Abstract
This paper concerns a discrete wild and sterile mosquito model with a proportional release rate of sterile mosquitoes. It is shown that the discrete model undergoes codimension-2 bifurcations with 1 : 2, 1 : 3, and 1 : 4 strong resonances by applying the bifurcation theory. Some numerical simulations, including codimension-2 bifurcation diagrams, maximum Lyapunov exponents diagrams, and phase portraits, are also presented to illustrate the validity of theoretical results and display the complex dynamical behaviors. Moreover, two control strategies are applied to the model.
1. Introduction
In recent years, the mosquito-borne diseases, including West Nile virus, dengue, malaria, Japanese encephalitis, and chikungunya, have attracted much attention. Until now, there are no available vaccines for many of these viruses, which constitute a significant threat to global health. To control the transmissions of mosquito-borne diseases, it is necessary to adopt some strategies to reduce the number of wild mosquitoes. One of effective and environment-friendly approaches is sterile insect technique (SIT). Generally, the mass male mosquitoes are exposed to radiation and thereby rendered sexually sterile. After being released into the desired area, the treated male mosquitoes will mate with wide female mosquitoes and their offspring is nonviable. In this process, choosing the number of sterile mosquitoes and suitable releasing strategy are particularly important.
To understand the interactive dynamics of wide mosquitoes and sterile mosquitoes, some mathematical models have been introduced, for example, [1–9]. Recently, Cai et al. [7] proposed three mathematical models which represent different releasing strategies of sterile mosquitoes. The existence and stability of equilibria were investigated, and the threshold dynamics were obtained. One of the models considered in [7] iswhere are the numbers of wild mosquitoes and sterile mosquitoes, respectively. All the coefficients are positive constants, is the number of offspring produced by per mate, is the linear release rate of sterile mosquitoes with coefficient , and and represent the density independent and dependent death rates of wild and sterile mosquitoes, respectively. It was shown in [10, 11] that continuous model (1) undergoes a sequence of codimensional-1 bifurcations (including saddle-node bifurcation, supercritical and subcritical Hopf bifurcation, and homoclinic bifurcation) and codimensional-2 bifurcation (Bogdanov–Takens bifurcation). When in (1) is replaced by , the system becomes a mosquito population model with nonlinear saturated releasing rate of sterile mosquitoes. Some complex bifurcation phenomena, such as Hopf bifurcation of codimension-3, nilpotent cusp of codimension-4, and Bogdanov–Takens bifurcation of codimension-2, were provided in [12] if and .
In mathematical modelling, difference equations (that is, discrete-time models) are also widely used to describe the population dynamics. Compared with continuous models, discrete-time cases have many advantages. The most obvious one is that discrete models are more convenient to formulate and simulate, especially when the populations have nonoverlapping generations [13–15]. Moreover, for several biological models, their discrete-time cases may produce much richer dynamical behaviors than the continuous cases.
In this paper, we study the complex dynamics of the following discrete-time model:which is established from (1) by the Euler forward difference method. are defined as in model (1). By the biological meaning of the model, we only consider system (2) in the region in the -plane. We will study the existence of codimension-2 bifurcations associated with 1 : 2, 1 : 3, and 1 : 4 strong resonances of model (2). The codimension-2 bifurcation diagrams, maximum Lyapunov exponent diagrams, and phase portraits are also presented to illustrate the validity of theoretical results and display the complex dynamical behaviors.
The bifurcation and chaotic behavior of discrete models have been widely investigated by many researchers, but most of them focus on codimensional-1 bifurcations, for example, [16–21]. The codimension-2 bifurcations for discrete models are considered very few. The codimension-2 bifurcations of a delayed discrete-time Hopfield neural network, discrete epidemic model with nonlinear incidence rate, and discrete SIS epidemic model with standard incidence are analyzed in [22–24]. A discrete Hindmarsh–Rose model undergoes codimension-2 bifurcations with 1 : 2 and 1 : 4 strong resonances [25], and 1 : 3 strong resonance [26]. 1 : 1 strong resonance bifurcation of a discrete predator-prey model with nonmonotonic functional response has also been studied in [27]. Moreover, Ren and Yu [28] investigated codimension-2 bifurcations of a discrete information diffusion model.
The main goal of this paper is to investigate the impact of the mating rate and the release rate coefficient on the interactive dynamics of wide mosquitoes and sterile mosquitoes. The rest of this paper is organized as follows. Preliminaries are given in Section 2. In Sections 3–5, some sufficient conditions of codimension-2 bifurcations associated with 1 : 2, 1 : 3, and 1 : 4 strong resonances are established, respectively. The numerical simulations and controller designs are presented in Sections 6 and 7. A brief discussion is given in Section 8.
2. Preliminaries
In this section, we first briefly introduce some known results on the existence of positive equilibria, which have been obtained in [7], and then provide some sufficient conditions for the existence of codimension-2 bifurcations with 1 : 2, 1 : 3, and 1 : 4 strong resonances.
Define the function
We can check that model (2) admits a positive equilibrium:if and only if has a positive root , which is also equivalent to the existence of positive solution to the equation with
Assuming that , we know the equation has two positive roots:
It follows that there exists a unique such that . Letthen the equation has two positive solutions with if and only if , and only one positive solution when . In summary, we have the following results on the existence of equilibria of model (2).
Lemma 1. Assume that . Then,(i)model (2) always has a boundary equilibrium (ii)if , then model (2) has no any positive equilibrium(iii)if , then model (2) has a unique positive equilibrium , where Moreover, and .(iv)if , then model (2) has two positive equilibria and , where
Moreover, and .
Denote by , the equilibrium of model (2), then the Jacobian matrix at isits trace is , and its determinant is , where
If the matrix has eigenvalues with , , or , then there may present a codimension-2 bifurcation with 1 : 2, 1 : 3, or 1 : 4 strong resonances at . For the equilibrium in Lemma 1, if codimension-2 bifurcation with 1 : 2, 1 : 3, or 1 : 4 strong resonance occurs, then and , or 0, which imply that and . It is a contradiction with . Similarly, since , we can also show that there does not exist the above codimension-2 bifurcations at the equilibrium . Thus, in this paper, we discuss the existence of codimension-2 bifurcation at .
Choosing and as bifurcation parameters, if , wherethen two eigenvalues of Jacobian matrix are . This shows that there may present a codimension-2 bifurcation with 1 : 2 strong resonance at .
If , wherethen two eigenvalues of the Jacobian matrix are . This shows that there may present a codimension-2 bifurcation with 1 : 3 strong resonance at .
Furthermore, if , wherethen two eigenvalues of the Jacobian matrix are . This shows that there may present a codimension-2 bifurcation with 1 : 4 strong resonance at . In the following, we will give the theoretical analysis and numerical simulations.
3. 1 : 2 Strong Resonance
In this section, by applying the bifurcation theory in [29], we analyze the codimension-2 bifurcation associated with 1 : 2 strong resonance of model (2) at the equilibrium .
We choose and as bifurcation parameters and assume that . Consider model (2) with and , where and are small perturbations of the parameters and . Let and , and model (2) becomes into the following form which has the equilibrium :
By expanding the right-hand side of (15) into the Taylor series at , we havewhere
For , the Jacobian matrix at iswhere
Take two linearly independent (generalized) eigenvectors of ,and that of the transposed matrix ,
It is easy to check that , where for any . Therefore, any vector can be represented in the formwhich implies that
Under new coordinates , map (16) can be written aswhere
By introducing a coordinate transformationmap (24) can be reduced into the following form:where
Next, we aim to eliminate all quadratic terms and some cubic terms in (27). To achieve it, we introduce the following transformation:where and will be determined later.
By applying transformation (29) and its inverse, map (27) becomeswhere
Assume that the following vanishing conditions hold:and we can obtain all the coefficients and satisfying . Moreover, by taking , we can determine the coefficients and for , see more details in [24, 25]. Thus, model (15) can be transformed into the normal form for 1 : 2 strong resonance bifurcation as follows:where and satisfy
Applying the results established in [29], we can get the existence of 1 : 2 strong resonance bifurcation.
Theorem 1. Assume that and . Then, model (2) has a 1 : 2 strong resonance bifurcation at when the parameters vary in a sufficiently small neighborhood of with fixed . If we further assume , then is a saddle; if , then is elliptic. determines the bifurcation scenario under small perturbations. Moreover, model (2) has the following bifurcation behaviors:(i)There is a pitchfork bifurcation curve , and there exist nontrivial equilibria for (ii)There is a nondegenerate Neimark–Sacker bifurcation curve (iii)There is a heteroclinic bifurcation curve
4. 1 : 3 Strong Resonance
In this section, we prove that there exists a codimension-2 bifurcation with 1 : 3 strong resonance of model (2) at . We choose and as bifurcation parameters of model (2) and assume that .
Similar to Section 3, we can also transform into and then expand the right-hand side of the map into Taylor series. For , the Jacobian matrix at iswithand it has two complex eigenvalues . The corresponding eigenvector and adjoint eigenvector of are defined as
Any vector can be represented as . Thus, for , model (16) can be written in the complex form as follows:where and
Next, we shall establish the normal form for 1 : 3 strong resonance bifurcation. We first eliminate some quadratic terms in (38) through the transformationwhere the coefficients satisfying will be determined later.
Using transformation (40) and its inverse transformation,map (41) is changed into the following form:where
Takethen transformation (40) is determined and map (42) becomes
In order to eliminate some cubic terms, we take the transformation
It is easy to check that the inverse transformation of (46) is
By applying (46) and (47), map (45) becomeswhere
Takethen transformation (46) is determined and map (48) reduces into the following form:
We define and . From the 1 : 3 strong resonance bifurcation analysis established in [29], we can get the following theorem.
Theorem 2. Assume that and . Then, model (2) undergoes a 1 : 3 strong resonance bifurcation at . determines the stability of the bifurcation invariant closed curve. Moreover, model (2) admits a number of the following complex codimension-1 bifurcation curves:(i)There is a nondegenerate Neimark–Sacker bifurcation at the trivial fixed point of (51)(ii)There is a saddle cycle of periodic three corresponding to saddle fixed points of map (51)(iii)There is a homoclinic structure formed by the stable and unstable invariant manifolds of the period three cycle intersecting transversally in an exponentially narrow parameter region
5. 1 : 4 Strong Resonance
In this section, we prove that there exists a codimension-2 bifurcation with 1 : 4 strong resonance of model (2) at . We choose and as bifurcation parameters of model (2) and assume that .
Similar to Section 3, we can also transform into the origin and then expand the right-hand side of the map into Taylor series. For , the Jacobian matrix at iswithand it has two complex eigenvalues . The corresponding eigenvector and adjoint eigenvector of are defined as
Any vector can be represented as . Thus, for , model (16) can be written in the following complex form:where and are defined as in (39) with replaced by , respectively.
Similar to Section 4, by (40) and (41) we can transform (55) into the following form:where
Takethen (56) becomes into the following form:
By (46) and (47), we can transform (59) into the following form as (51):
Let and . If , we denote . From the 1 : 4 strong resonance bifurcation analysis established in [29], we can obtain the following results.
Theorem 3. Assume that , , and . Then, model (2) has a 1 : 4 strong resonance bifurcation at . determines the bifurcation scenario near . Near the equilibrium , there exist two-parameter families of equilibria of order four bifurcation from , and one of them is unstable, or attractive or repelling invariant closed curves, which depends on the choices of parameters and . Moreover, there are many complex codimension-1 bifurcation curves of map (60) in a sufficiently small neighborhood of .
6. Numerical Simulations
In this section, we give a simulation illustration for codimension-2 bifurcations with 1 : 2, 1 : 3, and 1 : 4 strong resonances. The codimension-2 bifurcation diagrams, maximum Lyapunov exponents, and phase portraits of (2) are presented to illustrate the validity of theoretical results and display the complex dynamical behaviors.
Case 1. Choosing the values of the parameters as , we can numerically obtain and equilibrium . The corresponding eigenvalues of are . By further calculations, we have and . From Theorem 1, map (2) is equivalent to the normal form for 1 : 2 strong resonance bifurcation and is a 1 : 2 strong resonance bifurcation point.
Figure 1(a) shows 2-dimensional bifurcation diagram when and varies in , which is a neighborhood of . Figure 1(b) shows 3-dimensional bifurcation diagram when vary in a neighborhood of . Figures 2(a)–2(c) show the phase portraits of (2) near equilibrium for different parameters and . From the phase portraits, we know that there are periodic orbits and chaos at the corresponding parameter values.
The maximum Lyapunov exponents corresponding to the bifurcation diagram Figure 1(a) are presented in Figure 2(d). From Figure 2(d), we can see that some maximum Lyapunov exponents are positive and some ones are negative, which means that map (2) has chaotic behaviors near the 1 : 2 strong resonance point .
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Case 2. Choosing the values of the parameters as , we can numerically obtain and equilibrium . The corresponding eigenvalues of are . By further calculations, we have and . From Theorem 2, map (2) is equivalent to the normal form for 1 : 3 strong resonance bifurcation and is a 1 : 3 strong resonance bifurcation point.
Figure 3(a) shows 2-dimensional bifurcation diagram when and varies in , which is a neighborhood of . Figure 3(b) shows 3-dimensional bifurcation diagram when vary in a neighborhood of . Figures 4(a)–4(d) show phase portraits of map (2) near for different and . From Figure 4(a), we can see that there is a homoclinic structure formed by the stable and unstable invariant manifolds of the period three cycle intersecting transversally in a narrow parameter region.
The maximum Lyapunov exponents corresponding to the bifurcation diagrams Figures 3(a) and 3(b) are presented in Figures 5(a) and 5(b), respectively. We can see that some maximum Lyapunov exponents are positive and some ones are negative, which means that map (2) has chaotic behaviors near the 1 : 3 strong resonance point .
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Case 3. Choosing the values of the parameters as , we can numerically obtain and equilibrium . The corresponding eigenvalues of are . By further calculations, we have and . From Theorem 3, map (2) is equivalent to the normal form for 1 : 4 strong resonance bifurcation and is a 1 : 4 strong resonance bifurcation point.
Figure 6(a) shows 2-dimensional bifurcation diagram when and varies in , which is a neighborhood of . From Figure 6(a), we can see that there are flip bifurcations occur after Neimark–Sacker bifurcation near the 1 : 4 strong resonance point. Figure 6(b) shows 3-dimensional bifurcation diagram when vary in a neighborhood of . Figures 7(a)–7(d) show phase portraits of map (2) near for different and . When , the equilibrium is locally stable. When increase, map (2) has chaotic behaviors.
The maximum Lyapunov exponents corresponding to the bifurcation diagrams (Figures 6(a) and 6(b)) are presented in Figures 8(a) and 8(b), respectively. We can see that some maximum Lyapunov exponents are positive and some ones are negative, which means that map (2) has chaotic behaviors near the 1 : 4 strong resonance point .
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7. Controller Design
In this section, we apply two control strategies (tracking control and hybrid control) to (2). By using tracking control, we can annihilate wide mosquitoes and then prevent the transmission of mosquito-borne diseases. By applying hybrid control, we can advance the codimension-2 bifurcation and extend the range of parameter from the equilibrium to chaos.
7.1. Tracking Control
In this subsection, we design a tracking controller such that the number of wide mosquitoes vanishes gradually as in [23]. Consider a new controlled systemwhere is an exterior control signal. When , map (61) has chaos or period orbit under suitable parameter conditions. Let be our control aim, and we shall design a tracking controller such that as the number of iterations is sufficiently large.
Theorem 4. If the controller of controlled map (61) iswhereand the constants are positive constants satisfying all roots of lie in the unit circle, then as number of iterations tends to infinity.
Proof. Take the following transformationthen map (61) becomeswhereTaking (62) into (66), we can obtain the following map:By choosing suitable positive constants such that all roots of lie in the unit circle, map (61) after feedback is asymptotically stable. That is, as the number of iterations tends to infinity. Therefore, the trajectory of tracks an ideal state . This completes the proof.
We choose the values of the parameters as , and , and take and in . When the number of iterations is less than 2000, the chaotic behavior of for uncontrolled system (2) is presented in Figure 9(a). When , the controller is put on (2). From Figure 9(b), we can see that the trajectory of tracks ultimately an ideal state .
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7.2. Hybrid Control
In this section, we apply hybrid control, which is used in [28, 30], to (2). For any , consider a new controlled systemwhich has the same equilibria with (2). The Jacobian matrix at equilibrium isits trace is , and determinant is . By the analysis in Section 2, we know that only may be a 1 : 2, 1 : 3, or 1 : 4 strong resonance point.
Here, we only consider the codimension-2 bifurcation with 1 : 4 strong resonance. Taking , and in (68), we can numerically obtain and equilibrium , which is a 1 : 4 strong resonance point. The bifurcation diagram of controlled system (68) for fixed is presented in Figure 9(c) and is very similar to Figure 6(a) of the uncontrolled system. Recall that for the uncontrolled system, we have . Thus, by using hybrid control, we advance the corresponding codimension-2 bifurcation with 1 : 4 strong resonance. Compared with these two figures, we can also find the range of parameter from 1 : 4 strong resonance point to chaos in Figure 9(c) is about three times larger than that of Figure 6(a).
8. Discussion
In this paper, we analyzed codimension-two bifurcations with 1 : 2, 1 : 3, and 1 : 4 strong resonances of a discrete wild and sterile mosquito model, which is established from continuous case (1) proposed in [7] by the Euler forward difference method. In [10, 11], we have known that continuous model (1) undergoes a sequence of bifurcations such as saddle-node bifurcation, supercritical and subcritical Hopf bifurcation, homoclinic bifurcation, and Bogdanov–Takens bifurcation. Through the theoretical analysis and numerical simulations given in this paper, we find that corresponding discrete mosquito model (2) also has many complex dynamical behaviors, including codimension-1, codimension-2 bifurcations, and chaos. From the impact of the mating rate and the release rate coefficient on the dynamics of wide mosquitoes, we know that the release strategy plays an important role in controlling the number of wide mosquitoes. Moreover, two control strategies are applied to model (2). However, whether there is codimension-2 bifurcation with 1 : 1 strong resonance is an interesting problem worth considering.
Data Availability
All data generated or analyzed during this study are included in this article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (grant nos. 11801432, 11771373, and 11801429), China Postdoctoral Science Foundation (grant no. 2019M663610), Natural Science Basic Research Plan in Shaanxi Province of China (grant nos. 2018JM1011 and 2019JQ-136), and the Doctoral Scientific Research Foundation of Xi’an Polytechnic University.