Stability and periodicity in a mosquito population suppression model composed of two sub-models

In this paper, we propose a mosquito population suppression model which is composed of two sub-models switching each other. We assume that the releases of sterile mosquitoes are periodic and impulsive, only sexually active sterile mosquitoes play a role in the mosquito population suppression process, and the survival probability is density-dependent. For the release waiting period T and the release amount c, we find three thresholds denoted by T∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^*$$\end{document}, g∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g^*$$\end{document}, and c∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c^*$$\end{document} with c∗>g∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c^*>g^*$$\end{document}. We show that the origin is a globally or locally asymptotically stable equilibrium when c≥c∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\ge c^*$$\end{document} and T≤T∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\le T^*$$\end{document}, or c∈(g∗,c∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\in (g^*, c^*)$$\end{document} and T<T∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T<T^*$$\end{document}. We prove that the model generates a unique globally asymptotically stable T-periodic solution when either c∈(g∗,c∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\in (g^*, c^*)$$\end{document} and T=T∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T=T^*$$\end{document}, or c>g∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c>g^*$$\end{document} and T>T∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T>T^*$$\end{document}. Two numerical examples are provided to illustrate our theoretical results.

A wild female mosquito mating with a sterile mosquito will not be capable of producing offspring. Since the only function of the released males is to sterilize wild mosquitoes, only those sexually active sterile mosquitoes may play a role in the suppression process of wild mosquitoes. Based on these observations, the author in [17] first gave a novel modeling idea by treating the number of sterile mosquitoes as a given function instead of an independent variable satisfying a dynamical equation to explore the interactive dynamics of wild and sterile mosquitoes. This modeling technique mathematically reduces a planar system to a scalar differential equation, which makes mathematical analysis more tractable [18,19,25,26].
Due to the facts that mosquitoes have four metamorphic stages of development during their lifetime and the intraspecific competition mainly occurs in the aquatic stages and thus in the birth process [27,28], the author in [29] proposed the following model: where w = w(t) and g = g(t) are the numbers of wild and sterile mosquitoes at time t, respectively, a is the maximum number of survived offspring produced per mosquito, w w+g is the mating probability between wild mosquitoes, μ is the density-independent death rate of wild mosquitoes, ξ is the carrying capacity parameter such that 1 − ξw describes the density-dependent survival probability, and B(·) is the release rate of the sterile mosquitoes.
Applying the modeling idea in [17], the authors in [30] discarded the second equation of system (1.1) by treating g(t) as a given nonnegative continuous function in advance, and formulated the following nonautonomous ordinary differential equation model: The strategy of periodic and impulsive releases of sterile mosquitoes has been proposed and well studied in [18,19,25,26,31], where only sexually active mosquitoes are included in the models. LetT and T be the sexual lifespan of sterile mosquitoes and the waiting period between two consecutive releases, respectively. Accordingly, there are three possible release strategies: T <T , T =T , and T >T . For the release strategy of T =T , g(t) ≡ c, model (1.2) becomes With the introduction of the release amount threshold the authors in [30] gave the following theorem. (1) For the case when c ∈ (0, g * ), model (1.3) has three equilibria: the origin, denoted by w 0 , and two positive equilibria where A = a − μ aξ . Furthermore, w 0 and w 2 (c) are both asymptotically stable, and w 1 (c) is unstable. (2) For the case when c = g * , model (1.3) admits w 0 and w * = A 2 as the equilibria. w 0 is asymptotically stable, but w * is semi-stable, stable from the rightside and unstable from the left-side.
(3) For the case when c > g * , model (1.3) admits a unique equilibrium w 0 , which is globally asymptotically stable.
In this paper, we focus on the release strategy of T >T , which means that the sterile mosquitoes previously released have lost their mating competitiveness before the release of the next batch of sterile mosquitoes. We assume that a constant amount c of sterile mosquitoes are released at discrete time points . .. We find that g follows the following T -periodic and piecewise constant function: Then, model (1.2) is divided into the following two sub-equations: and which constantly switch each other, where i = 0, 1, 2, . . .. We assume a > μ throughout this paper, so that the wild mosquito population without the interference of sterile mosquitoes is stabilized at the population size A [25,32,33].
Define T * = a a − μT and c * = a − μ μξ as the release period threshold and the second release amount threshold, respectively. Obviously, we have c * > g * . In this paper, we investigate the existence, uniqueness and stability of T -periodic solutions to model (1.4)-(1.5) by using Poincaré map method. These results may be helpful for optimal control of mosquito vector or mosquito-borne diseases as shown in [34]. In Sect. 2, we derive the Poincaré map formulation by solving (1.4)-(1.5) with initial value. We prove in Sect. 3 that the origin is globally asymptotically stable for the case when c ≥ c * and T ≤ T * . The origin is locally asymptotically stable if and only if g * < c < c * and T < T * , and model (1.4)-(1.5) has a unique globally asymptotically stable T -periodic solution for the case when g * < c < c * and T = T * , or c > g * and T > T * . Two numerical examples are also given to illustrate our theoretical results. Finally, in Sect. 4, we provide some discussions about the current and future work.

Preliminaries
It is clear to see that model (1.4)-(1.5) admits w 0 as the unique equilibrium. For any initial condition Then, h(u) = w(T ;T ,h(u)).
To obtainh(u) and h(u), we need to solve initial value problem (1.4) with w(0) = u. Now, we assume c > g * . Equation (1.4) can be rewritten as: or, equivalently, Integrating (2.2) from 0 toT , we obtain which yields, from (2.1), and thus, we get, from (2.1), Define function sequences {h n } and {h n } as follows: We obtain, by induction, By using a similar argument to that in [25], we can easily prove the next lemma.

Main results
In this section, we will present our main results by dividing the study into two cases: T ≤ T * and T > T * .

Release more often with T ≤ T *
In this subsection, we consider the case of T ≤ T * . That is, we release sterile mosquitoes more often. Before presenting our first result, we give the following lemma.
Proof By (2.6), to prove (3.1), we only need to prove We divide the proof of (3.2) into two cases: T = T * and T < T * . For the case when T = T * , we let m = e − aξ αT . From (2.3) and (2.5), we have and Then, we get To prove (3.2), it suffices to prove By taking the derivative of H (u), we have Now, we show that (3.2) is still true. To this end, we assume by contradiction that (3.2) is not true. Then, Then, we obtain (3.9) which, by taking the derivative, yields which gives, by combining (3.8) with (3.9), Taking the derivative of both sides of (2.5) yields (3.11) Substituting (2.5) and (3.10) into (3.11), we have (3.12)

From (2.5) again, we get
which lead to, by (3.12), which yields (3.14) Let the quadratic polynomial Q(u) be defined as: Then, Q(û) ≥ 0 and Q(u) is concave up since B > 0. Moreover, we find that Q(A) = − a−μ+acξ a 2 ξ 2 mμA < 0. To sum up, we have which is impossible since 0 <û < A. Hence, (3.2) is also true for the case when T < T * , and thus (3.1) is true. The proof is complete.
The following result shows that if the release amount c ≥ c * , then the mosquito population will be eradicated eventually, which is guaranteed by the global asymptotical stability of w 0 .

Theorem 3.2
Assume that c ≥ c * and T ≤ T * . Then, the origin w 0 is globally asymptotically stable.
For the case when p = 1, there are two possible cases t 0 ∈ [0,T ] and t 0 ∈ (T , T ) to consider.
Since w(t) reaches its maximum at nT for t ∈ [(n − 1)T +T , nT +T ], n = 2, 3, . . ., it suffices to prove that We then give a numerical example to illustrate our theoretical results in Theorem 3.2.   Then, there is some which gives, along with (3.22) and the fact ofū < u 1 < Note thath(u) is strictly increasing, we haveh(u 2 ) > h(u 1 ), which leads to a contradiction. Hence, model (1.4)-(1.5) admits w(t; 0, u 1 ) as its unique T -periodic solution.
To prove the global asymptotical stability of w(t; 0, u 1 ), we only need to prove w 0 is unstable. From (3.20) and Lemma 2.1, we find that both {h n (u)} and {h n (u)} are strictly increasing for u ∈ (0,ū), which shows that w 0 is unstable. Hence, the proof of (1) is complete.
Then, by a similar argument to that of Theorem 3.2, we can prove that w 0 is asymptotically stable. The proof is finished.
For the critical release period T = T * , Theorem 3.3(1) shows that if g * < c < c * , then mosquito population eradication is impossible due to the existence of a unique globally asymptotically stable T -periodic solution. To see this, we give a numerical example in the following to illustrate our theoretical results in Theorem 3.3, together with an observation on the existence of exactly two periodic solutions for some T < T * and c ∈ (g * , c * ) with the theoretical proofs pending. The situation for the case when c ∈ (g * , c * ) and T < T * , however, is quite tricky. We can only prove the asymptotical stability of w 0 theoretically, but fail to count the exact number of T -periodic solutions of model (1.4)-(1.5). To numerically explore this problem, let c = 10,000 ∈ (g * , c * ) and T = 14.10 < T * . Then, w 0 is asymptotically stable from Theorem 3.3 (2). Moreover, there exists a asymptotically stable T -periodic solution as shown in panel (B) of Fig. 2. Thus, we deduce that model (1.4)-(1.5) has exactly two T -periodic solutions, the larger one being asymptotically stable, and the smaller one unstable, which has not been proved theoretically at the present time.
From Theorem 3.3, it is easy to see that the following corollary is true.

Release less often with T > T *
In this subsection, we consider the case of T > T * , which means that we release sterile mosquitoes less often. For this case, we have the following theorem. Proof From (2.9), we reach h (0) > 1, which indicates that h(u) > u for u ∈ (0, δ), where δ is sufficiently small. This, together with (2.6), means that there exists Assume by contradiction that model (1.4)-(1.5) has another T -periodic solution. Then, there is Finally, we verify that (3.27) is also impossible. To this end, let k − 1 > 0 be small enough such that (3.28) Substituting (3.28) in (3.13), we obtain which are equivalent to respectively, where Then, (3.29) becomes which leads to a contradiction, since B(k) > 0 and Q k (u) is a quadratic function. Hence, (3.27) is also impossible. This shows that T -periodic solution of model (1.4)-(1.5) is unique. By a similar arguments used in the proof of Theorem 3.3(1), we can prove the global asymptotical stability of the unique T -periodic solution. The proof is complete.
Combining Theorems 3.2 with 3.5, we obtain the following corollary.

Concluding remarks
Combining the modeling ideas for the interactive wild and sterile mosquitoes in [17], in which only the sexually active sterile mosquitoes are considered, in [29], in which the density dependent survival probability is assumed, and in [19,25,26], in which the releases of the sterile mosquitoes are impulsive, constant and periodic, we formulate and investigate model (1.4)-(1.5) in this paper. In [29], the author supposed that the release function g of sterile mosquitoes is an independent variable satisfying an independent dynamical equation, see also [1,32]. In [30], the authors treated the release function g as a given nonnegative continuous function in advance, especially, a constant function related to the release strategy of T =T .
Although model (1.4)-(1.5) looks simple, it has rich and complex dynamics including the local and global stability of the origin, the existence, stability and semistability of a unique T -periodic solution, and the existence and stability of exact two T -periodic solutions. We have found the corresponding threshold values such as T * for the waiting period between two consecutive releases, g * and c * for the release amount of the sterile mosquitoes each time. We have shown that the origin is asymptotically stable provided c ≥ c * and T ≤ T * , or c ∈ (g * , c * ) and T < T * in Theorem 3.2 and Theorem 3.3 (2), respectively, and model (1.4)-(1.5) admits a unique globally asymptotically stable T -periodic solution, provided c ∈ (g * , c * ) and T = T * or c > g * and T > T * in Theorem 3.3(1) and Theorem 3.5, respectively. Figure 1 verifies all conditions in Theorem 3.2 by four cases: (i) c > c * and T < T * , (ii) c = c * and T < T * , (iii) c > c * and T = T * , and (iv) c = c * and T = T * . Figure 2A, B verifies Theorem 3.3(1) and Theorem 3.3(2), respectively. We should mention here that for the case c ∈ (g * , c * ) and T < T * , it is possible for model (1.4)-(1.5) to have exact two T -periodic solutions, or a unique semi-stable T -periodic solution, or the origin is GAS. We leave this situation as our future work. Furthermore, it remains unknown for the case of c ∈ (0, g * ], which will be what we are going to study in the future, and by numerical simulations, we see that model (1.4)-(1.5) may have two T -periodic solutions, with the smaller one unstable and the bigger one stable.
Author contributions All authors contributed to the conception and design of the study. Material preparation was performed by Yantao Shi and Rong Yan, and analysis was performed by Zhongcai Zhu, Bo Zheng and Jianshe Yu. All authors read and approved the submission.

Conflict of interest
The authors declare that they have no conflict of interest.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/ by/4.0/.