A pest management model with state feedback control
- Qiong Liu^{1}Email author,
- Lizhuang Huang^{2} and
- Lansun Chen^{3}
https://doi.org/10.1186/s13662-016-0985-1
© Liu et al. 2016
Received: 6 May 2016
Accepted: 28 September 2016
Published: 15 November 2016
Abstract
In this paper, research of a class of state feedback control model which is mainly used in crop pests management, with Bendixson-Dulac discriminance, proves that this model has an unique and globally stable positive equilibrium under the weak time-delay kernel function. Also, we adopt the subsequent function method in the ordinary differential equation of the geometric theory to prove that a sufficient condition holds for the existence of an order one period solution in the system. At the same time, it also proves that the periodic solution is asymptotically stable.
Keywords
1 Introduction
Crop is an essential part of the human food resources for sustainable development. Therefore, crop yields directly affect social and economic development as well as social stability. In recent years, in order to improve the yield of crops and meet the needs of human survival, people used a lot of pesticide [1–4] to keep pest down to improve the yield of crops, but the excessive use of pesticides will make the pests drug-resistant, pollute the environment, and also do some harm to people and animals. So how to reduce the effect caused by pests but doing no harm to the living environment of human beings and animals is an issue which has attracted extensive concern of ecologists and biomathematics researchers. In [5–10], the authors established a specific pest management model based on the actual problem and analyzed pest management issues by the corresponding mathematical theory. We will explore the feasibility of the crop pest management from the mathematical model of state feedback control in this paper.
In recent years, the research on the pest management has made many achievements. It can be classified into two categories: one is using farm chemicals, but this method will strengthen the drug resistance of the pests and it is harmful to the environment. The other is adopting a biological treatment [11–13] such as taking advantage of the characteristics of inter-restriction between the beneficial organisms and pests, it can effectively maintain a long-term balance with the effects on the food chain. The latter treatment is a more extensive pest treatment at present. It is a better measure to control pest by cultivating natural enemies of the pests to destroy pests, but the delivery of natural enemies and use of pesticides are not regular, depending on the amount of pests. Therefore, we must give full consideration to the natural inhibition in the agricultural ecological system. Meanwhile, delivery of a natural enemy and use of pesticides are transient phenomena, which we call a pulse phenomenon. In biological control, we do not need to reduce the amount of pests to zero, but we usually control the indicator of the state of an illness under a certain indicator, and this cannot only boost the yields of crops, but also it has no negative influences on the environment. We call the indicator of the state of an illness the economic index (EI), also known as the control index.
2 Preparation
First of all the related definition is given [15].
Definition 2.1
For convenience in our discussion, the following definitions are used in this article in terms of the subsequent function.
Definition 2.2
The point of intersection between the path line L and the phase set \(u = ( 1 - \beta )\ln h\) is N, the point of intersection between the path line L and the pulse set \(u = \ln h\) is impulsing and its v coordinate difference to the phase point \(N_{1}\) on the phase set \(u = ( 1 - \beta )\ln h\), \(G ( N ) = N_{1v} - N_{v}\).
Lemma 2.2
Assume in the continuous dynamical system \((X,\Pi)\), there are two points, \(x_{1}\), \(x_{2}\) in pulse phase set, to make the successor functions, \(G(x_{1}) > 0\), \(G(x_{2}) < 0\), then there must be a point E between \(x_{1}\) and \(x_{2}\) to make \(G(E) = 0\). Thus, there must be a periodic solution of order one through E between \(x_{1}\) and \(x_{2}\).
Lemma 2.3
(Similar Poincaré norm)
Lemma 2.4
(Bendixson-Dulac discriminance [17])
3 Existence and stability of the periodic solution of order one for the impulsive differential equation of plant diseases and insect pests
Lemma 3.1
The positive equilibrium point of system (4) \(E ( u^{ *},v^{ *} ) = E ( \frac{ad}{c + bd},\frac{a}{c + bd} )\) has global stability.
Proof
We obtain \(\lambda_{1,2} = \frac{ - ( b + d ) \pm\sqrt{ ( b+ d )^{2} - 4 ( bd + c )}}{2}\), in which b, d, c are all constant.
- (1)
When \(\Delta= 0\), \(T < 0\), \(D > 0\), the two characteristic roots \(\lambda_{1} = \lambda_{2}\) is a multiple negative real root, at this point, the solution curve tends to equilibrium, called the degenerate node.
- (2)
When \(\Delta> 0\), \(T < 0\), \(D > 0\), the two characteristic roots \(\lambda_{1} \ne\lambda_{2}\) is a pair of negative real roots, at this point, the solution curve tends to equilibrium, called the stable node.
- (3)
When \(\Delta< 0\), \(T < 0\), \(D > 0\), the two characteristic roots \(\lambda_{1}\), \(\lambda_{2}\) have real parts and they are negative conjugate complex roots, at this point, the solution curve tends to equilibrium, and the equilibrium is called a focal point.
From Lemma 2.4 we know that on the whole plane the system has no limit cycle, and because there is only a balance in the system, so all the path lines take \(E(u^{*},v^{*})^{T}\) as the limit set, the system is stable at the balance point \(E(u^{*},v^{*})^{T}\) and on the plane \(R^{2}\). It is hereby proved. □
When the positive constants b, c, d meet \(( b + d )^{2} < 4 ( bd + c )\), the positive equilibrium is a stability focal point, namely when the real part of the two characteristic roots \(\lambda_{1}\), \(\lambda_{2}\) are negative conjugate complex roots, of which \(v_{1} = \frac{a - b\ln h}{c}\) is the intersection point \(D ( \ln h, v_{1} )\) between the pulse set \(u = \ln h\) and the line \(\frac{du}{dt} = 0\), and when \(h \ge1\), we have the following theorem.
Theorem 3.1
- (1)
If the pulse set \(0 < \ln h \le\frac{ad}{c + bd}\), then system (4) has a periodic solution of the order one.
- (2)If the pulse set \(\ln h > \frac{ad}{c + bd} > 0\), there are the following four conditions:
- (i)
When \(c_{1v} < v_{1}\), \(c_{2v} < v_{1}\), system (4) has a periodic solution of order one.
- (ii)
If the subsequent point \(B_{1}\) overlaps B, the other path line \(\Gamma_{1}\) has no intersection point with the pulse set, then system (4) has a periodic solution of order one.
- (iii)
If \(c_{2v} < v_{1}\), the system path line \(\Gamma_{1}\) has no intersection point with the pulse set lnh, then system (4) has no periodic solution of order one, but for any t, we have \(v(t) \le v_{1} +\frac{b\beta\ln h}{c}\).
- (iv)
If the system path lines \(\Gamma_{1}\), \(\Gamma_{2}\) have no intersection point with the pulse set lnh, then system (4) has no periodic solution of order one, but, for any t, we have \(v(t) \le v_{1} + \frac{c\ln h\beta}{c}\).
- (i)
Proof
From the equation \(\bigl \{\scriptsize{ \begin{array}{l} u = \ln h, \\ a - bu - cv = 0, \end{array}} \bigr.\) we obtain \(v_{1} = \frac{a - b\ln h}{c}\).
From the equation \(\bigl \{\scriptsize{ \begin{array}{l} u = ( 1 - \beta )\ln h, \\ a - bu - cv = 0, \end{array}} \bigr.\) we obtain \(v = \frac{a - bu}{c} = \frac{a - b ( 1 - \beta )\ln h}{c} = v_{1} + \frac{b\beta\ln h}{c}\).
According to Lemma 2.2, in the phase set \(u = ( 1 - \beta )\ln h\), there must be P which meets \(B_{v} < P_{v} < A_{v}\), to make \(G ( P ) = P_{v} - P_{1v} = 0\), namely there is a periodic solution of order one in system (4).
According to Lemma 2.2, in the phase set \(u = ( 1 - \beta )\ln h\), there must be ∃P (P), this meets \(B_{v} < P_{v} < A_{v}\), and it enables \(G ( P ) = P_{v} - P_{1v} = 0\), namely there is a periodic solution of order one in system (4).
(2) If the pulse set obeys \(\ln h > \frac{ad}{c + bd}\), under this condition, there are four situations as follows.
When \(( 1 - \beta )\ln h < u^{ *} = \frac{ad}{c + bd} < \ln h\), namely \(v_{1} < v^{ *} = \frac{a}{c + bd} < v_{1} + \frac{c\beta\ln h}{c}\).
According to Lemma 2.2, in the phase set \(u = ( 1 - \beta )\ln h\), P must meet \(B_{v} < P_{v} < A_{v}\), and this enables \(G ( P ) = P_{v} - P_{1v} = 0\), namely there is a periodic solution of order one in system (4).
When \(0 < h < 1\), namely the pulse set \(\ln h < 0 < \frac{ad}{c + bd}\), then there is no periodic solution of order one in system (4). This case has no real biological meaning, so it is no longer considered.
Theorem 3.2
When \(h \ge1\) and \(( b + d )^{2} < 4 (bd + c )\), the positive equilibrium is a stable focus and \(a - c\phi_{0} < b ( 1 - \beta )\ln h\), then the periodic path line I of order one in system (4) passing the point \((\ln h, \phi_{0})\) is asymptotically stable.
Proof
Theorem 3.3
When \(h \ge1\) and \(( b + d )^{2} < 4 ( bd + c )\), the positive equilibrium is a stable focus and \(a - c\phi_{0} < b ( 1 - \beta )\ln h\), then the periodic path line I of the order one in system (4) is asymptotically stable.
4 Conclusion
This paper focuses on an ecosystem model of class-A plant diseases and insect pests to analyze the stability of the equilibrium of the model; it discusses the sufficient condition for the existence of a periodic solution of order one of the system, and at the same time, it also proves the conditions under which the periodic solution is asymptotically stable. On the basis of the model, it carried on the qualitative and quantitate analysis, to draw some conclusions under certain conditions. That is to say, there is a periodic solution of order one in this paper and it is asymptotically stable, under this situation the number of pest is controlled to a certain extent, it will be finally controlled to a be a periodic solution of order one in which the once-occurring pulse finally controls the number of pest. It can be seen from the article that, according to the different crop growth cycle, the observation and record of the quantity of pests in production can help the crop-dusting to control pest populations, so as to protect crops.
Declarations
Acknowledgements
This work was supported by National Natural Science Foundation of China (10861003) and The Research Fund of Guangxi Higher Education of China (ZD2014137), Natural Science Foundation of Guangxi Province (No. 2016GXNSFAA380102).
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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