The dynamics of a stage structure population model with fixed-time birth pulse and state feedback control strategy
- Xiangsen Liu^{1, 2} and
- Binxiang Dai^{1}Email author
https://doi.org/10.1186/s13662-016-0852-0
© Liu and Dai 2016
Received: 11 September 2015
Accepted: 29 April 2016
Published: 11 May 2016
Abstract
In this paper, we study a stage structure population model with fixed-time birth pulse and state feedback control strategy. The stability of the trivial solution and the existence of periodic solutions are investigated. Sufficient conditions for the permanence of the system are obtained. Furthermore, some numerical simulations are given to illustrate our results. The superiority of the mixed control strategy is also discussed.
Keywords
stage structure birth pulse state feedback control strategy periodic solution permanence1 Introduction
Stage structure models have attracted much attention in recent years. In most cases, ordinary differential equations are used to build stage structure models [1, 2]. However, impulsive differential equations [3, 4] are also suitable for the mathematical simulation of evolutionary processes in which the parameters state variables undergo relatively long periods of smooth variation followed by a short-term rapid change in their values. Many results have been obtained for stage structure models described by impulsive differential equations [5–7].
In [16], the authors addressed some problems on system (1.1) such as the existence and stability of positive period-2T solutions and the existence of flip bifurcations by means of bifurcation theory.
Jiang et al. [17] investigated the periodic solutions and their relationship in an SIS epidemic model with fixed-time birth pulses and state feedback pulse treatments. Lin et al. [18] considered an SIS epidemic model with fixed-time birth pulses and state feedback pulse treatments. They investigated the existence of positive periodic solutions and permanence of the system. However, stage structure models with fixed-time birth pulses and state feedback control strategy have not been discussed. Motivated by this, we seek to analyze this problem in detail.
The remaining part of this paper is organized as follows. In the next section, we discuss the existence of positive periodic solutions of system (1.2). In Section 3, the stability of the trivial solution is considered. We study the permanence of system (1.2) in Section 4. In Section 5, some numerical simulations are given to illustrate our results. Finally, some concluding remarks are given.
2 The existence of periodic solutions
In this section, we investigate the existence of periodic solutions.
Theorem 2.1
Assume that condition (2.4) holds. Then there exists a period-T solution of system (1.2) where the initial point \((x_{0}^{\ast},y_{0}^{\ast})\) is as in (2.5).
A special period-2T solution that is subject to spraying pesticide once, and birth pulse two times per period 2T is investigated in the following. Set the initial point of system (1.2) as \(A_{0}(x_{0},y_{0})\). The trajectory originating from the initial point \(A_{0}\) reaches the point \(A_{1}(x_{11},y_{11})\) at \(t=T\) and jumps to the point \(A_{1}^{+}(x_{11}^{+},y_{11}^{+})\) due to the effect of birth pulse. Suppose that the trajectory reaches the line \(y=h\) at the point \(A_{2}(x_{21},y_{21})\) for \(t=T+t_{1}\), where \(0< t_{1}< T\), and jumps to the point \(A_{2}^{+}(x_{21}^{+},y_{21}^{+})\). The trajectory reaches the point \(A_{3}(x_{31},y_{31})\) at \(t=2T\) and jumps to the point \(A_{3}^{+}(x_{31}^{+},y_{31}^{+})\). Further, suppose that \(x_{31}^{+}=x_{0}\), \(y_{31}^{+}=y_{0}\). Then there exists a period-2T solution.
Suppose that \((x_{0},y_{0})=(\bar{x}_{0},\bar{y}_{0})\) is a solution of (2.9). Then there exists a period-2T solution of system (1.2) where the birth pulse occurs at the moments \(t=nT\), whereas the pesticide is sprayed at \(t=(2n-1)T+t_{1}\). The initial point is \((\bar{x}_{0},\bar{y}_{0})\). Then we obtain the following result.
3 The stability of the trivial solution
Now, we discuss the stability of the trivial solution of system (1.2).
Set \(p=\max\{p_{1},p_{2}\}\). Then \(\Delta N(t)=-N+p_{1}(N-y)+p_{2}y<-N+pN\).
- (H1)
The trajectory originating from the initial point \(A_{0}\) does not reach the line \(y(t)=h\) for \(0< t\leq T\).
- (H2)
The trajectory originating from the initial point \(A_{0}\) reaches the line \(y(t)=h\) once at time T for \(0< t\leq T\).
- (H3)
The trajectory originating from the initial point \(A_{0}\) reaches the line \(y(t)=h\) once at time \(t_{1}\) for \(0< t< T\) where \(0< t_{1}< T\).
- (H4)
The trajectory originating from the initial point \(A_{0}\) reaches the line \(y(t)=h\) k times for \(0< t< T\).
(H2) Suppose that the trajectory originating from the initial point \(A_{0}\) reaches the line \(y(t)=h\) at the point \(A_{21}(N_{21},y_{21})\) for \(t=T\). Then birth pulse occurs, and the pesticide is spayed. The trajectory jumps to the point \(A_{21}^{+}(N_{21}^{+},y_{21}^{+})\).
(H3) Suppose that the trajectory originating from the initial point \(A_{0}\) reaches the line \(y=h\) at the point \(A_{31}(N_{31},y_{31})\) at time \(t=t_{1}\), where \(0< t_{1}< T\), \(N_{31}=N_{0}\exp(-dt_{1})\), and \(y_{31}=h\). Then the pesticide is spayed, and the trajectory jumps to the point \(A_{31}^{+}(N_{31}^{+},y_{31}^{+})\). The trajectory reaches the point \(A_{32}(N_{32},y_{32})\) at \(t=T\) and jumps to \(A_{32}^{+}(N_{32}^{+},y_{32}^{+})\) due to the effect of birth pulse.
(H4) Suppose that the trajectory originating from the initial point \(A_{0}\) reaches the line \(y=h\) at the point \(A_{n}(N_{n},y_{n})\) at time \(t=t_{n}\), where \(0< t_{n}< T\), \(0< n\leq k\), and jumps to the point \(A_{n}^{+}(N_{n}^{+},y_{n}^{+})\). The trajectory reaches the point \(A_{k+1}(N_{k+1},y_{k+1})\) at \(t=T\) and jumps to the point \(A_{k+1}^{+}(N_{k+1}^{+},y_{k+1}^{+})\) due to the effect of birth pulse.
Theorem 3.1
The trivial solution of system (1.2) is locally asymptotically stable for \(0< b<\exp(dT)-1\).
4 Permanence
- (E1)
The trajectory originating from the initial point \(A_{0}\) does not reach the line \(y(t)=h\) for \(0< t\leq T\).
- (E2)
The trajectory originating from the initial point \(A_{0}\) reaches the line \(y(t)=h\) at time \(t_{1}\) where \(0< t_{1}\leq T\).
- (1)
\(N(t_{2}^{+})\geq D_{1}\).
- (2)
\(N(t_{2}^{+})< D_{1}\).
In conclusion, for any initial value \(N_{0}>0\), there exists \(t_{4}>0\) such that \(N(t)\geq m_{1}\) for \(t>t_{4}\).
In the following, we consider three cases.
(1) \(p_{2}h<\frac{\delta m_{1}}{\delta+d}<h\).
(2) \(\frac{\delta m_{1}}{\delta+d}\leq p_{2}h\).
It is well known that for large enough \(t>0\), \(\frac{\delta m_{1}}{\delta+d}\leq y(t)\leq h\).
(3) \(\frac{\delta m_{1}}{\delta+d}\geq h\).
Theorem 4.1
Assume that condition (4.2) holds, \(c>0\), \((1+b)^{2}\leq4b\exp (dT)\), and \((1+b)\exp(-dT)-b\exp(-(\delta+d)T)>1\). Then for any initial point \(A_{0}(x_{0},y_{0})\) in system (1.2) such that \(0< x_{0}+y_{0}<\frac{b\exp(\sigma T)}{c}\) and \(0< y_{0}\leq h\), there exists \(\bar{t}>0\) such that \(m_{1}\leq x(t)+y(t)\leq M_{1}\) and \(m_{2}\leq y(t)\leq h\) for \(t>\bar{t}\), where \(m_{1}\), \(M_{1}\), and \(m_{2}\) are given in (4.4), (4.5), and (4.7), respectively.
- (B1)
The trajectory originating from the initial point \(A_{1}\) does not reach the line \(y(t)=h\) for \(0< t\leq T\).
- (B2)
The trajectory originating from the initial point \(A_{1}\) reaches the line \(y(t)=h\) at time \(t_{7}\) where \(0< t_{7}\leq T\).
In the following, we discuss three cases.
(1) \(p_{2}h<\frac{\delta\bar{m}_{1}}{d}<h\).
(2) \(\frac{\delta\bar{m}_{1}}{d}< p_{2}h\).
It is well known that for large enough \(t>0\), \(\frac{\delta\bar {m}_{1}}{d}\leq y(t)\leq h\).
(3) \(\frac{\delta\bar{m}_{1}}{d}\geq h\).
Theorem 4.2
Assume that condition (4.8) holds and \(c=0\). Then for any solution of system (1.2), there exists \(t^{\ast}>0\) such that \(\bar{m}_{1}\leq x(t)\) and \(\bar{m}_{2}\leq y(t)\leq h\) for \(t>t^{\ast }\), where \(\bar{m}_{1}\) and \(\bar{m}_{2}\) are given in (4.10) and (4.11), respectively.
5 Numerical simulation
In our case, \(d=0.1\), \(\delta=0.2\), \(T=2\), \(c=0.2\), \(b_{0}=\exp(dT)-1=\exp(0.1\times 2)-1\approx0.2215\).
5.1 Species extinction
5.2 Species persistence
5.3 The existence of periodic solutions
5.4 The superiority of the mixed control strategy
It is seen from Figure 7(b) that the total number of spraying pesticide is 15 in the case of time-fixed pulse control strategy. However, \(y(t)\) is larger than the threshold value \(h=6\) at some points. It is seen from Figure 8(b) that the total number of spraying pesticide is seven times in the case of state feedback control strategy, and \(y(t)\) is controlled under the threshold value \(h=6\). In view of the total number of spraying pesticide and results of two control strategies, the state feedback control strategy is more effective than the time-fixed pulse control strategy.
6 Conclusion
In this paper, we study a stage structure population model with fixed-time birth pulse and state feedback control strategy. The stability of the trivial solution and the existence of periodic solutions are discussed. The sufficient conditions for the permanence of system (1.2) are obtained. It is shown that the maximum birth rate b plays an important role in the population dynamics. The trivial solution of system (1.2) is asymptotically stable for \(0< b<\exp(dT)-1\). For \(b>\exp(dT)-1\), the amount of mature pests can be controlled under the threshold value \(y=h\). There exist many kinds of periodic solutions of system (1.2). The period-T and period-2T solutions are discussed.
Declarations
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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