In this paper, we study some reaction-diffusion models of interactive dynamics of the wild and sterile mosquitoes. The well-posedness of the concerned model is proved. The stability of the steady states is discussed. Numerical simulations are presented to illustrate our theoretical results.

1. Introduction

Mosquito-borne diseases (e.g., malaria, dengue fever, and West Nile virus) remain a public health problem in developing countries despite many efforts of controls. The mortality and morbidity associated to this disease are still important for these countries. Many control strategies have been explored to control this disease among which are the development of effective insecticides and drug treatments, vaccine, and insecticide-treated bed nets. Nowadays, biological control of wild mosquito population by using genetically modified or transgenic mosquitoes is also used (see [1, 2]). This strategy control releases male mosquitoes (sterilized by radiological or chemical means) into the environment to mate with wild mosquitoes that are present in the environment. Female mosquitoes successfully mating with a sterile male will not breed or produce eggs that will not hatch. Repetitive releases of sterile mosquitoes or the release of a significantly large number of sterile mosquitoes can possibly wipe out or suppress a population of wild mosquitoes (see [3, 4]). Recently, several mathematical models have been proposed to measure the effectiveness of this control strategy (see [511]).

The model that we are studying by integrating the diffusion is obtained by assuming that the dynamics of the two mosquito populations follow a logistic growth, the birth rate of sterile mosquitoes being their release rate (see [5]):

The terms and are the numbers of wild and sterile mosquitoes consecutively, and , are the density independent and dependent death rates of the wild and sterile mosquitoes, respectively. The parameter is the number of matings per individual, per unit of time, with ; is the number of wild offspring produced per mate. designs the releasing rate of the sterile mosquitoes. Furthermore, it is well known that, among a number of factors that contribute to the upsurgence of malaria, we have vector immigration (see [12]). Mobility is well known to play an important role in the study of population dynamics (see [1316] and the references therein). Thus, it becomes important to take into account this aspect on the propagation of the disease. However, in the works cited above, we noticed that their authors do not take into account population mobility in space. Thus, motivated by the works in [5], we present a model that includes spatial diffusion of wild and sterile mosquitoes. This model is given by

Here, the parameters and represent the diffusion coefficients of wild and sterile mosquitoes, respectively. Symbol is the Laplacian operator, and is a connected and bounded spatial domain in with a smooth boundary . For model , we consider homogeneous Neumann boundary conditionsand initial conditions

The functions and are nonnegative continuous functions in and denotes the outward normal derivative on the boundary .

In this paper, we proposed a complete mathematical analysis of model (2). The analysis of model (2) in the case of constant releasing rate of sterile mosquitoes is studied Section 2. The well-posedness is established, and the stability of the equilibria is derived. Section 3 is devoted to the analysis of model (2) in the case of proportional releasing rate of sterile mosquitoes. In Section 4, numerical simulations are carried out to illustrate the theoretical results.

2. Existence and Positivity

In this paper, we study model (2) with a constant release rate of sterile mosquitoes, i.e., . We assume that the interaction rate is constant, and in order to simplify the writing, we always write instead of with ; then, system (2) becomeswith initial condition (4) and Neumann boundary condition (3).

We now derive the result about the existence and the positivity of the solution of system (5).

In this section, we establish the global existence, positivity, and boundedness of solutions of problem (5) under the conditions (3) and (4).

We state the following result.

Theorem 1. For any given initial data satisfying condition (4), there exists a unique solution of system (5) defined on , and this solution remains nonnegative and bounded for all .Proof. Let . System (5) can be rewritten abstractly in the Banach space as follows:where , , and andSince is locally Lipschitzian in , thus there exists a unique local solution of system (5) on the interval , where denotes the maximal existence time for solution of system (6) (see [7, 14, 15, 17]).
To achieve the global existence in , we need to prove the positiveness and boundedness of this solution.
Since and , for ,and


Then, integrating (10) over and using the fact that , we obtain

Hence, it follows thatwhere defines almost everywhere in .

Furthermore, from (5), we deduce that

By the principle comparison [18], it follows that , where is the solution of

Since for all , thus

In order to prove uniformly boundedness of , it is sufficient to show that is uniformly bounded (see [1], Theorem 2).



Since , it follows that

Hence,which implies thatwhere

Therefore, from [13], we deduce that there exists a positive constant that depends on such thatthat is, is .

Since and are bounded on , it follows from the standard theory for semilinear parabolic systems [19] that . Thus, the global existence follows. This completes the proof.

3. Analysis of the Equilibrium Points

Here, we are interested in the equilibrium points of model (5). Clearly, the presence of the reaction-diffusion term does not affect the number of the equilibrium. As in [5], for model (5), there is only one equilibrium point whose number of wild mosquitoes is zero, , where

Concerning the positive equilibrium of system (5), its components verifywhich implies

From (23), we deduce that

Hence, we introduce the following function:

Consequently, there is an equivalence between the fact that is an equilibrium point for system (5) and the fact that is the solution of the problem .

We havethat if is an extrema of with , then

Let us observe that

Thus, by defining the release threshold value of sterile mosquitoes as follows:then is equivalent to . Moreover, since and , then problem (26) admits one and only one positive solution if and only if , and two positive solutions, and where if and only if .

From the above discussion as in [5], we have the following result.

Theorem 2. Boundary equilibrium , see (22), is the unique equilibrium of (5) if and only if .
Model (5) admits a unique positive equilibrium withwhere given in (27), if and only if . For , we have two positive equilibria and for the model with

Using (27), we verify that does not depend on b; therefore, is also independent of . As soon as the mosquito release rate is higher than this rate, wild mosquitoes end up disappearing.

We now discuss the stability of the corresponding steady states of the model. Let us assume that be the eigenvalues of the operator on with the homogeneous Neumann boundary conditions and be the eigenfunction space corresponding to in . Let , be an orthonormal basis of and . Then,

Let be an arbitrary steady state of system (5). We define the following perturbation about :

By linearizing system (5) at , we obtainwhereand

We define the operator by

is invariant under the operator for all . Furthermore, is an eigenvalue of if and only if it is a root of the following equation:for some , and in this case, there exists an eigenvector in .

At the boundary , matrix becomesand in this case, characteristic (33) has only two roots , . So, is locally asymptotically stable.

It is easy to verify that at the equilibria and (when ), characteristic (33) takes the formwhere with or


Since (Cai et al.,2014), then for , we get , since for sufficiently large . Therefore, there is at least one positive root for . This yields that is unstable.

At the boundary , we have

Thanks to Cai et al., and choosing and such as ; it follows that . Moreover, using the fact that , we have

Therefore this choice of the parameters and allowed us to deduce that . Since and , then (40) has no positives roots; therefore, is locally asymptotically stable. We summarize the previous discussion in the following results.

Theorem 3. (i)For such that the boundary equilibrium is the only equilibrium for system (5), it is locally asymptotically stable(ii)For , the boundary equilibrium is a locally asymptotically stable node, is a unstable and moreover if , , then is stable(iii)For , the boundary equilibrium , see (26) is an unstable saddle node

4. Numerical Results

In this section, we give some numerical simulations to illustrate the theoretical results obtained in Section 3, we take one-dimensional spatial domain. and the grid sizes are and . Figure 1 presents the evolution of the population size of sterile and wild mosquito in system (5).

The threshold release value is . For , is a locally asymptotically stable node. We have two positive equilibria which is a saddle point an equilibrium which is a locally asymptotically stable. Solutions with different initial values approach either or as shown in the lower left and right figures in Figure 1, respectively. From these numerical simulations, we can, therefore, say that wild and sterile mosquitoes can coexist or although wild mosquitoes can disappear, depending on the initial quantity of the two populations.

Next, we take values and the other parameters remain unchanged. For this set of parameter values, the boundary equilibrium is the only equilibrium for system (5), and it is globally asymptotically stable and this is evident from Figure 2.

As shown by the results of the numerical simulations, the introduction of the diffusion in the model does not suppress the dynamics of wild mosquitoes or genetically modify, and one obtains globally the configurations that in the absence of the diffusion. However, the mathematical analysis becomes more complicated.

5. Discussion

In this work, we carried out a theoretical and numerical study of reaction-diffusion model describing the dynamic of the interaction between wild and sterile mosquitoes. We have established the theoretical and numerical effects of the introduction of the population diffusion on the stability of equilibrium states. On the other hand, when the number of sterile mosquitoes rejections is proportional to the population of wild mosquitoes, the results show that the solutions of the model exhibit a complex dynamic behavior. However, in order to have more realistic and more economical models for the release of sterile mosquitoes, it would be interesting to have a variable release rate that is proportional to the size of the wild mosquitoes while taking into account the mobility of the mosquitoes. This approach could be the subject of future studies [20].

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.