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 Open Access
Existence of order1 periodic solutions for a viral infection model with statedependent impulsive control
 Huilan Wang^{1},
 Binxiang Dai^{2}Email authorView ORCID ID profile and
 Qizhen Xiao^{1}
https://doi.org/10.1186/s136620191967x
© The Author(s) 2019
 Received: 14 September 2018
 Accepted: 15 January 2019
 Published: 29 January 2019
Abstract
It is well known that the drug treatment is always combined with the injection of immune factors. In this paper, a virus infection model with statedependent impulsive control is considered. Firstly, by deriving three categories of Bendixson domain and using the methods of geometry and successor function, we establish some criteria for the existence of positive order1 periodic solution for a general model, which extends the existing results in the literature. Further, the criteria are used to obtain the existence of positive order1 periodic solutions in the two cases that the positive equilibrium point is on the left or right side of the pulse line, respectively. Finally, an example is presented to illustrate our results.
Keywords
 Statedependent
 Impulse
 Periodic solution
 Successor function
 Bendixson domain
MSC
 37N25
 34A37
 34C25
1 Introduction
 (\(A_{1}\)):

\(f\in C^{1}(R,R)\), \(f(0)=\lambda>0\), \(f'(x)<0\) and there exists a positive number n such that \(f(n)=0\);
 (\(A_{2}\)):

\(g\in C^{1}(R,R)\), \(g(0)=0\), \(g'(x)>0\), and there exists a number \(m \in(0,n)\) such that \(g(m)=a\).
The principle of controlling the virus infection is either eradicating the virus or keeping the virus concentration at a low level while the ‘good’ cells at a high level. Since some classes of virus replicate so rapidly and irregularly that it is hardly possible to eradicate, the strategy of controlling the virus infection is to find a certain dynamical balance which does not lead to a disaster.
In fact, people often take measures to control the infected system before it reaches the worst case. For example, some scientists suggest that an HIV1 infected person should receive a corresponding treatment when the amount of CD4^{+}T decreases to 350 or 500 mm^{−3}. So the value 350 or 500 is regarded as one of the ‘therapy thresholds’. Theoretically, if the density of normal cells is always higher than the ‘therapy threshold’, we need not take the corresponding treatment measure. Otherwise, we must find an effective therapy to suppress the decline of ‘good’ cells. It is well known that a regular therapy for HIV infection is a continuous ART (antiretroviral therapy). However, in view of the viral reservoir it cannot be sufficiently targeted, the latent virus will be productive after a discontinuation of ART, which will lead to a burst of virus. Therefore, an integrate therapy is required. For example, in the treatment of HIV/SIV infection, the combination of Ad26/MVA vaccination and TLR7 stimulation results in decreased levels of viral DNA in lymph nodes and peripheral blood as well as in delays viral rebound for eight weeks following ART discontinuation [7].
Compared with the process of the disease, the impact of taking drugs or immune factors is short enough for it to be assumed that the therapy leads to an impulsive effect. On one hand, the drugs and immune injectors suppress the reproduction of the virus particles as well as the target cells at a certain rate; on the other hand, the immune injectors will stimulate the increasing of target cells.
The existence and the stability of positive periodic solutions are key issues on the study of mathematical biology models, so do for statedependent impulsive differential equations (see [8–21] and the references therein). In [13] and [14], the first integral of a system exists and therefore the Lambert W function is used to establish the existence of periodic solutions. However, if the first integral or the explicit solution of a system cannot be solved, then it is difficult to use the Lambert W function. For a start, Zeng, Chen and Sun [14] established a Poincaré–Bendixson ringdomain principle which is associated with a compression mapping. Some researchers considered such models by the geometric methods or successor function [15, 16, 18], and obtained some existence results of order1 periodic solution. The difficulty lies in the fact that the nontangent property is necessary to consider when we utilize the continuity of a successor function.
Motivated by the previous work, we are aiming to establish some criteria for different Bendixson domains, and hence to obtain an impulsive control strategy for system (1.3). We try to find the sufficient conditions that ensure the existence of order1 periodic solution which is superior to the ‘critical state’, or consider whether the control is required.
The structure of this paper is as follows. In Sect. 2, we begin with the qualitative analysis for system (1.1) without impulse, then introduce some notation and lemmas which will be used in the next sections. In particular, we derive three categories of Bendixson domain to deal with the tangent segment and to extend the existing Poincaré–Bendixson ringdomain principle in [14]. In Sect. 3, we obtain main results under two cases \(x^{*}< h\) and \(x^{*}>h\) (\(x^{*}=m\)). For the former, we consider the existence of positive order1 periodic solution by constructing an appropriate Bendixson domain. Under the case \(x^{*}>h\), we discuss how to determine an impulsive control based on the parameter q and the initial value. Finally, a conclusion and some examples are put forward in Sect. 4.
 (\(A_{3}\)):

\(\tau>ph\) (the stimulation of immune injectors on target cells is stronger than the suppression on them);
 (\(A_{4}\)):

\(m< h+(\tauph)< n\) (once the cells are infected, the stimulation of immune injectors is limited).
2 Preliminaries
First, we start from system (1.1). Set a Cartesian coordinate system \(xOv\), and let x axis be the horizontal axis. Denote any solution \((x(t),v(t) )\) of system (1.1) by \((x,v)\).
Lemma 2.1
Proof
Denote \(l_{1}:x=n\) and \(l_{2}:L(x,v)=0\), where \(L(x,v)=x+v (\frac {f(0)}{a}+n )\).
Lemma 2.2
Under the assumptions (\(A_{1}\)) and (\(A_{2}\)), the positive equilibrium \(E_{1}\) of system (1.1) is asymptotically stable and \(E_{0}\) is unstable.
Proof
From Lemma 2.1, it follows that the solutions of (1.3) are positive with positive initial values since \(\Delta x(t)= \taupx(t)\), \(\Delta v(t)=qv(t)\) and \(0< p<1\), \(0< q<1\).
Lemma 2.3
Proof
Similarly, \(h<\bar{h}\) implies \(v_{0}=\varphi(\bar{h})<\varphi(h)=\omega_{0}\). □
We define the positive orbit (or solution) starting from \(P (x(t),v(t) )\in\mathbb{R}_{+}^{2}\) by \(O^{+}(P)\) and the negative orbit arriving at it by \(O^{}(P)\). Obviously, the positive trajectories initiated from N will intersect with the impulse segment M or cannot approach it. Similarly, the negative trajectory initiated from M will be reachable or unreachable to the segment N. If \(O^{+}(P_{n})(\bar{h},v_{n})\) intersects firstly with M at point \(Q_{n}\triangleq(h,\tilde{v}_{n})\), we denote \(O^{+}_{M}(P_{n})=Q_{n}\).
For any \(A, B\in N\), if A lies above B, we denote \(A>B\). Moreover, we define \(AB=BA=v_{B}v_{A}\), where \(v_{A}\), \(v_{B}\) is the ordinates of A and B, respectively.
Lemma 2.4
[12] The successor function is continuous if it is well defined.
Based on the definition of orderk periodic solution for an impulsive dynamics system in [12], we give the definition of order1 periodic solution.
Definition 2.1
([12])
A trajectory \(O^{+}(P_{n})\) of system (1.3) together with the impulsive line \(\overline{Q_{n}P_{n}}\) is called an order1 cycle if \(v_{n+1}=v_{n}\).
From (2.3), \(F(P_{n})=0\) implies the existence of order1 periodic solution.
To ensure that the successor function is well defined, we consider three categories of Bendixson domain for system (1.3).
Definition 2.2
 (i)
there is no singularity in it;
 (ii)
trajectory \(L_{1}\) intersects with N, M at \(A_{0}\) and \(B_{0}\) in order; trajectory \(L_{2}\) intersects with N, M at \(A_{1}\) and \(B_{1}\) in order, respectively;
 (iii)
segments \(\overline{A_{0}A_{1}}\) and \(\overline{B_{0}B_{1}}\) cannot be tangent to trajectories of system (1.3) except at the end point.
If \(L_{1}\) is tangent to N at \(A_{0}\), and \(A_{0}>A_{1}\) gives \(B_{0}< B_{1}\), then we call the region D a subparallel trajectory rectangle (see Fig. 4(b));
If \(L_{1}\) is tangent with M at \(B_{0}\) and intersects with N at \(A_{0}\) and \(\bar{A}_{0}\) in order, \(L_{2}\) intersects with N at \(A_{1}\) and \(\bar{A}_{1}\) in order, and \(A_{0}>A_{1}\), then we call the region D a semiring domain (see Fig. 4(c)).
Lemma 2.5
Suppose a parallel or subparallel domain D is composed of \(\widetilde{A_{0}B_{0}}\), \(\overline{A_{0}A_{1}}\), \(\widetilde{A_{1}B_{1}}\) and \(\overline{B_{0}B_{1}}\) and with \(F(A_{0})F(A_{1})<0\). Then there exists an order1 periodic solution in D.
Proof
Since D is parallel or subparallel as defined above, we have \(O^{+}_{M} (A_{n})\neq\emptyset\) for any \(A_{n}\in\overline{A_{0}A_{1}}\). As the successor function \(F(A_{n})\) is continuous on \(A_{n}\in\overline {A_{0}A_{1}}\), it follows from \(F(A_{0})F(A_{1})<0\) that there must exist an \(A_{N}\in\overline{A_{0}A_{1}}\) such that \(F(A_{N})=0\), which implies the existence of an order1 periodic solution in D. □
Lemma 2.6
 (i)
if \(\mathfrak{F}(\overline{A_{0}A_{1}})\subseteq\overline{A_{0}A_{1}}\) or \(\mathfrak{F}(\overline{A_{0}A_{1}})\subseteq \overline{\bar{A}_{0}\bar{A}_{1}}\), then there exists an order1 periodic solution which is initiated from \(\overline {A_{0}A_{1}}\) or \(\overline{{\bar{A}_{0}\bar{A}_{1}}}\),respectively;
 (ii)
if \(\mathfrak{F}(\overline{A_{0}A_{1}})\subseteq \overline{\bar{A}_{0}A_{0}}\), then there is no order1 periodic solution in D.
Proof
(i) Obviously, if \(\mathfrak{F}(\overline {A_{0}A_{1}})\subseteq\overline{A_{0}A_{1}}\), then the continuous map \(\mathfrak{F}=IO^{+}_{M}\) is a compression mapping. Thus there exists a fixed point \(A_{n}\in\overline{A_{0}A_{1}}\) such that \(\mathfrak{F}(A_{n})=A_{n}\), which implies the existence of order1 periodic solution initiated from \(\overline{A_{0}A_{1}}\). If \(\mathfrak {F}(\overline{A_{0}A_{1}})\subseteq \overline{\bar{A}_{0}\bar{A}_{1}}\), then \(F(\bar{A}_{0})F(\bar{A}_{1})<0\), which implies the existence of order1 periodic solution initiated from \(\overline{\bar{A}_{0}\bar{A}_{1}}\).
(ii) If \(\mathfrak{F}(\overline{A_{0}A_{1}})\subseteq\overline{A_{0}\bar {A}_{0}}\), then all the trajectories initiated from \(\overline{A_{0}A_{1}}\) will be mapped onto \(\overline{A_{0}\bar{A}_{0}}\), from which the trajectories will not approach M since \(L_{2}\) is tangent to M at \(B_{0}\). Therefore, there is no order1 periodic solution in D. □
3 Main results
Suppose \(O^{+}_{M}(P_{0})=Q_{0}\triangleq (h,\tilde{v}_{0} )\) and denote the trajectories \(\widetilde{P_{0}Q_{0}}\), \(\widetilde{P_{1}Q_{1}}\) by functions \(v_{0}(x)\) and \(v_{1}(x)\), respectively. Then we have the following lemmas.
Lemma 3.1
Under the assumption (\(A_{1}\))–(\(A_{4}\)), if \(O^{+}_{M}(P_{0})\neq\emptyset\), then \(v_{0}(x)>\varphi(x)\) for \(x\in(h,\bar{h})\).
Proof
Provided that there exists an \(x_{0}\in(h,\bar{h})\) such that \(v_{0}(x_{0})=\varphi(x_{0})\), then the trajectory \(\widetilde {P_{0}Q_{0}}\) will intersect with the trajectory initiated from \((x_{0}, \varphi(x_{0}) ) \) which is tangent to the line \(x=x_{0}\). It will contradict the uniqueness of the solution to system (1.1). □
Lemma 3.2
Suppose \(O^{+}_{M}(P_{0})\neq\emptyset\). If \(F(P_{0})>0\), then there exists a point \(P\in N\) which lies above \(P_{0}\) such that \(F(P)\leq 0\).
Proof
Let \(O^{+}_{M}(P_{0})=Q_{0}\). Then all the trajectories initiated from N will approach M. \(F(P_{0})>0\) implies \(IO^{+}_{M}(P_{0})=P_{1}>P_{0}\). We consider two cases:
Case 1: \(x^{*}=m< h\).
We claim \(F(P_{k})< F(P_{k1})\) (\(k=1,2,\ldots\)).
Firstly, we prove that \(\overline{Q_{0}Q_{1}}<(1q)\overline{P_{0}P_{1}}\).
Case 2: \(h< x^{*}=m<\bar{h}\).
In this case, the trajectories reach the highest point at \(x=m\). Suppose \(O^{+}(P_{0})\cap\{x=m\}=Q'_{0}\). Since \((1q)Q_{0}>P_{0}\) and \(v_{Q'_{0}}>v_{Q_{0}}\), we have \((1q)Q'_{0}>(1q)Q_{0}>P_{0}\). Obviously, \(v_{0}(x)\) and \(v_{1}(x)\) are continuous on \([m,\bar{h}]\) and derivative on the open interval \((m,\bar{h})\). We apply the Cauchy mean theorem on interval \([m,\bar{h}]\). It follows from the proof of (i) that there exists a point \(P\in N\) which lies above \(P_{0}\) such that \((1q)Q'\leq P\), where \(Q'=O^{+}(P)\cap\{x=m\}\). Similarly, we have \(v_{Q'}>v_{Q}\), which gives \((1q)Q<(1q)Q' < P\), that is, \(F(P)\leq0\) (see Fig. 5(b)). □
In the following, we discuss the existence of periodic solutions in the cases of \(x^{*}< h\) and \(h< x^{*}<\bar{h}\).
3.1 \(x^{*}< h\)
If \(x^{*}=m< h\), then all the trajectories initiated from N intersect with M and cross it since the equilibrium \(E_{1}\) is asymptotically stable.
Theorem 3.1
Suppose \(x^{*}< h\) holds. Then there must exist an order1 periodic solution for (1.3) under the assumptions (\(A_{1}\))–(\(A_{4}\)).
Proof
We consider two possible cases according to \(F(P_{0})\).
Case 1. Suppose \(F(P_{0})>0\) holds. By Lemma 3.2, there exists a \(P\in N\) which lies above \(P_{0}\) such that \(F(P)\leq0\). Therefore, the domain composed of \(\widetilde{P_{0}Q_{0}}\), \(\overline{P_{0}P}\), \(\widetilde{PQ}\) and \(\overline{Q_{0}Q}\) is parallel, and thus \(F(P_{0})F(P)<0 \). By Lemma 2.5, there exists an order1 periodic solution initiated from \(\overline{P_{0}P}\).
Since \(P_{0}\) is the tangent point and \(P_{1}< P_{0}\), the region D composed of \(\widetilde{P_{0}Q_{0}}\), \(\overline{P_{0}P_{1}}\), \(\widetilde{P_{1}Q_{1}}\) and \(\overline{Q_{1}Q_{0}}\) is subparallel. Obviously, \(F(P_{1})>0\). Otherwise, it contradicts the fact that I is increasing. Hence, \(F(P_{0})F(P_{1})<0\), by Lemma 2.5, there exists an order1 periodic solution in D.
Claim
The periodic solution is initiated from \(\overline{P_{0}P_{1}}\) when \(F(P_{0})<0\). We need only to prove that \(P_{2}<\bar{P}_{1}\). Provided \(P_{2}>\bar{P}_{1}\), then \(\overline{P_{1}P_{2}}>\overline{P_{0}\bar{P}_{1}}\) as \(P_{1}< P_{0}\). However, by Lemma 3.2, we have \(\overline {Q_{0}Q_{1}}<\overline{P_{0}\bar{P}_{1}}\), which gives \(\overline{P_{1}P_{2}}=(1q) \overline {Q_{0}Q_{1}}<\overline{P_{0}\bar{P}_{1}}<\overline{P_{1}P_{2}}\). It comes to a contradiction (see Fig. 6(c)). □
Corollary 3.1
Proof
By Lemma 2.1, \(O^{+}_{M}(T_{\bar {h}})\triangleq S_{h}(h,s_{h})< T_{h}\). It follows from \(1q\leq\frac{t_{\bar{h}}}{t_{h}}\) that \(I(S_{h})< I(T_{h})=(1q) t_{h}< t_{\bar{h}}\), which gives \(F(T_{\bar{h}})<0 \).
If \(F(P_{0})>0\), then \(T_{\bar{h}}\) can be regarded as P in Theorem 3.1. Since \(v_{0}< t_{\bar{h}}\), the order1 periodic solution, which initiated from \(\overline{P_{0}T_{\bar{h}}}\), lies below the line \(x+v=\frac{f(0)}{a}+n\).
If \(F(P_{0})<0\), by Theorem 3.1 there must exist an order1 periodic solution initiated from \(\overline{P_{0}P_{1}}\), which lies below the line \(x+v=\frac{f(0)}{a}+n \). □
Remark 3.1
In fact, according to Lemma 2.5, if \(1q\leq\frac {t_{\bar{h}}}{s_{h}}\), then there must exist an order1 periodic solution under the line \(x+v=\frac{f(0)}{a}+n \) for system (1.3). Obviously, the condition \(1q\leq\frac{t_{\bar{h}}}{t_{h}}\) is stronger than \(1q\leq\frac{t_{\bar{h}}}{s_{h}}\) in the sense that \(s_{h}< t_{h} \). In view of the computation of \(t_{h}\) being more visible than \(s_{h}\), we prefer the former. On the other hand, if it does not hold, there maybe exists an order1 periodic solution above the line \(x+v=\frac{f(0)}{a}+n\). However, the state is not optimal because of the higher load of v.
3.2 \(h< x^{*}<\bar{h}\)
In this case, the trajectory \(O^{+}(P_{0})\) does not necessarily approach the line \(x=h\).
Lemma 3.3
 (i)
\(O^{}_{N}(W_{0})=\emptyset\Longleftrightarrow O^{+}_{M}(P_{0}) \neq\emptyset\);
 (ii)
if \(\omega_{0}>t_{h}\), then \(O^{}(W_{0})\) will intersect with \(x=\bar{h}\) at unique point \(W^{}_{0}\triangleq(\bar{h},\omega^{}_{0})\), and such that \(\omega^{}_{0}> t_{\bar{h}}\).
Proof
(i) If \(O^{}_{N}(W_{0})=\emptyset\), then the trajectory \(O^{}(W_{0})\) intersects with the isoline \(v=\varphi(x)\) at the point which lies on the left to \(P_{0}\) (see Fig. 1 and Fig. 2(b)). It is obvious that \(O^{+}(P_{0})\) will intersect with M, otherwise, \(O^{+}(P_{0})\) will pass through \(O^{}(W_{0})\) and approach \(E_{1}\), which contradicts the uniqueness of solution to system (1.1).
Suppose that \(O^{+}_{M}(P_{0})=Q_{0}(h,\tilde{v}_{0}) \neq\emptyset\). By Lemma 3.1, we have \(\tilde {v}_{0}=v_{0}(h)>\varphi(h)=\omega_{0}\). Therefore, the trajectory \(O^{}(W_{0})\) will intersect with the isoline \(v=\varphi(x)\) at the point that lies on the left to \(P_{0}\), which means \(O^{}_{N}(W_{0})=\emptyset\). The proof for (i) is completed.
(ii) We divided the proof into three steps.
Firstly, we prove that \(O^{}_{N}(W_{0})\neq\emptyset\). Assume that \(O^{}_{N}(W_{0})=\emptyset\). According to the result of (i), we have \(O^{+}_{M}(P_{0})=Q_{0}(h,\tilde{v}_{0})\neq\emptyset\) and \(\tilde {v}_{0}=v_{0}(h) >\varphi(h)=\omega_{0}\), which implies the trajectory \(O^{+}(P_{0})\) will go out from Ω. Thus \(O^{}_{N}(W_{0})\neq\emptyset\).
Next, we prove that \(O^{}_{N}(W_{0})>T_{\bar{h}}\). Otherwise, \(O^{}_{N}(W_{0})< T_{\bar{h}}\) will lead to a similar contradiction that the trajectory passing through \(W_{0}\) goes out from Ω.
Finally, we prove that \(O^{}(W_{0})\) intersects with \(x=\bar{h}\) at unique point. Assume that \(O^{}(W_{0})\) intersects with N at two points above \(T_{\bar{h}}\). The tangent point \(P_{0}\) will lie between the two intersected points, which means \(v_{0}>t_{\bar{h}}\), it is a contradiction to the fact that \(v_{0}< t_{\bar{h}}\).
Theorem 3.2
If \(\omega_{0}\geq t_{h}\), then there is no periodic solution below the line \(x+v=\frac{f(0)}{a}+n \) for system (1.3).
Proof
If \(\omega_{0}>t_{h}\), according to Lemma 3.3, \(O^{}(W_{0})\) will intersect with \(x=\bar{h}\) at unique point \(W^{}_{0}\), which implies that all the trajectories, initiated from the points under \(W^{}_{0}\) in N, will not hit the line \(x=h\). Further \(\omega^{}_{0}> t_{\bar{h}}\), therefore, there is no order1 periodic solution that lies in the domain Ω for system (1.3). □
Remark 3.2
Theorem 3.2 implies that we may take no measure to control the system (1.1) if \(\omega_{0}\geq t_{h}\) and the initial point \((x_{0}^{+},v_{0}^{+})\) lies in the domain \(\varOmega\cap\{(x,v)x>h\}\).
Theorem 3.3
 (i)
If \(O^{}_{N} (W_{0})=\emptyset\), then there must exist an order1 periodic solution; particularly, if \(1q<\frac {t_{\bar{h}}}{t_{h}}\), then the order1 periodic solution is below the line \(x+v=\frac {f(0)}{a}+n \) for (1.3).
 (ii)
If \(O^{}(W_{0})\) intersects with N at an unique intersected point \(W^{}_{0}(\bar{h},\omega^{}_{0})\) and \(1q<\frac{v_{0}}{t_{h}}\), then there is no order1 periodic solution below the line \(x+v=\frac{f(0)}{a}+n \) for system (1.3).
 (iii)
If \(O^{}(W_{0})\) intersects with N at two points \(\overline{W}^{}_{0} (\bar{h},\overline{\omega}^{}_{0} )\) and \(\underline{W}^{}_{0} (\bar{h},\underline{\omega}^{}_{0} )\), provided \(\frac{\overline{\omega}^{}_{0}}{\omega_{0}}<1q<\frac{t_{\bar{h}}}{t_{h}}\) or \(1q<\frac{\underline{\omega}^{}_{0}}{t_{h}}\), then there exists an order1 periodic solution initiated from \(\overline{W}^{}_{0}T_{\bar {h}} \) or from the line segment between \(\underline{W}^{}_{0} \) and \(I(W_{0})\), respectively.
Proof
(ii) From \(\omega_{0}< t_{h}\), it gives \(W_{0}< T_{h} \). Since \(O^{}(W_{0})\) intersects with N at an unique intersected point \(W^{}_{0}\), we have \(O^{+}(P_{0})=\emptyset\). Obviously, \(W^{}_{0}>P_{0}\), that is, \(\omega^{}_{0}>v_{0}\). It follows from \(1q<\frac {v_{0}}{t_{h}}\) that \(1q<\frac{\omega^{}_{0}}{t_{h}}\), which implies that all the points in \(\overline{W_{0}T_{h}}\) will be mapped onto the segment below \(W^{}_{0}\) by impulsive map I, and the trajectories initiated from segment under \(W^{}_{0}\) will not hit M any more. Therefore, there is no order1 periodic solution under the line \(x+v=\frac{f(0)}{a}+n \) (see Fig. 8(b)).
(iii) Since \(O^{}(W_{0})\) intersects with N at two points \(\overline{W}^{}_{0}\triangleq(h_{0}, \bar{\omega}^{}_{0})\) and \(\underline{W}^{}_{0}\triangleq(h_{0}, \underline{\omega}^{}_{0})\), \(\frac{\bar{\omega}^{}_{0}}{\omega_{0}}<1q<\frac{t_{\bar{h}}}{t_{h}}\) implies \(F(T_{\bar{h}})F(\overline{W}^{}_{0})<0\) and the domain composed of \(\widetilde{\bar{W}_{0}^{}W_{0}}\), \(\overline{W_{0}S_{h}}\), \(\widetilde{T_{\bar{h}}S_{h}}\) and \(\overline{T_{h}\bar{W}^{}_{0}}\) is parallel. By Lemma 2.5, there exists an order1 periodic solution which is initiated from \(\overline{W}^{}_{0}T_{\bar{h}}\). Similarly, it follows from \(1q<\frac{\underline{\omega}^{}_{0}}{t_{h}}\) that \(1q<\frac{\underline{\omega}^{}_{0}}{\omega_{0}}\), that is, \(I(W_{0})\triangleq U <\underline{W}^{}_{0}\). Denote \(O^{+}_{M}(U)=\tilde{U}\). Then the domain composed of \(\widetilde{\underline{W}_{0}^{}W_{0}}\), \(\overline{W_{0}\tilde{U}}\), \(\widetilde{U\tilde{U}}\) and \(\overline{U\underline{W}^{}_{0}}\) is semiring. It is obvious that \(I(W_{0}\tilde{U})\subseteq U\underline{W}^{}_{0}\). By Lemma 2.6, there is an order1 periodic solution which is initiated from \(U\underline{W}^{}_{0}\) (see Fig. 8(c)). □
Now, we will consider the stability of the order1 periodic solution for system (1.3).
Lemma 3.4
(Analog of Poincaré criterion [10, 11, 15])
Theorem 3.4
Proof
Suppose that \((X,V)\) intersects the sections M and N at points \(O (h,V(T) )\) and \(O^{+} (\bar{h}, (1q)V(T) )\), respectively.
4 Example
Proposition 4.1
If \(\frac{a}{\beta}< h\), then there must exist an order1 periodic solution for (4.1). Additionally, if \(1q<\frac{t_{\bar{h}}}{t_{h}}\), then the order1 periodic solution lies below the line \(x+v\leq\frac{\lambda}{a} +\frac{\lambda}{d}\).
Proposition 4.2
 (i)
if \(\frac{\lambdadh}{\beta h}> \frac{\lambda }{a}+\frac{\lambda}{d}h\), then there is no order1 periodic solution lies below the line \(x+v\leq\frac{\lambda}{a}+\frac{\lambda}{d}\);
 (ii)
if \(O^{+}_{N}(P_{0})\neq\emptyset\) and \(1q< \frac{\frac{\lambda }{a}+\frac{\lambda}{d}\bar{h}}{\frac{\lambda }{a}+\frac{\lambda}{d}h}\), then there must exist an order1 periodic solution below the line \(x+v\leq\frac{\lambda}{a} +\frac{\lambda}{d}\) for (4.1).
Moreover, \(g(x)=\beta x\) shows that \(g'(x)\geq\frac{g(x)}{x}\). Suppose \((X(t),V(t))\) is an order1 periodic solution of system (4.1). According to Theorem 3.4, we have the following.
Proposition 4.3
To verify the conditions of Propositions 4.1–4.3, we choose global parameters \(p=0.1\), \(\tau=0.5\) and \(h=3.5\), which implies \(\bar{h}=3.65\).
5 Conclusion and discussion
Theoretically, we are aiming to establish some criteria for the existence of order1 periodic solution based on the Bendixson domain types. Lemmas 2.5 and 2.6 can be extended to other models.
From the biological point of view, we are aiming to control the system when \(E_{1}\) is asymptotically stable since the natural state may lead to a disaster. We hope that the impulsive treatment can improve the natural state.
In the case \(x^{*}< h\), by Theorem 3.1, the impulsive treatment can prevent the deterioration since there always exists an order1 periodic solution between M and N. Further, when \(1q<\frac{v_{0}}{\tilde{v}_{0}}\), the periodic solution lies in a subparallel domain. The periodic solution lies in a parallel domain while \(1q>\frac{v_{0}}{\tilde{v}_{0}}\). Obviously, the former is superior to the latter because of the lower load of v and higher load of x. As is shown in Corollary 3.1, if \(1q<\frac{t_{\bar{h}}}{t_{h}}\), the periodic solution will lie under the line \(x+v=\frac{f(0)}{a}+n\). Therefore, we hold that \(1q\) is the smaller the better.
In the case \(h< x^{*}<\bar{h}\), if \(\omega_{0}>t_{h}\) and the initiate value of v is small enough, then there is no need to control the system in the sense that any trajectory cannot cross the line \(x=h\) or the natural state is superior to the critical state; if \(\omega_{0}< t_{h}\) and \(O^{+}_{M}(P_{0})\neq\emptyset\), then it is necessary to take the measure and let \(1q<\frac{t_{\bar{h}}}{t_{h}}\), so that there exist an order1 periodic solution that lies in Ω; if \(\omega_{0}< t_{h}\) and \(O^{+}_{M}(P_{0})=\emptyset\), as long as the impulsive point in N is close enough to \(P_{0}\), the impulsive control can prevent the trajectories from crossing the line \(x=h\). This also contributes to the fulfillment of the condition (3.8).
Declarations
Acknowledgements
The authors wish to thank the reviewers for their comments and suggestions on this work.
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Not applicable.
Funding
The work was supported by the National Natural Science Foundation of China (11871475), the Natural Science Foundation of Hunan Province (2018JJ2319) and the Natural Science Foundation of Youth Fund Project of Hunan Province (2018JJ3419).
Authors’ contributions
All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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