=Paper=
{{Paper
|id=Vol-2486/icaiw_ikit_3
|storemode=property
|title=Numerical Simulation of the Biological Control of the Chestnut Gall Wasp with T. Sinensis
|pdfUrl=https://ceur-ws.org/Vol-2486/icaiw_ikit_3.pdf
|volume=Vol-2486
|authors=Carlos Balsa,Margaux Citerici,Isabel Lopes,José Rufino
}}
==Numerical Simulation of the Biological Control of the Chestnut Gall Wasp with T. Sinensis==
https://ceur-ws.org/Vol-2486/icaiw_ikit_3.pdf
Numerical Simulation of the Biological Control
of the Chestnut Gall Wasp with T. Sinensis
Carlos Balsa1 , Margaux Citerici2 , Isabel Lopes3,4 , and José Rufino1
1
Research Centre in Digitalization and Intelligent Robotics (CeDRI), Instituto
Politécnico de Bragança, Campus de Santa Apolónia, 5300-253 Bragança, Portugal
{balsa,rufino}@ipb.pt
2
Université de Toulouse - Institut National Polytechnique de Toulouse, France
margaux.citerici@etu.enseeiht.fr
3
Applied Management Research Unit (UNIAG), Instituto Politécnico de Bragança,
Campus de Santa Apolónia, 5300-253 Bragança, Portugal
4
Centro ALGORITMI, Escola de Engenharia - Universidade do Minho,
Campus Azurém, 4800-058 Guimarães, Portugal
isalopes@ipb.pt
Abstract. Portugal is a country that produces chestnuts. However, in
recent years, a D. kuriphilus plague has invaded chestnut trees and sig-
nificantly affected the chestnut production. Studies in other countries,
such as Japan or Italy, have shown that the T. sinensis parasitoid can
achieve biological control and help to eradicate the invader. In this work,
the evolution of the density of D. kuriphilus and T. sinensis eggs across
time and space is studied through the numerical solution of mathematical
models previously proposed. It is concluded that the biological control
with T. sinenis operates and thus eradicates D. kuriphilus from the in-
fected area. However, the simulations also show that biological control is
not effective over time, as D. kuriphilus returns to the same area. There-
fore, it would be necessary to reinject T. sinensis periodically into the
infected zones for a sustained fight against D. kuriphilus.
Keywords: Biological control · Mathematical model · Numerical simu-
lations · Chestnut · Dryocosmus kuriphilus · Torymus sinensis
1 Introduction
The chestnut gall wasp Dryocosmus kuriphilus (D. kuriphilus) disrupts the growth
of chestnut trees and limits fruit production. Indeed, this species lays its eggs in
the buds of chestnuts in early summer. The larva spends all winter in this bud.
In the following spring, this induces the formation of galls on the buds and on
the leaves of the tree (Fig. 1). Therefore, this disrupter has a significant impact
on the production of chestnut, with further negative economical effects.
Originally from China, the pest spread to other Asian countries, like Japan,
where it was first detected in 1941. In the early 2000s, it arrived in the United
States and Europe, first in Italy, then in France and finally in Portugal, in
Copyright c 2019 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0)
2019 ICAI Workshops, pp. 230–243, 2019.
Numerical Simulation of the Biological Control of the Gall Wasp 231
(a) Wasp (b) Chestnut gall
Fig. 1. Chestnut gall wasp D. kuriphilus (a) and buds infected with D. kuriphilus (b)
2014 [1]. Solving the problem is very important for the economy of this sector,
specially for countries that are big producers, like Portugal, with an estimate of
25,000 to 30,000 tons of annual chestnut production.
Japan was the first country that tried to control the plague. Japanese re-
searchers started by using pesticides, but this was found to be noneffective.
Then they became interested in varieties of chestnut that could be resistant to
D. kuriphilus. Finally, they made tests with several species of parasitoids and
discovered that Torymus sinensis (T. sinensis), a parasite that also came from
China, could be used to effectively control the plague: T. sinensis lays its eggs in
the larvae of D. kyriphilus; this prevents them from developing and thus to latter
attack the chestnut tree and to further reproduce [7]. Besides Japan, experiments
with the biological control conducted in France and Italy have confirmed that
D. kuriphilus disappears entirely after a few years [9,5].
Researchers in Japan, Italy, and France began by studying this phenomenon
in the laboratory and then on a specific area of the country, to be able to ob-
serve and study the different effects of T. sinensis on D. kuriphilus. These field
studies often involve a considerable amount of time, human resources and logis-
tics. In this regard, computational simulations provide an economic alternative
to experimental testing or may be used to complement/validate experimental
studies/results. The numerical solutions of the mathematical models represent-
ing the evolution of two interacting species allow to obtain important information
that can be used to optimize the combat to plagues. In this context, carefully
choosing the mathematical models to be adopted is of fundamental importance.
As far as the authors of this work know, the only mathematical models of
the Biological Control of the Chestnut Gall Wasp by T. sinensis were devel-
oped by F. Paparella and co-authors [8]. The proposed models describe the
dynamics of the wasp population in the presence of the parasite population. The
interaction between the two species is a predator-prey type model based on the
Lotka–Volterra equations (see, for instance, [4]). However, the interaction be-
232 C. Balsa et al.
tween the two species is not direct, but through eggs. The parasite lays its eggs
inside the wasp larvae, thus causing its destruction. Therefore the mathematical
models also contemplate the quantity of eggs of the two species.
The mathematical models proposed by F. Paparella and co-authors [8] are
based on Ordinary Differential Equations (ODE) and Partial Differential Equa-
tions (PDE). In this work ODE and PDE are solved numerically by means of the
Octave [2] open source software. Several simulations were carried out to analyze
the evolutions of the two species over time and space. Finally, a study concerning
the improvement of the biological control is also presented.
The paper is organized as follows. Section 2 introduces the mathematical
models proposed by F. Paparella and co-authors [8] and presents the results
of numerical simulations that enable the estimation of the density of the two
species over the time. Section 3 is devoted to the mathematical models that
include the spatial variation of the two densities over time. Section 4 presents the
results of numerical simulations performed in order to evaluate the possibilities
of improving the biological control. Section 5 wraps up the paper with some final
considerations.
2 Density of the D. kuriphilus and T. sinensis Over
Time
In this section, the mathematical models that take into account the separate
evolution of D. kuriphilus and of T. sinensis, over time, are introduced, fol-
lowed by the complete mathematical model for the joint evolution of the two
species. These models are simplified because the spatial variation is not taken
into account, i.e., it is assumed that the two species do not change their territory
(for details see [8]). This first approach enables to estimate the evolution of the
density of the two species, and consequentially to know if the biological control
works or does not.
2.1 Model for D. kuriphilus
Let Un be the population of adult gall wasps carrying eggs during the summer
of the year n. Let Vn be the density of eggs laid in the chestnut buds (this will
allow to see the evolution of the population of D. kuriphilus over the years).
Considering the density βmax of chestnut buds and the maximum M of eggs
to be laid per bud, then the maximum density of eggs laid is:
Vmax = M βmax (1)
It is also possible to predict the emergence rate of this species, its mortality
rate, and its egg laying rate during one year. The expressions are next presented.
Let η be the survival rate during the overwintering; this will depend on the
geolocation of the species (e.g., it will not be the same in Japan and Portugal).
Numerical Simulation of the Biological Control of the Gall Wasp 233
Let Td be the length of the egg deposition season. The emergence rate is thus:
ηVn−1 (Td )
emergence rate = (2)
Td
Let a be the adult life span. Therefore, the death rate is:
Un (t)
death rate = − (3)
a
Finally, let rd be the optimum condition where each D. kuriphilus lays all its
eggs. This variable is worth Nd /a, with Nd being the maximum number of eggs
that can be laid by an adult. The laying rate is proportional to the density of the
laying rate of the location, which in the model is expressed as M βmax − Vn (t):
M βmax − Vn (t)
egg deposition rate = rd Un (t) (4)
Vmax
The combination of all these quantities results in the following formulation
that describes the evolution of Un and Vn during the season of the year n:
1 M.βmax −Vn (t)
∂
∂t Un (t) = − a
Vmax Un (t) − a1 Un (t) + η Vn−1
Td
(Td )
(5)
∂ V (t) = Nd M βmax −Vn (t) U (t)
∂t n a Vmax n
Non-dimensional variables are used in all the models of this work. For the
model of this section such implies the following definitions: un = Un / (ηVmax ),
vn = Vn /Vmax . Moreover, the new t, µ and Ed are defined as t = t/Td , µ = Td /a
and Ed = ηNd . Consequently, the new model to which the initial conditions are
added is given by: ∂
∂t un (t) = −µ(2 − vn (t))un (t)
∂
vn (t) = Ed µ(1 − vn (t))un (t)
∂t
(6)
un (0) = 0
vn (0) = 0
2.2 Model for T. sinensis
To obtain the mathematical model for the evolution of T. sinensis, the same
procedure that was previously used for the D. kuriphilus model is followed.
Initially, Pn is defined as the population of the egg-carrying T. sinensis fe-
males, of the year n, and Qn as the density of eggs laid, of the same year (likewise,
this allows to follow the evolution of the population of T. sinensis over the years).
Like before, it is possible to calculate the egg deposition rate. Let rt be the
optimum condition where each T. sinensis female lays all its eggs. This variable
234 C. Balsa et al.
evaluates to Nt /Tt , with Nt being the maximum number of eggs that can be laid
by an adult female during its life span Tt . Thus, the egg laying rate is given by:
ηVn−1 (Td ) − Qn (t)
egg deposition rate = rt Pn (t) (7)
Vmax
Properly combined, these quantities support a formulation that describes the
evolution of Pn and Qn during the season of the year n:
∂ 1 ηVn−1 (Td )−Qn (t)
∂t Pn (t) = − Tt
Vmax Pn (t)
(8)
∂ Q (t) = r ηVn−1 (Td )−Qn (t) P (t)
∂t n t Vmax n
Inserting in the previous formulation the non-dimensional variables
pn = Pn / (ηγVmax ) – with γ accounting for the sex ration of T. sinensis –,
qn = Qn / (ηVmax ), t = t/Tt , τ = Tt / (ηTd ), and adding the initial conditions,
such results in the new model for T. sinensis described in Equation (9).
∂ 1
∂t pn (t) = − τ (vn−1 (1) − qn (t))pn (t)
∂
qn (t) = Eτt (vn (1) − qn (t))pn (t)
∂t
(9)
pn (0) = qn−1 (ητ )
qn (0) = 0
2.3 Complete Model
The complete mathematical model enables to describe the evolution of D. ku-
riphilus when T. sinensis is injected on the same territory. It allows measuring
the impact of T. sinensis on D. kuriphilus and thus to see if it is possible to
limit the evolution of D. kuriphilus thanks to biological control.
To have a complete model, it is necessary to introduce the effect of T. sinensis
on D. kuriphilus. This effect will be that parasitized larvae of the gasp won’t give
rise to adults. This effect will have consequences on the emergence rate of D.
kuriphilus, that now becomes:
ηVn−1 (Td ) − Qn (Tt )
emergence rate = (10)
Td
This results in the complete model described by Equation (11), where initial
conditions are added:
Numerical Simulation of the Biological Control of the Gall Wasp 235
∂ 1
∂t pn (t) = − τ (vn−1 (1) − qn (t))pn (t)
Et
∂
∂t qn (t) = τ (vn (1) − qn (t))pn (t)
∂
un (t) = −µ(2 − vn (t))un (t) + vn−1 (1) − qn (ητ )
∂t
∂
vn (t) = Ed µ(1 − vn (t))un (t)
∂t
(11)
un (0) = 0
vn (0) = 0
pn (0) = qn−1 (ητ )
qn (0) = 0
2.4 Numerical Simulations
The system of Ordinary Differential Equations (11) was solved numerically by
the Octave built-in function ode45. This function combines the four and five
order Runge-Kutta method (see [6], for instance).
Many model parameters are fixed. There are two, however, that depend on
the geolocation of the species: the survival rate during overwintering of D. ku-
riphilus (η) and the accounts for the sex ration of T. sinensis (γ). Depending
on the value of these parameters, several types of evolution are possible. In this
work η = 0.9 and γ = 0.45. These values correspond to the reality of Portugal [8].
Fig. 2 depicts the evolution of the eggs density of the two species along the
time, in years. It is verified that a cycle appears: the density of D. kuriphilus
decreases for 2-3 years, which leads to the decline of T. sinensis. Then, the
density of D. kuriphilus rises and, when it is worth 1, that of T. sinensis goes
up there too. Notably, D. kuriphilus never disappears completely. To reduce
the density of D. kuriphilus it would be necessary to inject T. sinensis into the
environment, regularly. Indeed, T. sinensis lays its eggs in those of D. kuriphilus;
thus, if D. kuriphilus disappears from the area, T. sinensis does not lay eggs
and will also disappear from the area because it will not be able to reproduce.
To sum it up, although T. sinensis controls the evolution of D. kuriphilus, this
does not allow to definitely eradicate D. kuriphilus.
3 Two-Dimensional Model
Adding a spatial dimension to the previous complete model will take into account
the fact that the two species move over time from one area to another. To
accomplish that it is necessary to add a diffusivity term to the model, described
by Equation (11).
236 C. Balsa et al.
Fig. 2. Temporal evolution of D. kuriphilus and T. sinensis eggs density.
3.1 Mathematical Model
Let ∇2 un be the diffusivity of un , ∇2 pn be the diffusivity of pn , and δ represent
the diffusivity ratio between the two species. It was observed that D. kuriphilus
moves faster than the parasite and travels longer distances [3]. In this study, it is
assumed δ = 0.2, which corresponds to the speed of the T. sinensis front of about
6.4 km per season. Taking into account the new spacial-aware parameters, the
complete mathematical model with spacial dimension is given by Equation (12):
p (x, y, t) = δ∇2 pn (x, y, t) − τ1 (vn−1 (x, y, 1) − qn (x, y, t))pn (x, y, t)
∂
∂t n
∂ q (x, y, t) = Et (v (x, y, 1) − q (x, y, t))p (x, y, t)
∂t n n n n
τ
∂ un (x, y, t) = ∇2 un (x, y, t) − µ(2 − vn (x, y, t))un (x, y, t) + vn−1 (x, y, 1) − qn (x, y, ητ )
∂t
∂
vn (x, y, t) = Ed µ(1 − vn (x, y, t))un (x, y, t)
∂t
un (x, y, 0) = 0
v (x, y, 0) = 0
n
pn (x, y, 0) = qn−1 (x, y, ητ )
qn (x, y, 0) = 0
(12)
Numerical Simulation of the Biological Control of the Gall Wasp 237
3.2 Numerical Simulations
The numerical solution of the system of partial differential equations (12) implies
the discretization of the diffusive term. Using the finite difference method, as
stated in [6], the diffusivity of un is approached by
un (xi+1 , y, t) − 2un (xi , y, t) + un (xi−1 , y, t)
∇2 un (x, y, t) ≈
∆x2
(13)
un (x, yi+1 , t) − 2un (x, yi , t) + un (x, yi−1 , t)
+
∆y 2
The diffusivity of pn (approximation not shown) is discretized similarly. Once
the diffusive terms are discretized, the resulting system of ODEs is solved with
the built-in Octave function ode45, as in the non-spatial model. Moreover, after
implementing the algorithm to solve this problem, it is possible to look at the
evolution of the density of D. kuriphilus and T. sinensis over the years.
The results of the numerical solution of the model Equation (12), with
δ = 0.2, are presented in Fig. 3 for the years 3, 6, 9, 12, 15, 18, 21 and 23.
The simulation considered that the initial density of D. kuriphilus is maximal
everywhere except at a given place where it is zero, and the initial density of
T. sinensis is zero everywhere except at a certain place where is maximal. This
simulation allows to verify the impact of biological control once it is implanted
in a given location.
In Fig. 3, it may be seen that when there is an area where the density of
D. kuriphilus is maximal (Fig. 3 a), three years after its density is zero and T.
sinensis has appeared on the same area (Fig. 3 b). Thus, it can be deduced that
T. sinensis has eliminated the pest on that area. However, D. kuriphilus has
moved and has invaded another place. It spreads like a wave.
Regarding Fig. 3, the same cyclic phenomenon previously observed in the
non-spatial case is also observable. There is a cycle concerning the density of
the two species: when D. kuriphilus is maximum, T. sinensis appears; the later
controls D. kuriphilus, and therefore its density decreases; in turn, this leads to
the decreasing of the density of T. sinensis; as a result D. kuriphilus reappears
in the area. This cyclic phenomenon moves in the area over time.
So, biological control is a good way to control D. kuriphilus in a given area in
a few years. But when that control is over (because both species have disappeared
from the same area), D. kuriphilus returns and recolonizes that area. Thus, it
is necessary to look for a way to control D. kuriphilus more durably. It is also
important to know the time that D. kuriphilus and T. sinensis take to reappear.
3.3 Density of the Two Species Over Time in a Given Area
In this simulation, the density of the two species at the end of the year, in a
given space (matching the site shown in Fig. 3), is traced over time. The initial
conditions (year 0) sets the density of T. sinensis at its maximum (q0 = 1) while
that of D. kuriphilus is zero (v0 = 0). Fig. 4 depicts the densities over 40 years.
238 C. Balsa et al.
a) Year = 3 b) Year = 6
c) Year = 9 d) Year = 12
e) Year = 15 f) Year = 18
g) Year = 21 h) Year = 23
Fig. 3. D. kuriphilus and T. sinensis egg density after 3, 6, 9, 12, 15, 21, 23 years.
Numerical Simulation of the Biological Control of the Gall Wasp 239
Fig. 4. Eggs density of D. kuriphilus (vn ) and T. sinensis (qn ) in a given space over
the time
The observation of Fig. 4 permits to notice that a cycle appears. This is the
same cycle already observed in the previous numerical experiments. The density
of D. kuriphilus increases until it reaches a density of 1. Then it is the turn of the
density of T. sinensis to increase to reach 1 as well. When T. sinensis reaches a
certain density, 0.7 at the minimum, the density of D. kuriphilus begins to fall
sharply, reaching 0 in 2 years (on average). Similarly, when the density of D.
kuriphilus is very low, that of T. sinensis drops sharply so that it reaches 0. The
cycle is therefore composed of 4 stages that are repeated.
It may also be observed in Fig. 4 that as the years pass, the second stage of
the cycle takes more time to appear. Indeed, the more the cycles advance, the
more the density of D. kuriphilus remains at 1 before that of T. sinensis begins
to increase. For the 1st cycle the density of D. kuriphilus is 1 for 2.5 years. For
the 2nd cycle, its density is 1 for 5 years. Then for the last cycle, it is worth 1
at least 10 years. So T. sinensis takes more and more time to reappear in the
area.
Therefore, two questions concerning the release of T. sinensis must be an-
swered: i) should T. sinensis be reintroduced after several years? ii) if so, when
would it be necessary to reintroduce them to channel the density of D. kuriphilus
so that does not reach 1?
4 Improving the Biological Control
Along with the different numerical studies of the previous sections, it was noticed
that the biological control of D. kuriphilus by T. sinensis was effective, albeit
not totally. Indeed, although biological control can neutralize D. kuriphilus and
make it almost disappear from an area, the plague returns after a few years.
240 C. Balsa et al.
This happens for two reasons. The 1st is that D. kuriphilus moves from area to
area naturally, as it was observed in the spatial simulations. The 2nd is that T.
sinensis can not live without D. kuriphilus, with whom establishes a parasitic
relationship; thus, when D. kuriphilus is absent, T. sinensis dies and disappears.
Let’s briefly consider the two studies already presented. The first study looked
at the evolution of the density of the two species as a function of time without
taking into account the diffusive nature of the variables (Fig. 2). The last study
also focused on the density of the two species over time, but then considering
that the two species evolve in a two-dimensional space (Fig. 4).
On Fig. 2 and Fig. 4 the cycles mentioned earlier in this paper are easily
identifiable. Fig. 2 shows that once the density of D. kuriphilus reaches 10−9 , it
grows quickly, because T. sinensis has practically disappeared. In turn, Fig. 4
shows that T. sinensis takes more and more time to reappear in the area. This
is the reason why the hypothesis of reintroducing T. sinensis into a given area,
after a few years, was made in the previous section.
To verify the effect of reintroducing T. sinensis, the evolution of the density of
the two species, over 15 years, is now studied. The initial conditions are modified
so that, in the majority of the studied area, the density of T. sinensis is at 10−9 ,
as if it had been just released in this area and the density of D. kuriphilus is
zero as if it had just disappeared. In the rest of the study area, the density of
D. kuriphilus is maximal (v0 = 1) and the density of T. sinensis is minimal
(q0 = 0). Several cases will be considered in order to find when it is necessary to
reintroduce T. sinensis in function of D. kuriphilus density.
As a comparative baseline, that will allow to verify if injecting T. sinensis
into the area is useful, it was considered the evolution over 15 years of the two
densities, when T. sinensis is injected (q0 = 10−9 ) without D. kuriphilus inside
the considered area (v0 = 0). This baseline is presented in Fig. 5. This figure
shows that the introduction of T. sinensis into the zone when v0 = 0 gives rise
to a period of maximum density of D. kuriphilus for at least 10 years. Several
simulations were performed, with different initial values of the density of D.
kuriphilus (v0 ), and a T. sinensis density of q0 = 10−9 . It was observed that the
minimum period of maximum density of D. kuriphilus is reached with v0 = 10−9 .
The evolution of the densities of eggs of the two species, corresponding to
v0 = 10−9 , is depicted in Fig. 6. This figure shows that reintroducing T. sinensis
as soon as D. kuriphilus reaches its minimal value (v0 = 10−9 , see Fig. 2) and is
apparently extinct, reduces the years of the maximum density of D. kuriphilus to
4 years. This strategy is favorable for limiting the concentration of D. kuriphilus
in this zone for too long years.
However, it seems impossible to eliminate the density of D. kuriphilus. In
fact, it always comes to 1 after a while for several reasons. First, D. kuriphilus
propagates better in space than T. sinensis. Moreover, D. kuriphilus is a female
species, and so it can lay eggs much faster than T. sinensis, which is composed
of males and females. Finally, with T. sinensis laying its eggs in those of D.
kuriphilus, it depends on D. kuriphilus and therefore can not exist without it.
Numerical Simulation of the Biological Control of the Gall Wasp 241
Fig. 5. Eggs density of D. kuriphilus and T. sinensis when v0 = 0 and q0 = 10−9
Fig. 6. Density of D. kuriphilus and T. sinensis eggs when v0 = 10−9 and q0 = 10−9
5 Final Considerations
The previous numerical simulations suggest that the biological control of D. ku-
riphilus with T. sinensis works. Indeed, T. sinensis takes a few years to establish
itself on the territory. However, once this is done, the density of D. kuriphilus
decreases drastically. Indeed, in 2 to 3 years D. kuriphilus almost disappears
(only traces of its presence remain). However, it was also noticed that after such
disappearance, T. sinensis also vanishes from the same zone because it lays its
242 C. Balsa et al.
eggs in those of D. kuriphilus, who is now absent. After a few years, D. kuriphilus
recolonizes the same area, then T. sinensis resurges and a new cycle begins.
It was also observed that the more cycles pass, the more difficult it is for
T. sinensis to reappear in the same area (it takes more and more time), in
comparison to D. kuriphilus. This is due to several factors, such as the fact that
for D. kuriphilus only females exist, and the diffusion rate of both species differs.
Another important observation upon the simulations is that in an area where
D. kuriphilus and T. sinensis have disappeared, reinjecting T. sinensis from the
moment D. kuriphilus disappears will reduce the number of years during which
D. kuriphilus has a maximum density when it recolonizes that area. However,
reinjecting T. sinensis into an area does not completely eliminate D. kuriphilus.
This study is based on mathematical models that yield very precise results,
as to the value of the density of each species, of the order of 10−9 . But these
models depend on parameters that describe the reality. Their estimation would
be largely improved with the application of on-field instrumentation and sensors,
like those provided by the Internet of Things platforms. An integration of the
numerical simulations with these technologies would allow better tuning of the
mathematical models, making them more realistic and thus more useful.
Acknowledgement
UNIAG, R&D unit funded by the FCT – Portuguese Foundation for the Devel-
opment of Science and Technology, Ministry of Science, Technology and Higher
Education. Project n.o UID/GES/4752/2019.
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