Portugal is a country that produces chestnuts. However, in recent years, a D. kuriphilus plague has invaded chestnut trees and signi cantly a ected 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 infected area. However, the simulations also show that biological control is not e ective over time, as D. kuriphilus returns to the same area. Therefore, it would be necessary to reinject T. sinensis periodically into the infected zones for a sustained ght against D. kuriphilus.
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 signi cant impact on the production of chestnut, with further negative economical e ects.
Originally from China, the pest spread to other Asian countries, like Japan,
where it was rst detected in 1941. In the early 2000s, it arrived in the United
States and Europe, rst in Italy, then in France and nally in Portugal, in
(a) Wasp
(b) Chestnut gall
2014 [
Japan was the rst country that tried to control the plague. Japanese
researchers started by using pesticides, but this was found to be none ective.
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 e ectively 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 [
Researchers in Japan, Italy, and France began by studying this phenomenon in the laboratory and then on a speci c area of the country, to be able to observe and study the di erent e ects of T. sinensis on D. kuriphilus. These eld studies often involve a considerable amount of time, human resources and logistics. 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 representing 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
developed by F. Paparella and co-authors [
The mathematical models proposed by F. Paparella and co-authors [
The paper is organized as follows. Section 2 introduces the mathematical
models proposed by F. Paparella and co-authors [
Density of the D. kuriphilus and T. sinensis Over
In this section, the mathematical models that take into account the separate
evolution of D. kuriphilus and of T. sinensis, over time, are introduced,
followed by the complete mathematical model for the joint evolution of the two
species. These models are simpli ed 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 [
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 max of chestnut buds and the maximum M of eggs to be laid per bud, then the maximum density of eggs laid is:
Vmax = M max (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 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). Let Td be the length of the egg deposition season. The emergence rate is thus: Let a be the adult life span. Therefore, the death rate is:
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 max Vn(t): egg deposition rate = rd
M max
Vmax
Vn(t)
Un(t)
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: max Vn(t) Un(t) a Vmax a1 Un(t) +
Vn 1(Td)
Td
Non-dimensional variables are used in all the models of this work. For the model of this section such implies the following de nitions: un = Un= ( Vmax), vn = Vn=Vmax. Moreover, the new t, and Ed are de ned as t = t=Td, = Td=a and Ed = Nd. Consequently, the new model to which the initial conditions are added is given by: >> un(0) = 0 > > > > > >: vn(0) = 0 vn(t))un(t) vn(t))un(t) 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 de ned as the population of the egg-carrying T. sinensis females, 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 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: egg deposition rate = rt 1 Vn 1(Td) Qn(t) Pn(t)
Tt Vmax
Inserting in the previous formulation the non-dimensional variables pn = Pn= ( Vmax) { with accounting for the sex ration of T. sinensis {, qn = Qn= ( Vmax), t = t=Tt, = Tt= ( Td), and adding the initial conditions, such results in the new model for T. sinensis described in Equation (9).
1 (vn 1(1) >> pn(0) = qn 1( ) > > > > > >: qn(0) = 0
qn(t))pn(t) qn(t))pn(t) 2.3
The complete mathematical model enables to describe the evolution of D. kuriphilus 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 e ect of T. sinensis on D. kuriphilus. This e ect will be that parasitized larvae of the gasp won't give rise to adults. This e ect will have consequences on the emergence rate of D. kuriphilus, that now becomes: emergence rate = This results in the complete model described by Equation (11), where initial conditions are added: (7) (8) (9) >> un(0) = 0 > > > > > >>>> vn(0) = 0 > > > > >>>> pn(0) = qn 1( ) > > > > >: qn(0) = 0 (11) 2.4
The system of Ordinary Di erential Equations (11) was solved numerically by
the Octave built-in function ode45. This function combines the four and ve
order Runge-Kutta method (see [
Many model parameters are xed. There are two, however, that depend on
the geolocation of the species: the survival rate during overwintering of D.
kuriphilus ( ) and the accounts for the sex ration of T. sinensis ( ). Depending
on the value of these parameters, several types of evolution are possible. In this
work = 0:9 and = 0:45. These values correspond to the reality of Portugal [
Fig. 2 depicts the evolution of the eggs density of the two species along the time, in years. It is veri ed 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 de nitely eradicate D. kuriphilus. 3
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 di usivity term to the model, described
by Equation (11).
Let r2un be the di usivity of un, r2pn be the di usivity of pn, and represent
the di usivity ratio between the two species. It was observed that D. kuriphilus
moves faster than the parasite and travels longer distances [
qn(x; y; t))pn(x; y; t) qn(x; y; t))pn(x; y; t) (2 vn(x; y; t))un(x; y; t) + vn 1(x; y; 1) qn(x; y; ) vn(x; y; t))un(x; y; t)
The numerical solution of the system of partial di erential equations (12) implies
the discretization of the di usive term. Using the nite di erence method, as
stated in [
x2 2un(x; yi; t) + un(x; yi 1; t) y2 (13)
The di usivity of pn (approximation not shown) is discretized similarly. Once the di usive 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 = 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
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.
a) Year = 3 b) Year = 6 c) Year = 9 d) Year = 12 e) Year = 15 f) Year = 18 g) Year = 21 h) Year = 23
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 answered: 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
Along with the di erent numerical studies of the previous sections, it was noticed that the biological control of D. kuriphilus by T. sinensis was e ective, 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.
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 brie y consider the two studies already presented. The rst study looked at the evolution of the density of the two species as a function of time without taking into account the di usive 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 identi able. 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 e ect of reintroducing T. sinensis, the evolution of the density of the two species, over 15 years, is now studied. The initial conditions are modi ed 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 nd 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 gure 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 di erent 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 gure 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. The previous numerical simulations suggest that the biological control of D. kuriphilus 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 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 di cult 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 di usion rate of both species di ers.
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- eld 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.
UNIAG, R&D unit funded by the FCT { Portuguese Foundation for the Development of Science and Technology, Ministry of Science, Technology and Higher Education. Project n.o UID/GES/4752/2019.