digestion rates, ovary development, and gonotrophic
cycle duration in adult females (Eldridge, 1965; Madder et al.,
1983).
C. pipiens biological characteristics. The females lay their eggs only
upon standing water, and the eggs are not drying resistant. In cool
temperate areas, C. pipiens hibernate as nulliparous, inseminated
females that enter a facultative reproductive diapause (Mitchell,
1983). The adult diapause in females is induced by shorter day
length and the low temperature experienced during larval and
pupal development (Spielman, 2001).
A. vexans biological characteristics. A. vexans is a floodwater
mosquito and it is known that fluctuations in the abundance of
the larvae of the genus Aedes are influenced by the flood regime of
their breeding sites (Maciá et al., 1995). Females of this mosquito
species lay their eggs in moist substrates without standing water, with eggs usually being resistant to desiccation and hatching
when flooded. If the environmental conditions are unfavorable, the
eggs are dormant and can hatch 5–7 years later (Kliewer, 1961).
This species overwinters as eggs. The egg diapause is an adaptation to the seasonality of climatic conditions, with a winter egg
diapause being typical for mosquito species occurring in temperate zones. The photoperiod and temperature are the main environmental factors responsible for the induction of an egg diapause in
mosquitoes, and the photoperiod is the major diapause-inducing
stimulus (Vinogradova, 2007).
2.3. Matrix dimension determination
The mosquito population of both species is divided into three
immature stages: eggs, larva, and pupae; and six adult stages:
one nulliparous stage in which females do not reproduce, and
five parous stages or five gonotrophic cycles in which females
reproduce (Fig. 2). It is well known that temperature changes
have a significant effect on the duration of the immature stages
(de Meillon et al., 1967); therefore the developmental stages are
further divided. The dimensions of each projection submatrix
for developmental stages were determined as the maximal time
required to complete all stages at low mean daily temperatures,
according to data of Kamura (1959). The egg stage is therefore
further divided into durations of 20 days, as this is supposed to
be the maximum time needed for eggs to develop into larvae.
Fig. 2. Overview of the mosquito life-cycle as used in the models. The mosquito
life-cycle is divided into nine stages, and each stage is further divided into days
of duration according to the maximum time required to complete development
in each stage at low mean daily temperatures. Egg stage (E) is divided into a
20-day duration, the larval (L) stage into a 26-day duration, the pupal (P) stage
into a 5-day duration, and the nulliparous (N) stage and all five gonotrophic cycles
(GCs) into 8-day durations. The life-cycle graph shows possible transitions between
developmental stages.
Larval stages are not separated to instars, but this stage was further
divided into a 26-day duration. The pupal stage was further divided
into 5 days as this is the maximum time needed for development.
In C. pipiens mosquitoes, the average duration of the gonotrophic
cycle is 5.54 ± 1.73 days (Faraj et al., 2006), so the nulliparous
cycle and all five gonotrophic cycles were further divided into
8-day durations. The projection matrix had dimensions 99 × 99,
and distinguished individuals according to their stages and time
spent in each stage. The matrix dimensions were determined
according to literature data available, and the same matrix
dimensions were used for both mosquito species. As the model was
constructed for a three-year period (2008–2010), 1096 projection
matrices were constructed for every day in a year, and then
the matrices were combined to one three-dimensional projection
matrix (TDPM), with dimensions 99 × 99 × 1096.
2.4. C. pipiens model
2.4.1. Temperature-dependent transition probabilities
Transition probabilities were calculated based on the data of
stage durations at various temperatures for C. pipiens mosquitoes
(Kamura, 1959). First, the data were fitted and the functions were
chosen based on correlation coefficients. Using those functions, we
could calculate stage duration (SD) values for each developmental
stage at different mean daily temperatures (T ). The functions
used for modeling the stage durations at various mean daily
temperatures, their general form, and function parameters are
listed in Table 1.
We assumed that the calculated values of stage duration at
mean daily temperature in time (t) represent the time at which
50% of the population in each of stage develop to the next stage;
and therefore that value can be used as an inflection point of a twoparameter logistic function that we used for modeling transition
probabilities:
Pi (t ) =
1
1 + exp(−b(ln(TS (t )) − ln(SD(t ))))
,
(1)
Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94
85
Table 1
Best-fit functions and function parameters used for calculations of stage durations at various mean daily temperatures (T ). The same functions were used for both mosquito
species. Two different parameterizations (1, 2) of the Weibull model exist within the drc package (dose-response curve) under an R software environment, and they do not
yield the same fitted curve for a given dataset (Seber and Wild, 1989).
Stage
Function
General form
Function parameter values
b
Egg
Larva
Pupa
Adult
Four-parameter Weibull (1)
Four-parameter Weibull (1)
Four-parameter logistic
Four-parameter Weibull (2)
SDE (t ) = c + (d − c ) exp(− exp(b(ln(T (t )) − ln(e))))
SDL (t ) = c + (d − c ) exp(− exp(b(ln(T (t )) − ln(e))))
SDP (t ) = 1/1 + exp(b(ln(T (t ) − ln(e))))
SDA (t ) = c + (d − c )(1 − exp(− exp(b(ln(T (t ) − ln(e))))))
c
2.978 1.916
9.872 11.932
43.246 2.999
−5.561 2.585
Correlation (r)
d
e
8.343
26.251
5.000
25.398
15.485
20.533
18.999
13.802
0.9989
0.9999
0.9999
0.9843
where q(t ) is Leslie’s density-dependent factor, λ(t ) is the
dominant eigenvalue of the projection matrix in time t, nLAR (t ) is
the sum of all larvae present in time t, and K is the carrying capacity
for larvae.
2.4.3. Fecundity
The fecundity of C. pipiens females changes with the season
and with the female’s age through each subsequent gonotrophic
cycle. For modeling procedures, the seasonal changes in fecundity
experimental data of Sichinava (1978) were best fitted using the
four-parameter Brain–Cousen function:
Fec (t ) = 81.13 +
Fig. 3. Probabilities of moving from the larval to the pupal stage, based on the
time spent in the larval stage and the mean daily temperature. It is not possible
for larvae to move to the pupal stage before day 7, as this is the minimal time
needed for morphogenesis to be completed. The transition probability increases
with increasing mean daily temperature and increasing time spent in the larval
stage.
where Pi(t ) is the probability of moving from stage i to stage i + 1,
TS (t ) is time spent in the developmental stage (days) and b is
slope of the two-parameter logistic function. We assumed that
the slope (b) is equal in functions describing stage durations and
transition probabilities. The probability of moving from stage i to
stage i + 1 thus depends on the time spent in stage i and the
mean daily temperature (Fig. 3). As the temperature increases,
the transition probability increases, while the development time
decreases. So, during one projection interval, an individual can
continue development through the same stage (becoming one day
older), or can ‘‘jump’’ directly to the next developmental stage,
which depends on the mean daily temperature.
The dependence of the developmental rates on ambient temperature introduces complications in the population models
(Rueda et al., 1990), as the transition probabilities describing
development (e.g., probability of egg becoming larva) are held
constant. However, in our model, the transition probabilities are
temperature dependent, and the projection interval is one day, so
the elements of the projection matrix are changing daily.
2.4.2. Density dependence
In 1948, Leslie introduced a density-dependent model to illustrate limited population growth. Several authors have reported
that the density dependence in the mosquito populations occurs
during early larval stages (Gilpin and McClelland, 1979; Service,
1985; Juliano, 2007), and that the density-dependent competition among larvae is an important factor regulating the growth
of mosquito populations (Agnew et al., 2000). For those reasons,
the population carrying capacity (K ) was chosen to be in the larval
stage.
q(t ) =
K + (λ(t ) − 1) 
K
nLAR (t ),
(2)
−57.52 + 1.67ODY
,
1 + exp(ln(ODY ) − ln(216.71))
(3)
where ODY is the Ordinal Day of Year.
The fecundity also changes with every subsequent gonotrophic
cycle. Using experimental data (Rouband, 1944), egg raft ratios
were calculated with respect to the first gonotrophic cycle.
During the projection interval, the fecundity in each gonotrophic
cycle was multiplied by its associated ratio. As females do not
reproduce continuously in time, the fecundities are distributed at
8-day intervals in the first four gonotrophic cycles, with females
ovipositing at the beginning of each, and at the end of the last
gonotrophic cycle. The general form of the fecundity vector is
F=
1
60 F
···0
2
68 F
···0
3
76 F
···0
4
84 F
···0
0 · · · 599 F ,

(4)
where ab F
is the fecundity (average number of eggs) laid by a female
b days old, and in the ath gonotrophic cycle.
2.4.4. Rainfall dependence
Frequent rainfalls increase the abundance of habitats available
for the mosquito females to oviposit their eggs. If rainfall is absent
for a long period of time, the females have nowhere to deposit their
eggs; so in the model, we assumed that the fecundity is rainfall
dependent. To model the influence of rainfall on fecundity, we used
the cumulative number of rainy days in an 8-day period:
rf ( t ) =
t −8

rd · rpf ,
(5)
t
where rf (t ) is the rain factor which incorporates the cumulative
number of rainy days (rd ) during the past 8 days. The rain potential
factor (rpf ) accounts for the strength of the rainfall influence
on the Culex mosquito population. The rain factor decreases as
the number of rainy days in the 8-day interval decreases and,
accordingly, the fecundity decreases:
Fec (t ) = Fec (t )
rf
10
+ rrf ,
(6)
where rrf is the rest-rain fecundity, which denotes the minimum
number of eggs laid by one female in extremely unfavorable
environmental conditions. This parameter was introduced to the
model because, although there might not be any rain, even for a
long period of time, there are always places where some females
will be able to deposit their eggs (i.e. fecundity is never 0).
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Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94
Fig. 4. Modeling changes of carrying capacity in the A. vexans model. With increase of the Danube’s water level, the flooded area increases and thus the carrying capacity
for Aedes larvae increases. Flooding starts when the Danube’s water level is above 200 cm.
2.4.5. Diapause induction
The C. pipiens model includes an overwinter period for adult
females during unfavorable environmental conditions (i.e., the
temperature and photoperiod are below threshold values).
Temperature diapause induction
The threshold temperature, below which no development occurs, is 8 °C (Farghal et al., 1987), so this temperature was used
as the threshold value in the model (trshTEMP ). In the model, when
the mean daily temperature is below the threshold value, the values on the main diagonals of the adult projection submatrices are
set to 1, and all other values in the projection matrix are set to 0
(e.g., all females remain in their stages, and all other stages die).
During those unfavorable temperature conditions, females do not
oviposit, so the fecundity is also set to 0.
Photoperiod diapause induction
Adult females enter a diapause in response to a shorter day
length. As photoperiods shorter than 12 h induce a reproductive
diapause in all females (Spielman and Wong, 1973), we used that
value as the threshold value (trshPHP ) at which all females enter
a diapause. In the model, when the photoperiod is below the
threshold value, the values on the main diagonals of the adult
projection submatrices are set to 1, and all other values in the
projection matrix are set to 0 (e.g., all females remain in their stages
and all other stages die, as no other stages are present during the
overwinter period). The fecundity is also set to 0.
2.5. A. vexans model
In the A. vexans model, the same transition probabilities and the
same fecundity changes were used as in the C. pipiens model.
2.5.1. Flooding dynamics of A. vexans breeding sites
Flooding dynamics has several important influences on Aedes
mosquitoes. An increase in the surface of the flooded area enhances
egg hatching and subsequent larval survival which in effect leads
to increased adult mosquito abundances. In the A. vexans model,
we modeled the variable carrying capacity that is dependent on
the Danube’s water level. If we assume a conic shape of the
periodically flooded areas (i.e., mosquito breeding sites), then, with
every unit of increase in the Danube’s water level, the surface of
the flooded area increases exponentially. For those reasons, the
carrying capacity coefficient was modeled using an exponential
function:
Kf (t ) = 1 + 0.4391 exp(0.005WLD (t )),
(7)
where Kf (t ) is the carrying capacity coefficient at time t, and
WLD (t ) is the Danube’s water level (cm) at time t. The
parameters of the function (7) were obtained empirically based
on environmental experiments and monitoring programs (Merdić,
2002). The Danube’s flooding threshold is 200 cm, so an increase
of the Danube’s water level above that value causes an increase
of the flooded area, and the carrying capacity (K ) for A. vexans
larvae increases (Fig. 4). The carrying capacity for larvae at each
time interval (t) is calculated using
K (t ) = K0 Kf (t ),
(8)
where K0 is the carrying capacity when there is no flood (i.e., the
Danube’s water level is ≤200 cm), and Kf is the carrying capacity
coefficient at time t.
The density-dependent factor, which is set only for larvae, is
therefore modified accordingly to account for the variable carrying
capacity:
q(t ) =
K (t ) + (λ(t ) − 1) 
K (t )
nLAR (t ).
(9)
The second important characteristic of A. vexans mosquitoes is the
resistance of the eggs to desiccation and freezing. Eggs can hatch
after several years of estivation; so, during one season, eggs from a
previous season are hatching too, given the right environmental
conditions. To model this influence we also used the Danube’s
water level. With the increase of water level, the flooded area
increases, and thus more of the previously laid eggs hatch. This
influence is modeled using a second-degree polynomial function.
The parameters of the function are obtained empirically based on
monitoring programs and by model calibration.
NEGG (t ) = 0.0264(WLD )2 (t ) + 9.431WLD (t ) − 80.128,
(10)
where NEGG (t ) is the number of eggs from previous seasons that
are flooded and that can hatch. At each time interval, the number
of ‘‘older’’ eggs is summed with one-day-old eggs in the population
vector at time t.
2.5.2. Diapause induction
A. vexans mosquitoes overwinter as eggs, and an egg diapause
in mosquitoes can be induced by temperature and photoperiod
decrease.
Temperature diapause induction
Eggs of this mosquito species are completely dormant at
temperatures below 8 °C (Gjullin et al., 1950), so this temperature
was set as the threshold value (trshTEMP ) below which an
egg diapause is induced. In the model, when the mean daily
temperature is below the threshold value, the values on the main
diagonal of the egg projection submatrix are set to 1, and all other
values in the projection matrix are set to 0 (e.g., all eggs remain in
their stages, and all other stages die).
Photoperiod diapause induction
As previously stated, the photoperiod is the major diapauseinducing stimulus for the induction of an egg diapause in
mosquitoes. A photoperiod of ≤12 h is set as the threshold value
(trshPHP ) at which all eggs are diapausing. In the model, when the
photoperiod is below the threshold value, the values on the main
diagonal of the egg projection submatrix are set to 1, and all other
values in the projection matrix are set to 0 (e.g., all eggs remain in
their stage and all other stages die, as no other stages are present
during the overwinter period). The fecundity is also set to 0.
2.6. Structure of the models
The C. pipiens model has the general form
n(t + 1) = TDPM (T , R, Php)q−1 n(t ),
(11)
Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94
87
Fig. 5. General structure of the model. Projection matrices (A) were constructed for every day in a year, for a three-year period (2008–2010), based on the climatological
data, biological, and ecological characteristics of the modeled mosquito species. The matrices were then combined into one three-dimensional projection matrix (TDPM).
For every projection matrix within the TDPM, the finite rate of the increase (λ) was calculated. n(t ) shows the general structure of the population vector.
and the A. vexans model has the form
n(t + 1) = TDPM (T , Php, WLD )q
−1
n(t ),
(12)
where t is time measured in days and n is a vector with the number
of individuals in each stage and ‘‘age’’. Every projection matrix
within the three-dimensional projection matrix (99 × 99 × 1096)
in the C. pipiens model is a nonlinear function of the mean daily
temperature (T ), rainfall (R), and photoperiod (Php). In the A.
vexans model, the projection matrices are functions of mean daily
temperature (T ), photoperiod (Php), and the Danube’s water level
(WLD ). q−1 is the reciprocal of Leslie’s density-dependent factor
(based on the number of larvae present at time t). The general
structure of the three-dimensional projection matrix and of the
population vector is shown in Fig. 5. Abbreviations, variables, and
parameters used in the models are listed in Table 2.
2.7. Transient model analysis
The model analysis becomes more complicated with an increase
in projection matrix size and increase in the number of projection
matrices. Also, as individuals in the models are separated according
to their stage and ‘‘age’’, interpretation of the sensitivity and
elasticity results would be of little practical value. For those
reasons, before analyzing the model, the projection matrices (A(t ))
were reduced to size 4 × 4 to correspond to the four developmental
stages of mosquitoes: eggs, larvae, pupae, and adults. The elements
of the reduced projection matrices (S(t )) were calculated using
projection matrices A(t ), population vectors n(t ), and generation
vectors (sum of all individuals in stage at time t) g(t ) and g(t + 1).
After the projection matrices were reduced in size, the transient
sensitivities of n(t + 1) to the elements of S(t ) were calculated for
every matrix (Caswell, 2007):
dg(t + 1)
dg(t )
=S
+ (gT (t ) ⊗ I),
(13)
dvecT S
dvecT S
where ⊗ denotes Kronecker product, I is the identity matrix, T is
matrix transpose, and the vec operator is used to stack columns
of a matrix into column vector. The result of this calculation is a
matrix that contains elements of the vector g(t + 1) sensitive to
the elements of S(t ).
The transient elasticities of g(t + 1) to the elements of S(t ) were
calculated (Caswell, 2007):
dg(t + 1)
diag [S] ,
(14)
dvecT S
where diag[x] is a matrix with x on the diagonal and zeros elsewhere.
The transient population growth rate at time t was calculated
from the model outputs:
diag [g(t + 1)]−1
r (t ) = log
N (t + 1)
,
N (t )
where N is total population size in time t + 1 and t.
(15)
2.8. Model validation
To validate the models, we compared the 7-day smoothed
model outputs with data on human bait collection (HBC) obtained
from the Public Health Institute of Osijek-Baranja County. Absolute
mosquito population sizes are very difficult, if not impossible, to
estimate from the field data, so model outputs are often compared
to field data by looking for a good correlation or agreement
rather than absolute numeric agreement (Lord, 2007). Our model
validation scaled both field observations and 7-day smoothed
model outputs, dividing all number of bites with the maximum
number of bites. The C. pipiens and A. vexans 7-day smoothed model
outputs were also divided by the associated maximum output
from the model. As the mosquito bite data were not categorized
according to mosquito species, we compared those values with the
values from the C. pipiens and A. vexans models and with the sum.
2.9. Weather input data
Mean daily temperature, rainfall (measurement station: Osijek), and Danube’s water level (measurement station: Batina) data
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Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94
Table 2
Symbols for abbreviations and parameters used in the model.
T
R
Php
SD(t )
TS (t )
A(t )
S(t )
TDPM
S(t )
ODY
Pi(t )
q (t )
rf (t )
rd
rpf
rrf
WLD
K f (t )
K (t )
K0
K
Fec (t )
nLAR (t )
n(t )
g(t )
NEGG (t )
trshTEMP
trshPHP
Mean daily temperature (°C)
Rain
Photoperiod (hours of daylight)
Stage duration of different developmental stages at different mean daily temperatures at time t calculated from best-fit functions (Table 1)
Time spent in developmental stage at time t
Projection matrix in time t
Reduced projection matrix
Three-dimensional projection matrix
Sensitivity matrix in time t
Ordinal Day of Year
Two-parameter logistic function used for calculation of transition probabilities at time t for every developmental stage
Leslie’s density-dependent factor at time t
Rain factor at time t
Rainy day (denotes days when it was raining)
Rain potential factor (denotes rain influence on mosquito population)
Rest-rain fecundity (denotes minimum number of eggs laid by one female in extremely unfavorable environmental conditions (i.e., long periods without rain))
Danube’s water level (cm)
Carrying capacity coefficient at time t calculated based on the Danube’s water level
Carrying capacity for A. vexans larvae in time t
Carrying capacity for A. vexans larvae when there is no flood
Carrying capacity for C. pipiens larvae
Five-parameter Brain–Cousens function that calculates fecundities in time (t)
Number of larvae present in time t
Population vector at time t
Generation vector in time t (obtained by summing all individuals of different ‘‘age’’ that are in the same stage)
Second-degree polynomial function which calculates the number of eggs from the previous season that are flooded and that can hatch
Temperature threshold value (8 °C) below which no development occurs
Photoperiod threshold value (12 h) below which a diapause is induced
were obtained from the Croatian Meteorological and Hydrological
Service in Zagreb, Croatia.
2.10. Computational methods
Computations, simulations, and plotting were performed using
R (version 2.11.1), an open-source language and environment for
statistical computing and graphics (R Development Core Team,
2010, Vienna, Austria), an implementation of S-language (Ihaka
and Gentleman, 1996). Experimental data were fitted using the drc
(dose-response curve) package under an R software environment
(Ritz and Streibig, 2005). The elasticity matrices were plotted using
the Plotrix package under an R software environment (Lemon,
2006).
3. Results
Simulations were computed for a three-year period (2008–
2010) for both mosquito species, with initial parameters for C.
pipiens K = 1000, rrf = 5, rpf = 1, N (adults) = 100,
and for A. vexans K = 10000, N (egg) = 100 (Fig. 6). In the
A. vexans model, the time delay between the first appearances
of the different developmental stages at the beginning of the
reproduction season is evident. The number of eggs than can
hatch increases during early spring, although no adults are
present. Those are the eggs laid in previous seasons that can
hatch when flooded, if ambient temperatures are suitable. The
population dynamics of the A. vexans adults starts to change
15–30 days later, depending on environmental conditions. The
population dynamics of C. pipiens mosquitoes starts to change
during mid-spring, when the photoperiod is above the threshold
value (12 h) and the adult diapause is terminated. Both mosquito
populations have several peaks during the seasons, with the first
peak usually occurring in the early to mid spring, followed by
several others depending on environmental conditions. The adult
population of the A. vexans mosquitoes at the beginning of
reproduction seasons in 2008 and 2009 peaked approximately
10–15 days before the adult C. pipiens population, while in 2010 the
adult population of C. pipiens peaked first, approximately 10 days
before A. vexans adults. The population dynamics of both modeled
species during late fall and winter (i.e., during the overwinter
season) is invariable, but the number of overwinter population
(A. vexans eggs and C. pipiens adults) between the two reproduction
seasons changes.
Daily growth rates (r (t )) were calculated for both mosquito
species from the model outputs as log(N (t + 1)/N (t )) (Fig. 7). The
growth rates show a response of mosquito populations to immediate environmental conditions. A. vexans and C. pipiens mosquitoes
have different growth rate patterns during all three reproduction
seasons, as they have different ecological characteristics and their
growth and development depends on various different environmental factors.
The transient elasticities show a proportional effect on the
population structure from the proportional changes in each value
of the projection matrix for every day in the three-year period. Both
mosquito population structures (i.e. population stages) reacted
to perturbation of the same matrix elements (Fig. 8). Egg stage
reacted to perturbation of matrix element S1,1 (probability of egg
remaining in the same stage), and matrix element S1,4 (fecundity).
The larval stage reacted only to the perturbation of matrix element
S2,2 (probability of larvae remaining in the same stage). Adults
reacted to the perturbation of matrix elements S4,3 (probability
of pupa moving to the adult stage) and S4,4 (probability of adults
remaining in the stage). Pupae reacted to the perturbation of
matrix element S3,3 (probability of pupae remaining in the same
stage). Values in all other matrix elements were zero at all times,
except for matrix elements S2,1 (probability of egg moving to
the larval stage) and S3,2 (probability of larvae moving to the
pupal stage), whose values were zero or very close to zero
(e.g., less than 10−10 ). To determine which stage has the greatest
proportional sensitivity, i.e., elasticity, results in both mosquito
species were analyzed by examining how many days the elasticity
values for a given stage were smaller or greater than the median
elasticity value during the three-year reproduction seasons. The
median value was calculated using all elasticity values from three
reproduction seasons (i.e., from 1 April to 30 September) for each
mosquito species. In the C. pipiens model, the elasticities of adults
to matrix element S4,4 were above the median value during most
of the reproduction seasons (503 days), the elasticities of larvae
Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94
89
Fig. 6. 7-day smoothed model outputs: A. vexans eggs (a_egg), larvae (a_lar), pupae (a_pup), and adults (a_ad), and C. pipiens eggs (c_egg), larvae (c_lar), pupae (c_pup), and
adults (c_ad).
Fig. 7. Growth rates r (t ) of C. pipiens and A. vexans mosquitoes during the three-year reproduction seasons. Periods from 1 April to 30 September of each year (2008–2010)
are shown.
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Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94
Fig. 8. Elasticities of population structure n(t + 1) to reduced projection matrix elements S in the C. pipiens and A. vexans models. Only periods from 1 April to 30 September
of each year (2008–2010) are shown.
Fig. 9. Frequencies showing the number of days during the three-year reproduction season (552 days total) when elasticity values for a given stage were smaller or
greater than the median elasticity value in the C. pipiens model (C.ele1.1—elasticities of eggs to matrix element S1,1 (i.e., probability of egg remaining in the same stage);
C.ele1.4—elasticities of eggs to matrix element S1,4 (i.e., fecundity); C.ell2.2—elasticities of larvae to matrix element S2,2 (i.e., probability of larvae remaining in the same
stage); C.elp3.3—elasticities of pupae to matrix element S3,3 (i.e., probability of pupae remaining in the same stage); C.elad4.3—elasticities of adults to matrix element S4,3
(i.e., probability of pupa moving to the adult stage); C.elad4.4 — elasticities of adults to matrix element S4,4 (i.e., probability of adults remaining in the stage)) and the A.
vexans model (A.ele1.1—elasticities of eggs to matrix element S1,1 (i.e., probability of egg remaining in the same stage); A.ele1.4—elasticities of eggs to matrix element S1,4
(i.e., fecundity); A.ell2.2—elasticities of larvae to matrix element S2,2 (i.e., probability of larvae remaining in the same stage); A.elp3.3—elasticities of pupae to matrix element
S3,3 (i.e., probability of pupae remaining in the same stage); A.elad4.3—elasticities of adults to matrix element S4,3 (i.e., probability of pupa moving to the adult stage);
A.elad4.4—elasticities of adults to matrix element S4,4 (i.e., probability of adults remaining in the stage)). The median elasticity value was calculated using all elasticity values
from a three-year reproduction season (i.e. from 1 April to 30 September 2008–2010) for each mosquito species.
to matrix element S2,2 were above the median value for 498 days,
and the elasticities of pupae to matrix element S3,3 were above the
median value for 491 days. In the A. vexans model, the elasticities of
larvae to matrix element S2,2 were above the median value during
most of the reproduction seasons (516 days), the elasticities of
pupae to matrix element S3,3 were above the median value for
508 days, and the elasticities of adults to matrix element S4,4 were
above the median value for 496 days (Fig. 9).
When the model outputs were compared to the field data,
a general correlation or agreement between model outputs and
field data existed. The A. vexans model had a more significant
agreement with field data in the sense of accurately predicting
timing and maximum adult population values (Fig. 10). Also, the
model accurately predicted the number of population peaks during
seasons. In 2008, the model’s first population peak underestimated
abundances, but other increased adult abundances were accurately
predicted. The C. pipiens model outputs had less agreement. The
model outputs during 2008 and 2009 overestimated the adult
abundances, but the timing of the peaks showed relatively good
agreement with field data. The summed values of the C. pipiens and
A. vexans model outputs had the best agreement with field data.
4. Discussion
The climate limits the distribution of infectious diseases, and
the weather affects the dynamics and intensity of disease out-
Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94
91
Fig. 10. A. vexans (a_ad), C. pipiens(c_ad) adult model outputs and summed model outputs (summ) compared to field data on human bait collection (HBC) obtained from
the Public Health Institute of Osijek-Baranja County. Both model outputs and field observations were scaled by dividing each dataset by the corresponding maximum value.
All model outputs are 7-day smoothed.
breaks; hence the prediction of environmental conditions that lead
to an increase in mosquito populations is essential in the prevention of possible disease outbreaks and maximization of control efficiencies. In temperate areas, during certain periods, population
growth is limited and the main factors responsible are the photoperiod, low ambient temperatures, and dry seasons. Overwintering periods are usually neglected in the other models using
a similar approach in model construction (Shaman et al., 2006;
Erickson et al., 2010; Ahumada et al., 2004; Schaeffer et al., 2008),
but they can significantly influence the dynamics and abundance of
mosquito populations in subsequent reproduction seasons, especially in species that overwinter as eggs. In both models presented
in this paper all developmental stages of mosquitoes are included
as well as all environmental factors that influence the population
dynamics of the two modeled mosquito species.
Mosquitoes have four developmental stages, and certain time
segments are characterized by the presence of only one or several
developmental stages. To adequately describe the population dynamics of mosquitoes and other ecologically similar insect species,
especially in strong seasonal environments, it is important that the
model includes and achieves time delay, because not all developmental stages are present at the same time. Modeling those populations by using only stage or age structured matrix population
models is very difficult. Our research design implemented models that were structured according to the individual’s stage and the
length of each developmental stage (i.e., ‘‘age’’ of individual in each
stage). During one projection interval, every individual can continue its development through the same stage becoming one day
older, or it can move directly to the next developmental stage. The
transition probabilities depend on the mean daily temperatures
and the time spent in the developmental stage. The older the individual gets, the higher the probability of moving to the next developmental stage. So during one or even several projection intervals,
depending on environmental conditions, none of the individuals
have to move to the next developmental stage, which creates a delay in the model. The length of those time delays between the developmental stages is temperature dependent. At lower mean daily
temperatures that delay is longer, as morphogenesis of individuals
within the stage is not completed. At high ambient temperatures,
this delay is shorter, because morphogenesis at high ambient temperatures is completed in a shorter period of time. Although this
approach requires the use of quite large dimension projection matrices, modern computer and mathematical software enable these
complicated and demanding calculations.
Periodic changes of various environmental conditions can be
the cause of changes in carrying capacity for some mosquito
species. Various theoretical frameworks have shown that the
average total biomass of a population in a periodic environment
can be greater than or less than the average total biomass in
the associated constant average habitat (Henson and Cushing,
1997). The fact that changes in the carrying capacity can have a
significant influence on the population dynamics of some insects
has been proven experimentally. Jillson’s (1980) experiment
with flour beetles (Tribolium castaneum) showed that the total
population numbers in the periodically fluctuating environment
can be more than twice of those in the constant environment,
even though the average flour volume in which the flour beetles
were grown was the same in both cases. For those reasons, it
is important to include variations of carrying capacity in the
models constructed for most of the mosquito species. In our work,
the A. vexans model includes variable carrying capacity which
is influenced by flooding dynamics (i.e. changes in the Danube’s
water level). It is known that fluctuations in the abundance of
the larvae of the genus Aedes are basically influenced by the
flood regime of their breeding sites (Maciá et al., 1995), as the
increased surface wetness favors mosquito reproduction and larval
survival, which can subsequently lead to an increase in flood
and swamp water mosquito abundances (Shaman et al., 2006).
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Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94
Because mosquitoes have a rather short life span, and immature
developmental stages can last from several days to at most two
weeks, it is clear that changes in the flooding dynamics of floodwater mosquitoes’ breeding sites can significantly influence adult
mosquito abundance. This is even more important for those
mosquito species that overwinter in the egg stage, as those eggs
can hatch several years later, given the right conditions (i.e. flood
and temperature), and also contribute to increased adult mosquito
abundances. However, in the C. pipiens model, the mosquito
carrying capacity was held constant, as this mosquito has very
different survival strategy: it lays its eggs in virtually any receptacle
containing water that is rich in decomposing organic material; so
we assumed that the carrying capacity for this mosquito species in
urban areas remains constant.
Mosquito development, especially in the immature stages, is
very dynamic, so the population response to changes in the environmental conditions is very rapid. Development can be accelerated or slowed down within several days, so estimating the
population growth over the entire cycle (i.e., whole reproduction
season) cannot provide precise information on immediate population states. As mosquitoes are organisms with vital rates that
can vary significantly within period of several days, we determined population growth rates r (t ) from the model outputs as
log(N (t + 1)/N (t )), as those values give more precise indications
on population states and show their response to immediate environmental conditions. In both mosquito species, the population
growth rate is determined mostly by the ambient temperature.
However, for the C. pipiens mosquito, the rain pattern and frequency are important for reproduction and egg hatching, while for
the A. vexans mosquitoes the important factor is the flooding dynamics. From the growth rates (Fig. 7) we can see that in 2010 the A.
vexans growth rate was zero or slightly negative during all of May,
while becoming positive in June. During 2010 the Danube’s water level was mostly below the flooding threshold (200 cm), which
caused delayed egg hatching and resulted in the A. vexans peaking after the C. pipiens mosquitoes, which was not the case in 2008
and 2009.
The matrix sensitivity analysis is usually based on determining
the sensitivity of the asymptotic growth rate to changes in
matrix elements aij . Such analysis assumes that the distribution
of the population’s age, stage, or size remains stable through
time and that the population grows according to a constant rate
(i.e., λ). Taylor (1979) concluded that many (possibly even most)
insect species growing in seasonal environments never experience
a stable age distribution, and various empirical evidence also
suggests that stable population states rarely occurs in nature
(Bierzychudek, 1999; Clutton-Brock and Coulson, 2002). Several
authors point out the importance of transient dynamics in
population management (Ezard et al., 2010; Stott et al., 2011),
and show that transient dynamics can be very different from the
asymptotic dynamics (Koons et al., 2005; Buhnerkempe et al.,
2011). Considering the characteristics of mosquito populations,
especially in temperate climate regions, it is highly unlikely that
those populations ever experience stable stage distributions, so,
in order to gain proper knowledge on the mosquito population
characteristics, a transient model analysis was implemented.
Different methods are available today for transient model analysis
(Fox and Gurevitch, 2000; Yearsley, 2004; Caswell, 2007).
Our model used large projection matrices to enable variable
durations of different developmental stages, and we also used
different projection matrices for every day, which caused the
analysis of the model to become computationally very demanding,
even not possible at all. Also, the interpretation of results would be
of little practical value as individuals in the models were separated
according to their stage and ‘‘age’’. For example, if five- or sixday-old larvae were the most sensitive, that would imply that
control measures applied to those individuals would have had the
greatest impact in reducing the total population size. However,
those individuals cannot be separated from the rest of the larvae
population. Therefore, the projection matrices were reduced to
biologically meaningful forms. The reduced matrices describe the
developmental stages of the mosquitoes: eggs, larvae, pupae, and
adults. Transient analysis of those matrices can give us biologically
relevant results that can be easily interpreted and used for
practical purposes, especially in mosquito control management.
Transient analysis showed that both mosquito species reacted
to perturbation of same matrix elements; however, not all of
those perturbations are important at the same time (Fig. 8).
The analysis shows that elasticities are constantly changing due
to population intrinsic properties and changing environmental
conditions. If control measures are planned in some period of time,
it is important to know which developmental stages the control
measures should be aimed at. Control measures implemented
in stages with highest proportional sensitivity, i.e., elasticity,
will have the best result in reducing the total population
size. In both mosquito models, larval and adult stages have
greatest proportional sensitivities during most of the reproduction
season (Fig. 9), so implementation of control measures at these
developmental stages seems optimal. However, it is important to
note that, as the elasticities of both mosquito species are changing
with different dynamics, control measures applied at a certain time
will have a diverse impact on different mosquito species.
When comparing our models to the field data, the A. vexans
model showed relatively good compliance with field data, while
the C. pipiens model fitted to field data to a lesser extent. The reason
for such a difference in model compliances could be due to the fact
that the experimental data did not target specifically these two
species specifically, but all nuisance mosquito species in the city
and its surrounding area. As A. vexans makes up the majority of
mosquito fauna, it is reasonable to assume that most of the human
bait collection data can be attributed to this mosquito species.
This is probably the reason for better agreement between the A.
vexans model and field data. The C. pipiens mosquito, according
to available data, makes up only 5–10% of the mosquito fauna
in the city of Osijek. Also, almost all mosquito control efforts are
aimed at this mosquito species. Mosquito control measures can
alter the population dynamics, and thus can be the reason for
wrong estimates, especially for the C. pipiens mosquito.
Mechanistic models can be valuable tools in predicting increased mosquito abundances and thus possible disease outbreaks.
The implementation of these principles during the model design and development, based on the population’s biological and
ecological characteristics, allows us to predict their dynamics in
different environmental conditions and closely examine intrinsic
characteristics of the entire population. This would not be possible if the models were constructed through correlating weather
pattern data with historical mosquito abundance data. The main
advantage of this model is that it can be easily modified for implementation in different geographical areas, or modified in order
to describe the population dynamics of different mosquito species,
even when historical data about mosquito abundances are unavailable. The model can also be modified for prediction of the risk of
mosquito-borne diseases.
References
Agnew, P., Haussy, C., Michlakis, Y., 2000. Effects of density and larval competition
on selected life history traits of Culex pipiens quinquefasciatus (Diptera:
Culicidae). J. Med. Entomol. 37 (5), 732–735.
Ahumada, J.A., Lapointe, D., Samuel, M.D., 2004. Modeling the population dynamics
of Culex quinquefasciatus (Diptera: Culicidae), along an elevational gradient in
Hawaii. J. Med. Entomol. 41 (6), 1157–1170.
Akritidis, N., Boboyianni, C., Pappas, G., 2010. Reappearance of viral hemorrhagic
fever with renal syndrome in northwestern Greece. J. Infect. Dis. 14, e13–e15.
Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94
Bernardelli, H., 1941. Population waves. J. Burma Res. Soc. 31, 1–18.
Bierzychudek, P., 1999. Looking backwards: assessing the projections of a transition
matrix model. Ecol. Appl. 9, 1278–1287.
Bogh, C., Pedersen, E.M., Mukoko, D.A., Ouma, J.H., 1998. Permethrin-impregnated
bednet effects on resting and feeding behaviour of lymphatic filariasis vector
mosquitoes. Kenya. Med. Vet. Entomol. 12, 52–59.
Buhnerkempe, M.G., Burch, N., Hamilton, S., Byrne, K.M., Childers, E., Holfelder, K.A.,
McManus, L.N., Pyne, M.I., Schroeder, G., Doherty Jr., P.F., 2011. The utility of
transient sensitivity for wildlife management and conservation: bison as a case
study. Biol. Cons. 144, 1808–1815.
Caswell, H., 2007. Sensitivity analysis of transient population dynamics. Ecol. Lett.
10, 1–15.
Chevalier, V., de la Rocque, S., Baldet, T., Vial, L., Roger, F., 2004. Epidemiological
processes involved in the emergence of vector-borne diseases: West Nile fever,
Rift Valley fever, Japanese encephalitis and Crimean–Congo haemorrhagic
fever. Revue scientifique et technique (International Office of Epizootics) 23,
pp. 535–555.
Clements, A.N., 1992. Growth and development. In: Clements, A.N. (Ed.), Biology of
Mosquito. Chapman & Hall, London, pp. 150–165.
Clutton-Brock, T.H., Coulson, T., 2002. Comparative ungulate dynamics: the devil is
in the detail. Philos. Trans. R. Soc. London, Ser. B 357, 1285–1298.
Confalonieri, U., Menne, B., Akhtar, R., Ebi, K.L., Hauengue, M., Kovats, R.S.,
Revich, B., Woodward, A., 2007. Human health. In: Parry, M.L., Canziani, O.F.,
Palutikof, J.P., van der Linden, P.J., Hanson, C.E. (Eds.), Climate Change 2007:
Impacts, Adaptation and Vulnerability. Contribution of Working Group II to the
Fourth Assessment Report of the Intergovernmental Panel on Climate Change.
Cambridge University Press, Cambridge, UK, pp. 391–431.
de Meillon, B., Sebastian, A., Khan, Z.H., 1967. The duration of egg, larval and pupal
stages of Culex pipiens fatigans in Rangoon. Burma. Bull. World. Health. Organ.
36 (1), 7–14.
Eldridge, B.F., 1965. The influence of environmental factors on blood-feeding and
hibernation in mosquitoes of C. pipiens complex, Ph.D. Thesis. Purdue Univ.,
p. 96;
Vinogradova, A., 2000. Culex Pipiens Pipiens Mosquitoes: Taxonomy, Distribution, Ecology, Physiology, Genetics, Applied Importance and Control. In: Pensoft
Series Parasitologica, vol. 2.. Pensoft Publishers, p. 100.
Erickson, R.A., Presley, S.M., Allen, L.J.S., Long, K.R., Cox, S.B., 2010. A stagestructured, Aedes albopictus population model. Ecol. Modell. 221, 1273–1282.
Ezard, T., Bullock, J., Dalgleish, H., Millon, A., Pelletier, F., Ozgul, A., Koons, D., 2010.
Matrix models for a changeable world: the importance of transient dynamics in
population management. J. Appl. Ecol. 47, 515–523.
Faraj, C., Elkohli, M., Lyagoubi, M., 2006. Cycle gonotrophique de C. pipiens (Diptera:
Culicidae), vecteur potentiel du virus West Nile, au Maroc: estimation de la
durée en laboratoire. Bull. Soc. Pathol Exot. 99 (2), 119–121.
Farghal, A.I., Morsy, M.A.A., Ali, A.M., 1987. Development of Culex pipiens molestus
Forsk, under constant and variable temperature. Assuit J. Agric. Sci. 18 (1),
141–150.
Focks, D.A., Haile, D.G., Daniels, E., Mount, G.A., 1993. Dynamic life table model
for Aedes aegypti (Diptera: Culicidae): analysis of the literature and model
development. J. Vector Ecol. 30 (6), 1004–1017.
Fouque, F., Baumgartner, J., 1996. Simulating development and survival of A. vexans
(Diptera: Culicidae) preimaginal stages under field conditions. J. Med. Entomol.
33, 32–38.
Fox, G.A., Gurevitch, J., 2000. Population numbers count: tools for near-term
demographic analysis. Am. Nat. 156, 242–256.
Gilpin, M.E., McClelland, G.A.H., 1979. System analysis of the yellow fever mosquito
Aedes aegypti. Fortschr. Zool. 25 (2–3), 355–388.
Goddard, L., Roth, A., Reisen, W., Scott, T., 2002a. Vector competence of California
mosquitoes for West Nile virus. Emerg. Infect. Dis. 8, 1385–1391.
Goddard, L.B., Roth, A.E., Reisen, W., Scott, T.W., 2002b. Vector competence of
California mosquitoes for West Nile virus. Emerg. Infect. Dis. 8 (12), 1385–1391.
Gjullin, C.M., Yates, W.W., Stage, H.H., 1950. Studies on Aedesvexans (Meig.) and
Aedessticticus (Meig.), flood-water mosquitoes, in the lower Columbia River
Valley. Ann. Entomol. Soc. Am. 43, 262–275.
Hales, S., de Wet, N., Maindonald, J., Woodward, A., 2002. Potential effect of
population and climate changes on global distribution of dengue fever: an
empirical model. The Lancet 360 (9336), 830–834.
Henson, S.M., Cushing, J.M., 1997. The effect of periodic habitat fluctuations on a
nonlinear insect population model. J. Math. Biol. 36, 201–226.
Hongoh, V., Berrang-Ford, L., Scott, M.E., Lindsay, L.R., 2012. Expanding geographical
distribution of the mosquito, C. pipiens, in Canada under climate change. Appl.
Geogr. 33 (2012), 53–62.
Hu, W., Tong, S., Mengersen, K., Oldenburg, B., 2006. Rainfall, mosquito density
and the transmission of Ross River virus: a time-series forecasting model. Ecol.
Modell. 196, 505–514.
Ihaka, R., Gentleman, R., 1996. R: a language for data analysis and graphics.
J. Comput. Graphical Stat. 5, 299–314.
Jillson, D., 1980. Insect populations respond to fluctuating environments. Nature
288, 699–700.
Juliano, S.A., 2007. Population dynamics. J. Am. Mosq. Control. Assoc. 23, 265–275.
Kallio-Kokko, H., Uzcategui, N., Vapalahti, O., Vaheri, A., 2005. Viral zoonoses in
Europe. FEMS Microbiol. Rev. 29 (5), 1051–1077.
Kamura, T., 1959. Studies on the C. pipiens group of Japan. 4, ecological studies on
the Nagasaki molestus. Endemic. Dis. Bull. Nagasaki. Univ. 1 (1), 51–59;
Vinogradova, A., 2000. Culex Pipiens Pipiens Mosquitoes: Taxonomy, Distribution, Ecology, Physiology, Genetics, Applied Importance and Control. In: Pensoft
Series Parasitologica, vol. 2. Pensoft Publishers, p. 100.
93
Kelly-Hope, L., Kay, B., Purdie, D., 2002. The risk of Ross River and Barmah Forest
virus disease in Queensland: implications for New Zealand. Aust. N. Z. J. Public
Health 26, 69–77.
Kliewer, J.W., 1961. Weight and hatchability of Aedes aegypti eggs. Ann. Entomol.
Soc. Am. 54, 912–917.
Knight, R.L., Walton, V.E., O’Mearac, G.F., Reisen, W.K., Wass, R., 2003. Strategies
for effective mosquito control in constructed treatment wetlands. Ecol. Eng. 21,
211–232.
Koons, D.N, Grand, J.B, Zinner, B., Rockwell, R.F., 2005. Transient population
dynamics: relations to life history and initial population state. Ecol. Model. 185,
283–297.
Lefkovitch, L.P., 1965. The study of population growth in organisms grouped by
stages. Biometrics 21, 1–18.
Lemon, J., 2006. Plotrix: a package in the red light district of R. R-News 6 (4), 8–12.
Leslie, P.H., 1945. On the use of matrices in certain population mathematics.
Biometrika 33, 183.
Leslie, P.H., 1948. Some further notes on the use of matrices in population
mathematics. Biometrika 35 (3–4), 213–224.
Lewis, E.G., 1942. On the generation and growth of a population. Sankhya 6, 93–96.
Lindsay, M., Broom, A., Wright, A., Johansen, C., Mackenzie, J., 1993. Ross River virus
isolation from the mosquitoes in arid regions of western Australia: implication
of vertical transmission as a means of persistence of the virus. Am J. Trop. Med.
Hyg. 49, 686–696.
Lord, C.C., 2007. Modeling and biological control of mosquitoes. AMCA Bull. 7,
252–264.
Maciá, A., García, J.J., Campos, R.E., 1995. Bionomía de Aedes albifasciatus y Ae.
crinifer (Diptera: Culicidae) y sus enemigos naturales en Punta Lara Buenos
Aires. Neotropica 41, 43–50.
Madder, D.J., Surgeoner, G.A., Helson, B.V., 1983. Induction of diapause in C. pipiens
and Culex restuans (Diptera, Culicidae) in southern Ontario. Can. Ent. 115 (8),
877–883.
Martens, W.J.M., Jetten, T.H., Rotmans, J., Niessen, L.W., 1995. Climate change and
vector-borne diseases: a global modelling perspective. Global. Environ. Chang.
5 (3), 195–209.
Merdić, 2002. Seadm godina monitoringa i istraživanja komaraca u Osijeku, Zbornik
radova 17. Savjetovanja u DDD i SUZPP. Zagreb.
Merdić, E., Sudarić-Bogojević, M., Boca, I., Turić, N., 2010. Determined and estimated
mosquito (Diptera, Culicidae) fauna in the city of Osijek, Croatia, using dry-ice
baited CDC traps. Period. Biol. 112 (2), 201–205.
Mitchell, C.J., 1983. Differentiation of host seeking behavior from blood-feeding
behavior in overwintering C. pipiens (Diptera: Culicidae) and observations on
gonotrophic dissociation. J. Med. Entomol. 20, 157–163.
Molaei, G., Andreadis, G., 2006. Identification of avian-and mammalian-derived
bloodmeals in A. vexans and culiseta melanura (Diptera: Culicidae) and its
implication for West Nile virus transmission in Connecticut. USA J. Med.
Entomol. 43 (5), 1088–1093.
Mori, A., Wada, Y., 1978. The seasonal abundance of Aedes albopictus in Nagasaki.
Trop. Med. 20, 29–37.
Ogden, N.H., St-Onge, L., Barker, I.K., Brazeau, S., Bigras-Poulin, M., Charron, D.F.,
Francis, C.M., Heagy, A., Lindsay, L.R., Maarouf, A., Michel, P., Milord, F.,
O’Callaghan, C.J., Trudel, L., Thompson, R.A., 2008. Risk maps for range expansion
of the Lyme disease vector, Ixodes scapularis, in Canada now and with climate
change. Int. J. Health. Geogr. 22 (7), 24.
Polley, L., 2005. Navigating parasite webs and parasite flow: emerging and reemerging parasitic zoonoses of wildlife origin. Int. J. Parasitol. 1279–1294.
Ratte, H.T., 1985. Temperature and insect development. In: Hoffmann, K.H.
(Ed.), Environmental Physiology and Biochemistry of Insects. Springer, Berlin,
pp. 33–66.
R Development Core Team, 2010. R: a language and environment for statistical
computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-161
900051-07-0. URL http://www.R-project.org.
Reisen, W.K., Cayan, D., Tyree, M., Barker, C.M., Eldridge, B., Dettinger, M., 2008.
Impact of climate variation on mosquito abundance in California. J. Vector Ecol.
33 (1), 89–98.
Reisen, W.K., Milby, M.M., Presser, S.B., Hardy, J.L., 1992. Ecology of mosquitos and
St. Louis encephalitis virus in the Los Angeles basin of California, 1987–1990.
J. Med. Entomol. 29, 582–598.
Ritz, C., Streibig, J.C., 2005. Bioassay analysis using R. J. Statist. Softw. 12 (5).
Rouband, E., 1944. Sur la fécondité du moustique commun (C. pipiens), L’evolution
humaine et les adaptations biologiques. Ann. Sc. iNat (Zool) 16, 5–168.
Rueda, L.M., Patel, K.J., Axtell, R.C., Stinner, R.E., 1990. Temperature-dependent
development and survival rates of Culex quinquefasciatus and Aedes aegypti
(Diptera: Culicidae). J. Med. Entomol. 27, 892–898.
Schaeffer, B., Mondet, B., Touzeau, S., 2008. Using a climate-dependent model to
predict mosquito abundance: application to Aedes (Stegomyia) africanus and
Aedes (Diceromyia) furcifer (Diptera: Culicidae). Inf. Genet. Evol. 8, 422–432.
Seber, G.A.F., Wild, C.J., 1989. Nonlinear Regression. Wiley and Sons, New York,
pp. 338–339.
Service, M.W., 1985, Population dynamics and mortalities of Mosquito preadults, in:
Lounibos, L.P., Rey, J.R., Frank, J.H., (Eds), Ecology of Mosquitoes: Proceedings of a
Work Shop, Vero Beach, FL, Florida Medical Entomology Laboratory, p. 185–201.
94
Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94
Shaman, J., Spiegelman, M., Cane, M., Stieglitz, M., 2006. A hydrologically driven
model of swamp water mosquito population dynamics. Ecol. Modell. 194,
395–404.
Shaman, J., Stieglitz, M., Stark, C., Le Blancq, S., Cane, M., 2002. Using a dynamic
hydrology model to predict mosquito abundances in flood and swamp water.
Emerg. Infect. Dis. 8, 6–13.
Shone, S.M., Curriero, F.C., Lesser, C.R., Glass, G.E., 2006. Characterizing population
dynamics of Aedes sollicitans (Diptera: Culicidae) using meteorological data.
J. Med. Entomol. 43, 393–402.
Sichinava, Sg.G., 1978. Feeding relationship between blood-sucking mosquitoes of
Georgia and homoiothermal and poikilothermal animals in nature and in the
experiment. Izvestiya AN GSSR (ser. biol.) 4 (1), 113–115 (in Russian, English
summary).
Spielman, A., 2001. Structure and seasonality of nearctic C. pipiens populations. Ann.
NY Acad. Sci. 951, 220–234.
Spielman, A., Wong, J., 1973. Environmental control of ovarian diapause in C. pipiens.
Ann. Ent. Soc. Am. 66 (4), 905–907.
Stott, I., Townley, S., Hodgson, D.J., 2011. A framework for studying transient
dynamics of population projection matrix models. Ecol. Lett. 14 (9), 959–970.
Taylor, R., 1979. On the applicability of current population models to the growth of
insect populations. Rocky Mount. J. Math. 9, 149–151.
Thomson, M.C., Doblas-Reyes, F.J., Mason, S.J., Hagedorn, R., Connor, S.J., Phindela,
T., Toma, T., Sakamoto, S., Miyagi, I., 1982. The seasonal abundance of Aedes
albopictus in Okinawajima, the Ryukyu archipelago. Japan. Mosq. News. 42,
179–183.
Thomson, M.C., Doblas-Reyes, F.J., Mason, S.J., Hagedorn, R., Connor, S.J., Phindela,
T., Morse, A.P., Palmer, T.N., 2006. Malaria early warnings based on seasonal
climate forecasts from multi-model ensembles. Nature 439 (7076), 576–579.
Townson, H., Nathan, M.B., 2008. Resurgence of chikungunya. Trans. R. Soc. Trop.
Med. Hyg. 102, 308–309.
Turell, M.J., Sardelis, M.R., O’Guinn, M.L., Dohm, D.J., 2002. Potential vectors of West.
Vinogradova, E.B., 2007. Diapause in aquatic insects, with an emphasis on
mosquitoes. In: Alekseev, V.R., et al. (Eds.), Diapause in Aquatic Invertebrates.
Springer, pp. 83–113.
Yang, G., Brook, B.W., Bradshaw, C.J.A., 2009. Predicting the timing and magnitude
of tropical mosquito population peaks for maximizing control efficiency. PLoS
Negl. Trop. Dis. 3, e385.
Yearsley, J.M., 2004. Transient population dynamics and short-term sensitivity
analysis of matrix population models. Ecol. Model. 177, 245–258.
Yildirim, A., Inci, A., Duzlua, O., Biskina, Z., Icaa, A., Sahin, I., 2011. A. vexans and
C. pipiens as the potential vectors of Dirofilaria immitis in Central Turkey. Vet.
Parasitol. 178 (2011), 143–147.