Alexis Leculier
email: alexis.leculier@u-bordeaux.fr

Nga Nguyen

A control strategy for Sterile Insect Techniques using exponentially decreasing releases to avoid the hair-trigger effect

Keywords: sterile insect technique, hair-trigger effect, traveling waves

In this paper, we introduce a control strategy for applying the Sterile Insect Technique (SIT) to eliminate the population of Aedes mosquitoes which are vectors of various deadly diseases like dengue, zika, chikungunya... in a wide area. We use a system of reactiondiffusion equations to model the mosquito population and study the effect of releasing sterile males. Without any human intervention, and due to the so-called hair-trigger effect, the introduction of only a few individuals (eggs or fertilized females) can lead to the invasion of mosquitoes in the whole region after some time. To avoid this phenomenon, our strategy is to keep releasing a small number of sterile males in the treated zone and move this release forward with a negative forcing speed c to push back the invasive front of wild mosquitoes. By using traveling wave analysis, we show in the present paper that the strategy succeeds in repulsing the population while consuming a finite amount of mosquitoes in any finite time interval even though we treat a moving half-space {x > ct}. Moreover, we succeed in constructing a 'forced' traveling wave for our system moving at the same speed as the releases. We also provide some numerical illustrations for our results.

1 Introduction

The biological motivation

Pest and disease vector controls have become a global issue because of the spread of these species all around the world causing crop losses and disease epidemics. For example, the oriental fruit fly is a serious pest of a wide variety of fruit crops in Asia and has also invaded a number of other countries and is a very damaging pest wherever it occurs (PMP-FFM, 2004). It was first detected in French Polynesia in 1996 and invaded Africa in 2004. Few individuals have been detected in Italy in 2018 and hence southern Europe is at high risk. Similarly, according to the World Health Organization, the global incidence of dengue has grown dramatically with about half of the world's population now at risk. It was first identified in the 1950s during dengue epidemics in Philippines and Thailand due to the travel and invasion of its vectors, female mosquitoes of the species Aedes aegypti and Ae. albopictus. They are also vectors of chikungunya, yellow fever, Zika viruses..., and, until now, there is neither effective treatment nor vaccine for these diseases. So pest/vector controls play an important role in getting rid of these problems. The classical control method based on insecticides induces resistance, which reduces its own efficiency and is detrimental to the environment. Among others, the Sterile Insect Technique (SIT) aiming at reducing the size of the insect population recently gathered much attention. The SIT is a biological method where people release sterile individuals (modified in laboratories) of pest species to introduce sterility into the wild population, and thus control it (see [START_REF] Dyck | Sterile Insect Technique: Principles and Practice in Area-Wide Integrated Pest Management[END_REF] for an overall presentation of SIT). It is a promising control method against many agricultural pests and disease vectors, most notably screw worms and fruit flies (see [START_REF] Dyck | Sterile Insect Technique: Principles and Practice in Area-Wide Integrated Pest Management[END_REF]), and recently mosquitoes of genus Aedes. This technique has been applied successfully for Aedes mosquitoes in the field in many different countries, for instance, in Italy [START_REF] Caputo | A bacterium against the tiger: preliminary evidence of fertility reduction after release of aedes albopictus males with manipulated wolbachia infection in an italian urban area[END_REF], Cuba [START_REF] Gato Armas | Sterile insect technique: Successful suppression of an aedes aegypti field population in cuba[END_REF], and China [START_REF] Zheng | Incompatible and sterile insect techniques combined eliminate mosquitoes[END_REF]. In our work, we focus on applying SIT in a vast region using the idea of the "rolling carpet": a large number of sterile insects are released near the front of the invasion, and as soon as this area is free from wild insects, we move the front of release and continue to release a few sterile individuals in the already treated area (see [START_REF] Dyck | Sterile Insect Technique: Principles and Practice in Area-Wide Integrated Pest Management[END_REF]). The purpose of these small releases at the back is to prevent reinvasion by the so-called hair-trigger effect (where the existence of just a few individuals leads to the total invasion of the territory). The notion of 'hair-trigger' was first introduced in [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF] to refer to the persistence in long-time of the solution with respect to any non-trivial initial data. In our case, it has been observed in [START_REF] Dyck | Sterile Insect Technique: Principles and Practice in Area-Wide Integrated Pest Management[END_REF] that the mosquitoes reinvade the treated territory without this small amount of releases of sterile males. By implementing such a process, we succeed in eradicating wild insects, preventing reinvasion, and keeping the number of released sterile insects below a threshold in a finite time interval [0, T ]. It is in our interest to consume as few sterile males as possible since it is one of the main costs of the strategy. We propose in the present work to study a mathematical model of such release strategy used in the field for Aedes mosquitoes.

Our model and the spreading results

Following ideas in e.g. [START_REF] Almeida | Analysis of the "Rolling carpet" strategy to eradicate an invasive species[END_REF], [START_REF] Strugarek | On the use of the sterile insect release technique to reduce or eliminate mosquito populations[END_REF], we model the mosquito population by a partially degenerate reactiondiffusion system for time t > 0, position x ∈ R:

                     ∂ t E = β F 1 - E K -(ν E + µ E )E, ∂ t F -D∂ xx F = rν E E M M + γM s -µ F F, ∂ t M -D∂ xx M = (1 -r)ν E E -µ M M, ∂ t M s -D∂ xx M s = Λ(t, x) -µ s M s ,
(E, F, M, M s )(t = 0, x) = (E 0 , F 0 , M 0 , M 0 s )(x).

(1)

In this system, we have:

• E, M, M s and F denote respectively the number of mosquitoes in the aquatic phase, adult males, sterile adult males and fertilized adult females depending on time t and position x;

• Λ(t, x) is the number of sterile mosquitoes that are released at position x and time t;

• the fraction M M+γM s corresponds to the probability that a female mates with a fertile male, and parameter γ models the competitivity of sterile males;

• β > 0 is a birth rate; µ E > 0, µ M > 0, and µ F > 0 denote the death rates for the mosquitoes in the aquatic phase, for adult males and for adult females, respectively;

• K is an environmental capacity for the aquatic phase, accounting also for the intraspecific competition;

• ν E > 0 is the rate of emergence;

• D > 0 is the diffusion rate;

• r ∈ (0, 1) is the probability that a female emerges, then (1r) is the probability that a male emerges;

• the initial data (E 0 , F 0 , M 0 , M 0 s ) ≥ (0, 0, 0, 0) (component by component).

We introduce the basic offspring number as follows

R 0 = β rν E µ F (ν E + µ E ) . ( 2
)
When there is no regulation of sterile males, our model becomes

       ∂ t E = β F 1 -E K -(ν E + µ E )E, ∂ t F -D∂ xx F = rν E E -µ F F, ∂ t M -D∂ xx M = (1 -r)ν E E -µ M M, (3)
It is obvious that (0, 0, 0) is an equilibrium of [START_REF] Almeida | Optimal control of bistable travelling waves: looking for the best spatial distribution of a killing action to block a pest invasion[END_REF]. When the basic offspring number R 0 > 1, this system has a second equilibrium (E * , F * , M * ) where

E * =K β rν E -µ F (ν E + µ E ) β rν E > 0, F * =K β rν E -µ F (ν E + µ E ) β µ F > 0, M * =K 1 -r r β rν E -µ F (ν E + µ E ) β µ M > 0. ( 4
)
Note that, the positive equilibrium (E * , F * , M * ) is stable and (0, 0, 0) is unstable. Thus, in the case without sterile males, the following result shows the spread of the population toward the positive equilibrium and provides the existence of the spreading speed for the solution of the system (3).

Proposition 1. If the basic offspring number R 0 > 1, then there exists a spreading speed c * > 0 such that for any positive ε, the solution (E, F, M) of system (3) satisfies

• if the initial data (E 0 , F 0 , M 0 ) is compactly supported and 0 ≤ (E 0 , F 0 , M 0 ) < (E * , F * , M * ), then lim t→+∞ max |x|≥t(c * +ε) max(E, F, M)(t, x) = 0, (5)
• if the initial data (E 0 , F 0 , M 0 ) < (E * , F * , M * ) and if there exists a set with a positive measure Ω ⊂ R, such that max(min

x∈Ω E 0 , min x∈Ω F 0 ) > 0 then lim t→+∞ max |x|≤t(c * -ε) max(E * -E, F * -F, M * -M)(t, x) = 0. ( 6
)
We present in Appendix A a proof for this result based on the result in the work of Lui [START_REF] Lui | Biological growth and spread modeled by systems of recursions. I. mathematical theory[END_REF] with an extension for reaction-diffusion system in Weinberger et al. [START_REF] Weinberger | Analysis of linear determinacy for spread in cooperative models[END_REF] for a monostable system. We also underline that, with only females at the initial time (i.e. E 0 ≡ 0, M 0 ≡ 0, and F 0 > 0 in some ball), invasion still occurs. This is due to the fact that in our model we consider F to be the fertilized females. Therefore, if F 0 > 0 on a set with a positive measure, then the same holds for the aquatic phase at any t > 0 in the whole domain R, and the dynamics of invasion start to occur.

The main result in the present work shows that when a release function of sterile males Λ moving with a certain speed c < 0 is imposed in the system, we can succeed in suppressing the mosquitoes and in avoiding reinvasion. In the present work, we consider the release function

Λ(t, x) = 0 for x -ct ≤ 0, Ae -η(x-ct) for x -ct > 0, (7)
with constants A > 0, η > 0.

Theorem 2. If the basic offspring number R 0 > 1, (E 0 , F 0 , M 0 ) ≤ (E * , F * , M * ), (E 0 , F 0 , M 0 ) |R + = (0, 0, 0) and M 0 s ≥ φ s , where φ s is the solution of -cφ s -φ s = Ae -ηx 1 {x>0} -µ s φ s and φ s (±∞) = 0
then for any speed c < 0, there exist A c > 0, η c > 0 such that for any A ≥ A c , 0 < η ≤ η c , we have the solution (E, F, M, M s ) of system ( 1) and ( 7) satisfies

lim t→+∞ sup x>ct max(E, F, M)(t, x) = 0.
From this result, we can see that if the initial data is compactly supported in R -and below (E * , F * , M * ), the invasion does not occur: the equilibrium (0, 0, 0) invades the positive equilibrium (E * , F * , M * ). We also remark that the number of sterile males released in the field in a finite time interval [0, T ] is T × A η finite even though the space is infinite. However, if T → +∞, the total amount of released mosquitoes also tends to +∞. Finally, we point out that in the above results, the number of sterile males released (A, η) depends on the speed c of the rolling carpet. This can be observed more precisely in the proof in section 4 and discussed in section 2 with some numerical illustrations. However, finding A, η that minimizes the number of released mosquitoes each time remains a challenge.

State of the art

Based on biological knowledge, mathematical modeling and numerical simulations can be additional and useful tools to prevent failures, improve protocols, and test assumptions before applying the SIT strategy in the field. Many works have been done using mean-field temporal models to assess the SIT efficiency for a long-term period (see e.g. [START_REF] Bliman | Implementation of Control Strategies for Sterile Insect Techniques[END_REF], [START_REF] Strugarek | On the use of the sterile insect release technique to reduce or eliminate mosquito populations[END_REF] and references therein). Only a few works exist modeling explicitly the spatial component due to the lack of knowledge about vectors in the field. Moreover, from the mathematical point of view, the studies of spatial-temporal models are more sophisticated. A reaction-diffusion equation was first used in [START_REF] Manoranjan | On a diffusion model for sterile insect release[END_REF] to model the spreading of a pest in the SIT model. Then, the model was completed by considering the release of sterile females in [START_REF] Lewis | Waves of extinction from sterile insect release[END_REF]. In this article, the author assumed that the same amount of sterile insects is released in the whole field (i.e. Λ ≡ constant). It follows that if the number of released sterile insects is large enough, the reaction term becomes strictly negative, and the extinction of the wild population follows. However, this hypothesis is unrealistic in a large area since the number of sterile insects to release tends to infinity as the size of the domain increases. The main contribution of our work is to tackle this problem by following what has been done in the field experiment: we assume that the releases are not homogeneous. By considering only releases supported in R + with exponential decay, the amount of sterile males released in a finite time interval is constant. In [START_REF] Seirin Lee | Modelling Aedes aegypti mosquito control via transgenic and sterile insect techniques: Endemics and emerging outbreaks[END_REF], the authors studied SIT control with barrier effect using a system of two reaction-diffusion equations for the wild and the sterile populations. Recently, a sex-structured system including the aquatic phase of mosquitoes has been studied in [START_REF] Anguelov | On the use of Traveling Waves for Pest/Vector elimination using the Sterile Insect Technique[END_REF]. Using the theory of traveling waves, the authors proved, for a similar system to (1), the existence of natural invading traveling wave when {M s = 0} and the system is either monostable or bistable. They also provide some numerical implementation of the SIT but only for the bistable case.

In the bistable case, one can release the mosquitoes in a compact set since the equilibrium 0 is stable. The main result in [START_REF] Almeida | Analysis of the "Rolling carpet" strategy to eradicate an invasive species[END_REF] shows that if the initial wild mosquitoes distribution behaves as 1 R -and we release enough sterile males in some compact set (ct, L + ct) with a speed c < 0, then the wild population remains close to 0 in the set {x > L + ct} thanks to the assumed natural dynamics of the mosquitoes. We also quote [START_REF] Almeida | The sterile insect technique used as a barrier control against reinfestation[END_REF][START_REF] Almeida | Wave blocking in a bistable system by local introduction of a population: application to sterile insect techniques on mosquito populations[END_REF], which was done before [START_REF] Almeida | Analysis of the "Rolling carpet" strategy to eradicate an invasive species[END_REF] where the authors studied the analogous system of reaction-diffusion equations to (1) in a bistable context taking into account the strong Allee effects. They proved that for large enough constant releases in a bounded interval, there exists a barrier that blocks the invasion of mosquitoes. However, for the monostable case, they obtain numerically that there is no blocking. The so-called "hair-trigger effect" makes the monostable case become more complicated since one can not rely on the natural dynamics of the mosquitoes. So the main purpose of the present work is to study an efficient strategy for the SIT to deal with the difficulty in this case.

The control of sterile insect techniques in a bistable context in a bounded domain is studied in [START_REF] Trélat | Allee optimal control of a system in ecology[END_REF][START_REF] Trélat | Optimal Population Control Through Sterile Males[END_REF]. We also quote [START_REF] Bressan | On the optimal control of propagation fronts[END_REF][START_REF] Almeida | Optimal control of bistable travelling waves: looking for the best spatial distribution of a killing action to block a pest invasion[END_REF] that focus on the optimal form to stop or repulse an invading traveling wave by spreading a killing agent (such as insecticide). In [START_REF] Bressan | On the optimal control of propagation fronts[END_REF] the authors study the optimal shape of spreading in order to repulse an invasion. In [START_REF] Almeida | Optimal control of bistable travelling waves: looking for the best spatial distribution of a killing action to block a pest invasion[END_REF], the authors study the optimal shape of spreading in order to block an invasion but consider more constraints on the spreading area than in [START_REF] Bressan | On the optimal control of propagation fronts[END_REF]. The key argument in these works is to consider that the reaction term is bistable. In the present work, we propose a way to deal with the difficulty of the monostable case with a finite amount of control agents (such as sterile insects or insecticides, or other kinds of control) in any finite time interval.

The traveling wave results

Another natural question that arises in the study of our model of reaction-diffusion equations is the existence of a traveling wave solution. First, we study the traveling wave problem for the system (3) in which there are no sterile male. Then, by imposing a control function Λ that moves with a speed c < 0, we will construct a traveling wave for the main system (1) moving with the same speed.

Recall that a traveling wave solution of (3) with any speed c is the pair (U, c) where U = (E, F, M) T and U(xct) is a nontrivial and bounded solution of (3). We say (U, c) is a wavefront if U(±∞) exist and U(-∞) = U(+∞). The existence of such wavefronts for reaction-diffusion systems has been studied widely in the literature. In our case, the nonlinearity is monostable and it is well-known that there exists a minimal speed such that the monostable system admits traveling wave solutions with any speed larger than this minimum value. For example, in the book [START_REF] Volpert | Traveling Wave Solutions of Parabolic Systems[END_REF], the authors studied the existence of minimal speed and the stability of wavefronts for the non-degenerate system. However, our systems are partially degenerate because the first stage E is quiescent (does not diffuse). The paper [START_REF] Fang | Monotone Wavefronts for Partially Degenerate Reaction-Diffusion Systems[END_REF] studied monotone wavefronts for partially degenerate systems and they proved that the spreading speed of the solution is the minimal wave speed of monotone wavefronts in the monostable cooperative case. The authors of [START_REF] Anguelov | On the use of Traveling Waves for Pest/Vector elimination using the Sterile Insect Technique[END_REF] proved the same result for a similar system to (3) and for the sake of completeness, we present it in the following Proposition 3. Let c * be defined in Proposition 1, then for each c + ≥ c * , system (3) has a nonincreasing wavefront U(xc + t) connecting (E * , F * , M * ) and (0, 0, 0). While for any c + ∈ (0, c * ), there is no wavefront connecting (E * , F * , M * ) and (0, 0, 0).

The general system (1) (with M s > 0) is not cooperative at first glance. Some works in the recent literature have tackled the lack of comparison principle for non-cooperative Fisher-KPP systems (see e.g. [START_REF] Girardin | Non-cooperative Fisher-KPP systems: traveling waves and long-time behavior[END_REF][START_REF] Girardin | Non-cooperative Fisher-KPP systems: Asymptotic behavior of traveling waves[END_REF][START_REF] Girardin | A Liouville-type result for non-cooperative Fisher-KPP systems and nonlocal equations in cylinders[END_REF]). However, due to the fact that the system (1) in the present paper is partially degenerate, that is, it does not satisfy that min D ii > 0 where D is the diffusive matrix, we can not apply these results in our work. Fortunately, the system (1) can be put in the setting of cooperative systems by the change of variable ( M s = C -M s with C a large constant). With this change of variable, we define a new order for the solutions (E, F, M, M s ) of (1) such that

(E 1 , F 1 , M 1 , M 1 s ) ≥ (E 2 , F 2 , M 2 , M 2 s ) if E 1 ≥ E 2 , F 1 ≥ F 2 , M 1 ≥ M 2 , M 1 s ≤ M 2 s .
In section 4.1, we present more precisely the comparison principle used in our problem.

One of the main interests of this article is the establishment of a 'forced' traveling wave solution for (1) with a control function Λ as in [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF]. Dealing with the whole system of ODE-PDE like (1) is by no means an easy task, so our first idea is to try to simplify the system to a single reaction-diffusion equation by adding some assumptions and then find a general strategy to study the full model. When we assume that the equilibrium of the aquatic phase is attained instantaneously (i.e. ∂ t E = 0) then from the first equation of (1), one has

E = β F β F K + ν E + µ E
. Thus, if the number of females F is equal to the number of males M, and the sterile males are assumed to be equal to Λ in the treating time interval [0, T ], using the second equation of (1), we end up with only a single equation :

∂ t F -D∂ xx F = F F + Λ β F β F K + ν E + µ E -µ F F. ( 8
)
The model of a scalar reaction-diffusion equation was used widely in the literature studying SIT (see e.g. [START_REF] Lewis | Waves of extinction from sterile insect release[END_REF], [START_REF] Zhu | Stability and periodicity in a mosquito population suppression model composed of two sub-models[END_REF]) or in other contexts, for e.g. in climate change [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate?[END_REF], [START_REF] Berestycki | Forced waves of the Fisher-KPP equation in a shifting environment[END_REF]. In our case, the source term Λ(t, x) moving with a certain speed c < 0, we can construct the 'forced' traveling wave solution of (8) moving with the same speed. Equation ( 8) having the form

∂ t u -∂ xx u = f (x -ct, u) with f (s, u) : R × R + → R
is asymptotic of F-KPP type as s → ±∞ and was studied in the literature (see e.g. [START_REF] Berestycki | Forced waves of the Fisher-KPP equation in a shifting environment[END_REF] and references therein). In the present work, even if it has been already studied, we provide in Section 3 an explicit construction of the 'forced' wave for [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate?[END_REF] which can help to grasp the general idea of the proof for the whole system. Indeed, back to the main model of the present paper, we infer from the scalar model that the main difficulty lies in the construction of the super-solution of (1). The forced wave has the form

(E, F, M, M s )(t, x) = (φ E , φ F , φ M , φ s )(x -ct), where c < 0 is the forced speed and (φ E , φ F , φ M , φ s ) is the profile satisfying                  -cφ E = β φ F 1 - φ E K -(ν E + µ E )φ E , -cφ F -Dφ F = rν E φ E φ M φ M + γφ s -µ F φ F , -cφ M -Dφ M = (1 -r)ν E φ E -µ M φ M , -cφ s -Dφ s = φ -µ s φ s , (9)
where φ (xct) = Λ(t, x) in [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF]. To overcome the difficulty of the construction of the super-solution, we use the fact that the dynamic is governed, in some sense, by F. Thanks to what was observed for the scalar equation, we have a natural candidate to be the super-solution for F. More in detail, by denoting φ E , φ F , φ M respectively the super-solution of φ E , φ F , φ M , we proceed as follows:

• Step 1. Fix φ s = φ s = C s e -ηx 1 {x>0} . We insert φ F = F * 1 {x≤0} + 1 {x>0} e -λ
x in the first equation of (9),

• Step 2. We prove that with such a φ F the associated φ E satisfies φ E ≤ Ce -λ x for x large enough,

• Step 3. We insert φ E = E * 1 {x≤0} + 1 {x>0} e -λ
x in the third equation of ( 9),

• Step 4. We prove that with such a φ E the associated φ M satisfies φ M ≤ Ce -λ x for x large enough

• Step 5. We define φ M = M * 1 {x≤0} + 1 {x>0} e -λ x ,
• Step 6. We prove that (φ E , φ F , φ M , φ s ) is a super-solution of ( 9), where φ s is the solution of the last equation with φ s (±∞) = 0.

We present precisely this construction of a super-solution in section 4.3 and we also construct a subsolution in section 4.4. Therefore, we obtain the main result:

Theorem 4. If the basic offspring number R 0 > 1, then for any speed c < 0, there exist A c > 0, η c > 0 such that for any A ≥ A c , 0 < η ≤ η c , system (9) with φ (xct) = Λ(t, x) defined in ( 7) admits a solution

(φ E , φ F , φ M , φ s ) such that (φ E , φ F , φ M ) converges to (E * , F * , M * ) at -∞ and to (0, 0, 0) at +∞.
We underline that to obtain the exact limits at -∞ is technical since the sterile males diffuse, φ s > 0 everywhere and the system is not heterogeneous in R -(contrary to the super-solutions). Using a perturbation of the equilibrium (E * , F * , M * ), we succeed in obtaining a sub-solution (φ E , φ F , φ M , φ s ). However, this sub-solution satisfies lim

x→-∞ (φ E , φ F , φ M , φ s ) = (E * -ε E , F * -ε F , M * -ε M , ε 0 ) (where ε E,F,M,0
are small positive constants) so we can not deduce directly the limit of (φ E , φ F , φ M , φ s ) at -∞. We prove that the solution of (9) satisfies the desired limit at -∞ by contradiction (see Section 4.5).

Outline of the paper

The outline of the rest of this paper is the following: section 2 is devoted to showing some numerical illustrations to support our theoretical results. Next, in section 3 we provide the technical details for the results stated for the simplified model. Finally, section 4 is devoted to the technical details that allow proving Theorems 2 and 4. As mentioned in the introduction, the results for the case without any sterile males (Propositions 1 and 3) are applications of former works. For the sake of completeness, we present the proofs in Appendix A.

Numerical illustrations 2.1 The numerical scheme

In this section, we present some numerical illustrations for our theoretical results using a simple finite difference scheme. Since we study the model in one-dimensional space, we use a semi-implicit secondorder scheme for space discretization, and a first-order explicit scheme for time discretization, with the time step following a CFL condition. We use Neumann boundary conditions on the boundary of a very large spatial interval. It is well-known that such a spatial domain approximates correctly R, or at least regarding spreading properties of reaction-diffusion systems.

Observations

The values of parameters are chosen following [START_REF] Dufourd | Impact of environmental factors on mosquito dispersal in the prospect of sterile insect technique control[END_REF] for mosquitoes of species Aedes albopictus and presented in Table 1. With these parameters, we first verify that the basic offspring number R 0 ≈ 30.77 > 1, thus the condition in our theorems is satisfied. The positive equilibrium is (E * , F * , M * ) ≈ (193.5, 77.4, 55.3).

In our plots, the time unit is a day, space unit is 1 km. We consider the domain [-50, 50] of width 100 km discretized by 500 points, with a 60-day time interval. We show in Figure 1 the dynamics of the female population over time and space. In this simulation, the initial data are taken as compactly

β K ν E µ E µ F µ M µ s γ r D Values
10 200 0.08 0.05 0.1 0.14 0.14 1 0.5 0.5 supported functions. When there is no SIT control, the wave of mosquitoes invades the space (see Figure 1a) and approaches the steady state F * = 77.4. This illustrates the invasion phenomenon in Proposition 1.

To stop this invasion, we keep releasing sterile mosquitoes over time with a release function that decays exponentially on half of the space Λ(t, x) = Ae -η(x-ct) 1 {x>ct} .

In practice, the number of sterile males to release is usually fixed and one can adjust the speed of the releases to obtain the best result. To illustrate our result, first, we fix A = 600, η = 0.2 and vary the speeds c ≤ 0 to observe the dynamics of mosquitoes while applying SIT. When we do not move the release (c = 0), we observe in Figure 1b that the wave is blocked near x = 0 and cannot pass through the release zone. Then, by moving this release domain to the left with velocity c = -0.3, we succeed to push back the wave to the left (see Figure 1c), and there is no mosquito behind the releases which illustrates the main result in Theorems 4 and 2. However, we observe in Figure 1d that if we move the releases faster to the left with velocity c = -0.7, there is a reinvasion on the right of the zone. It seems that the faster we move the release domain, the faster we push back the mosquito waves, but we need to release more sterile males in the treated zone to prevent reinvasion. Indeed, when the speed c = -0.7 is fixed and the number of sterile males is increased by taking A = 800 (see 1e), and η = 0.1 (see 1f), one can see that the reinvasion in the treated zone gets slower and disappears.

3 Study of the simplified model

The simplified model

From [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate?[END_REF] we study in this section the following scalar equations:

     ∂ t u -∂ xx u = u u + Λ β u β u K + δ -µu, for x ∈ R,t > 0, u(t = 0, x) = u 0 (x). (10)
where β , δ , µ, K are parameters, u is the density of mosquitoes, and the function Λ(t, x) is the control (i.e. the number of sterile males released). In order to ensure the existence of a non-trivial steady state, we need the following assumption:

Assumptions 1. The parameters β , δ , µ, K are positive and βµδ > 0.

We first treat briefly the case without any control (i.e. Λ = 0) and then we explain how to obtain a similar result to Theorem 2.

The case without control

In this case, when Assumption 1 holds, the equation has two equilibria u 0 = 0 and u * = K(βµδ )

β µ > 0.
The reaction term f (u) := β u µu = f (0)u. Then, from the result in [START_REF] Kolmogorov | A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem[END_REF], there exists a number c * > 0 such that [START_REF] Bliman | Implementation of Control Strategies for Sterile Insect Techniques[END_REF] possesses "natural" traveling wave solutions u(t, x) = v N (xc + t) for all speed c + ≥ c * with v N solutions of

β u K + δ -µu > 0 for any u ∈ (0, u * ), f (0) = β δ -µ > 0, and f (u) < β u δ - (
     -c + v N -v N = β v N β v N K + δ -µv N , v N (-∞) = u * , v N (+∞) = 0.
Hence, when t → +∞, the positive state u = u * invades the extinction state u = 0 (see [START_REF] Aronson | Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation[END_REF]Theorem 4.1] for more details). We recall the following classical result Proposition 5. [6, Theorem 4.1] For any positive initial data u 0 , the solution of (10) with Λ ≡ 0 satisfies

∀c + ≥ c * , lim t→+∞ sup |x|<c + t |u(t, x) -u * | = 0, .
Remark 1. Depending on the initial data, the front can go faster and even accelerate (see [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF]). But, in any case, the steady-state u * invades the steady state 0 at least with a speed c * .

The controlled case

In this case, function Λ is considered as in ( 7), and we prove the existence of a forced traveling wave moving with the same speed as Λ satisfying

     -cv -v = v v + φ β v β v K + δ -µv, v(-∞) = u * , v(+∞) = 0, (11)
with φ (xct) = Λ(t, x) and speed c negative. The result is the following: Theorem 6. For any c < 0, there exists constants A, η > 0 such that for any A ≥ A, 0 < η ≤ η, and the release function φ (xct) = Λ(t, x) defined in [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF], problem (11) admits a solution v.

Then, we have the following result for the space-time model [START_REF] Bliman | Implementation of Control Strategies for Sterile Insect Techniques[END_REF] (which is an analog to Theorem 2): Theorem 7. For any initial data u 0 ≥ 0 with u 0 ≤ u * and u 0|R + = 0 and c ≤ 0, there exist constants A, η > 0 such that for any A ≥ A, 0 < η ≤ η, and the release function φ (xct) = Λ(t, x), one has that the solution u of (10) satisfies, with any ε > 0, that

lim t→+∞ sup x>(c+ε)t u(t, x) = 0.
By imposing a control with exponential decay, we succeed in suppressing the insects in the region behind the release. It is contrary to what happens naturally (when the stable steady state u * invades the unstable steady state 0). Notice that the hypothesis on the initial data u 0 takes into account any positive and compactly supported initial data bounded by u * (up to a translation of the support in R -).

In the following section, we construct a super-solution for [START_REF] Bressan | On the optimal control of propagation fronts[END_REF] in Proposition 8, then we can apply this result to prove Theorem 7. The existence of a sub-solution of ( 11) is presented in Proposition 10 in section 3.3. Finally, by using comparison principle for a scalar reaction-diffusion equation, we prove Theorem 6.

Construction of a super-solution for the simplified model

The existence of super-solution for [START_REF] Bressan | On the optimal control of propagation fronts[END_REF] is shown in the following proposition Proposition 8. For any fixed speed c and any fixed parameter α ∈ 0, δ µ β , there exists a constant r(α) < 0 depending on α, c such that the function

w(x) = u * when x < 0, u * e r(α)x when x ≥ 0, ( 12
)
is a super-solution of [START_REF] Bressan | On the optimal control of propagation fronts[END_REF] with

φ (x -ct) = Λ(t, x) for any η ∈ [0, -r(α)] and A ≥ u * α -u * > 0.
Proof of Proposition 8. For a constant c < 0, we study the following problem

   -cw -w = αβ δ -µ w on [0, +∞), w > 0 on [0, +∞), w(+∞) = 0. ( 13
)
Consider the characteristic polynomial r 2 + cr + αβ δ µ = 0, since αβ δ µ < 0 then for any c < 0, the polynomial admits two distinct roots

r ± = -c ± c 2 -4 αβ δ -µ 2
where r + > 0 and r -< 0.

Since we look for a solution w of ( 13) with w(+∞) = 0, then the solution of ( 13) is

w(x) = u * e r(α)x for x > 0, (14)
with r(α

) = r -= -c -c 2 -4 αβ δ -µ 2 < 0.
Now, remarking that Assumption 1 provides δ µ β ≤ 1, it follows for any α ∈ (0, δ µ β ) and any constant η ∈ [0, -r(α)] and A ≥ u * α u * > 0, one defines φ (xct) = Λ(t, x) as in [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF], then for all x ∈ [0, ∞), one

has w(x) w(x) + φ (x) = u * e r(α)x u * e r(α)x + Ae -ηx = u * u * + Ae -(η+r(α))x ≤ α. We deduce that -cw -w - w w + φ β w β w K + δ + µw ≥ -cw -w - αβ δ -µ w = 0.
For any x < 0, one has w(x) = u * and

-cw -w - w w + φ β w β w K + δ + µw = - β u * β u * K + δ + µu * = 0.
Moreover, we have lim

x→0 -w (x) = 0 > r(α)u * = lim x→0 + w (x).
Hence, function w as in ( 12) is a supersolution of [START_REF] Bressan | On the optimal control of propagation fronts[END_REF] with any φ (xct) = Λ(t, x) of the form in [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF].

From the existence of this super-solution we have the following proof of Theorem 7:

Proof of Theorem 7. Let u(t, x) = w(xct), Λ(t, x) = φ (xct), and w, φ provided by Proposition 8 with a certain speed c < 0. It is clear that with such a choice of Λ(t, x), we have that u is a supersolution of [START_REF] Bliman | Implementation of Control Strategies for Sterile Insect Techniques[END_REF]. Thanks to the definition of w, we have u 0 (x) ≤ u(t = 0, x), therefore, the comparison principle implies that for any t > 0, x ∈ R, u(t, x) ≤ u(t, x). For any ε > 0, x > (c + ε)t, one has u(t, x) ≤ u * e r(α)εt → 0 when t → +∞, then the result follows.

Construction of a sub-solution for the simplified model

We are going to construct this sub-solution by part. In the part where φ ≡ 0, we recall f (s) = β s

β s K + δ -µs
which corresponds to the reaction term of [START_REF] Bressan | On the optimal control of propagation fronts[END_REF] with φ ≡ 0. Consider the following system

   -w = f (w) in R -, w(0) = 0; lim x→0 -w (x) = -2 u * 0 f (s)ds. (15)
We have the following Lemma Lemma 9. System (15) admits a solution w ≥ 0 such that for any x < 0 w (x) < 0 and lim

x→-∞ w(x) = u * .
Proof. By Cauchy-Lipschitz theorem, problem (15) admits a solution w ≥ 0 in [-L 0 , 0) for some L 0 ∈ (0, +∞]. Multiplying the first equation of ( 15) by w and integrating in (-L, 0) for some L ∈ (0, L 0 ], we have

- 0 -L (w ) 2 2 dx = 0 -L f (w)w dx, then w (-L) 2 2 - w (0) 2 2 = - w(-L) 0 f (s)ds.
From ( 15), we have w (0

) 2 = 2 u * 0 f (s)ds then w (-L) 2 2 = u * w(-L) f (s)ds. ( 16
)
Since f is monostable, then w (-L) = 0 if and only if w(-L) = u * . Define

L := inf{x > 0 : w (-x) = 0} = inf{x > 0 : w(-x) = u * } ≤ +∞. (17)
If L < +∞, from the definition of L one has w (-L) = 0 and w(-L) = u * . However, u * is a stable equilibrium of equation -w = f (w), so w(-L) = u * implies that w ≡ u * . This is contradictory to the fact that w(0) = 0. Hence, L = +∞. So we have w (x) < 0 and w(x) < u * for any x < 0. We can deduce from this bound that w converges when x → -∞. Since lim x→-∞ w(x) < w(0) = 0, then w converges to u * . Now, we can use the solution w of (15) to construct a sub-solution of [START_REF] Bressan | On the optimal control of propagation fronts[END_REF].

Proposition 10. For any c < 0, problem (11) has a sub-solution w which is defined as follows

w(x) = w(x) when x < 0, 0 when x ≥ 0, ( 18
)
with φ (xct) = Λ(t, x).

Proof. For any c < 0, for any x < 0, one has φ (x) = 0, w(x) = w(x), w (x) < 0, then

-cw -w - w w + φ β w β w K + δ + µw = -cw -w -f (w) = -cw < 0. Moreover, lim x→0 -w(x) = -2 u * 0 f (s)ds < 0 = lim x→0 + w(x).
Hence, w is a sub-solution of [START_REF] Bressan | On the optimal control of propagation fronts[END_REF].

Construction of a traveling wave solution for the simplified model

We construct a solution from the above sub-and super-solutions.

Proof of Theorem 6. According to Propositions 8 and 10, for the control function φ (xct) = Λ(t, x), problem [START_REF] Bressan | On the optimal control of propagation fronts[END_REF] has the super-solution w as in ( 12) and the sub-solution w as in [START_REF] Fang | Monotone Wavefronts for Partially Degenerate Reaction-Diffusion Systems[END_REF]. Moreover, the suband super-solutions are well-ordered : w ≤ w (see Figures 2). By applying the classical technique of sub-and super-solution (see e.g. [START_REF] Smoller | Shock Waves and Reaction-Diffusion Equations[END_REF]), there exists a classical solution of [START_REF] Bressan | On the optimal control of propagation fronts[END_REF]. Moreover, we have

R φ (x)dx = C s +∞ 0 e -λ x dx = C s λ < +∞.
4 Study of the whole system

In subsection 4.1, we provide some preliminary results such as a comparison principle adapted to system (1). In subsection 4.2, we prove the main Theorem 2 by introducing a super-solution. The proof of the result which states that it is indeed a super-solution is postponed to subsection 4.3. Subsection 4.4 is devoted to the establishment of a sub-solution of (9). Finally, in subsection 4.5, we provide the proof of Theorem 4.

Preliminary results

In this part, we focus on studying the existence of traveling wave solutions for system (1) and then apply it to prove Theorem 2. In the rest of the paper, we study this system in the subset {E ≤ K} of the positive cone since we have the following property.

Lemma 11. On the positive cone

{E ≥ 0, F ≥ 0, M ≥ 0, M s ≥ 0}, the subset {E ≤ K} is time invariant, that is, if 0 ≤ E 0 ≤ K, then E(t, •) ≤ K for all t > 0.
Proof. Assume that there exists a time t 0 > 0 such that E(t 0 , x) > K for some x. Since 0 ≤ E 0 ≤ K, and E is continuous over time, we can deduce that there exists a time t 1 ∈ (0,t 0 ) such that E(t 1 , x) > 0 and

∂ t E(t 1 , x) > 0.
However, we also have

∂ t E(t 1 , x) = β F(t 1 , x) 1 - E(t 1 , x) K -(ν E + µ E )E(t 1 , x) < 0.
This contradiction proves the result.

We recall that in the subset {E ≤ K}, system (1) is not cooperative due to the introduction of sterile males M s > 0. Indeed, from the second equation of (1), we have the reaction term

g(E, F, M, M s ) := rν E E M M + γM s -µ F F, and
∂ g ∂ M s = - γrν E EM (M + γM s ) 2
< 0 on the positive cone. Hence, we introduce a new comparison principle that can be applied to system (1) in the following part. We first define the nonlinear vector-valued function

f(E, F, M; ψ) =   f 1 (E, F, M) f 2 (E, F, M) f 3 (E, F, M)   =   β F 1 -E K -(ν E + µ E )E rν E E M M+γψ -µ F F (1 -r)ν E E -µ M M   , (19)
where ψ(t, x) is a fixed function. Denote U(t, x) = (E, F, M)(t, x) ∈ R 3 + then we obtain the following system

∂ t U -D∂ xx U = f(U; ψ). (20)
Next, we introduce the following theorem Theorem 12 (Comparison principle for (1)).

Consider two functions M 1 s , M 2 s ∈ L 1 loc ((0, +∞) × R) such that 0 ≤ M 2 s (t, x) ≤ M 1 s (t,
x) for all t ≥ 0, x ∈ R. Suppose that

• (E 1 , F 1 , M 1
) is a sub-solution of system [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF] with ψ ≡ M 1 s ,

• (E 2 , F 2 , M 2 ) is a super-solution of system (20) with ψ ≡ M 2 s , • (E 1 , F 1 , M 1 )(t = 0) ≤ (E 2 , F 2 , M 2 )(t = 0), for any x ∈ R, then (E 1 , F 1 , M 1 )(t, x) ≤ (E 2 , F 2 , M 2 )(t, x),
for all t > 0, x ∈ R.

Proof. Recall that system (20) with ψ(t, x) fixed is a cooperative system. Indeed,

∂ f 1 ∂ F = β 1 - E K > 0, ∂ f 1 ∂ M = 0, ∂ f 2 ∂ E = rν E M M + γψ > 0, ∂ f 2 ∂ M = γψrν E E (M + γψ) 2 > 0,
and

∂ f 3 ∂ E = (1 -r)ν E > 0, ∂ f 3 ∂ F = 0.
On the other hand, from the assumption of Theorem 12, one has 0 ≤ M 2 s (t, x) ≤ M 1 s (t, x) for any t > 0, x ∈ R, we deduce that f(U;

M 1 s ) ≤ f(U; M 2 s ) for any U ∈ R 3 + . Hence, recalling that U 1 = (E 1 , F 1 , M 1 ) is a sub-solution of system (20) with ψ ≡ M 1 s , it follows ∂ t U 1 -D∂ xx U 1 -f(U 1 ; M 2 s ) ≤ f(U 1 ; M 1 s ) -f(U 1 ; M 2 s ) ≤ 0.
This inequality deduces that U 1 is also a sub-solution of system [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF] with ψ ≡ M 2 s . From assumptions in Theorem 12, we also have U 2 = (E 2 , F 2 , M 2 ) is a super-solution of this system. Moreover, U 1 (t = 0) ≤ U 2 (t = 0). Therefore, by applying the comparison principle for this cooperative system (see e.g. [START_REF] Volpert | Traveling Wave Solutions of Parabolic Systems[END_REF], Chapter 5, §5), we obtain that

(E 1 , F 1 , M 1 )(t, x) ≤ (E 2 , F 2 , M 2 )(t, x) for any t > 0, x ∈ R.
Next, we will use Theorem 12 for studying system (1) and prove the main result in Theorem 2.

Proof of Theorem 2

Before treating the main system, we first fix the distribution of sterile males by assuming that the sterile males neither die nor diffuse, and we assign M s (t, x) = φ s (xct) where

φ s (x) = 0 for x < 0, C s e -ηx for x ≥ 0, (21)
with constants C s > 0, η > 0. We consider the traveling wave solution (E, F, M)(t, x) = (φ E , φ F , φ M )(xct) where (φ E , φ F , φ M ) satisfies the following system

               -cφ E = β φ F 1 - φ E K -(ν E + µ E )φ E , -cφ F -Dφ F = rν E E φ M φ M + γφ s -µ F φ F , -cφ M -Dφ M = (1 -r)ν E φ E -µ M φ M , (φ E , φ F , φ M )(-∞) = (E * , F * , M * ), (φ E , φ F , φ M )(+∞) = (0, 0, 0). ( 22
)
with speed c < 0. Note that, system ( 22) is cooperative on the positive cone {E ≥ 0, F ≥ 0, M ≥ 0}, thus we can apply directly the comparison principle for a cooperative system (see e.g. [START_REF] Volpert | Traveling Wave Solutions of Parabolic Systems[END_REF], Chapter 5, §5). Our idea is to construct a super-solution for the system [START_REF] Lewis | Waves of extinction from sterile insect release[END_REF] where the sterile males' distribution is fixed and then deduce a super-solution for the main system [START_REF] Berestycki | Forced waves of the Fisher-KPP equation in a shifting environment[END_REF]. First, we need to show that the solution φ s of ( 9) is larger than φ s in the whole R. It will follow that the solution M s (t, x) of the Cauchy problem with appropriate initial data is also larger than φ s Lemma 13. For a certain speed c < 0 and function φ (xct) = Λ(t, x) defined in [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF], there exists a solution φ s of equation -cφ s -Dφ s = φµ s φ s , φ s (±∞) = 0, such that for A > C s large enough and η > 0 small enough, one has φ s ≤ φ s in R.

Moreover, if M 0 s ≥ φ s then the solution M s of

∂ t M s -D∂ xx M s = Λ -µ s M s , M s (t = 0) = M 0 s ( 23
)
satisfies M s (t, x) → φ s (xct) uniformly with respect to time and M s (t, x) ≥ φ s (xct).

Proof. Denote σ ± = -c ± c 2 + 4Dµ s 2D two roots of the characteristic polynomial of equation -cφ s -Dφ s + µ s φ s = 0, then we have σ -< 0 < σ + . Assume that 0 < η < -σ -, and define A s := A -Dη 2 + cη + µ s , then we have solution

φ s (x) = B + e σ + x + B -e σ -x
for x ≤ 0, A + e σ + x + A -e σ -x + A s e -ηx for x > 0, for some A ± , B ± . Since we have φ s (±∞) = 0, then B -= A + = 0. To ensure that φ s is C 1 , we need B + = A -+ A s , σ + B + = σ -A -+ ηA s . Hence, we obtain that

A -= η + σ + σ --σ + A s < 0, B + = η + σ - σ --σ + A s > 0, since 0 < η < -σ -. Now for any x ≤ 0, one has φ s (x) = η + σ - σ --σ + A -Dη 2 + cη + µ s e σ + x > 0 = φ s (x). Otherwise, if x > 0, one has φ s (x) = η + σ + σ --σ + A s e σ -x + A s e -ηx > η + σ - σ --σ + A -Dη 2 + cη + µ s e -ηx >
C s e -ηx if A > C s large enough.

For the second claim, we split the solution M s into two parts:

M s = M 1 s + M 2 s solutions of ∂ t M 1 s -D∂ xx M 1 s = -µ s M 1 s , M 1 s (t = 0) = M 0 s -φ s and ∂ t M 2 s -D∂ xx M 2 s = Λ -µ s M 2 s , M 2 s (t = 0) = φ s
By linearity, it is clear that M 1 s + M 2 s is a solution of [START_REF] Lui | Biological growth and spread modeled by systems of recursions. I. mathematical theory[END_REF]. Moreover, we have M 1 s = H * (M 0 sφ s ) e -ct (where H stands for the heat kernel in R × [0, +∞[ and * stands for the convolution) and

M 2 s (x -ct) = φ s (x -ct). Since (M 0 s -φ s ) ∈ C 0 b (R) ∩ L 1 (R)
we deduce that M 1 s converges uniformly to 0 as t → +∞. Finally, remarking that (M 0 sφ s ) ≥ 0 we deduce that M 1 s ≥ 0 and M s ≥ φ s > φ s .

The next Proposition shows that we can construct a super-solution of (9) by studying system [START_REF] Lewis | Waves of extinction from sterile insect release[END_REF] Proposition 14. Assume that the basic offspring number R 0 > 1, then for any speed c < 0 and the control function φ s defined in [START_REF] Kolmogorov | A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem[END_REF] with C s > 0 large enough and η > 0 small enough, there exists a non-negative super-solution (φ E , φ F , φ M ) of system [START_REF] Lewis | Waves of extinction from sterile insect release[END_REF] such that φ E ≤ E * , φ F ≤ F * , φ M ≤ M * , and when x → +∞, (φ E , φ F , φ M ) converges to (0, 0, 0). Hence, we deduce that (φ E , φ F , φ M , φ s ) is a super-solution of (9) where φ s is defined in Lemma 13.

The proof of Proposition 14 is long and technical therefore, we postpone it to section 4.3. We finally provide the details of the proof of Theorem 2.

Proof of Theorem 2. We define [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF] with ψ(t, x) = φ s (xct) and φ s defined in [START_REF] Kolmogorov | A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem[END_REF]. Denote (E, F, M, M s ) solution of system (1) with Λ defined in [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF]. Then (E, F, M) is a sub-solution of system [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF] with ψ ≡ M s . From Lemma 13, we can choose A > C s such that M s (t, x) ≥ φ s (xct) for any t > 0 and x ∈ R. Moreover, by the construction of (φ E , φ F , φ M ) in Proposition 14 (see Section 4.3), we have (E 0 , F 0 , M 0 )(x) ≤ (E, F, M)(t = 0, x). Now, we apply the comparison principle in Theorem 12 and we obtain that (E, F, M)(t, x) ≤ (E, F, M)(t, x) for any time t > 0 and x ∈ R. Since (φ E , φ F , φ M )(x) → (0, 0, 0) when x → +∞, we conclude that

(E, F, M)(t, x) = (φ E , φ F , φ M )(x -c t) where c < c < 0, (φ E , φ F , φ M ) is defined in Proposition 14 with a speed c . It is clear that (E, F, M) is a super-solution of system
lim t→+∞ sup x<ct (E, F, M)(x,t) ≤ lim t→+∞ sup x<ct (E, F, M)(x,t) = lim t→+∞ sup x<ct (φ E , φ F , φ M )(x -c t) ≤ lim t→+∞ Ce (c-c )t (1, 1, 1) = (0, 0, 0).
In the following parts, we construct super-and sub-solutions for [START_REF] Berestycki | Forced waves of the Fisher-KPP equation in a shifting environment[END_REF], then conclude by proving Theorem 4.

Construction of a super-solution for (9)

We first remark that if (φ E , φ M , φ F ) is a super-solution of ( 22) then (φ E , φ M , φ F , φ s ) is a super-solution of [START_REF] Berestycki | Forced waves of the Fisher-KPP equation in a shifting environment[END_REF]. Indeed, by applying Lemma 13, we have φ s ≥ φ s in R, thus, we have

-cφ F -Dφ F -rν E φ E φ M φ M + γφ s + µ F φ F ≥ -cφ F -Dφ F -rν E φ E φ M φ M + γφ s + µ F φ F ≥ 0.
Following the idea we used with the simplified model, we construct super-solutions for [START_REF] Lewis | Waves of extinction from sterile insect release[END_REF] established by two parts, a constant part on (-∞, x * ] and a tail on (x * , +∞) that decays to 0 at +∞, with some x * ≥ 0. We start by considering φ F as follows φ F (x) = F * when x ≤ 0, F * e -λ x when x > 0, [START_REF] Manoranjan | On a diffusion model for sterile insect release[END_REF] with some λ > 0. Next, we construct the tails for φ E and φ M , and clarify the value of x * . After that, we provide proof of Proposition 14.

• Construction of function φ E : First, on R + , we consider function φ E (x) such that

   -c φ E = β F * e -λ x 1 -φ E K -(ν E + µ E ) φ E , φ E > 0, lim x→+∞ φ E = 0, φ E (0) = E * . ( 25
)
Hence, for any x ≥ 0, we obtain φ E of the form

φ E (x) = e δ (x) - β F * c x 0 e -λ s-δ (s) ds + E * > 0, ( 26
)
where

δ (x) = - β F * λ cK e -λ x + ν E + µ E c x + β F * λ cK . One has δ (0) = 0 and lim x→+∞ δ (x) = -∞. We have the following lemma Lemma 15. Assume that λ + ν E + µ E c < 0, then there exists a constant C E > E * such that φ E (x) ≤
C E e -λ x for any x ≥ 0.

Proof. Since λ + ν E + µ E c < 0 and c < 0, for any x ≥ 0, we obtain that δ (x)

≤ ν E + µ E c x ≤ -λ x.
Therefore, e δ (x) ≤ e ν E +µ E c

x ≤ e -λ x . On the other hand, one has

δ (x)
x 0 e -λ s-δ (s) ds ≤ e

ν E +µ E c x x 0 e -λ s-ν E +µ E c s e -β F * cλ K (1-e -λ s ) ds ≤ -e -β F * cλ K λ + ν E +µ E c e -λ x .
Then one has

C E := E * + β F * c e -β F * cλ K λ + ν E +µ E c > E * .
This induces the result of the lemma.

From Lemma 15, we can deduce that lim x→+∞ φ E (x) = 0. Moreover, we define

x E := sup{x ≥ 0 : φ E (x) = E * } < +∞, (27)
and φ E (x) < E * for any x > x E . We define function φ E as follows

φ E (x) = E * when x ≤ x E φ E (x) when x > x E . (28)
Then for any x, we have φ E (x) ≤ min{E * ,C E e -λ x }, lim x→+∞ φ E (x) = 0, and lim

x→x - E φ E (x) = 0 ≥ φ E (x E ) = lim x→x + E φ E (x).
• Construction of function φ M : Next, on R + , we consider function φ M which satisfies

   -c φ M -D φ M = (1 -r)ν E C E e -λ x -µ M φ M , φ M (x) > 0, lim x→+∞ φ M (x) = 0, φ M (0) = M * . ( 29
)
Consider the characteristic polynomial -Dδ 2cδ + µ M = 0 with two roots

δ ± = -c ± c 2 + 4Dµ M 2D ,
where δ + > 0, δ -< 0. Then any solution of ( 29) has the form φ M (x) = C M e -λ x +C 1 e δ -x +C 2 e δ + x , where

C M = (1 -r)ν E C E -Dλ 2 + cλ + µ M . ( 30
)
Since we look for lim

x→+∞ φ M (x) = 0, then C 2 = 0. Moreover, M * = φ M (0) = C M +C 1 , thus C 1 = M * -C M .
Assume that λ + δ -< 0, so we have µ M > -Dλ 2 + cλ + µ M > 0 and

C M > (1 -r)ν E C E µ M = M * C E E * ≥ M * .
Moreover, since δ -< -λ , then for any x > 0, we have

C M e -λ x > φ (x) = C M e -λ x + (M * -C M )e δ -x > M * e δ -x > 0.
and we have lim [START_REF] Trélat | Allee optimal control of a system in ecology[END_REF]. We define

x→+∞ φ M (x) = 0, so φ M is a solution of problem
x M = sup{x ≥ 0 : φ M (x) = M * } < +∞, (31)
and

φ M (x) = M * when x ≤ x M φ M (x) when x > x M . ( 32
)
Again we have φ M (x) ≤ min{M * ,C M e -λ x } for any x, lim x→+∞ φ M (x) = 0, and lim

x→x - M φ M (x) = 0 ≥ φ M (x M ) = lim x→x + M φ M (x).
Now we prove that for C s large enough, (φ E , φ F , φ M ) defined as above is a super-solution of [START_REF] Lewis | Waves of extinction from sterile insect release[END_REF].

Proof of Proposition 14. Fix a positive parameter α such that α <

µ F F * rν e C E = E * C E < 1. Then, we choose a positive constant λ such that λ ≤ min    - ν E + µ E c , c + c 2 + 4Dµ M 2D , c + c 2 + 4Dµ F 1 -α C E E * 2D    . ( 33
)
Recalling C M defined respectively in [START_REF] Volpert | Traveling Wave Solutions of Parabolic Systems[END_REF], we take η < λ and C s large enough such that

C s C M ≥ 1 γ 1 α -1 .
Then for any x > 0,

φ s φ M ≥ C s e -ηx C M e -λ x + (M * -C M )e δ -x ≥ C s e -ηx C M e -λ x ≥ C s C M , thus we obtain that φ M φ M + γφ s = 1 1 + γ φ s φ M ≤ α.
We now check that (φ E , φ F , φ M ) is a super-solution of [START_REF] Lewis | Waves of extinction from sterile insect release[END_REF].

• Checking for φ E : For any

x ≤ x E , since φ E (x) = E * , φ F (x) ≤ F * , then -cφ E -β φ F 1 - φ E K + (ν E + µ E )φ E ≥ -β F * 1 - E * K + (ν E + µ E )E * = 0,
and for x > x E > 0, one has

-cφ E -β φ F 1 - φ E K + (ν E + µ E )φ E = -c φ E -β F * e -λ x 1 - φ E K + (ν E + µ E ) φ E = 0.
• Checking for φ F : For any x ≤ 0, we have φ

F = F * , φ E ≤ E * , then -cφ F -Dφ F -rν E φ E φ M φ M + γφ s + µ F φ F ≥ -rν E E * + µ F F * = 0.
For any x > 0, we have

φ E (x) ≤ C E e -λ x , φ F (x) = F * e -λ x , φ M φ M +γφ ≤ α. From (4), we note that µ F F * E * = rν E , thus -cφ F -Dφ F -rν E φ E φ M φ M + γφ s + µ F φ F ≥ F * e -λ x -Dλ 2 + cλ -µ F α C E E * + µ F ≥ 0 since 0 < λ ≤ c + c 2 + 4Dµ F 1 -α C E E * 2D . • Checking for φ M : For any x ≤ x M , one has φ M (x) = M * , φ E (x) ≤ E * , thus -cφ M -Dφ M -(1 -r)ν E φ E + µ M φ M ≥ -(1 -r)ν E E * + µ M M * = 0.
On the other hand, when x > x M , one has φ E (x) ≤ C E e -λ x , φ M (x) = φ M (x) with φ M defined in [START_REF] Trélat | Allee optimal control of a system in ecology[END_REF] thus

-cφ M -Dφ M -(1 -r)ν E φ E + µ M φ M ≥ -c φ M -D φ M -(1 -r)ν E C E e -λ x + µ M φ M = 0.
In conclusion, for λ > 0 small such that (33) holds, (φ E , φ F , φ M ) defined as above is a super-solution of [START_REF] Lewis | Waves of extinction from sterile insect release[END_REF] where C s is large enough and 0 < η < λ . Then, we deduce that (φ E , φ F , φ M , φ s ) is a super-solution of (9).

Construction of a sub-solution for (9)

First, we remark that the sub-solution is established only to prove Theorem 4. Therefore, according to Theorem 12, we need to establish

(φ E , φ F , φ M , φ s ) such that φ s ≥ φ s and              -cφ E ≤ β φ F 1 - φ E K -(ν E + µ E )φ E , -cφ F -Dφ F ≤ rν E φ M φ M + γφ s φ E -µ F φ F , -cφ M -Dφ M ≤ (1 -r)ν E φ E -µ M φ M .
The first difficulty is that the sterile males diffuse so φ s > 0 on R. It is clear that φ s (x) -→ |x|→+∞ 0 uniformly.

Therefore, we deduce that

∀ε > 0, ∃x ε = inf{x ∈ R : φ s (x) < ε} and x ε < +∞.
Moreover, taking ε small enough, we can consider x ε ≤ 0. Then, we take

φ s (x) = ε for x < x ε , φ s for x > x ε . (34)
The second difficulty is that (E * , F * , M * ) is no more an equilibrium if we impose φ s (-∞) = ε. Nevertheless, thanks to the implicit function theorem, we obtain Proposition 16. There exists ε 0 > 0 such that for any ε ∈ [0, ε 0 ), there exists a strictly positive solution

(E * ε , F * ε , M * ε ) of        β F * ε 1 -E * ε K -(ν E + µ E )E * ε = 0, rν E E * ε 1 1+ε -µ F F * ε = 0, (1 -r)ν E E * ε -µ M M * ε = 0. ( 35
)
Moreover, one has

(E * ε , F * ε , M * ε )
is decreasing continuously with respect to ε, and

(E * 0 , F * 0 , M * 0 ) = (E * , F * , M * ). Proof. We define f 2 (E, F, M, ε) =    β F 1 -E K -(ν E + µ E )E rν E E 1 1+ε -µ F F (1 -r)ν E E -µ M M    .
According to the explicit writing of (E * , F * , M * ) in ( 4) and since R 0 > 1, we have that

det(D E,F,M f 2 (E * , F * , M * , 0)) = -µ M [β rν E -µ F (µ E + ν E )] < 0.
Then, the implicit function theorem provides the existence of ε 0 . Still thanks to the implicit function theorem, there holds

  ∂ ε E * ε ∂ ε F * ε ∂ ε M * ε   = -(D E,F,M f 2 ) -1 • ∇ ε f 2 (E * ε , F * ε , M * ε , ε) = rν E E * ε (1 + ε) 2 (det D E,F,M f 2 (E * ε , F * ε , M * ε , ε))         β 1 - E * K µ M β F * ε K + µ E + ν E µ M β F * ε K + µ E + ν E µ F         . Recalling, that det(D E,F,M f 2 (E * , F * , M * , 0)) < 0, we deduce by continuity that   ∂ ε E * ε ∂ ε F * ε ∂ ε M * ε   <   0 0 0   and the conclusion follows.
Because of our choice of φ s , we construct a subsolution that converges to (E ε , F ε , M ε ) for some positive ε. We construct a sub-solution (φ E , φ F , φ M ) for system [START_REF] Lewis | Waves of extinction from sterile insect release[END_REF] by two parts. The first part of

(φ E , φ F , φ M ) is equal to 0 on [x ε , +∞) and the second part on (-∞, x ε ) converges to (E * ε , F * ε , M * ε ) when x → -∞.
The construction of the sub-solution on (-∞, x ε ) is the third difficulty. To cope with this problem, we use the fact that φ E ≤ E ε . We present the result of the existence of a sub-solution as follows Proposition 17. For a speed c < 0, there exists ε 1 ∈ (0, ε 0 ) and ε < ε 1 a constant small enough such that for the control function ψ = φ s defined in (34), there exists a non-negative sub-solution (φ E , φ F , φ M ) of system [START_REF] Lewis | Waves of extinction from sterile insect release[END_REF]

such that φ E ≤ E * ε 1 , φ F ≤ F * ε 1 , φ M ≤ M * ε 1 . Moreover, when x → -∞, (φ E , φ F , φ M )(x) converges to (E * ε 1 , F * ε 1 , M * ε 1 ).
Proof. We fix c < 0 and ε 1 ∈ (0, ε 0 ) (where ε 0 is defined in Proposition 16). Then, we consider ( Ê, F, M) a solution of the following linear system in R -

       -c Ê = β F 1 - E * ε 1 K -(ν E + µ E ) Ê, -c F -D F = rν E 1+ε 1 Ê -µ F F, -c M -D M = (1 -r)ν E Ê -µ M M, (36) with Ê(-∞) = E * ε 1 , F(-∞) = F * , M(-∞) = M * .
Now, we will study this linear system by denoting

U =       Ê F M F M       . Then system (36) becomes U = BU where B =        ν E +µ E c -µ F (ν E +µ E ) β rν E /(1+ε 1 ) 0 0 0 0 0 0 1 0 0 0 0 0 1 -rν E D(1+ε 1 ) µ F D 0 -c D 0 -(1-r)ν E D 0 µ M D 0 -c D        , since 1 - E * ε 1 K = µ F (ν E + µ E ) β rν E /(1 + ε 1 ) . Hence, the characteristic polynomial is det(B -λ I) = λ λ 2 + c D λ - µ M D P M (λ ) -λ 2 + ν E + µ E c - c D λ + ν E + µ E c + µ F D P F (λ )
.

It is clear that λ 0 = 0 is an eigenvalue associated to the eigenvector

U 0 =       E * ε 1 F * ε 1 M * ε 1 0 0      
. Denote eigenvalues λ + M > 0, λ - M < 0 which are the roots of P M (λ ), λ + F > 0, λ - F < 0 which are the roots of P F (λ ). We aim at building a solution U(x) that converges to U 0 when x → -∞, then we construct U of the following form

U(x) = U 0 + e λ + M x U + M + e λ + F x U + F ,
where U + M ,U + F the corresponding eigenvectors of λ + M , λ + F . We consider the following cases:

Case 1: λ + M = λ + F : Since λ + M is a root of P M (λ ), then U + M =       0 0 a 0 aλ + M       for some a ∈ R. Denote U + F =       b 1 b 2 b 3 b 4 b 5      
an eigenvector associated to λ + F . We have

BU + F = λ + F U + F ,
* ε 1 , F * ε 1 , M * ε 1 , we have b 2 = b 1 F * ε 1 E * ε 1 1 - cλ + F ν E + µ E , b 3 = b 1 M * ε 1 E * ε 1 µ M -D[(λ + F ) 2 + c D λ + F -µ M D ] , b 4 = λ + F b 2 . ( 37
)
For any x < 0, we have

Ê(x) = E * ε 1 + b 1 e λ + F x , F(x) = F * ε 1 + b 2 e λ + F x , M(x) = M * ε 1 + ae λ + M x + b 3 e λ + F x . We choose b 1 = -E * ε 1 , a = -M * ε 1 -b 3 . Thus we obtain that Ê(0) = 0, M(0) = 0, F(0) = cF * ε 1 λ + F ν E + µ E < 0.
Then, there exists a unique constant y F < 0 such that F(y F ) = 0. Claim: For any x < 0, one has Ê(x)

< E * ε 1 , F(x) < F * ε 1 , M(x) < M * ε 1 .
Indeed, since b 1 < 0, we deduce from (37) that b 2 < 0, then for any x < 0, Ê(x) < E * ε 1 , F(x) < F * ε 1 . It remains to show that M(x) < M * ε 1 for any x < 0. One has

M(x) = M * ε 1 (1 -e λ + M x ) + b 3 (e λ + F x -e λ + M x ).
We only need to show that b 3 (e λ + F xe λ + M x ) < 0 for any x < 0. Indeed,

• if λ + F < λ + M , then e λ + F x -e λ + M x > 0 for any x < 0 and (λ + F ) 2 + c D λ + F -µ M D < 0.
From (37), we deduce that b 3 < 0;

• if λ + F > λ + M , we have e λ + F xe λ + M x < 0 for any x < 0 and b 3 > 0.

Case 2: λ + F = λ + M = λ + : Now λ + has one-dimensional eigenspace generated by

U + =       0 0 a 0 aλ +      
for some constant a. The solution of U = BU becomes U(x) = U 0 + xe λ + x U + + e λ + x V + , with V + some vector to be determined. Plugging this U(x) into the equation yields

e λ + x U + + λ + xe λ + x U + + λ + e λ + x V + = U (x) = BU = λ + xe λ + x U + + e λ + x BV + . Hence, (B -λ + I)V + = U + . Denote V + =       b 1 b 2 b 3 b 4 b 5       , one has                    b 2 = b 1 F * ε 1 E * ε 1 1 - cλ + ν E + µ E , b 4 = λ + b 2 , b 5 = λ + b 3 + a, a = - M * ε 1 E * ε 1 1 1 + 2Dλ + µ M b 1 .
Then, we have

Ê(x) = E * ε 1 + b 1 e λ + x , F(x) = F * ε 1 + b 2 e λ + x , M(x) = M * ε 1 + axe λ + x + b 3 e λ + x . We choose b 1 = -E * ε 1 , b 3 = -M * ε 1 , then Ê(0) = 0, M(0) = 0, F(0) = cF * ε 1 λ + ν E + µ E < 0.
Thus, there exists a unique constant y F < 0 such that F(y F ) = 0.

Since we have a = M *

ε 1 1 + 2Dλ + µ M
> 0 , we obtain that for any x < 0, Ê(x)

< E * ε 1 , F(x) < F * ε 1 , M(x) < M * ε 1 .
Hence, in both cases, we constructed solution ( Ê(x), F(x), M(x)) of (36) such that Ê(x)

< E * ε 1 , F(x) < F * ε 1 , M(x) < M * ε 1 . Moreover, ( Ê, F, M) converges to (E * ε 1 , F * ε 1 , M * ε 1
) at -∞. Now, we use these functions to construct a sub-solution for [START_REF] Lewis | Waves of extinction from sterile insect release[END_REF]. Construction of a sub-solution:

Now, we construct φ E , φ F , φ M a sub-solution of system [START_REF] Fang | Monotone Wavefronts for Partially Degenerate Reaction-Diffusion Systems[END_REF] taking φ s = φ s defined in (34) with ε ∈ (0, ε 1 ) that will be fixed later on. Due to a translation of space, without loss of generality, we can assume here that x * ε 1 = 0. Next, we define

φ E (x) = Ê(x) when x ≤ 0, 0 when x > 0, φ F (x) = F(x) when x ≤ y F , 0 when x > y F , φ M (x) = M(x) when x ≤ 0, 0 when x > 0. ( 38
)
Note that, by the definitions of φ s , the fraction φ M φ M + γφ s is well-defined in R. We now check the sub-solution inequalities for (φ E , φ F , φ M ). We can see that for any x > 0, the inequalities are trivial.

• Checking for φ E (x): For any x ≤ y F < 0, since φ E ≤ E * ε 1 , thus

-cφ E -β φ F 1 - φ E K + (ν E + µ E )φ E ≤ -c Ê -β F 1 - E * ε 1 K -(ν E + µ E ) Ê = 0,
For any x such that y F < x ≤ 0, we have

-cφ E -β φ F 1 - φ E K + (ν E + µ E )φ E = -c Ê + (ν E + µ E ) Ê = β F 1 - E * ε 1 K < 0,
since F < 0 on (y F , 0]. At x = 0, we also have lim

x→0 -φ E (x) = Ê (0) = -λ + E * ε 1 < 0 = lim x→0 + φ E (x).
• Checking for φ F (x): First, we consider the case where x ≤ y F . Taking ε < min ε 1 2 , 1 γ ε 1 min

x≤y F φ M
it follows, recalling that in this case, φ s = ε, one has

-cφ F -Dφ F -rν E φ E φ M φ M + γφ s + µ F φ F = rν E φ E 1 1 + ε 1 - M M + γε ≤ 0,
due to the fact that ( Ê, F, M) is a solution of (36). For y F < x ≤ 0, we have φ F (x) = 0, φ E (x) = Ê(x) ≥ 0, φ (x) = A, φ M (x) = M(x) > 0, thus

-cφ F -Dφ F -rν E φ E φ M φ M + γφ + µ F φ F = -rν E Ê M M + γA ≤ 0.
At x = y F , we have lim

x→y - F φ F (x) = F (0) = -λ + F * ε 1 1 - cλ + ν E + µ E < 0 = lim x→y + F φ F (x).
• Checking for φ M (x): For any x ≤ 0, one has

-cφ M -Dφ M -(1 -r)ν E φ E + µ M φ M = -c M -D M -(1 -r)ν E Ê + µ M M = 0.
Similarly, at x = 0, in both cases lim

x→0 -φ M (x) = M (0) < 0 = lim x→0 + φ M (x).
It finishes the establishment of the sub-solution.

Construction of the traveling wave for (9) (Proof of Theorem 4)

As mentioned above, we prove the existence of traveling wave solutions using the sub and the super solutions constructed before. We underline the following Remark 2. For a certain speed c < 0 and function φ (xct) = Λ(t, x) defined in [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF], there exists a solution φ s of equation -cφ s -Dφ s = φµ s φ s , φ s (±∞) = 0, such that for A > C s large enough and η > 0 small enough, one has φ s ≤ φ s ≤ φ s in R.

Thanks to Remark 2, we are able to prove Theorem 4.

Proof of Theorem 4. First, notice that since the equation for φ s is independent from the other equations, we deduce that φ s exists and is provided in the proof of Lemma 2. Next, in section 4.3, we obtained that (φ E , φ F , φ M , φ s ) is a super-solution of the original system. In section 4.4, we obtain that (φ E , φ F , φ M , φ s ) is a sub-solution of the original system. Moreover, we have by construction that (φ E , φ F , φ M ) ≤ (φ E , φ F , φ M ) and according to Remark 13, we have φ s ≤ φ s ≤ φ s . By applying the comparison principle for the cooperative system (9), we deduce that there exists a traveling wave solution (φ E , φ F , φ M , φ s ) for system [START_REF] Berestycki | Forced waves of the Fisher-KPP equation in a shifting environment[END_REF] with

(φ E , φ F , φ M ) ≤ (φ E , φ F , φ M ) ≤ (φ E , φ F , φ M )
Thus (φ E , φ F , φ M ) converges to 0 at +∞, and at -∞, one has

(E * ε 1 , F * ε 1 , M * ε 1 ) ≤ (φ E , φ F , φ M ) < (E * , F * , M * ).
It only remains to prove by contradiction that

(φ E , φ F , φ M ) -→ x→-∞ (E * , F * , M * ).
Assume it is not the case, we denote

(E * , F * , M * ) = ( lim inf x→-∞ φ E (x), lim inf x→-∞ φ F (x), lim inf x→-∞ φ M (x)). It follows max E * -E * , F * -F * , M * -M * > 0.
Next, we introduce

ε 2 = inf{ε > 0 : (E ε , F ε , M ε ) ≤ (E * , F * , M * )}.
Notice that by assumption ε 2 > 0. The end of the proof is split into three claims:

1. Prove by contradiction that F * -F * ε 2 > δ F (where δ F is a small positive constant), 2. Prove by contradiction that E * -E * ε 2 > δ E (where δ E is a small positive constant), 3. Prove by contradiction that M * -M * ε 2 > δ M (where δ M is a small positive constant).

Then the three steps above are in contradiction with the definition of ε 1 . Indeed, if the claims are true since the dependence of (E ε , F ε , M ε ) with respect to ε is continuous, we deduce the existence of

ε 3 < ε 2 such that (E * ε 2 , F * ε 2 , M * ε 2 ) < (E * ε 3 , F * ε 3 , M * ε 3 ) ≤ (E * , F * , M * ).
Therefore, if the claims are true, the contradiction follows and the proof is achieved.

• Claim 1. Assume by contradiction that F * = F * ε 2 . It follows the existence of a decreasing and unbounded sequence x n such that φ

F (x n ) < F * ε 2 + 1/n, φ F (x n ) = 0 and -φ F (x n ) ≤ 0.
Such sequence exists because if φ F does not change its sign, it follows that φ F converges and this is absurd since it can only converge to F * . Notice that we also have by definition of ε 2 that φ [START_REF] Almeida | The sterile insect technique used as a barrier control against reinfestation[END_REF]. Inserting these inequalities in the equation that φ F satisfies, we obtain

E (x n ) > E * ε 2 + o 1 (n) and φ M (x n ) > M * ε 2 + o n (
rν E φ E (x n ) 1 1 + γφ s (x n )/φ M (x n ) -µ F φ F (x n ) = cφ F (x n ) -∆φ F (x n ) ≤ 0. Since φ M (x n ) ≥ M * + o n (1) and φ s (x n ) = o n (1)
, we deduce, thanks to (35),

rν E φ E (x n ) 1 1 + γφ s (x n )/φ M (x n ) -µ F φ F (x n ) > rν E E * ε 2 1 1 + o n (1) -µ F F * ε 2 + o n (1) > rν E E * ε 2 1 1 + o n (1) -rν E E * ε 2 1 1 + ε 2 + o n (1) > rν E E * ε 2 ε 2 -o n (1) (1 + ε 2 )(1 + o n (1)) + o n (1)
> 0.

Taking n large enough, it follows that F * -F * ε 2 > δ F for some positive constant δ F . • Claim 2. Assume by contradiction that E * = E * ε 2 . It follows the existence of a decreasing and unbounded sequence x n such that φ

E (x n ) < E * ε 2 + 1/n, φ E (x n ) = 0.
Inserting this inequality in the equation satisfied by φ E , we obtain as above

0 = -cφ E (x n ) = rν E φ F (x n ) 1 - φ E (x n ) K -(µ E + ν E )φ E (x n ) > rν E φ F (x n ) 1 - E * ε 2 K -rν E F * ε 2 1 - E * ε 2 K + o n (1) > rν E 1 - E * ε 2 K φ F (x n ) -F * ε 2 + o n (1).
Recalling that E * ε 2 < E * < K (since R 0 > 1 and by the definition of E * ) and using claim 1, it follows the following contradiction by taking n large enough such that o n (1) is small enough

rν E 1 - E * ε 2 K φ F (x n ) -F * ε 2 + o n (1) > rν E 1 - E * ε 2 K δ F + o n (1) > 0.
We conclude to the existence of a positive constant δ E such that E * -E * ε 2 > δ E . • Claim 3. Assume by contradiction that M * = M * ε 2 . It follows the existence of a decreasing and unbounded sequence x n such that φ M (x n ) < M * ε 2 + 1/n, φ M (x n ) = 0 and -φ M (x n ) ≤ 0. Inserting these inequalities in the equation satisfied by φ M , we obtain as above

0 ≥ -cφ M (x n ) -φ M (x n ) = (1 -r)ν E φ E (x n ) -µ M φ M (x n ) > (1 -r)ν E φ E (x n ) -(1 -r)ν E E * ε 2 + o n (1) > (1 -r)ν E φ E (x n ) -E * ε 2 + o n (1)
. Recalling claim 2, it follows the following contradiction by taking n large enough such that o n (1) is small enough:

(

1 -r)ν E φ E (x n ) -E * ε 2 + o n (1) > (1 -r)ν E δ E + o n (1) > 0. We conclude to the existence of a positive constant δ M > 0 such that M * -M * ε 1 > δ M . It concludes the proof.

A Proofs of Propositions 1 and 3

We recall [START_REF] Weinberger | Analysis of linear determinacy for spread in cooperative models[END_REF]Theorem 4.2] which shows the estimate of the spreading speed c * for the monostable system in discrete setting

u n+1 = Q[u n ]
where the vector-valued function u n (x) = (u 1 n (x), u 2 n (x), ..., u k n (x)) represents the population densities of the populations of k species at the point x and the time nτ, with τ a fixed generation time. Then in section 4 of this work, the authors showed how to apply the results to a reaction-diffusion system by letting Q be its time τ map. That is, replacing Q by Q τ where Q τ [u 0 ] := u(x, τ). Next, we recall the result of this work and apply it to the system (3).

Consider the system of reaction-diffusion equations

∂ t u i -d i ∂ xx u i = f i (u), with 1 ≤ i ≤ k and denote f = ( f 1 , f 2 , . . . f k ).
The reaction function f needs to satisfy the following assumptions.

Assumptions 2.

i. f(0) = 0 and there is a vector u 0 such that f(u) = 0 which is minimal in the sense there are no v other than 0 and u such that f(v) = 0 and 0 v ≤ u.

ii. The system is cooperative, that is, f i (u) is nondecreasing in all components of u with the possible exception of the i th one.

iii. f(u) is continuous and piecewise continuously differentiable at u for 0 ≤ u ≤ u and differentiable at 0.

iv. The Jacobian matrix f (0) is in Frobenius form. The principal eigenvalue η 1 (0) of its upper left diagonal block is positive and strictly larger than the principal eigenvalues η σ (0) of its other diagonal blocks, and there is at least one nonzero entry to the left of each diagonal block other than the first one.

For any positive parameter µ, if the initial data are of the form e -µx u 0 then the solution of this system has the form e -µx v, where the vector-valued function v is the solution of the system of ordinary differential equations with constant coefficients ∂ t v = C µ v, with v(0) = u 0 . The coefficient matrix is given by

C µ = diag d i µ 2 + f (0), (39)
and denote γ σ (0) the principal eigenvalue of the σ th diagonal block of the matrix C µ . We introduce the constant

c := inf µ>0 γ 1 (µ) µ . ( 40
)
Let µ ∈ (0, ∞] again denote the value of µ at which this minimum is attained, and let ζ (µ) be the eigenvector of C µ which correspond to the eigenvalue γ 1 (µ). Then, the following theorem presents the main result Theorem 18 (Theorem 4.2 in [START_REF] Weinberger | Analysis of linear determinacy for spread in cooperative models[END_REF]). Suppose that f satisfies the Assumptions 2. Assume that either (a) µ is finite, γ 1 (µ) > γ σ (µ) for all σ > 1, (41)

and f(ρζ (µ)) ≤ ρf (0)ζ (µ), (42)
for all positive ρ; or (b) There is a sequence µ ν µ such that for each ν the inequalities (41) and (42) with µ replaced by µ ν are valid.

Then the system has a unique speed c * = c with c * defined in Proposition 1.

Now we apply this theorem to system (3) with f(E,

F, M) =    β F 1 -E K -(ν E + µ E )E rν E E -µ F F (1 -r)ν E E -µ M M  
, and we provide the proof as follows Proof of Propositions 1 and 3. First, we need to show that f satisfies Assumptions 2. With β rν E > µ F (ν E + µ E ), we can deduce that f has two zeros (0, 0, 0), (E * , F * , M * ), and satisfies (i). When E ≤ K, one has f is cooperative, thus f satisfies (ii). It is easy to see that f satisfies (iii). Now we only need to check the assumption (iv). The Jacobian of f at (0, 0, 0)

f (0) =   -ν E -µ E β 0 rν E -µ F 0 (1 -r)ν E 0 -µ M   (
-(ν E + µ E )e ± + β f ± = -(ν E + µ E + µ F ) ± (ν E + µ E -µ F ) 2 + 4β rν E 2 e ± .
So

β f ± = ν E + µ E -µ F ± (ν E + µ E -µ F ) 2 + 4β rν E 2 e ± .
Since ν E + µ Eµ F -(ν E + µ Eµ F ) 2 + 4β rν E 2 < 0, then e -and f -always have different signs. Hence, η + is the only eigenvalue that has the corresponding positive eigenvector, and it is the principal eigenvalue of B 1 . Moreover, due to the assumption β rν E > µ F (ν E + µ E ), one has η 1 (0) = η + > 0 > -µ M = η 2 (0). This concludes that f satisfies (iv). Now, one has the matrix

C µ =   -ν E -µ E β 0 rν E Dµ 2 -µ F 0 (1 -r)ν E 0 Dµ 2 -µ M   .
Similarly to the matrix f (0), the principal eigenvalue of the first block of C µ is

γ 1 (µ) = Dµ 2 -ν E -µ E -µ F + (Dµ 2 + ν E + µ E -µ F ) 2 + 4β rν E 2
By the assumption β rν E > µ F (ν E + µ E ) and D > 0, we have γ 1 (µ) > 0. It is easy to see that γ max(E, F, M)(t, x) = 0, if the initial data (E 0 , F 0 , M 0 ) is compactly supported and 0 ≤ (E 0 , F 0 , M 0 ) (E * , F * , M * ). Furthermore, for any strictly positive constant ω, there is a positive R ω with the property that if min(E 0 , F 0 , M 0 ) ≥ ω on an interval of length 2R ω , then max(E * -E, F * -F, M * -M)(t, x) = 0. Moreover, Proposition 3.3 in the work of Lui [START_REF] Lui | Biological growth and spread modeled by systems of recursions. I. mathematical theory[END_REF] provides, in a discrete setting, some conditions in which the constant R ω can be chosen to be arbitrarily small and independent of ω. This result can be transposed to the continuous case like what has been done in section 4 of [START_REF] Weinberger | Analysis of linear determinacy for spread in cooperative models[END_REF] and it is simple to verify that, when min E 0 > 0 or min F 0 > 0, our system satisfies those conditions so we leave it to the readers. Hence, by applying this result, we deduce that if the initial data E 0 or F 0 are strictly positive on a set with a positive measure, then the result in Proposition 1 holds. Now, to prove Proposition 3, the paper [START_REF] Fang | Monotone Wavefronts for Partially Degenerate Reaction-Diffusion Systems[END_REF] provides some conditions in which the spreading speed estimated in [START_REF] Weinberger | Analysis of linear determinacy for spread in cooperative models[END_REF] of the monostable system is the minimum speed of the traveling wave. The authors in [START_REF] Anguelov | On the use of Traveling Waves for Pest/Vector elimination using the Sterile Insect Technique[END_REF] have checked all the conditions for the same system as (3), hence we obtain the same result for our system.

7 Figure 1 :

 71 Figure 1: Dynamics of the female density in system (1).

Figure 2 :

 2 Figure 2: Control function φ and super-, sub-solutions.

  and since rank(B-λ + F I) = 4 then all entries b 2 , b 3 , b 4 , b 5 depend explicitly on b 1 ∈ R. More precisely, using the formula of E

1 (µ) µ ∼ 1 µ 1 

 111 when µ → 0 + , and γ 1 (µ) µ ∼ µ when µ → +∞. Hence, one can deduce that there exists a finite constant µ ∈ (0, +∞) such that γ the eigenvector corresponding to the eigenvalue γ 1 (µ) of C µ , where e fis the positive eigenvector associated to the principal eigenvalue γ 1 (µ) of the first diagonal block. So m > 0 if and only ifγ 1 (µ) > γ 2 (µ) = Dµ 2µ M , that is 2µ M -Dµ 2ν Eµ Eµ F + (Dµ 2 + ν E + µ Eµ F ) 2 + 4β rν E > 0. (44) Hence, whenever the parameters satisfy condition (44), the inequality (41) holds, the eigenvector ζ (µis positive, and for any positive ρ, f(ρζ (µ))ρf (0)ζ (µ) = ρ < 0, then (42) holds. Now, applying the result of Theorem 18, we obtain the spreading speed c * = c. By applying Theorem 4.1 in [31], the solution of (3) satisfies lim t→+∞ max |x|≥t(c * +ε)

Table 1 :

 1 Parameters for the numerical illustration Parameters

  43) is in Frobenius form with two diagonal blocksB 1 = -ν Eµ E β rν E -µ F and B 2 = -µ M . There is a positive entry (1r)ν E to the left of B 2 .The block B 1 has two eigenvaluesη ± = -(ν E + µ E + µ F ) ± (ν E + µ Eµ F ) 2 + 4β rν E 2. Denote e ± f ± the eigenvectors corresponding to eigenvalues η ± of B 1 . Then, one has

Acknowledgement

Both authors want to sincerely thank Luis Almeida and Nicolas Vauchelet for all the fruitful discussions and their precious advice, and Pierre-Alexandre Bliman for a careful reading of the manuscript. We also thank the editor and the reviewers for the time and effort they dedicated to providing feedback on our manuscript and are grateful for the insightful comments to improve our paper. The first author has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 740623). The second author has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 945332.