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Ecological Modelling 493 (2024) 110740
Available online 4 May 2024
0304-3800/© 2024 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Contents lists available at ScienceDirect
Ecological Modelling
journal homepage: www.elsevier.com/locate/ecolmodel
Dynamic Energy Budget approach for modeling growth and reproduction of
Neotropical stink bugs
Evridiki Klagkou a, Andre Gergs b, Christian U. Baden b, Konstadia Lika a,∗
aDepartment of Biology, University of Crete, Heraklion, 70013, Greece
bBayer AG, Crop Science Division, Alfred-Nobel Straße 50, 40789 Monheim, Germany
ARTICLE INFO
Keywords:
Dynamic Energy Budget theory
Insect energetics
Nymphal growth
Pentatomidae
Euschistus heros
Nezara viridula
ABSTRACT
Global warming and other climate change drivers have a significant impact on the life cycle of insects which
in turn affect their population dynamics and geographic distribution. Among these insects are stink bugs
(Hemiptera: Pentatomidae), which are hemimetabolous agricultural pests that cause major damage on crops in
many countries. Their life cycle consists of three stages: egg, nymph, and adult. The nymph grows and molts
through a small number of instars until reaching emergence, at which point growth ceases. The Dynamic
Energy Budget (DEB) theory provides the framework for modeling the life cycle of stink bugs. Nevertheless,
determining the most accurate model for nymphal growth and energy allocation during the adult stage
remains a challenge, as the best model is not always evident. For two species of the Pentatomidae family,
we parameterized and evaluated four DEB models that differ in nymphal growth and the energy allocation
rules in adults: isomorphic or V1-morphic nymphal growth combined with the energy allocation scheme in
adults that follows the 𝜅-rule or the scheme that the 𝜅-rule is not operational in adults. Overall, all models
fit the data, but those with isomorphic nymphal growth reproduce the growth trajectories and instar duration
better. However, with the available data we cannot conclude which rule is most likely the energy allocation
in adults. Using these models, we further studied the effects of temperature and food availability on individual
dynamics and its life cycle. Simulations show that temperature has a higher impact on the duration of life
stages and survival, while food availability affects egg production. When simulating a variable environment in
terms of temperature for both a summer and a fall generation, the results reveal that the summer generation
has higher feeding and egg production rates in comparison to the fall generation. DEB models are valuable
tools for simulating individual under various environmental conditions and when combined with appropriate
modeling approaches, they can predict population traits and even the potential distribution of a species.
1. Introduction
Insects belong to one of the most diverse class in the Animal
Kingdom. One million different species have been identified, while
80% remain to be discovered (Stork,2018). Despite this diversity,
they all have some common features, like a chitinous exoskeleton, a
three-part body (head, thorax and abdomen), three pairs of jointed
legs, compound eyes and one pair of antennae. Their postembryonic
development follows three primary modes: ametaboly, hemimetaboly
and holometaboly. Hemimetaboly evolved from ametaboly and may
have emerged as a consequence of wing emergence in Pterygota (Belles,
2019). The evolution of flight initiated a shift in development, lead-
ing to distinct juvenile and adult stages. Truman (2019) provides a
review of insect metamorphosis. In summary, evolution started with
a modification of growth after hatch, leading to distinct nymph–adult
∗Corresponding author.
E-mail address: lika@uoc.gr (K. Lika).
differences observed in hemimetabolous species. The next step in-
volved the transition to complete metamorphosis, holometaboly, result-
ing in larva, pupa and adult stages. The evolution of holometabolous
larva involved fundamental changes in embryogenesis. These changes
are linked with hormonal systems, the ecdysteroids and the juvenile
hormones, which regulate growth and development acting through
the stage-specification genes Kruppel homolog 1 (Kr-h1), broad and
Ecdysone-inducible protein 93F (E93) (Truman,2019). Broad expression
is a feature of nymphal and pupal stages and is maintained by the ju-
venile hormone acting though Kr-h1. E93 expression suppresses broad
and determines the adult stage. It is most likely that the pupa–adult
molt for holometabolous species corresponds to the nymph–adult molt
in hemimetabolous species.
Insects have evolved and adapted over millions of years to thrive
in different environments and ecological conditions. The development
https://doi.org/10.1016/j.ecolmodel.2024.110740
Received 2 February 2024; Received in revised form 24 April 2024; Accepted 26 April 2024
Ecological Modelling 493 (2024) 110740
2
E. Klagkou et al.
of wings (Jockusch and Nagy,1997) allowed many insect species to
exploit new habitats and food sources, the evolution of hardiness for
cold climates (Marshall et al.,2020), and the development of resistance
mechanisms against toxicants (Dawkar et al.,2013) are some examples
of insects’ ability to adapt to changing environments. This has led to an
increasing challenge over recent decades: the introduction of numerous
pest species into new regions. Temperature, precipitation, and food
availability can impact the distribution and abundance of insect species.
Many experiments have been conducted to better understand of how
insects respond to environmental changes.
Insect pests represent a group of insects that can play a significant
role in the transmission of diseases, affecting both plants and humans,
and are one of the main factors to crop destruction, in addition to
the influence of weather conditions, such as heat waves, droughts and
heavy storms (Bartholomew et al.,2003). One very important category
of pests are the stink bugs, members of the Pentatomidae family. Stink
bugs feed by inserting their stylets into the plant tissue to extract
nutrients, causing damage in the process. This feeding process may
also transmit plant pathogens, which will further increase the damage.
Some species of stink bugs feed on many different species of plants,
often with the potential of becoming major pests (Panizzi,2000;Pal
et al.,2023). Temperature increase can have a direct impact on pest’s
reproduction, survival, distribution, and population dynamics as well
as on the relationships between pests, their environment, and their
natural enemies (Skendžić et al.,2021). Increased temperature has been
shown to increase consumption, accelerate development and enhance
movement (Skendžić et al.,2021), which can escalate the damage
potential of a pest. Climate change, however, not only results to higher
temperature, it also leads to increased CO2and altered precipitation
patterns. Increased CO2levels changes the carbon-to-nitrogen ratio in
the plants, which makes them less nutritious, while fluctuations in
precipitation can either decrease or increase the survival of several
species (Skendžić et al.,2021). Understanding the impact of environ-
mental factors on the developmental stages, growth, and reproduction
of insect pests is of great importance, since these traits ultimately shape
the dynamics of their populations and their interactions with crops.
Mathematical and computational models have long been recognized
as a powerful research tool to understand physiological and ecological
processes or to predict the tendencies in the dynamics of populations.
These models are increasingly used to model population dynamics of
pests (Gilioli et al.,2016;Rossini et al.,2020), interactions between
insect and crop (Misra and Yadav,2024), or biological control of a
species (Campos et al.,2016). Nijhout and Wheeler (1996) used the
Gompertz growth model and a competition model, that treats the
growth of imaginal structures explicitly as occurring in a closed system
with a limited amount of nutrient, to study the complex allometric
growth in holometabolous insects and Liu and Meng (2007) the day–
degree model, the logistic equation, and the Wang model to describe
the relationships between temperature and development rate at con-
stant conditions. Several studies model effects of temperature on vari-
ous traits such as growth rate (Golizadeh and Zalucki,2012;Rebaudo
and Rabhi,2018), development, survival and reproduction (Régnière
et al.,2012;Azrag et al.,2017) or potential distribution (Fand et al.,
2014;Otieno et al.,2019). Damos and Savopoulou-Soultani (2012)
provide a review of representative temperature-driven models for in-
sect development. Temperature-dependent models based on life table
parameters have been developed for various Pentatomidae species with
the aim to study development, survival or reproduction. Examples in-
clude the coffee antestia bug, Antestiopsis thunbergii (Azrag et al.,2017),
the brown marmorated stink bug, Halyomorpha halys (Baek et al.,
2017;Mermer et al.,2023) and the two-spotted stink bug, Bathycoelia
distincta (Muluvhahothe et al.,2024). Most of these models are based
on empirical relationships between the response variable and temper-
ature. Empirical models are limited because they lack a conceptual
basis, which hinders their extrapolation to other different conditions.
Moreover temperature-dependent models face additional limitations to
predict scenarios where the food availability varies. Nisbet and Gurney
(1983) introduced a mechanism that dynamically varies instar duration
to regulate insect population dynamics where the transition from one
instar to the next is triggered by weight rather than age. Aguirre-
Zapata et al. (2022) proposed a semi-physical growth model based on
first principles to describe quantitatively the different stages of Lobesia
botran. These models, however, consider the macroscopic view of each
larva stage as part of a whole, rather than focusing on the specific
physiological processes occurring within a single individual.
Given the inherent differences between species, the most construc-
tive way to model them is to use a framework that represents the
complete life cycle of an individual organism as a function of tem-
perature and food availability. Dynamic Energy Budget (DEB) theory
provides the conceptual and quantitative framework to study the entire
life cycle of an individual organism in a dynamic environment, while
making explicit use of energy and mass balances (Kooijman,2010).
Models based on DEB theory have been developed for a wide range
of organisms, including fish, mammals, reptiles, echinoderms, molluscs
and others (AmP,2023). The simplest model derived from DEB theory,
named the standard DEB model, applicable to various organisms, such
as birds, turtles and lizards, assumes isomorphy throughout the entire
life of an individual, while the organism gives a fixed fraction of
mobilized energy to somatic maintenance plus growth, and the rest to
maturity maintenance and maturation or reproduce (for adults). The
standard DEB model assumes that growth continues after reproduction
has started. However, insects do not grow in the adult stage, and
isomorphy is not typical during the nymphal stage. DEB theory has
also been used to model insect species, both hemimetabolous and
holometabolous, (Llandres et al.,2015;Maino,2015;Maino and Kear-
ney,2015a,b;Maino et al.,2016a,b;Matyja et al.,2020;Gergs and
Baden,2021), but its application is limited in the AmP collection (AmP,
2023) to 29 species from the orders of Thysanura, Ephemeroptera, Or-
thoptera, Phasmatodea, Hemiptera, Diptera, Trichoptera, Lepidoptera,
Hymenoptera and Coleoptera. To the best of our knowledge there is no
DEB model developed for any species in the Pentatomidae family.
The aim of this study is to develop a DEB-based modeling frame-
work for studying the life cycle of Neotropical stink bugs (Hemiptera:
Pentatomidae), using Euschistus heros and Nezara viridula for
parametrization and validation. E. heros and N. viridula are among
the most damaging pests of soybean globally, resulting in substantial
economic losses (Musser et al.,2011). E. heros feeds mainly on soybean
and is mostly spread in Brazil (Panizzi,2008). It is currently the
most important stink bug species in the Neotropical region, due to
its increasing abundance over the decades (Panizzi et al.,2022). N.
viridula, on the other hand, feeds on various plants, with preference for
legumes, and it has been distributed around the world (Pavlovčič et al.,
2008;Squitier,2020). Despite its polyphagy and global distribution,
populations of this species have declined over the decades, making
it more sensitive to environmental changes (Panizzi et al.,2022).
Therefore, both species are important pest species and it is worth
studying them further. Sting bugs have a nymphal stage, during which
they grow and molt through a small number of instars until reaching
emergence, after which growth ceases. Identifying the most accurate
model for nymphal growth and energy allocation scheme in the adult
stage remains challenging.
We have started by developing DEB models, each based on two
distinct assumptions concerning energy allocation in the adult stage
and two mechanisms governing nymphal growth. The four resulting
models were parameterized separately for E. heros and N. viridula,
using data derived from the literature and our own research. We then
compared these models based on the results for each species and
discuss their plausibility, aiming to identify the best model to describe
nymphal growth and energy allocation in the adult stage. Because
data availability was different between the two species, using the two
species allowed us to mitigate the risk of the type of data influencing
model choices. Subsequently, the model which performed better was
Ecological Modelling 493 (2024) 110740
3
E. Klagkou et al.
Fig. 1. The typical life cycle of a stink bug. The outer circles illustrate the morphological life stages: egg, nymph (comprising five instars), and adult. The inner circle shows the
functional stages as modeled in DEB models: embryo (which does not feed nor reproduce, only matures), juvenile (which feeds, with the exception of the 1st instar, matures and
does not reproduce) and adult (which feeds, does not mature further, and reproduces). Black arrows indicate life stage events.
used to explore the potential impact of various environmental scenarios
on life history traits of the species. Temperature and food availability
were chosen as the two most important factors that impact the life
cycle of the species. Understanding the effects of these factors may
facilitate the identification of differences between the species in a
stressful environment, which could potentially explain their distinct
distribution patterns.
2. Materials and methods
2.1. The life cycle of hemimetabolous species
In the present study, we focus on Neotropical stink bugs and use
Euschistus heros and Nezara viridula as study species. These species are
hemimetabolous insects, meaning they undergo incomplete metamor-
phosis. Hemimetabolous species hatch as nymphs, which are morpho-
logical similar to the adults but lack wings (Belles,2019;Truman,
2019). Nymphs grow progressively though a series of instars, until
the last nymphal instar, which molts into the adult stage and stops
growth. This pattern is indicated by the juvenile hormone, which is
high though the penultimate nymphal stage, drops during the last
nymphal stage and is absent through the molt to the adult (Truman,
2019). Adults differ from nymphs by having wings and no longer molt
but also because they possess functional reproductive organs (Belles,
2019). We acknowledge the variation in terminology for the immature
stage of these insects, but to avoid confusion, we follow the terminology
in Truman (2019) and use the term ‘‘nymph’’ to refer to all instars.
Fig. 1 illustrates a typical life cycle of a hemimetabolous insect
adapted to the life cycle of a stink bug which is characterized by five
instars and a non-feeding first instar (Panizzi and Meneguim,1989;
Panizzi and Oliveira,1998). Additionally, the diagram shows the link
between insect morphological and DEB functional stages discussed in
the next section.
2.2. DEB models
DEB theory provides a modeling framework to explore the interac-
tions between an organisms’s metabolic processes and its environment
across its entire life cycle. The standard DEB model, the simplest model
derived from DEB theory, assumes three life stages: embryo, juvenile,
and adult. Each of these stages captures the properties of corresponding
morphological stages, which may exhibit as eggs, larvae (in the case of
insects), chicks (for birds), among others. Consequently, the processes
within each life stage are different. Key processes modeled by DEB
theory include assimilation, growth, maturation, maintenance, and
reproduction. The embryo does not feed but uses stored energy received
from the mother for growth, development (termed as maturation in
DEB context) and maintenance. As a juvenile, the organism starts
feeding and uses the energy derived from food for growth, maturation
and maintenance. As an adult, the organism no longer allocates energy
for further maturation, but rather gives this energy for reproductive
processes, such as formation of gametes and egg laying.
Each individual organism can be described using four state vari-
ables: reserve (𝐸), structure (𝑉), energy invested into reproduction
(𝐸𝑅), and energy invested into maturation (𝐸𝐻). The first three vari-
ables contribute to the organism’s biomass, while the fourth tracks its
developmental stage (Kooijman,2010). Reserve is a buffer, supplying
the energy needed for all physiological processes. Energy in reserve can
be acquired through feeding or from the mother (during the embryo
stage). Maturity can be described as the energy utilized to increase
the complexity of the organism. A higher value of maturity describes
an individual capable of performing various functions such as feeding
and reproduction. The energy invested into maturation is not stored
as biomass; instead, it dissipates as heat. Maturity level is the trigger
that links life stages: the maturity level is zero at fertilization, hits
a threshold (𝐸𝑏
𝐻) at birth (which marks initiation of feeding) and
increases until puberty (𝐸𝑝
𝐻) (which marks start of reproduction).
Ecological Modelling 493 (2024) 110740
4
E. Klagkou et al.
Fig. 2. Conceptual representation of the standard DEB model. The boxes represent
state variables and the arrows energy fluxes: assimilation, 𝑝𝐴; mobilization, 𝑝𝐶; somatic
maintenance, 𝑝𝑆; maturity maintenance, 𝑝𝐽; growth, 𝑝𝐺; reproduction, 𝑝𝑅. After food
uptake, energy is assimilated, added into reserve and then mobilized before use for
metabolic processes. A fixed fraction 𝜅of the mobilized flux is allocated to somatic
maintenance and growth, and the remaining 1 − 𝜅to increase and maintain maturity
or to reproduction.
The state variables, the main metabolic processes and the dynamics
of the standard DEB model are summarized in Table 1. The main DEB
parameters are presented in Table 2; each parameter is linked to a
metabolic process. The flow of energy within an organism, as defined
by DEB theory for the standard DEB model, is presented in Fig. 2. Food
uptake is assumed to follow a functional response relationship with
food density and to be proportional to organism’s surface area. The
scaled functional response, denoted as 𝑓, represents the feeding rate
as a fraction of the maximum feeding rate achievable by an individual
of a particular size eating the same food. It ranges between 0 and
1, with 0 indicating a state of starvation and 1 unrestricted feeding,
capturing the concept of food availability. Subsequently, food uptake
is converted into reserves with a constant efficiency, which depends
on the type of food being consumed. Assimilation, 𝑝𝐴, is the process
that transforms food into reserves; an embryo uses reserves stored in
the egg to grow and develop ( 𝑝𝐴= 0). Assimilated energy is mobilized
from reserves to provide the energy for all other metabolic processes.
Specifically, a fixed fraction 𝜅of the mobilization flux, 𝑝𝐶, is allocated
to the somatic branch of metabolic processes and the remaining 1 −
𝜅is directed towards the reproduction branch. The somatic branch
fuels growth, 𝑝𝐺, which accounts for the increase in an organism’s
structure, after subtracting the energy needed for somatic mainte-
nance, 𝑝𝑆. Somatic maintenance processes include the maintenance of
concentration gradients across membranes, the turnover of structural
body proteins, a certain level of muscle tension and movement, and
the production of hairs, feathers, scales, leaves (of trees) (Kooijman,
2010). The 1−𝜅branch fuels maturity maintenance and either increase
maturity (prior to puberty) or reproduction (post puberty). As with
somatic maintenance, the costs associated with maturity maintenance,
which relate to maintain current state of maturity (e.g. maintaining
defense systems) (Kooijman,2010), are paid first and only the surplus
is invested in maturation or reproduction.
The abstract state variables of reserves and structure can be linked
to commonly measured quantities, such as length and mass. An assump-
tion of the auxiliary DEB theory is that the ratio of the volumetric
structural length 𝐿(defined as 𝐿=𝑉1∕3) to the physical observed
length 𝐿𝑤is constant, i.e. 𝛿𝑀=𝐿
𝐿𝑤
. The shape coefficient 𝛿𝑀depends
on the specific length measure used (e.g. total length or head length)
and, in principle, this value could differ between the life stages of the
organism. Body mass of an individual has contributions from structure
(𝑉), reserve (𝐸), and (for reproducing adults) energy reserve in repro-
duction buffer (𝐸𝑅). Mass quantified as wet weight (𝑊𝑤) is given by
𝑊𝑤=𝑑𝑉 𝑤𝑉+ (𝐸+𝐸𝑅)𝑤𝐸 𝑑 𝑑𝑉 𝑤
𝜇𝐸𝑑𝑉 𝑑
, where 𝑑𝑉 𝑑 and 𝑑𝑉 𝑤 are the specific
densities of dry and wet structure (g/cm3), 𝑤𝐸𝑑 the molecular weight
of reserve (g/mol) and 𝜇𝐸the chemical potential of reserve (J/mol)
(for details see Section 3.2.1 in Kooijman (2010)).
Mass fluxes of organic (food, faeces, reserves and structure) and
mineral (e.g., CO2, nitrogenous waste) compounds can be written as
weighted sum of three basic fluxes: assimilation ( 𝑝𝐴), dissipation ( 𝑝𝐷)
and growth ( 𝑝𝐺) (Kooijman,2010, Chapter 4). Dissipation accounts
for processes that exclude assimilation and somatic growth overheads.
For adults, it is given as 𝑝𝐷=𝑝𝑆+𝑝𝐽+ (1 − 𝜅𝑅)𝑝𝑅and for the non-
reproductive stages as 𝑝𝐷=𝑝𝑆+𝑝𝐽+𝑝𝑅. For the non-reproductive
stages, the energy invested to maturation is excreted into the environ-
ment in the form of heat and metabolites and does not contribute to
the total weight.
The standard DEB model assumes isomorphy, which implies that
surface area is proportional to structural volume to the power 2/3. If
food and temperature are constant, the standard DEB model reduces
to von Bertalanffy growth in volumetric length 𝐿=𝑉1∕3. Many
animal species grow isomorphically with simple life stages and the
standard DEB model could be applied (AmP,2023). To accommodate
complex life stages and various forms of metabolic acceleration, the
standard DEB model was extended and a set of typified models were
derived (Marques et al.,2018). A typical deviation from the DEB model
is metabolic acceleration, often indicated by an upward in length-at-
age curve, a short period after birth. Metabolic acceleration means
that the isomorphic individual temporarily switches to the V1-morphic
growth mode, meaning that the surface area is proportional to (struc-
tural) volume (Kooijman,2014). This has as effect that, during the
acceleration period, both specific assimilation rate {𝑝𝐴𝑚}and energy
conductance 𝑣 increase with a (structural) length measure. At the end
of this period, which frequently coincides with metamorphosis, i.e. a
sudden change in morphology, both parameters remain constant again
(apart from effects of temperature), but differ from the original value by
the acceleration factor 𝑠𝑀, which depends on food availability during
the acceleration period. The acceleration factor equals one before the
acceleration period, 𝑠𝑀= (𝐿∕𝐿1)1∕3 during the acceleration period
and 𝑠𝑀= (𝐿2∕𝐿1)1∕3 after the acceleration period, with 𝐿1being the
structural length at the start of acceleration (usually coincides with
birth, 𝐿1=𝐿𝑏) and 𝐿2at the end of acceleration, which may occur
at puberty (𝐿2=𝐿𝑝) or before puberty (𝐿2=𝐿𝑗).
Growth in the nymphal stage
The standard DEB model is categorized within the class of typified
models called s-models and those that exhibit metabolic acceleration
during part of their life cycle, following the rules for V1-morphy,
are classified under the a-models. Within the s- and a-models, there
exists a sub-type of models where growth ceases at puberty: the ‘‘sbp’’
model and the ‘‘abp’’ model (Marques et al.,2018). These models
have been applied to harpacticoid copepod Nitocra spinipes (Koch and
De Schamphelaere,2019) and in this paper we will investigate their
applicability to stink bugs.
In most insects, growth during the nymphal stage is characterized
by a rapid and typically exponential development. In DEB theory,
this type of growth is referred to as a V1-morph. To model our test
species, we initially used the abp model, where metabolic acceleration
occurs between birth and puberty and that after acceleration there
is no growth. For these insects birth coincides with hatching and
puberty with emergence. However, the available data for both species
showed growth patterns closer to the von Bertalanffy growth curve.
This observation prompted us to explore also the sbp model.
Ecological Modelling 493 (2024) 110740
5
E. Klagkou et al.
Table 1
Energy fluxes linked to metabolic processes, state variables, and dynamics of the
standard DEB model.
State variables Dynamics
Energy in reserve, 𝐸𝑑𝐸
𝑑𝑡 =𝑝𝐴−𝑝𝐶
Structural body volume, 𝑉𝑑𝑉
𝑑𝑡 =𝑝𝐺
[𝐸𝐺]
Energy invested into maturation, 𝐸𝐻
𝑑𝐸𝐻
𝑑𝑡 =𝑝𝑅(𝐸𝐻< 𝐸𝑝
𝐻)
Energy invested into reproduction, 𝐸𝑅
𝑑𝐸𝑅
𝑑𝑡 =𝜅𝑅𝑝𝑅(𝐸𝐻≥𝐸𝑝
𝐻)
Metabolic process Energy flux
Assimilation 𝑝𝐴= { 𝑝𝐴𝑚}𝑓 𝑉 2∕3
Mobilization 𝑝𝐶=𝐸[𝐸𝐺]𝑣𝑉 −1∕3+[ 𝑝𝑆]
𝜅[𝐸]+[𝐸𝐺]
Somatic maintenance 𝑝𝑆=𝑝𝑀= [ 𝑝𝑀]𝑉+ { 𝑝𝑇}𝑉2∕3
Growth 𝑝𝐺=𝜅 𝑝𝐶−𝑝𝑆a
Maturity Maintenance 𝑝𝐽=
𝑘𝐽𝐸𝐻
Maturation/Reproduction 𝑝𝑅= (1 − 𝜅)𝑝𝐶−𝑝𝐽b
Dissipation 𝑝𝐷=𝑝𝑆+𝑝𝐽+𝑝𝑅
𝑝𝐷=𝑝𝑆+𝑝𝐽+ (1 − 𝜅𝑅)𝑝𝑅c
Modifications to the standard DEB models for the proposed stink bugs models.
a𝑝𝐺= 0 during the adult stage.
bFor the no 𝜅-rule energy allocation scheme during the adults stage: 𝑝𝑅=𝑝𝐶−𝑝𝑆−𝑝𝐽.
cFor the 𝜅-rule energy allocation scheme during the adults stage: 𝑝𝐷=𝜅 𝑝𝐶+𝑝𝐽+ (1 −
𝜅𝑅)𝑝𝑅, the energy intended for growth dissipates.
Table 2
Main parameters of the standard DEB model.
Symbol Units Description
{𝑝𝐴𝑚}aJ/d cm2Max spec assimilation rate
{
𝐹𝑚}L/d cm2Max spec searching rate
𝜅𝑋– Digestion efficiency of food to reserve
𝜅𝑃– Faecation efficiency of food to faeces
𝑣acm/d Energy conductance
𝜅– Allocation fraction to soma
𝜅𝑅– Reproduction efficiency
[𝑝𝑀]J/d cm3Vol-spec somatic maint
{𝑝𝑇}J/d cm2Surf-spec somatic maint
𝑘𝐽1/d Maturity maint rate coefficient
[𝐸𝐺]J/cm3Spec cost for structure
𝐸𝑏
𝐻J Maturity at birth
𝐸𝑝
𝐻J Maturity at puberty
ℎ𝑎1/d2Weibull aging acceleration
𝑠𝐺– Gompertz stress coefficient
aIn the abp models, these parameters are multiplied with the acceleration factor
𝑠𝑀= max(1,min(𝐿, 𝐿𝑝)∕𝐿𝑏), where 𝐿is the structural length, 𝐿𝑏and 𝐿𝑝are the
structural lengths at birth and puberty, respectively.
In both models, the adult stage is characterized by no growth,
i.e. 𝑑𝑉
𝑑𝑡 = 0. Both models have the same number of parameters as the
standard model (Table 2). However, in the abp model, since V1-morphy
only concerns the relationship between surface area and structural
volume, changes in shape affect only the specific maximum assimilation
rate, {𝑝𝐴𝑚}, and the energy conductance, 𝑣. Specifically, these param-
eters are multiplied by the acceleration factor which for this model is
𝑠𝑀= max(1,min(𝐿, 𝐿𝑝)∕𝐿𝑏), where 𝐿=𝑉1∕3 is the structural length, 𝐿𝑏
and 𝐿𝑝are the structural length at birth and puberty, respectively.
The growth of the immature insects is characterized by a series of
distinct developmental stages, known as instars. Dyar’s Law postulates
that the ratio of the lengths of the head capsule at two consecutive
molts, i.e. 𝐿𝑖+1
𝐿𝑖
(or any power of it), is constant. In insects, the surface
area of the head, which controls food acquisition, and that of the gut,
which controls assimilation, may not grow synchronously. Empirical
observations show that stink bugs are not constantly eating (Panizzi
et al.,2021, Chapter 4), which will be the case if they are limited by
their morphology. However, due to lack of detailed information and
data, in this study, we make the approximation that both surface area
of the head and that of the gut are propositional to the structural length
squared. Thus, the molt occurs when 𝐿2(𝑡) = 𝑠𝑖𝐿2
𝑖where 𝐿𝑖is the
structural length at the start of instar 𝑖and surface area is proportional
to 𝐿2
𝑖. So, for each instar, a constant 𝑠𝑖is introduced, with 𝑠𝑖=𝐿2
𝑖+1
𝐿2
𝑖
.
Based on the biology of the species modeled in this study, we as-
sume that during the first instar, the nymph does not feed (Panizzi and
Meneguim,1989;Panizzi and Oliveira,1998). We choose to classify
the first instar as a (non-feeding) juvenile rather than an embryo,
as typically would have been in the DEB context (Fig. 1). This deci-
sion was motivated by our aim to develop a general framework for
hemimetabolous insects with feeding adult stage, which not all have a
non-feeding first instar (for example from the orders Orthoptera (Delvi
and Pandian,1972;Kaufmann,1972) or Phasmatodea (Berger,2004),
which are also modeled as abp (AmP,2023)). Moreover, within the
DEB context embryos grow isomorphically, while we also explored V1-
morphic larvae growth. In terms of energy utilization, the first instar
still uses the egg reserves to fuel metabolism, and the end of first
instar is controlled by the molting rules instead of the investment to
maturation.
Energy allocation in adult stage
One key assumption in DEB theory is the 𝜅-rule, as shown in Fig. 2.
The 𝜅-rule implies that a fraction 𝜅of mobilized energy is allocated for
somatic maintenance and growth, and the remaining fraction (1 − 𝜅)
is allocated for maturity maintenance and reproduction (post puberty).
However, in the case of sbp and abp models, where there is no fur-
ther growth after puberty, there arises a question about the energy
allocation during the adult stage. Up to now the two models have
been applied under the assumption that the 𝜅-rule is not operational
in adults.
In this paper, we investigate two allocation schemes during the
adult stage: (1) The energy allocation scheme illustrated in Fig. 3 (left)
(referred to as 𝜅-rule) follows the 𝜅-rule, and the energy that would
have been allocated for further growth is dissipated. (2) The energy
allocation scheme illustrated in Fig. 3 (right) (referred to as no 𝜅-
rule) assumes that the 𝜅-rule is not operational in adults; the mobilized
reserves first are used to cover (somatic and maturity) maintenance
costs and the remaining energy goes into reproduction.
Combining the two assumed nymphal growth patterns with the two
energy allocation schemes in adult stage, four DEB models are resulted:
isomorphic nymphal growth with the no 𝜅-rule (sbp model), isomorphic
nymphal growth with the 𝜅-rule (sbp-𝜅model), V1-morphic nymphal
growth with the no 𝜅-rule (abp model), and V1-morphic nymphal
growth with the 𝜅-rule (abp-𝜅model).
Survival
The aging module of DEB theory introduces two extra parameters,
the Weibull aging acceleration
ℎ𝑎and the Gompertz stress coefficient
𝑠𝐺(Kooijman,2010, Section 6.1). In addition to hazard
ℎ(𝑡)due to
aging, to model the nymphal and adult survival, two stage-related
hazard rates were introduced:
ℎ𝑗(juvenile) and
ℎ𝑎(adult). The hazard
rates relate to the survival probability according to the differential
equation 𝑑𝑆
𝑑𝑡 = −(
ℎ+
ℎ𝑠𝑡𝑎𝑔𝑒)𝑆, with 𝑆(0) = 1 and
ℎ𝑠𝑡𝑎𝑔𝑒 the hazard rate
at the given stage.
Temperature effects
In DEB theory, to account for the effects of temperature on
metabolic processes, the Arrhenius function is used. The simplest form
of this function uses a single parameter, the Arrhenius temperature
𝑇𝐴, which is estimated for each species. This function introduces a
temperature correction factor that quantifies how a biological rate at
a given temperature 𝑇compares to that at a reference temperature
𝑇𝑟𝑒𝑓 (Kooijman,2010;Stavrakidis-Zachou et al.,2023)
For the species-specific Arrhenius temperature 𝑇𝐴, the rate of a
physiological process
𝑘(𝑇)at temperature 𝑇is given by
𝑘(𝑇) =
𝑘(𝑇𝑟𝑒𝑓 ) exp( 𝑇𝐴
𝑇𝑟𝑒𝑓
−𝑇𝐴
𝑇)
Ecological Modelling 493 (2024) 110740
6
E. Klagkou et al.
Fig. 3. Alternative energy allocation schemes for adults. Left: energy allocation follows the 𝜅-rule, while the energy that would have been allocated for further growth dissipates.
Right: the 𝜅-rule is not operational in adults; the mobilized reserves first are used to cover (somatic and maturity) maintenance costs and the remaining energy goes into
reproduction. Solid arrows represent energy fluxes: assimilation, 𝑝𝐴; mobilization, 𝑝𝐶; somatic maintenance, 𝑝𝑆; maturity maintenance, 𝑝𝐽; reproduction, 𝑝𝑅.
However, for temperatures near the thermal limits of a species, the
Arrhenius function needs to be extended to consider the reduction
of rates at low and high temperatures, 𝑇𝐿/𝑇𝐻, of the thermal toler-
ance range and incorporates species-specific Arrhenius temperatures,
𝑇𝐴𝐿/𝑇𝐴𝐻 , which control the rate of decrease at the boundaries. In our
case, due to lack of data at extreme temperatures, we used the simple
version of the Arrhenius function.
2.3. Study species and data
In the present study, we focused on Neotropical stink bugs and used
Euschistus heros and Nezara viridula as study species. These species feed
mainly on soybean (significant part of their diet during their feeding
stages), and cause major economic damage. Similar to most insects, it
is observed that these stink bugs do not grow during their flying stage
(imago stage). In our modeling, we will use the DEB functional life
stages, referring to the egg stage as the embryo, the nymphal stage as
the juvenile, and the imago stage as the adult (Fig. 1).
For both species, two types of experiments were conducted in order
to determine the effects of different feeding levels and temperature
on various endpoints over the life cycle. In the first experiment, the
influence of food supply on the development and egg production was
studied. Newly hatched (<24 h) larvae were placed in separate petri
dishes for each food supply at 24.5 ◦C and 55% humidity. For E. heros,
different concentration (100%, 80%, 50% and 20%) of food sachets
with nutrient solution were available. For N. viridula different schedules
(ad libitum: 100% (i.e. 24 h/7 days), diet 1: 63%, diet 2: 49%, diet 3:
35% of the time) were the food (carrots, cabbage and sunflower seeds)
was available were studied. Measurements of head length and weight
were made every 2–5 days from 20 randomly chosen animals. For N.
viridula after the animals reached the adult stage, they were transferred
(each together per diet) into a new dish with filter paper in it, to collect
the eggs. For E. heros, the study was terminated at 37 days due to high
mortality and no eggs could be collected.
In the second experiment, the influence of temperature on age at
hatch was studied. The experiment set up involved, for E. heros, nine
trials at each temperature within the range 18–34 ◦C with increments
of 2 ◦C, and for N. viridula, two trials at each temperature within the
range 18–28 ◦C with increments of 2 ◦C. For each temperature and
trial, 50 eggs were placed in climate cabinets without light according
to the given temperature. Every day, the number of hatched individuals
was recorded. After all the eggs hatched, the average age at hatch was
computed. Detailed description of the experiments and the results are
available in the Supplementary Material.
To parameterize the models, we combined the data from this study
with data available in the scientific literature. For E. heros, data on
the duration of instars were digitized from Malaguido and Panizzi
(1999), Oliveira et al. (2015) and Costa et al. (1998) and Frugeri et al.
(2023), cumulative number of eggs and respiration rate from Haddi
et al. (2016), and data on adult survival from Malaguido and Panizzi
(1999). For N. viridula, data for duration of instars and embryo growth
rate were digitized from Hiromitu (1960) and data on nymphal survival
from Medina et al. (2022).
To validate the model predictions, additional data were obtained
from various sources: age at hatch (Daane et al.,2022) and age at emer-
gence (Alie and Ewiess,1977;Musolin and Numata,2003;Chanthy
et al.,2015;Daane et al.,2022) for N. viridula and age at emer-
gence (Hayashida et al.,2018) and egg production rate (Silva et al.,
2008;Hayashida et al.,2018) for E. heros.
2.4. Parameter estimation
The procedure for the parameter estimation is described in detail
in Marques et al. (2019). Parameters are estimated simultaneously
based on zero- and uni-variate data sets to minimize the symmetric
bounded loss function, which is a function of data, predictions, and
weight coefficients: 𝐹=∑𝑛
𝑖=1 ∑𝑛𝑖
𝑗=1 𝑤𝑖𝑗
(𝑑𝑖𝑗 −𝑝𝑖𝑗 )2
𝑑2
𝑖+𝑝2
𝑖
. In this function, 𝑛is the
number of different data sets, 𝑛𝑖is the number of data points within
data set 𝑖,𝑤𝑖𝑗 ’s are weight coefficients, 𝑑𝑖𝑗 ’s are data-points, 𝑝𝑖𝑗 ’s are
predicted values. The mean value and mean predicted value for data
set 𝑖are calculated as: 𝑑𝑖=𝑛−1
𝑖∑𝑛𝑖
𝑗=1 𝑑𝑖𝑗 ,𝑝𝑖=𝑛−1
𝑖∑𝑛𝑖
𝑗=1 𝑝𝑖𝑗 . In addition
to real data, the estimation procedure incorporates pseudo-data to
prevent ill-posed situations, due to lack of information in the data, and
minimize the risk of arriving at biologically unreasonable parameter
values. Pseudo-data consist of a predetermined set of parameter values
for a generalized animal species that do not depend on maximum body
size (Lika et al.,2011;Marques et al.,2019). They are treated as data
with small weight coefficients to make sure that if data determine
parameters well, pseudo-data hardly contribute.
The following parameters were used in our estimation procedure
as pseudo-data: the energy conductance 𝑣 =0.02 cm/d, the allocation
fraction to soma 𝜅=0.8, the volume-specific somatic maintenance
[𝑝𝑀] = 18 J/d cm3, growth efficiency 𝜅𝐺= 0.8and the maintenance
ratio 𝑘=
𝑘𝑗
𝑘𝑀
= 1, where
𝑘𝑀=[𝑝𝑀]
[𝐸𝐺]the somatic maintenance rate
coefficient. The selection of the last parameter as pseudo-data, is based
on the implied property of the model that when 𝑘= 1, stage transitions
occur at fixed structural length. This property aligns with biological
observations that show that length at which stage transitions occur does
Ecological Modelling 493 (2024) 110740
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E. Klagkou et al.
Table 3
DEB parameters of the typified models abp and sbp for Nezara viridula and Euschistus heros at 𝑇𝑟𝑒𝑓 = 20 ◦C.
Symbol N.viridula E. heros Units Description
Value (abp/sbp) Value (abp/sbp)
{𝑝𝐴𝑚}a93.05/377.29 55.39/336.48 J/d cm2Spec max assimilation rate
𝑣a0.030/0.015 0.032/0.015 cm/d Energy conductance
𝜅0.99/0.94 0.87/0.84 – Allocation fraction to soma
[𝑝𝑀]86.62/463.90 53.70/530.60 J/d cm3Vol-spec somatic maint
𝑘𝐽0.020/0.108 0.012/0.116 1/d Maturity maint rate coefficient
[𝐸𝐺]4432/4225 4323/4455 J/cm3Spec cost for structure
𝐸𝑏
𝐻0.03/0.03 0.60/0.13 J Maturity at birth
𝐸𝑝
𝐻1.93/6.00 14.05/6.09 J Maturity at puberty
ℎ𝑎8.4 ⋅10−6/ 1.5 ⋅10−5 1.1 ⋅10−6 /4.7 ⋅10−6 1/d2Weibull aging acceleration
𝑇𝐴6508/12 690 9986/9518 K Arrhenius temperature
𝛿𝑀1.4/0.9 1.6/1.0 – Head shape coefficient
ℎ𝑗0.011/0.009 0.010/0.010 1/d Juvenile hazard rate
ℎ𝑎– 0.005/0.005 1/d Adult hazard rate
𝑠11.12/1.43 1.06/1.27 – Constant for first molt
𝑠23.30/4.07 1.66/2.84 – Constant for second molt
𝑠30.51/2.28 1.40/1.47 – Constant for third molt
𝑠43.94/1.76 1.57/1.49 – Constant for fourth molt
aIn abp model, these parameters are multiplied with the acceleration factor 𝑠𝑀which is 1 (during embryo stage) increases linear with structural
length (until puberty) to 4.04 for N. viridula and to 2.87 for E. heros at abundant food, after which it remains constant.
not change much across different levels of food availability for many
insect species (Jucker et al.,2017;Bawa et al.,2020).
Due to lack of data, some parameters could not be estimated and we
used the generalized animal (Kooijman,2010;Lika et al.,2011). Those
parameters that are involved in the estimation include the reproduction
efficiency (𝜅𝑅= 0.95), the Gompertz stress coefficient (𝑠𝐺= 0.0001), the
chemical potential of reserve (𝜇𝐸= 550 kJ/mol) the molecular weight
of reserve (𝑤𝐸𝑑 = 23.9g/mol), and the specific densities (𝑑𝑉 𝑤 = 1
g/cm3and 𝑑𝑉 𝑤 = 0.17 g/cm3).
Two goodness-of-fit measures were used to evaluate the overall
model performance: the mean relative error (MRE) and the symmetric
mean squared error (SMSE) (Marques et al.,2019). MRE can take values
in the interval [0,∞), while SMSE has values in the interval [0, 1].
Values of MRE and SMSE close to 0 mean that the model predictions
are close to the data.
3. Results and discussion
3.1. Parameter estimation & model predictions
The results of the parameter estimation for the abp and sbp models,
assuming the no 𝜅-rule for the allocation of mobilized energy, applied
to both species, N. viridula and E. heros, are presented first. Table 3
gives the values of the parameter estimates of both models for the two
stink bug species. While many parameters have similar values between
the two models, there are some notable differences within both species.
Specific maximum assimilation rate, {𝑝𝐴𝑚}, volume specific somatic
maintenance, [𝑝𝑀], and maturity maintenance rate coefficient,
𝑘𝐽, have
larger values in sbp model, while the energy conductance, 𝑣, is larger in
abp model. This pattern is present to both species. These differences are
attributed to the different structure of the two models, and that {𝑝𝐴𝑚}
and 𝑣 are multiplied by 𝑠𝑀= min(𝐿, 𝐿𝑝)∕𝐿𝑏in the abp models, which
depends on the value of the functional response 𝑓.
The uni- and zero-variate data used in the estimation with the model
predictions are given in Figs. 4 &5and Table 4 for N. viridula and in
Figs. 6 &7and Table 5 for E. heros. Overall, the goodness-of-fit metrics
have lower values when the sbp model was used to model N. viridula
(abp: MRE =0.408, SMSE =0.146; sbp: MRE =0.175, SMSE =0.055).
In the E. heros case, however, the metrics have similar values for both
models (abp: MRE =0.134, SMSE =0.051; sbp: MRE =0.104, SMSE =
0.026).
The head length, wet weight, and cumulative number of eggs as
function of age at different feeding experiments, for N. viridula, are
presented in Fig. 4. In each experiment, animals were provided with the
same amount of food, but the food was available for a certain time at
different time schedules. The data indicate that for all trials head length
reached the same value at emergence, approximately 0.26 cm, while
the weight and cumulative numbers differ among trials, particularly
between that of ad libitum food and the rest. The trials, where food was
present 63% and 42% of the experiment duration, resulted in similar
weights and cumulative number of eggs, while that of 35% shows
slightly smaller weight and fewer eggs. To model the differences in the
feeding experiments and to capture the concept of food availability,
we introduced separate scaled functional response parameters for each
experiment. For N. viridula, the estimated values for the four feeding
experiments were 1, 0.68, 0.80, 0.66 for the abp model and 1, 0.65,
0.78, 0.66 for the sbp model, respectively. The first value, in each set,
corresponds to the experiment that food was continuously available
during the experiment, while the other three values vary only in terms
of the duration that food was available. We do not observe a decrease
in the value of 𝑓with the decrease in the duration of food availability,
which would have been expected. One possible explanation is that
the insects may not require frequent feeding. This observation aligns
with the common practice, where insects are often fed every three
days on average. Both abp and sbp models resulted in predictions that
match well the data. However, growth during the nymphal stage for the
various feeding experiments, was captured better from the sbp model.
Fig. 5 illustrates embryo and nymphal development at different
temperatures for N. viridula from various sources. Both models gen-
erally capture the trends with some deviation. Specifically, the sbp
model tends to underestimate embryo developmental time and, thus,
overestimate embryo growth rate, while the abp model does the oppo-
site. Additionally, both models capture equally the nymphal survival at
23 ◦C, with a slight overestimation at the early stages.
The model predictions and the data analysis for E. heros are pre-
sented next. Fig. 6 shows the evolution of the head length and wet
weight of nymphs in various feeding experiments with different food
solution concentrations (100%, 80%, 50% and 20%). The reduction in
food availability notably affected the wet weight at emergence, while
the impact on head length was less pronounced. The scaled functional
response estimates that account for food availability were 1, 0.95, 0.77,
0.57 for the abp model and 1, 0.89, 0.69, 0.45 for the sbp model,
respectively. The decrease in the estimated values of the functional
responses correspond to the different food availability (food solution
100%, 80%, 50% and 20%) in the experiments. Similarly to the N.
viridula analysis, both models predict nymphal growth with accuracy
Ecological Modelling 493 (2024) 110740
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E. Klagkou et al.
Fig. 4. Growth and reproduction trajectories for Nezara viridula: nymphal head length (first column), nymphal wet weight (second column) and cumulative reproduction (third
column) for four different food availabilities (rows: food was available from top 100%, 63%, 49%, 35% of the experiment duration). Model prediction for the abp (blue dotted
line) and sbp (red solid line) models, and observations (black dots).
Source: Data from current study.
for all experiments, with the sbp model capturing the curvatures of the
data more effectively. Additionally, reproduction data from the liter-
ature were integrated with the growth data from this study (Fig. 7a).
While both models perform well in predicting the initial part of the
cumulative reproduction, the sbp model tends to overestimate the last
part.
Embryo and nymphal development at different temperatures for E.
heros from various sources are shown in Fig. 7. Embryo developmental
time and instar duration decrease with temperature increase, within the
thermal range of the species, and both models generally capture the
trends with some small deviation. Additionally, both models capture
equally the nymphal and adult survival at 26 ◦C. Both models gave
close predictions for various life history traits (Table 5), with the sbp
model predicting duration of instars with smaller error. The single
value for respiration was better predicted by the sbp model.
For both stink bug species, the abp typified DEB model, i.e. V1-
morphic nymphal growth (metabolic acceleration) and no 𝜅-rule, al-
though had an overall good fit, it did not capture as well the shape
of the growth trajectories along all feeding levels. Koch and De Scham-
phelaere (2019) arrived at similar conclusions when they compared the
abp and sbp models for copepod Nitocra spinipes. Both energy allocation
rules share a limitation in that asymptotic length, that the insect would
Ecological Modelling 493 (2024) 110740
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E. Klagkou et al.
Fig. 5. Results for Nezara viridula: model predictions from the abp (blue dotted line) and sbp (red solid line) and univariate data (black dots). Top: age at hatch (data used from
current study), duration of instars, and embryo growth rate at different temperatures, Hiromitu (1960). Bottom: nymphal growth rate at different temperatures (Hiromitu,1960)
and nymphal survival (Medina et al.,2022).
Table 4
Zero variate data and predictions from abp and sbp models for Nezara viridula.
Data Value (abp) Value (sbp) Units Description
6 9.67 4.81 d Age at hatch at 25 ◦Ca
34 35.47 37.94 d Time at emergence at 25 ◦Ca
75 74.82 75.15 d Life span as imago at 25 ◦Cb
0.6 0.78 0.53 mm Head length at hatchc
0.47 2.55 0.75 mg Wet weight at hatchc
187 168 164 mg Wet weight at emergenceb
12.63 8.46 11.7 #/d Eggs/female/day at 25 ◦Cb
5.5 4.89 4.57 d Duration of instar 1 at 25 ◦Cd
6.5 6.21 6.98 d Duration of instar 2 at 25 ◦Cd
7.5 7.18 8.15 d Duration of instar 3 at 25 ◦Cd
7.8 7.39 8.58 d Duration of instar 4 at 25 ◦Cd
8.8 8.27 9.67 d Duration of instar 5 at 25 ◦Cd
Data sources.
aHiromitu (1960).
bFortes et al. (2006).
cCurrent study.
dAlie and Ewiess (1977).
have if growth continued, is not observable, since insects stop growing
after emergence. This complicates the estimation of the allocation
fraction 𝜅and the volume-specific somatic maintenance [𝑝𝑀].
A model that combines metabolic acceleration in the very early
instars with isomorphic growth in the later instars could potentially
offer a more accurate representation of the life cycle of stink bugs, but
this approach would increase by one the number of parameters to be
estimated. To further investigate whether metabolic acceleration occurs
throughout the entire nymphal stage or only during part of it, it would
be necessary to include individual data such as respiration and food
uptake over time with high temporal resolution. Existing DEB models
for insects assume exponential growth at the nymphal stage (Llandres
et al.,2015;Gergs and Baden,2021), and only recently entries in the
AmP collection for insect species combine exponential and isomorphic
growth during the nymphal stage (see for example Augustine and Gergs
(2019) and Maino and Kooijman (2022)). A challenge in this modeling
Table 5
Zero variate data and predictions from the abp and sbp models for Euschistus heros.
Data Value (abp) Value (sbp) Units Description
6 6.29 5.81 d Age at hatch at 26 ◦Ca
31.4 30.15 30.74 d Age at emergence at 26 ◦Ca
120 120 122 d Life span as imago at 26 ◦Cb
0.55 0.63 0.51 mm Head length at hatchc
0.74 1.32 1.00 mg Wet weight at hatchb
72.85 31.08 46.56 mg Wet weight at emergenceb
4.133 3.61 3.66 #/d Eggs/female/day at 27 ◦Cd
3.1 2.8 2.92 d Duration of instar 1 at 26 ◦Ca
6 5.67 6.00 d Duration of instar 2 at 26 ◦Ca
3.5 3.38 3.49 d Duration of instar 3 at 26 ◦Ca
4.7 4.48 4.61 d Duration of instar 4 at 26 ◦Ca
8.1 7.5 7.91 d Duration of instar 5 at 26 ◦Ca
0.0287 0.0216 0.0275 ml CO2/h Respiration rate at 27 ◦Ce
Data sources.
aOliveira et al. (2015).
bMalaguido and Panizzi (1999).
cCurrent study.
dMichereff et al. (1999).
eHaddi et al. (2016).
approach is to estimate the point at the switch from exponential to
isomorphic stage.
3.2. Energy allocation rules
To assess the impact of the different energy allocation rules in the
adult stage, we re-estimated the parameters using the abp-𝜅and the
sbp-𝜅models. These models assume that the 𝜅-rule operates also in
adults, where growth ceases, and the energy that would have been
allocated for further growth, dissipates. Both energy allocation rules
capture well the data for the embryo and juvenile stage (the results are
shown in the Supplementary Material). We here focus in comparing the
different models only in the adult stage and primarily on the results of
E. heros, but similar conclusions were obtained for N. viridula as well.
Ecological Modelling 493 (2024) 110740
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Fig. 6. Growth trajectories for Euschistus heros: nymphal head length (first column) and wet weight (second column) for four different food availabilities (rows from top 100%,
80%, 50%, 20%). Model prediction for the abp (blue dotted line) and sbp (red solid line) models, and observations (black dots).
Source: Data used from current study.
The predictions for the two energy allocation rules are similar,
particular in terms of cumulative number of eggs as shown in Fig. 8.
However, when comparing the model predictions with the observed
respiration rate (0.0287 ml CO2/h Haddi et al.,2016), the abp-𝜅
model significant overestimates the observed value (0.0744 ml CO2/h),
whereas the abp model slightly underestimates it (0.0216 ml CO2/h).
Both allocation schemes with isomorphic nymphal growth, give respira-
tion rates close to the observed data (0.0297 ml CO2/h for the sbp-𝜅and
0.0275 ml CO2/h for the sbp model). In terms of goodness of fit criteria,
the overall data fit for both nymphal growth models, is similar in both
allocation rules. Specifically, the abp model (MRE =0.134, SMSE =
0.051) is almost identical to the abp-𝜅(MRE =0.131, SMSE =0.043).
Ecological Modelling 493 (2024) 110740
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E. Klagkou et al.
Fig. 7. Results for Euschistus heros. Model predictions from the abp (blue dotted line) and sbp (red solid line) and univariate data (black dots). Top: age at hatch at different
temperatures (data from current study), duration of instars 2–5 at different temperatures (Malaguido and Panizzi,1999;Oliveira et al.,2015;Costa et al.,1998;Frugeri et al.,
2023), cumulative number of eggs (Haddi et al.,2016). Bottom: nymphal and adult survival (Malaguido and Panizzi,1999).
Fig. 8. Comparison of model predictions with data for cumulative number of eggs for Euschistus heros. Left: Predictions from the abp (red solid line) and abp-𝜅(blue dashed line)
models. Right: Predictions from the sbp (red solid line) and sbp-𝜅(blue dashed line) models.
Source: Data (black dots) from Haddi et al. (2016).
Similarly, the two allocation schemes with isomorphic nymphal growth
show almost equal performance: MRE =0.123, SMSE =0.034 for the
sbp-𝜅model and MRE =0.104, SMSE =0.026 for the sbp model.
The existing data do not have sufficient information to choose
between the energy allocation rules in the adult stage. Data such as CO2
production, O2consumption or food uptake throughout the nymphal
and adult stages will be needed to support any choice. However, con-
sidering that the no 𝜅-rule performs equally well for both the abp and
the sbp models and since both models have been applied to species with
no growth during the adult stage (AmP,2023), to maintain consistency
we will adopt the no 𝜅-rule as the chosen energy allocation rule.
3.3. Variability in data
For the estimation of model parameters, the data available from this
study for head length and weight were the means from 20 randomly
chosen animals. The data, however, show considerably variability in
weight with coefficient of variation up to 60% in some trials for N.
viridula and 74% for E. heros. We explored potential sources of the
observed variability by modifying selected model parameters of the
sbp model. We found that variability was best reproduced by adding
scatter to the scaled functional response 𝑓. To simulate this source of
variability, for each experiment, random numbers were generated from
a log-normal distribution with mean the estimated value of 𝑓for that
experiment and coefficient of variation 𝐶𝑉 = 15%. Data also show
some variability in head length at hatch, with 𝐶𝑉 = 4% for N. viridula
and 𝐶𝑉 = 2.5% for E. heros. For this source of variability, random
numbers from a log-normal distribution with mean the predicted value
for head length at hatch for each species and coefficient of variation the
value observed in the data of that species. Figures 5 and 6 in the Sup-
plementary Material show the results of 200 Monte Carlo simulations
for all feeding levels. Overall, the model with the proposed sources of
scatter were able to reproduce the variability in both head length and
weight. This suggests that varying food availability for each individual
contributes to the observed variability. Because of the high mortality
observed in all experiments, the provided data regarding the time at
emergence is insufficient to explore its variability. However, the sim-
ulated results indicate increased variability as food decreases (Tables
Ecological Modelling 493 (2024) 110740
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E. Klagkou et al.
Fig. 9. Age at hatch, nymphal stage duration and egg production rate for E. heros (blue dotted line) and N. viridula (red solid line), the dots represent data not used for the
parameter estimation. Left (Daane et al.,2022), middle (Alie and Ewiess,1977;Musolin and Numata,2003;Chanthy et al.,2015;Hayashida et al.,2018;Daane et al.,2022),
right (Silva et al.,2008;Hayashida et al.,2018).
12 and 13 in Supplementary Material), which as shown in Gergs and
Baden (2021) is expected when insects face unfavorable environmental
conditions.
3.4. Effects of environmental changes
Climate change can have a significant impact on all organisms,
including insects. To investigate possible effects of global warming, sim-
ulations were conducted across a large range of constant temperatures
(15–30 ◦C), for both species. In these scenarios, we use the sbp model,
i.e. isomorphic nymphal growth with no operational 𝜅-rule in adults,
based on the performance of the different models.
Fig. 9 illustrates the predictions and available observations for age
at hatch, nymphal duration, and egg production rate for both species.
The observed data for each species were not used for the parameters
estimation. The age at hatch and nymphal duration decrease, while
the egg production rate increase as the temperature increases, within
the range from 15 to 30 ◦C. Additional data are required to predict
the effects of temperature at extreme temperature values. The impact
of temperature on age at hatch is quite similar for both species. N.
viridula’s nymphal duration is more sensitive to temperature changes,
especially in the lower temperatures. This trend is in agreement with
findings in the literature for various other species (Sanderson and
Durhmn,1910;Davidson,1944;Nayar,1972;Porter et al.,1991;
Régnière et al.,2012;Wu et al.,2015). When it comes to the egg
production rate, N. viridula has a more pronounced response to tem-
perature increase compared to E. heros. Temperature has been shown
to have a significant effect in longevity, survival and reproductive
performance, while the weight does not change much (Chanthy et al.,
2015;Frugeri et al.,2023). In Chanthy et al. (2012) it has been shown
that N. viridula adults cannot survive temperatures higher than 42 ◦C
or lower than −10 ◦C for 2 h.
Temperature does not have only direct but also indirect effects
on organisms. The indirect effects are via altering food availability
and quality. To explore these effects, simulations were performed on
hatchlings across a range of constant temperatures (15−30 ◦C) and
food availabilities (𝑓= 0.4−1) for both species. E. heros’s age at
emergence is more sensitive to food availability changes, particular to
low temperatures, as indicated by the sharp changes in color gradient
(compare Fig. 10a and b). E. heros’s reproduction rate is more affected
by changes in food availability across to a larger range of high temper-
atures (Fig. 10c), while N. viridula’s reproduction rate is more sensitive
to temperature changes across different feeding levels (Fig. 10d). This
could be one of the reasons why N. viridula has a more widespread
global distribution, while E. heros tends to be more localized (Squitier,
2020;Panizzi,2008). N. viridula has expanded its range in Japan over
the last decades due to warmer temperatures that are more favorable
for its overwintering survival, which is not possible in temperatures
below 3 ◦C (Tougou et al.,2009;Musolin,2003). In a simulated climate
change scenario (Takeda et al.,2010) assumes +2.5 ◦C increase in
temperature, during winter, spring and fall, conditions were favorable
resulting in accelerated nymphal development, larger sizes and higher
survival rates. However, in the summer, the extreme temperature led
to slower nymphal developed, reduced egg production, and increased
mortality (Musolin et al.,2010;Takeda et al.,2010). These patterns
can be predicted from the DEB models that incorporate the extended
Arrhenius function, which is not applied in the present study due to
lack of information at extreme temperatures. E. heros is also affected
by warmer temperatures, its range withing Brazil is shifting depending
on the month in order to optimize its reproductive performance, while
the most favorable range is 25–28 ◦C (Rodrigues et al.,2023).
Furthermore, it is important to study the effects of a variable
environment, considering that those species have multiple generations
in a year. According to studies conducted on N. viridula and E. servus
in Georgia, USA, two distinct generations, one in June and another
in September, were identified (Herbert and Toews,2011,2012). To
simulate these experiments, temperature data were collected from the
study area during the experiment from WU (2023) and food availability
was assumed to be ad libitum (𝑓= 1). Simulations were performed for
150 days for the June generation and 210 for the September generation.
For both species and generations, the effect of temperature on feeding
and egg production rate and growth were analyzed, aiming to predict
which generation poses a higher risk to the crops.
The results of the simulations are shown in Figs. 11 and 12 for N.
viridula and E. heros, respectively. For both species, there is a delay in
age at hatch and, more noticeable, in age at emergence from the June to
September generation, which is attributed to the temperature decrease
following the September generation.
For both species, the June generation had egg production and
feeding rate one order higher compared to the September one. This
indicates that during the summer months, they consume food and
reproduce at a faster rate. In both generations, the ultimate length
(reached at emergence) is the same. However, the rate at which they
reach it is affected by the temperature changes, which also affects the
feeding and egg production rate. These effects are due to temperature
changes solely, since food is assumed to be the same in both gen-
erations. These results are in agreement with findings from studies
on several species. The effect of a range of constant temperatures
(18–35 ◦C) was studied for Spodoptera exigua (Lemoine et al.,2014)
and other insects from the families Lepidoptera, Hymenoptera and
Coleoptera (Lee and Roh,2010). They both observed higher consump-
tions rates in elevated temperatures. In Schlesener et al. (2020) and
Ponsonby and Coplad (1998), the effects of constant temperatures
(13–33 ◦C) were studied for Drosophila suzukii and Chilocorus nigritus,
respectively, were warmer temperatures resulted to higher egg produc-
tion rate for both species. Fluctuating temperatures that remain within
thermal ranges of the species generally improve performance (Colinet
et al.,2015).
Ecological Modelling 493 (2024) 110740
13
E. Klagkou et al.
Fig. 10. Age at emergence and egg production rate for both species, scaled to their maximum values. 𝑥-axis shows the change in functional response, 𝑦-axis shows the change in
temperature and the color bar shows the changes in the values.
Moreover, our model predictions are consistent with several pub-
lications that highlight the increased damaging potential of insects in
a warmer environment with higher levels of atmospheric CO2(Child,
2007;DeLucia et al.,2008). In these model simulations, though, we
assumed ad libitum food, which does not reflect the natural fluctuations
in food quantity and quality that occur in nature driven by changes in
environmental conditions. Furthermore, even if food availability is ad
libitum during the initial phase of crop infestation, stink bugs can cause
damage on plants, compromising the quality of the food. This degrada-
tion may occur as early as seven days into infestation, emphasizing the
rapid impact of stink bug activity on food quality (Negrón and Riley,
1987;Ni et al.,2010;Cissel et al.,2015;Zobel et al.,2016). Given data
on food availability, a similar analysis could provide valuable results on
the effects of a variable environment, including both temperature and
food availability.
4. Conclusions
We focused on one category of pest insect species, that of neotrop-
ical stink bugs, for which we analyzed dynamic energy budget models
for their full life cycle. However, we have developed the models in a
way that ensures their applicability to other hemimetabolous species
with feeding adult stage. With extensive life history data for E. heros
and N. viridula we parameterized and evaluated four DEB models,
which all assume no growth during the adult stage, but differ in
nymphal growth and the energy allocation rules in adults: isomorphic
or V1-morphic nymphal growth combined with the energy allocation
scheme in adults that follows either the 𝜅-rule or the scheme that the
𝜅-rule is not operational in adults. If food and temperature are constant,
isomorphy results in a von Bertalanffy growth curve in length and
weight up till puberty, while V1-morphy (metabolic acceleration) in
exponential growth. All models were able to fit the data well, with those
with isomorphic nymphal growth reproducing the growth trajectories
and instar duration better. We concluded that V1-morphic nymphal
growth combined with the 𝜅-rule in adults – where the energy from
the 𝜅fraction of mobilized energy not used for somatic maintenance
dissipates – is unlikely to hold, since its predictions for CO2production
rate were twice of that reported in literature (Haddi et al.,2016).
Metabolic acceleration frequently occurs in the animal kingdom and
developed several times in various animal lineages (Kooijman,2014;
Marques et al.,2018). Metabolic acceleration is present in Actinoptery-
gii with varying degrees of acceleration observed among different taxa,
but it is absent in Chondrichthyes (Lika et al.,2022). With the available
entries in the AmP collection, the basic insects, such as ephemeropter-
ans and odonata, exhibit a moderate degree of metabolic acceleration,
while holometabolic insects accelerate to a much greater extend (Mar-
ques et al.,2018). However, the AmP collection currently includes a
limited number of insect species, and further work is required to classify
them as either accelerating or non accelerating species.
Proceeding with the typified sbp DEB model, which our current
analysis has shown to reproduce a more accurate representation of the
life cycle and the growth trajectories, we studied the effects of different
constant temperatures and food availability. The study shows that tem-
perature has more pronounced impact on stage duration in comparison
to food availability. In a climate change scenario characterized by high
temperatures and reduced food availability, one can expect a decrease
in the duration of life stages, leading to an increase in the number
of generations within a year. Model simulations, assuming a variable
temperature for a summer and a fall generation, have shown higher
egg production and feeding rate during the summer generation for both
species.
These changes in the individual dynamics and its life cycle can have
significant implications for population growth and the development of
pest management strategies, especially in the context of climate change.
The DEB model developed in this study could be combined with other
modeling approaches to further enhance our understanding and address
these challenges. The DEB model could be combined with population
models, such as the IBMs (Individual Based Models) (Martin et al.,
2012), to investigate how the individual dynamics contribute to the
population dynamics. Moreover, the Mechanistic Niche models (Kear-
ney et al.,2012) combined with the results of the DEB model can
predict the possible distribution of a species, which can be a crucial
information amid climate change.
Ecological Modelling 493 (2024) 110740
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E. Klagkou et al.
Fig. 11. Feeding and egg production rates and length (solid black lines) at variable temperatures (orange dashed lines) for Nezara viridula for the June (left) and September
generations (right).
Code availability
The code underlying the parameter estimation for the sbp model is
available in the Add-my-Pet library (AmP,2023) at https://bio.vu.nl/
thb/deb/deblab/add_my_pet/entries_web, and can be accessed directly
by using the species names (Nezara viridula and Euschistus heros).
CRediT authorship contribution statement
Evridiki Klagkou: Writing – original draft, Software, Methodology.
Andre Gergs: Writing – review & editing, Funding acquisition, Concep-
tualization. Christian U. Baden: Writing – review & editing, Data cu-
ration. Konstadia Lika: Writing – review & editing, Writing – original
draft, Supervision, Software, Funding acquisition, Conceptualization.
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Data availability
Data are included in the Supplementary Material.
Acknowledgments
The authors would like to thank Bas Kooijman and Michael Kearney
for the helpful discussions on modeling insects in the DEB context
during the DEB2023 School and Symposium and K. Kuhl for conducting
the feeding experiments. We also thank the anonymous reviewers for
their careful reading of our manuscript and the insightful suggestions
and comments.
Funding
This work has been funded by Bayer AG, CropScience Division.
Appendix A. Supplementary data
Supplementary material associated with this article can be found in
the online version and include (1) data for the feeding experiments for
both species Nezara viridula and Euschistus heros, (2) the results for the
Ecological Modelling 493 (2024) 110740
15
E. Klagkou et al.
Fig. 12. Feeding and egg production rates and length (solid black lines) for Euschistus heros at variable temperatures (orange dashed lines) for June (left) and September generation
(right).
abp-𝜅and sbp-𝜅models for both species, and (3) supplementary results
on simulations for data variability.
Supplementary material related to this article can be found online
at https://doi.org/10.1016/j.ecolmodel.2024.110740.
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