Abstract

Water hyacinth and its ecological invasion have negative impacts on diversity of indigenous species and ecosystems, which becomes one of the hotspots in current ecological research. In this paper, a water hyacinth ecological system with two state-dependent impulse controls is studied. Firstly, we define the successor functions of semicontinuous dynamic system and give existence theorems of order-1 periodic solution and order-2 periodic solution of such system. Secondly, we analyze singular points of the system without impulsive state feedback control qualitatively and get the condition for focus point. Thirdly, we obtain the sufficient condition under which the system has an order-1 or order-2 periodic solution through the method of successor function and prove the stability of the order-1 or order-2 periodic solution by the analogue of Poincaré’s criterion. Furthermore, some examples and numerical simulations are given to illustrate our results.

1. Introduction

The water hyacinth, a water plant with a showy purple flower, is a native of the Amazon Basin and is now treated as the most important nuisance aquatic plant worldwide. Rapid spread of water hyacinth causes serious environmental problems, which is recognized throughout the world. Water hyacinth, known as one of the top ten creatures into the en-roach on grass, has spread by waters in the south of China in recent years, becoming a great ecological and social harm. The expansive water hyacinth sheets have blocked shipping lanes, prevented commercial ships from entering ports and local fishing vessels from leaving the shore, and interfered with the intake of water by industries. Meanwhile, the invasive weed has created anoxic conditions in the lake, thus raising toxicity and disease levels. The resulting stagnant water is an ideal breeding ground for malarial mosquitoes and schistosomiasis and a detriment for the functioning of fishing industries. As for the ecological harm, biological control, development, and utilization of water hyacinth, see [15] and the reference cited therein.

Nowadays, biological control measures are widely used to control water hyacinth. Biological control is to introduce natural enemies from the origin, establish populations, and control water hyacinth in the long term. The biological control of water hyacinth has the advantages of long-lasting prevention, low cost, and safe environment.

Recently, impulsive state feedback control differential equations have aroused scholars’ enormous interest (see [620]). Chen et al. [6] and Chen [7] starting from a practical problem of the pest control established a class of pest prevention model with impulsive feedback control. The geometric theory of semicontinuous dynamical systems is presented, and by applying this theory, it is proved for this model to have at least one order-one periodic solution. Nie et al. [1618] and Tian et al. [19] put forward a class of predator-prey models with state-dependent impulsive effects. By using Poincaré map and Lambert function, the criteria for the existence and stability of a semitrivial solution and positive periodic solution of system are obtained. Zhao et al. [20] using successor functions and Poincaré-Bendixson theorem of impulsive differential equations studied the existence of periodical solutions to a predator-prey model with two state impulses. By stability theorem of periodic solution to impulsive differential equations, the stability conditions of periodic solutions to the system are obtained. As a continuation, a water hyacinth ecological system with Kuznets curve effect and two state-dependent impulse controls is put forward and numerical simulations are carried out based on the qualitative analysis on the proposed model.

This paper is organized as follows. In Section 2, a water hyacinth ecological system with Kuznets curve effect and two state-dependent impulses control is put forward, and some basic definitions are given. In Section 3, we first discuss the qualitative analysis of system (3) and then discuss the existence and stability of periodic solutions. In Section 4, we analyze our theoretical results by numerical simulations and give a brief discussion. Finally, conclusions are presented in Section 5.

2. Model Formulation and Preliminaries

2.1. Model Formulation

In ecosystems, water hyacinth can be farmed with fish. Water hyacinth can not only serve as the diet of fish but can also purify the water quality, which controls the pond water quality well. But there are also disadvantages. Too much water hyacinth, covering the water surface, will reduce the activity space of the fish. Most importantly, it consumes the oxygen in the water and prevents the entry of oxygen in the air; the fish are suffocated, causing the extinction of fish populations in water ecological system. Considering the water hyacinth as feed in fish farms and its characteristic of the ecological damage caused by its rapid spread, inspired by [2124], we establish the water hyacinth ecological system with Kuznets curve effect. where , , , , , and are all constants; and denote the water hyacinth (prey) and fish (predator) population, respectively, at any time ; is the intrinsic growth rate of water hyacinth; is the carrying capacity of water hyacinth; is the coefficient of interspecific competition; denotes the death rate of fish; and is the Kuznets curve effect function (see Figures 13). When the population of water hyacinth is less than , it is conducive to the growth of fish; when the population of water hyacinth is greater than , it will hinder the growth of fish.

The rapidly propagating water hyacinth will cover the water surface, causing intense intraspecific competition, leading to death of rot, pollution of water body, and aggravate eutrophication of water body, thus hindering the growth of other aquatic creatures and causing the imbalance of the ecological chain, irreversible damage to the ecosystem, loss of biodiversity, and frequent ecological disasters. The key to solving water hyacinth problem is to develop and make proper use of water hyacinth to avoid harm. At present, many places are trying to develop and utilize water hyacinth, using it as feed for livestock, poultry, grass carp, and so on, so as to turn harm into profit.

According to the characteristics of water hyacinth ecosystem, the population size of water hyacinth is set at a controllable economic threshold (ET). When the population size of water hyacinth is less than , it will be detrimental to the growth of fish population. We consider using the methods of manually releasing water hyacinth and fishing fish population to control the ecological balance. When the population size of water hyacinth is greater than , it will lead to the extinction of fish population. We consider using the methods of manually fishing water hyacinth and releasing fish population to control the ecological balance.

Now, we take the integrated control tactics into account for water hyacinth ecological system. Once the density of the water hyacinth population reaches the economic threshold (ET), we introduce impulsive state feedback control. Thus, we obtain the following system:

This model describes that in productive practice for controlling water hyacinth, people always take such a strategy that when the water hyacinth arrives at a given ET or , they began to use pulses to control the population of water hyacinths. ET denotes the economic value with a small population of water hyacinth, and denotes the economic value with a large population of water hyacinth . When the population of water hyacinth is equal to and the population of fish is greater than , the method of impulse control is adopted to obtain economic benefits by harvesting fish, and meanwhile to increase the prey of fish, promoting the population of fish by releasing water hyacinth. represents the harvest coefficient of fish, and represents the quantity of water hyacinth. When the population of water hyacinth is equal to and the population of fish is less than , the mass reproduction of water hyacinth is likely to lead to the extinction of fish. At this time, the method of impulse control is adopted to reduce the population of water hyacinth by fishing it and at the same time to increase the population of fish by releasing fish. represents the harvest coefficient of water hyacinth, and is the release amount of fish.

Considering the biology background of system (2), we all assume that and we only discuss system (2) in the region .

For system (2), if , we change it to , , , , , and . So we can derive equivalent system (3) from system (2) (let new variables be , , and ).

On the basis of the ecological significance, system (3) should meet , , and . We can get the relationship between the phase set and isocline as shown in the following figures (see Figures 46).

We first discuss the same position, i.e., (see Figure 5); we change and . So we can derive equivalent system (4) from system (3).

2.2. Preliminaries

In this section, we first introduce some definitions and lemmas of the geometric theory of the semicontinuous dynamic system, which will be useful for the latter discussion. And for more details, we can see [615] and the references cited therein.

Definition 1. A triple is said to be a semidynamical system if is a metric space, is the set of all nonnegative real, and is a continuous map such that (1) for all ; (2) is continuous for and ; and (3) for all and , . Sometimes, a semidynamical system is denoted by .

Definition 2. Define the impulsive set and ; let ; is a closed subset. Define the continuous functions and ; thus, the phase set can be defined . Then, is called an impulsive semidynamic system.

Definition 3. Define the periodic solution of the system (3) (see Figure 7): (1)When there is a point at phase set and a such that and , then is called order-1 periodic solution(2)When there is a point at phase set and a such that and , then is called order-1 periodic solution(3)When there is a point at phase set and a such that and , there exists a such that and , then is called order-2 periodic solution

Next, we will give the definition of the successor function of system (3). Let be a coordinate axis defined at ; the origin point is the intersection point of line with axis ; the positive direction is consistent with the positive direction of axis , then we obtain a number axis . For any , let be the coordinate of point .

Definition 4 (see Figure 8). (1)When there is a point at phase set and a such that and , then is called the successor function of the point (2)When there is a point at phase set and a such that and , then is called the successor function of the point (3)When there is a point at phase set and a such that and , there exists a such that and , then is called the successor function of the point

Remark 1 (see Figure 7). (1)If the successor function , the trajectory with initial point is an order-1 periodic solution of system (3)(2)If the successor function , the trajectory with initial point is an order-2 periodic solution of system (3)

Lemma 1. The successor function and is continuous.

Lemma 2. (1)Let the continuous dynamical system be ; if there are two points and at the phase set, they make , then there must exist a point between and so that ; thus, there must exist an order-1 periodic solution by point (2)Let the continuous dynamical system be ; if there are two points and at the phase set, they make , then there must exist a point between and so that ; thus, there must exist an order-2 periodic solution by point

Lemma 3 (analogue of Poincaré’s criterion) [25]. The -periodic solutions and of the system are orbitally asymptotically stable if the Floquet multiplier satisfies the condition , where with and , , , , , , , and are calculated at the point , and , , and are a sufficiently smooth function with grad , and are the moments for the jumps.

For the convenience of discussion, we need to define the following (see Figure 9).

, , , , , and .

The intersection point of and is denoted by .

The intersection point of and is denoted by . The phase point of at the phase set is denoted by . If of system (4), we define a function , where . We have . We can choose enough large, such that , so the straight line is nontangent. The intersection point of and is denoted by .

Assume that , , , and are dividing lines of the saddle point , where and are the stable flows of and and are the unstable flows of . The intersection point of and is denoted by . The intersection point of and is denoted by .

3. Model Analysis

On the basis of the ecological significance, system (3) should meet , , and . In this section, we will first study the qualitative characteristic of system (3) without the impulsive effect. Secondly, if , the sufficient conditions for the existence and stability of order-1 positive periodic solutions are analyzed. Thirdly, if , the sufficient conditions for the existence and stability of order-2 positive periodic solution are analyzed. Finally, if , the sufficient conditions for the existence of order- () positive periodic solution are analyzed.

3.1. Simple Qualitative Analysis of System (3) without Impulsive Effect

From system (3) without the impulsive effect, we obtain the following system:

By and , we can obtain two boundary equilibriums and in system (8).

We can see that (i) there is no positive equilibrium if ; (ii) there is one positive equilibrium if ; and (iii) there are two positive equilibriums and if .

The Jacobian matrix at equilibrium is given by

Theorem 1. is the saddle point; if holds, then is the saddle point; if or holds, then is the stable node point.

Theorem 2. is the stable node point, if ; is the saddle point, if .

Theorem 3. There exists no limit cycle in system (8) in .

Proof 1. Let the straight line ; we have , then the straight line segment is nontangent, and the path line of system (8) thereon is always to the left direction. Define a function , where . We have . We can choose enough large, that is such that , so the straight line is nontangent, the trajectory of system (8) from the upper right of through into the lower left. The straight line intersects axis and at points and , respectively.
For system (8), we construct a Bendixson ring including . Define , , , and as the length of line , , , and , respectively. Since and are the length of orbit of system (8), the orbits of system (8) go through into the interior of the Bendixson ring from the outer of and . Hence, we know that system (8) is uniformly bounded.
Let the Dulac function ; we have ; by the Bendixson-Dulac theory, system (8) does not exist a limit cycle in . The proof is completed.

The illustration of vector fields of system (8) can be seen in Figures 13.

3.2. Existence and Stability of Order-1 Periodic Solution of System (4)

Theorem 4. If and , there exists an order-1 periodic solution in system (4).

Proof 2. If (see Figure 10), there exists a point satisfying . The trajectory of system (4) over with at point , which is pulsed to and the phase point, is , so .
There exists another point , satisfying . The trajectory of system (4) over with at point , which is pulsed to and the phase point, is , so .
Therefore, there must exist a point , which satisfies , so that . By Lemma 2, we know that system (4) has an order-1 periodic solution. Thus, the proof is completed.

Theorem 5. For system (4), if , the order-1 periodic solution to it is stable.

Proof 3. According to Lemma 3 (see Figure 10), let and be a periodic solution to system (4) and and ; and ; we have and Then Since is the periodic solution of system (4), if and conditions of Theorem 4 hold, then . Therefore, the order-1 periodic solution of system (4) is stable, if . Thus, the proof is completed.

Theorem 6. If and , there exists an order-1 periodic solution in system (4).

Proof 4. If (see Figure 11), there exists a point satisfying . The trajectory of system (4) over with at point , which is pulsed to and the phase point, is , so .
There exists another point , satisfying . The trajectory of system (4) over with at point , which is pulsed to and the phase point, is , so .
Therefore, there must exist a point and it satisfies , so that . By Lemma 2, we know that system (4) has an order-1 periodic solution. Thus, the proof is completed.

Theorem 7. For system (4), if , the order-1 periodic solution to it is stable.

Proof 5. According to Lemma 3 (see Figure 11), let and be a periodic solution to system (4) and and ; and ; we have and Then Since is the periodic solution of system (4), if and conditions of Theorem 6 hold, then . Therefore, the order-1 periodic solution of system (4) is stable, if . Thus, the proof is completed.

3.3. Existence and Stability of Order-2 Periodic Solution of System (4)

Theorem 8. If and , there exists an order-2 periodic solution in system (4).

Proof 6. If and (see Figure 12), there exists a point , satisfying . The trajectory of system (4) over with at point , which is pulsed to and the phase point, is . We know that ; the trajectory of system (4) over with at point , which is pulsed to and the phase point, is , so .
There exists another point , satisfying . The trajectory of system (4) over with at point , which is pulsed to and the phase point, is . We know that ; the trajectory of system (4) over with at point , which is pulsed to and the phase point, is , so .
So there must exist a point , satisfying , so that . By Lemma 2, we know that system (4) has an order-2 periodic solution. The proof is completed.

Theorem 9. For system (4), if and , the order-2 periodic solution to it is stable.

Proof 7. According to Lemma 3 (see Figure 12), let and be a periodic solution to system (4) and and ; and ; we have , , , and . We also have and Then Obviously, , if and and conditions of Theorem 8 hold. Therefore, the order-2 periodic solution of system (4) is stable, if and . Thus, the proof is completed.

3.4. Existence of Order- () Periodic Solution of System (3)

Case I. (see Figure 13).
For notation simplicity, let the intersection of the phase set and be . The intersection point of and is denoted by .

Theorem 10. If and , there exists an order-1 periodic solution in system (3).

Proof 8. The proof is similar to that of Theorem 4 and omitted thereby.

Theorem 11. If and , there exists an order-1 periodic solution in system (3).

Proof 9. The proof is similar to that of Theorem 6 and omitted thereby.

Theorem 12. If and , there exists an order-2 periodic solution in system (3).

Proof 10. The proof is similar to that of Theorem 8 and omitted thereby.

Case II. (see Figure 14).
For notation simplicity, let the intersection of the phase set and be . The intersection point of and is denoted by .

Theorem 13. If and , there exists an order-1 periodic solution in system (3).

Proof 11. The proof is similar to that of Theorem 4 and omitted thereby.

Theorem 14. If and , there exists an order-1 periodic solution in system (3).

Proof 12. The proof is similar to that of Theorem 6 and omitted thereby.

Theorem 15. If and , there exists an order-2 periodic solution in system (3).

Proof 13. The proof is similar to that of Theorem 8 and omitted thereby.

4. Numerical Simulations and Discussion

In this section, a specific example is presented to verify the theoretical results obtained in the previous section by considering the change of the control parameters , , and . From Figures 1525, we know that the numerical simulation results are consistent with the theoretical results.

In system (3), let , , , , , , , , , and ; we get the following system:

Obviously (see Figure 9), we know that and are saddle points and and are stable node points. Let ; we have , , , , and .

4.1. Verification for the Case

Case I. Let , , and , then , and the impulsive set and the phase set are and . According to Theorem 4, there exists an order-1 periodic solution in system (16), which can be seen in Figure 15. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 8).

Case II. Let , , and , then , and the impulsive set and the phase set are and . According to Theorem 6, there exists an order-1 periodic solution in system (16), which can be seen in Figure 16. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 8 and 9).

Case III. Let , , and , then , and the impulsive set and the phase set are and . According to Theorem 8, there exists an order-2 periodic solution in system (16), which can be seen in Figure 17. We can observe that there exists an order-2 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 9).

4.2. Verification for the Case

Case I. Let , , and , then , and the impulsive set and the phase set are and . According to Theorem 10, there exists an order-1 periodic solution in system (16), which can be seen in Figure 18. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 7.5).

Case II. Let , , and , then , and the impulsive set and the phase set are and . According to Theorem 11, there exists an order-1 periodic solution in system (16), which can be seen in Figure 19. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 7.5 and 8).

Case III. Let , , and , then , and the impulsive set and the phase set are and . According to Theorem 12, there exists an order-2 periodic solution in system (16), which can be seen in Figure 20. We can observe that there exists an order-2 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 8).

Case IV. Let , , and , then , and the impulsive set and the phase set are and . According to Theorem 12, there exists an order-2 periodic solution in system (16), which can be seen in Figure 21. We can observe that there exists an order-2 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 8).

4.3. Verification for the Case

Case I. Let , , and , then , and the impulsive set and the phase set are and . According to Theorem 13, there exists an order-1 periodic solution in system (16), which can be seen in Figure 22. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 8.5).

Case II. Let , , and , then , and the impulsive set and the phase set are and . According to Theorem 14, there exists an order-1 periodic solution in system (16), which can be seen in Figure 23. We can observe that there exists an order-1 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 8.5 and 9).

Case III. Let , , and , then , and the impulsive set and the phase set are and . According to Theorem 15, there exists an order-2 periodic solution in system (16), which can be seen in Figure 24. We can observe that there exists an order-2 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 9).

Case IV. Let , , , then , and the impulsive set and the phase set are and . According to Theorem 15, there exists an order-2 periodic solution in system (16), which can be seen in Figure 25. We can observe that there exists an order-2 periodic solution of system (16) which lies between the phase set and the impulse set (i.e., between 6 and 9).

From Figures 1, 1525, we know that at the initial point (7.21, 0.9) in the system without pulse control, fish species will be extinct, which is harmful to biodiversity. In order to maintain biodiversity, we introduce the state feedback control and put forward the economic threshold (ET) of ecological control to effectively control the amount of water hyacinth population which can prevent the extinction of fish and can fully employ water hyacinth as feedstuff for continuous development of fishery production, changing the harm into benefit.

5. Conclusions

In the system with pulse control, we gain order-1 periodic solutions and order-2 periodic solution of the system, respectively, verifying the accuracy of Theorems 4, 6, 8, and 1015. At the same time, by using the comprehensive control strategies, such as periodically dropping and harvest, we achieve the diversity of species in water hyacinth ecological system.

In the future, from the viewpoint of optimization, we will further consider the costs and profits under domination of the market economy and further improve the model, thus realizing the effective protection of the diversity of species in water hyacinth ecology and obtaining the biggest economic profits.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

No potential conflict of interest was reported by the authors.

Acknowledgments

This work is supported by the Program for New Century Excellent Talents in University of Fujian Province (NCETFJ2018) and the Natural Science Foundation of Fujian Education Department (JA13370).