- Research
- Open Access
Periodic solution and control optimization of a prey-predator model with two types of harvesting
- Jianmei Wang^{1},
- Huidong Cheng^{1}Email author,
- Hongxia Liu^{1, 2} and
- Yanhui Wang^{1, 2}
https://doi.org/10.1186/s13662-018-1499-9
© The Author(s) 2018
- Received: 27 October 2017
- Accepted: 18 January 2018
- Published: 25 January 2018
Abstract
In this work, a prey-predator model with both state-dependent impulsive harvesting and constant rate harvesting is investigated, where the replenishment rate of prey and the harvesting rate are linearly related with the selected threshold. By first using the successor function method and differential equation geometry theory, the existence, uniqueness and asymptotic stability of the order-1 periodic solution are discussed. And then numerical simulations with an example are given to illustrate the feasibility of the theorem-related results. Moreover, in order to increase the total profit, the optimization strategy is presented and the optimal threshold is obtained.
Keywords
- prey-predator model
- state-dependent impulse
- order-1 periodic solution
- optimization
- stability
MSC
- 34C25
- 34D20
- 92B05
- 34A37
1 Introduction
Fishery is the natural source and basis of fishery production, and it is also one of the important food sources for human beings. If fishery resources are used properly, it can adapt to the natural regeneration ability of the resource and maintain the optimum sustainable yield. If people harvest fish unrestricted, it will lead to the extinction of the species [1–4]. Therefore, looking for a reasonable harvest strategy to ensure the sustainable development of fishery resources has become the focus of research.
In the past few decades, various harvest strategies have been proposed and implemented in fishing industry. In general, if a species is harvested frequently and regularly, we can adopt the strategy with constant rate harvesting [5–8]. And due to the seasonal and economic reasons, periodic harvesting is an effective harvesting strategy for the infrequent harvesting. This periodic harvesting can be described by impulsive differential equations [9–16]. There are some papers studying the effects of periodic impulse harvesting strategy to the species resource. For example, Pei et al. [17] proposed a continuous impulsive harvesting strategy for a prey-predator system with stage structure and time delay, and analysed the global attractivity of extinction periodic solution of the mature predator. Jiao et al. [18] considered a periodic impulsive harvesting prey-predator system with prey hibernation, and they obtained the conditions of the global asymptotic stability criterion for the predator-extinction boundary and the permanent conditions. However, these two methods of harvesting are carried out without knowing the number of species, which can lead to overexploitation and even depletion of resources.
Recently, state-dependent impulse feedback control has attracted the attention of many scholars [19–23], a novel strategy based on state-dependent impulse feedback control is proposed and applied in the harvest [24–27] and pest management [28–33]. The procedure goes like this: when the number of species reaches a specific requirement, the harvesting strategy is implemented, otherwise the harvesting behavior is suppressed. Some other related studies can be found in [34–39] and the references therein.
The main contents of this work are organized as follows. In Section 2, some main definitions and lemmas are provided. In Section 3, the existence, uniqueness and asymptotic stability of the order-1 periodic solution of system (2) are mainly discussed under some conditions. The theoretical results are then verified by numerical simulations, and the optimization problem is presented and solved for obtaining the maximum harvesting profits in Section 4. This work ends with a conclusion.
2 Preliminaries
Definition 2.1
([43])
Remark 2.1
Based on system (2), we get \(M=\{(x, y)\mid x=h, y\geq0\}\), \(N=\{(x, y)\mid x=(1-p_{h})h+\tau_{h}, y\geq 0\}\), for any point \((x,y)\in M\), when \(x=h\), we get \(I: (h, y)\in M\rightarrow((1-p_{h})h+\tau_{h}, (1-q_{h})y)\in N\). For this article, the coordinate of the arbitrary point \(A\in R_{2}^{+}\) is marked as \((x_{A},y_{A})\).
Definition 2.2
([44])
If there exist a point \(A\in N\) and a time T such that \(g(A,T)=B\in M\) and \(I(B)=I(g(A, T))=A\in N\), then \(g(A, t)\) is defined as an order-1 periodic solution of system (4) with period T.
Definition 2.3
([45])
Assume \(\Gamma=g(A, t)\) is an order-1 periodic solution of system (4). The order-1 periodic solution Γ is orbitally asymptotically stable if for any \(\varepsilon>0\), there must exist \(\delta>0\) and \(t_{0}\geq0\), such that, for any point \(A_{1}\in U(A, \delta)\cap N\) and \(t>t_{0}\), we have \(\rho(g(A_{1},t),\Gamma)< \varepsilon\).
Definition 2.4
([46])
3 Dynamical analysis of system (2)
The dynamical properties of the order-1 periodic solution of system (2) are mainly investigated in this section. Before these discussions, we firstly analyze the qualitative characteristics of system (2) without control, and the conditions that system (2) without control has no closed orbit are discussed.
3.1 Qualitative analysis of system (2) without control
It is easy to see that \(Det(J(E_{1}))>0\) and \(Det(J(E_{2}))<0\), thus \(E_{1}(x_{E_{1}},y_{E_{1}})\) is an elementary but not saddle-type positive equilibrium, and \(E_{2}(x_{E_{2}},y_{E_{2}})\) is a saddle.
According to the Bendixson-Dulac theorem, the closed orbit of system (5) does not exist in the plane \(R_{+}^{2}\). In conclusion, the following theorem is obtained.
Theorem 3.1
3.2 Existence, uniqueness and stability of order-1 periodic solution
Theorem 3.2
System (2) exists a unique order-1 periodic solution if the conditions \((H_{1})\), \((H_{2})\) and \(0< h<(1-p_{h})h+\tau_{h} <K\) hold.
Proof
For different threshold h, let us consider three cases as follows.
Case I. \(0< h< x_{E_{1}}<(1-p_{h})h+\tau_{h} <x_{E_{2}}\).
There exists a threshold \(h\in(0,x_{E_{1}})\) such that \(q_{h}\in [q_{\min}, q_{\max}]\), due to impulsive effects, point C jumps to a point \(D_{1}\in\overline{D_{0}D'}\subset N\), then \(y_{D_{0}}< y_{D_{1}}=(1-q_{h})y_{C}< y_{D'}\). Besides, the orbit of system (2) starting from point \(D_{1}\) must pass through a point \(C_{1}\in M\), then jumps back to a point \(D_{2}\in N\). Because distinct orbits are disjoint, then \(y_{C'}< y_{C_{1}}< y_{C}\) and \(y_{D_{2}}=(1-q_{h})y_{C_{1}}<(1-q_{h})y_{C}=y_{D_{1}}\), thus the successor function of point \(D_{1}\) is \(g(D_{1})=y_{D_{2}}- y_{D_{1}}<0\).
Moreover, another point \(D_{\epsilon}\in\overline{D_{0}D'}\) is selected and satisfies \(y_{D_{\epsilon}}=y_{D_{0}}+\epsilon\) (\(\epsilon>0\) sufficiently small). There must be an orbit starting from point \(D_{\epsilon}\) and passing through point \(C_{\epsilon}\in M\), and point \(C_{\epsilon}\) is next to point C, due to impulsive effects, point \(C_{\epsilon}\) jumps to a point \(D_{\epsilon+1}\in N\). Because distinct orbits are disjoint, we know \(y_{C_{1}}< y_{C_{\epsilon }}< y_{C}\) and \(y_{D_{\epsilon }}< y_{D_{2}}=(1-q_{h})y_{C_{1}}<(1-q_{h})y_{C_{\epsilon }}=y_{D_{\epsilon+1}}\). Then we have \(g(D_{\epsilon})=y_{D_{\epsilon +1}}-y_{D_{\epsilon}}>0\).
We can easily get \(g(D_{1})g(D_{\epsilon})<0\), there is a point \(S\in \overline{D_{\epsilon}D_{1}}\) such that \(f(S)=0\) by Lemma 2.2, i.e. the order-1 periodic solution is existent.
For any point \(S_{1}\in\overline{DD'}\), the orbit of system (2) starting from point \(S_{1}\) intersects a point in the line \(x=h\) which is denoted as \(S_{1}^{-}\), then jumps to a point \(S_{1}^{+}\in N\) after impulsive effects. Because distinct orbits are disjoint, then \(y_{C'}< y_{S_{1}^{-}}< y_{C}\) and \(y_{S_{1}^{+}}=(1-q_{h})y_{S_{1}^{-}}<(1-q_{h})y_{C}=y_{D_{1}}<y_{S_{1}}\), thus we get \(g(S_{1})=y_{S_{1}^{+}}-y_{S_{1}}<0\), which says there is no order-1 periodic orbit passing through point \(S_{1}\in\overline {DD'}\). In addition, for any point \(S_{2}\in\overline{BD_{0}}\), the orbit starting from point \(S_{2}\) eventually passes through the line \(y=0\) and unaffected by any impulse, namely, there is no order-1 periodic orbit passing through point \(S_{2}\).
Case II. \(x_{E_{1}}\leq h<(1-p_{h})h+\tau_{h} \leq x_{E_{2}}\).
Case III. \(x_{E_{1}}< h< x_{E_{2}}<(1-p_{h})h+\tau_{h} <K\).
For this subcase, the stable flow of \(E_{2}\) intersects the line \(x=(1-p)h_{h}+\tau_{h}\) at point \(G_{1}\), and the unstable flow of \(E_{2}\) intersects the line \(x=(1-p)h_{h}\) at point \(H_{1}\). We can select a point \(G_{\epsilon}\) satisfying \(y_{G_{\epsilon }}=y_{G_{1}}+\epsilon\), there must exist an orbit starting from point \(G_{\epsilon}\) and passing through point \(H_{\epsilon}\in M\), and point \(H_{\epsilon}\) is next to point \(H_{1}\). By impulsive effects, point \(H_{\epsilon}\) jumps to a point \(G_{\epsilon+1}\in N\) which is above \(G_{\epsilon}\). Then \(g(G_{\epsilon})=y_{G_{\epsilon +1}}-y_{G_{\epsilon}}>0\).
Furthermore, we can select another orbit that is far from the stable flow and unstable flow of \(E_{2}\) which passes through point \(G_{2}\in N\), and reaches point \(H_{2}\in M\), then jumps back to the line \(x=(1-p_{h})h+\tau_{h} \) at point \(G_{3}\), and point \(G_{3}\) is below point \(G_{2}\), then \(g(G_{2})=y_{G_{3}}-y_{G_{2}}<0\).
For any point \(S_{3}\in\overline{G_{1}B}\), the orbit starting from point \(S_{3}\) eventually passes through the line \(y=0\) and unaffected by any impulse, namely, there is no order-1 periodic orbit passing through point \(S_{3}\in\overline{G_{1}B}\). □
In this paper, we assume that the order-1 periodic solution of system (2) is \(\widehat{SS^{-}S}\), where \(S\in N\) and \(S^{-}\in M\). Next we prove the stability of the periodic solution \(\widehat {SS^{-}S}\). Since the methods used in the above three cases are similar, we only prove Case II.
Theorem 3.3
The periodic solution \(\widehat{SS^{-}S}\) is orbitally asymptotically stable if \(x_{E_{1}}< h<(1-p_{h})h+\tau_{h} <x_{E_{2}}\) and \(\frac {a}{b}(1-q_{h}) ( 1-\frac{h}{K} ) \geq\frac{H}{\lambda b[(1-p_{h})h+\tau_{h} ]-d}\) hold under Theorem 3.2.
Proof
Therefore, all the successor points in the segment \(\overline{D'D_{0}}\) are attracted to point S after the corresponding impulsive effect, then the periodic solution \(\widehat{SS^{-}S}\) is orbitally asymptotically stable. That completes the proof. □
4 Simulations and optimization
4.1 Numerical simulations
4.2 Determination of optimal threshold h
The practical significance of studying the order-1 periodic solution is that it provides the possibility to determine the replenishment rate of prey fish and the harvesting rate of predator fish, which makes the impulsive control to be not a real-time monitoring of fisheries, but rather a periodic one. In order to maintain the ecological balance of fisheries, further determine the optimal replenishment rate of prey fish and the optimal harvesting rate of predator fish, and make sure the harvest period is shortest and the profit is highest, we consider the following optimization problem to find the optimal threshold.
5 Conclusion
This work presents a prey-predator system with both state-dependent impulsive harvesting and constant rate harvesting, where the harvesting frequency of constant harvesting is more frequent than that of impulse harvesting. Moreover, the combination of these two harvesting methods is more practical which provides higher commercial value and avoids the exhaustion of resources. Meanwhile, the existence, uniqueness and stability of the order-1 periodic solution are proved by using the method of successor functions and differential equation geometry theory. Numerical simulations with a specific example are given to verify feasibility of the impulsive strategy. Furthermore, to maximize economic benefit, we provide an optimization strategy for the pisciculture and obtain the optimal threshold. However, the optimization results have some deviations which need to be further improved.
Declarations
Acknowledgements
The paper is supported in part by the National Natural Science Foundation of China (No. 11371230, 11501331), in part by Shandong Provincial Natural Science Foundation, China (No. S2015SF002), in part by SDUST Research Fund (2014TDJH102), and in part by Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province of China.
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors claim 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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