Dynamic Complexity of a Phytoplankton-Fish Model with the Impulsive Feedback Control by means of Poincaré Map

Li, Dezhao; Liu, Yu; Cheng, Huidong

doi:https://doi.org/10.1155/2020/8974763

Complexity

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 8974763 | https://doi.org/10.1155/2020/8974763

Dynamic Complexity of a Phytoplankton-Fish Model with the Impulsive Feedback Control by means of Poincaré Map

Dezhao Li,¹Yu Liu,²and Huidong Cheng¹

Academic Editor: Lingzhong Guo

Received15 Jan 2020

Accepted21 Mar 2020

Published15 Apr 2020

Abstract

The phytoplankton-fish model for catching fish with impulsive feedback control is established in this paper. Firstly, the Poincaré map for the phytoplankton-fish model is defined, and the properties of monotonicity, continuity, differentiability, and fixed point of Poincaré map are analyzed. In particular, the continuous and discontinuous properties of Poincaré map under different conditions are discussed. Secondly, we conduct the analysis of the necessary and sufficient conditions for the existence, uniqueness, and global stability of the order-1 periodic solution of the phytoplankton-fish model and obtain the sufficient conditions for the existence of the order- periodic solution of the system. Numerical simulation shows the correctness of our results which show that phytoplankton and fish with the impulsive feedback control can live stably under certain conditions, and the results have certain reference value for the dynamic change of phytoplankton in aquatic ecosystems.

1. Introduction

Fisheries can provide people with quality food resources for the survival and development of human beings. Therefore, the healthy development of fishery resources is the focus of attention. If we cannot regulate the capture of fish regularly, it will lead to the depletion of fishery resources. People would be faced with increasing shortages of fish resources. Therefore, it is imperative to formulate effective fishing strategies, maintain the ecological balance of fishery resources, and protect the ecological environment [1–4].

Due to the interaction of energy conversion and the nutrient cycle between plankton and herbivores (such as fish), they play an significant role in the most terrestrial and aquatic ecological system. In [5], a phytoplankton and zooplankton model is established by the interaction of nutrients, and the dynamics properties such as limit cycle are investigated in this paper. In [6], a commercially valuable model of phytoplankton and zooplankton predation is proposed, which analyzes the stability of the equilibrium point, explores methods to maintain the ecological balance of the population at different harvest levels, and discusses the impact of selective harvesting on fisheries.

Recently, threshold state pulsed dynamic systems have been widely used [7–14]. The geometric theory of impulse dynamical system has been well-developed [15–23]. Many pulse equations have been studied which simulate the ecological processes of populations [24–32]. However, with the further development of the state feedback control model, we need more new methods to find out the complete dynamic properties and control strategies of dynamic systems and to discuss its biological significance. The primary purpose of this paper is to provide a comprehensive qualitative analysis of the global dynamics through analyzing a phytoplankton-fish model with impulse feedback control using the Poincaré map.

The central arrangement of this paper is as follows. In Section 2, we establish a phytoplankton-fish model for catching fish based on impulse feedback control. In Section 3, the Poincaré map for the phytoplankton-fish model is defined and the properties of monotonicity, continuity, and differentiability of the Poincaré map are analyzed. In Section 4, we discuss the existence, uniqueness, and global stability of the order-1 periodic solution of the model and obtain the conditions for the existence of order- periodic solution of the model. In Section 5, we perform numerical simulations.

2. Model Establishment

In the ecosystem of lakes, phytoplankton is considered to be the most favorable source of food for fish or other aquatic animals. Wang et al. [33] propose a predator model of continuously harvested phytoplankton and herbivorous fish:where and represent the population density of phytoplankton and fish at the moment, respectively, and is the effort for continuous harvesting. For details of other parameters, see [34–36].

System (1) shows that, without considering the number of phytoplankton and herbivore, continuous capture will lead to resource depletion. We will seek an integrated capture strategy to achieve ecological stability, for which we make the following assumption:(i)Assuming that phytoplankton and fish stocks are evenly distributed within the lake(ii)Let be the intrinsic growth rate of phytoplankton, be the absorption rate of phytoplankton by fish, be the conversion rate of biomass, and denote the mortality rate of fish(iii)Formula represents the death number of fish due to the distribution of phytoplankton toxicants, where represents semisaturation constant and represents the rate at which phytoplankton releases toxins

Let denote the threshold at which the fish are allowed to be caught, that is, when the density of the fish is lower than the threshold , it is unreasonable to catch fish; however, only when the density of fish reaches, the predetermined value can catch the fish. The amount of phytoplankton is affected when the fish is caught. The numbers of the fish and the amount of phytoplankton are updated to and , respectively.

Here, represents the maximum fishing rate, represents the half-saturation constant, indicates the number of phytoplankton reductions, and represents the morphology parameters, where , , , and [37].

We have established the following impulse feedback control model based on the abovementioned assumptions:

System (2) is called a semicontinuous dynamic system [38–40].

When there is no impulse, system (2) becomes

Lemma 1 (see [37]). System (3) has two equilibrium points, O(0,0) and , where the boundary equilibrium point is the saddle point and the internal balance point is a stable center:

It is easy to calculate, the first integral of system (3) iswhere is the constant related to the initial value. For ease of expression, two isoclines are defined as

To study the dynamic properties of system (2), the following research methods are given.

3. Poincaré Map of System (2) and Its Properties

We intersect the dynamic properties of model (2) in based on biological significance. In order to accurately define the impulse set and phase set of the pulsed semidynamic system (2), the following set is given first:

Apparently, within , and represent two straight lines of the vertical -axis, respectively, and we assume and intersect the line at point and point , respectively, since the internal balance point is the center point. According to the size of , we can divide the pulse set and phase set into the following cases:

Case I:

When , the trajectory must intersect the at point (see Figure 1(a)). In this case, impulse sets and the corresponding phase set is

Case II:

When , if the trajectory intersects the at a point, we assume this intersect is (see Figure 1(b)). In this case, the impulse set and the phase set are the same as those in Case I. It is easy to infer the impulse setand the corresponding phase set

(a)

(b)

(c)

(a)

(b)

(a)

(b)

If the trajectory and does not intersect (see Figure 1(c)), the trajectory intersects the at point and point , respectively, where .

In this case, the impulse setand the corresponding phase set

Based on the abovementioned discussion, we define the Poincaré map as follows.

Letwhere ; the trajectorywhich goes through point will reach the at pointafter time , and here

Then, there iswhich indicates that the ordinate is determined by .

Since point is on the impulse set, jumps to pointwhere

Consider the scalar differential equation of model (3):

Let and , then we have in .

We define

According to model (21),

From (20) and (23), the Poincar map expression of system (2) is

The properties of the Poincar map is discussed below.

Theorem 1. Let , trajectory intersects with line at point and point , respectively, where . Poincar map has the following properties:(i)The domain of is , is monotonically decreasing on , and monotonically increasing on .(ii) is continuously differentiable on .(iii)When , there is a unique fixed point on (see Figure 2(b)). When , there is a unique fixed point on (see Figure 2(a)). When , is the fixed point of .

Proof. (i)Since is the center point and , take any point on , and the trajectory of point will reach the impulse set at point , so the domain of is . For any and , from the uniqueness of the solution of the differential equation, it can be known that and by the definition of the Poincar map we can get . Therefore, is monotonically increasing on . When , , where . The trajectories start from point , and point will pass through the isocline and intersect the set at point and point , respectively, where and . It can be seen from the uniqueness of the solution of the differential equation that because and , so ; therefore, is monotonically decreasing on . Note 1. Because is monotonically decreasing on and monotonically increasing on , it is easy to know that is true for any , and if and only if .(ii)It is easy to know that is continuously differentiable in the first quadrant from (21), and from the continuous differentiability theorem of differential equations, that is, the Cauchy and Lipschitz theorem with parameters, we know that the Poincar map is continuously differentiable in the first quadrant.(iii)Considering the Poincar map at the position of image of , there are three cases:(a)When , then is the fixed point of function .(b)When , that is, because , then That is, . So, it has at least one satisfies , according to the continuous differentiable of the closed interval.(c)When , let , we know that is monotonically decreasing on , so , and because , so there is at least one which satisfies , according to the continuous differentiable of the closed interval.In conclusion, has at least one fixed point. Then, we prove the fixed point is unique.
We assume that system (2) has two fixed points, and , that is to say and . Let , we defineThen, take the derivative of the abovementioned formula:LetThen,sothat is,andFrom system (2),that is,It is easy to know that if , soIt is contradictory with , so the fixed point is unique.
Note 2. When and trajectory is an intersect to line , the Poincar map of system (2) has similar properties to case (a).

Theorem 2. Let , and trajectory does not intersect line and trajectory intersects the straight line at point and point , respectively, where . Poincar map has the following properties:(i)The domain of is , where is monotonically decreasing on and monotonically increasing on .(ii) is continuously differentiable on and , respectively.(iii)When , there is a unique fixed point on (see Figure 3(a)). When , there is no fixed point (see Figure 3(b)).

Proof. (i)Since is the center point. If and trajectory does not intersect line , trajectory intersects the straight line at two points, that is, and , respectively, where , then we take any point in . If , the trajectory of point will reach the impulse set at point , if the trajectory of point has no intersection with the impulse set . So, the domain of is . For any and , from the uniqueness of the solution of the differential equation, it can be known that and by the definition of the Poincar map we get . Therefore, is monotonically increasing on . When , , and where . The trajectories start from point and point will pass through the isocline which intersects the at point and point , respectively. Then, and . It can be seen from the uniqueness of the solution of the differential equation that ; therefore, is monotonically decreasing on .(ii)From (21), we can know that is continuously differentiable in the first quadrant; from the continuous differentiability theorem of differential equations, that is, the Cauchy and Lipschitz theorem with parameters, we know that the is continuously differentiable on and , respectively.(iii)Considering the Poincar map at the position of image of , there are two cases:(a)When (see Figure 3(a)), we assume , and we know that is monotonically decreasing on , so , and because , so there is a point , and it satisfies , according to the continuous differentiability of the closed interval.(b)When (see Figure 3(b)), there is no which satisfies .For any , the trajectory of point is tangent to the straight line at point ; will be pulsed to . Easy to get , that is, , so there is no and satisfies .
In a word, when , there is a unique fixed point on . When , there is no fixed point.

4. The Order- Periodic Solution of the Semicontinuous Dynamic System (2) and Its Stability

From Theorem 1, we know that system (2) has a unique fixed point under certain conditions. That is, system (2) has a unique order-1 periodic solution, and the following are the dynamic properties of system (2).

Theorem 3. and are established, then the order-1 periodic solution of system (2) is globally asymptotically stable.

Proof. From (iii) of Theorem 1, we know that when , has a unique fixed point on , i.e., . For any point in , where , the trajectory of the point will intersect the impulse set, then reach the point , which is , repeating the abovementioned process:that is,Furthermore,The following two cases are discussed below according to the size of :1)When , since and is monotonously increasing on , let , then Repeating the abovementioned process, we can obtain From the monotonous boundedness of the sequence, we can obtain(2)When , from the known conditions, Furthermore, By mathematical induction, Thus,(3)When , since and is monotonously decreasing on , so for any there is . So, we can conclude that , and we can convert this to two cases based on the size of . When , this is the same as case (1) above; when , this is the same as case (2) above. In both cases,The order-1 periodic solution of system (2) is globally asymptotically stable.

Theorem 4. When , and are true, and then the semicontinuous dynamical system (2) has a stable order-1 periodic solution or order-2 periodic solution.

Proof. Take a point on the phase set, here , since is the center of system (2). The trajectory that goes through point must intersect at point , where . will reach point by an impulse, so , repeating this process to get the sequenceWhen , we can know that is monotonically increasing on and there is no fixed point on from (i) of Theorem 1. Therefore, there must be a positive integer , which satisfiesWhen , the trajectory through the initial point will pass through the isocline and intersects at point , where ; this translates into the abovementioned situation.
This means that, for any , there is always be a positive integer which satisfiesSo, we only need to take account of the initial point , where . Since is monotonically decreasing on , soLet and , that is, the trajectory with the initial point is not the order-1 periodic solution or order-2 periodic solution of system (2). We consider the following four situations: Case I: . According to the monotonicity of the Poincar map, Furthermore, Thus, there is Proved by mathematical induction, Case II: . Because is monotonically decreasing on . We can obtain Then, so Case III: . In the same way, we can deduce Case IV: .We can perform the procedure similar to Case I, which can yield\Considering Cases I and III, there must exist so thatwhich means that system (2) has a stable order-1 periodic solution.
For Cases I and IV, there are two points andThis shows that system (2) has a stable order-2 periodic solution with the initial points and.
Theorem 4 illustrates that system (2) has a stable order-1 or order-2 periodic solution under certain conditions. However, sufficient and necessary conditions for global stability are not given. Then, we give below theorem.

Theorem 5. Let and , then the necessary and sufficient conditions for the global stability of the order-1 periodic solution of system (2) is , for any .

Proof. Sufficiency: It can be seen from Theorem 4 that when , there exists which satisfies .
For any , let and because of , and from the monotonicity of , we can get ; furthermore, , so .
By the monotonic boundness of the sequence,Therefore, the order-1 periodic solution of system (2) is globally asymptotically stable.
Necessity:
We assume that the order-1 periodic solution of system (2) is globally asymptotically stable. Then, the following part proves that is right for any , which is proved by the counterevidence method below.
If is not true for any , then there exists a maximum , which satisfies . For any , there exist and , which is true for any from Theorem 4. From the continuity of on the interval , it follows that there is at least one number and , which indicate that the trajectory with as the initial point is the order-2 periodic solution, and this is contradictory.

Theorem 6. If and there exist , when , then system (2) has an order-3 periodic solution.

Proof. If , there is a unique order-1 periodic solution in , i.e., , where . Since the Poincar map is continuous over the closed interval , and , then there exist and . Furthermore, ; on the contrary, ; according to the nature of the continuous function on the closed interval, at least one value of satisfies . This means that system (3) has an order-3 periodic solution with an initial point of .
In a similar way, if , where , then system (2) has an order- periodic solution.

5. Simulations

Figures 4(a) and 4(b), respectively, show the order-1 periodic solutions simulated under the two conditions of Figures 2(a) and 2(b), in which the blue line represents the trajectory when the order periodic solution is not reached and the red line represents the trajectory when the order periodic solution is reached. This suggests that populations of phytoplankton and fish can be kept within a stable range with the impulsive feedback control.

(a)

(b)

We simulated the order-1 periodic solution and order-2 periodic solution in the case of Figure 3(a) (see Figure 5). In the simulated Poincar map, we found the order-1 periodic solution (the intersection point of the red and yellow line and blue line in Figure 5(a)) and the order-2 periodic solution (the intersection point of the black line and red and yellow line in Figure 5(a)). Figure 5(b) shows the motion trajectories of the order-1 periodic solution and the order-2 periodic solution, in which the blue line is the motion trajectory of the order-2 periodic solution and the red and yellow line is the motion trajectory of the order-1 periodic solution.

(a)

(b)

Figures 6(a) and 6(b) are schematic diagrams of the number of phytoplankton and the number of fishes and time in the order-2 periodic solution in Figure 5(b), respectively. Figures 6(c) and 6(d) are schematic diagrams of the planktonic quantity and the fish quantity and time in the order-1 periodic solution in Figure 5(b), respectively. As it is revealed in the figure, the number of fish in the order-2 periodic solution and the order-1 periodic solution both change periodically, with a period of one. The number of phytoplankton in the order-2 periodic solution and the order-1 periodic solution are both changing periodically, in which the number of plankton in the order-2 periodic solution has a period of two. The period of the number of phytoplankton in the order-1 periodic solution is one.

(a)

(b)

(c)

(d)

In Figure 7, the blue and red lines indicate the trajectory of the system with or without pulses. This shows that the number of phytoplankton and fish can be kept in a stable range.

(a)

(b)

(c)

As can be revealed in Figure 8, different initial points will eventually converge to the same order-1 periodic solution and tend to be stable. This indicates the global asymptotic stability of the order-1 periodic solution.

6. Conclusion

How to rationally develop and utilize fishery resources has become an essential issue for the sustainable development of lake resources. In this paper, we introduce the impulse feedback control into the phytoplankton-fish model. It is of great significance to study the dynamics of phytoplankton and fish.

Compared with [1], we use the Poincar map as a tool to give a more comprehensive qualitative analysis and to prove the dynamics of the phytoplankton-fish model, for example, the necessary and sufficient conditions for the global asymptotic stability of the order-1 periodic solution and the existence conditions of the order- periodic solution.

For the biological significance, the order- periodic solution of system with state impulsive feedback control indicates that the number of phytoplankton and fish populations can maintain periodic oscillations under certain conditions with proper capture of fish, that is, the number of phytoplankton and fish can be kept in a stable range. These results have some reference values to the dynamic changes in aquatic ecosystem research of fish and phytoplankton.

Data Availability

The data used to support the findings of this study are available upon request from the corresponding author.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11371230), Shandong Provincial Natural Science Foundation of China (no. ZR2019MA003), SDUST Research Fund (no. 2014TDJH102), and Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, SDUST Innovation Fund for Graduate Students (no. SDKDYC170225).

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Copyright © 2020 Dezhao Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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