Dynamical Properties of a Herbivore-Plankton Impulsive Semidynamic System with Eating Behavior

Wang, Yufei; Cheng, Huidong; Li, Qingjian

doi:https://doi.org/10.1155/2020/5726895

Complexity

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 5726895 | https://doi.org/10.1155/2020/5726895

Dynamical Properties of a Herbivore-Plankton Impulsive Semidynamic System with Eating Behavior

Yufei Wang,¹Huidong Cheng,¹and Qingjian Li²

Academic Editor: Oveis Abedinia

Received29 Dec 2019

Accepted09 Mar 2020

Published28 Apr 2020

Abstract

In this paper, an impulsive semidynamic system of the relationship between plankton and herbivore is established, and the Poincaré map method is used to extend the new properties of the model. We define the Poincaré map of the impulsive point series in phase concentration and analyze the characteristics. A comprehensive and detailed analysis of the periodic solution is performed. In addition, the numerical simulations illustrate the correctness of our arguments. The results show that plankton and herbivore can survive stably under effective control.

1. Introduction

In terrestrial and aquatic ecosystems, the energy conversion and nutrient cycle between plankton and herbivores play an important role [1]. Since plankton can provide energy for herbivores, studies on the population model of plankton and herbivores have drawn increasing attention from scholars [2–4]. In Zhong et al.’s paper [5], the interaction model of nutrient solution, plankton, and herbivores was established, and the limit cycle and other dynamic properties were studied. Sharma et al. [6] put forward a predator model of plankton and herbivores, made a qualitative analysis of the model system, discussed the stability of the equilibrium point, and explored ways to maintain the ecological balance of the population at different harvest levels.

Recently, the impulsive semidynamical system with threshold has been widely applied [7–11], such as in the diagnosis and treatment of the antivirus system [12–15], the integrated control strategy of ecological resources and plant diseases and insect pests [16–20], vaccination strategies, and epidemic control [21–25]. The research and application of geometric theory in pulsed semidynamic systems have also yielded good results [26–30]; the existence, uniqueness, and stability of system periodic solutions are discussed. Also, the successor function, Lyapunov function, and coincidence theory were used to study the dynamics of biological systems [31–36]. However, with the further development of the state feedback control model and the complexity of global dynamics, there are still many unresolved problems in model research. For example, when the control parameters are changed, the global dynamics are not well processed; the periodic solutions of the model are not thoroughly studied; and the biological meanings of the complex dynamics are not well analyzed and revealed. In order to solve these problems, more advanced qualitative techniques and new methods are needed to reveal the complete dynamic properties and control strategies of biological dynamic systems. Therefore, we propose a herbivore-plankton pulse semidynamic system, which uses the Poincaré map method to comprehensively analyze and study the complex dynamics of the model.

In previous studies, the main assumptions are the number of organisms fished or destroyed during the control process is a linear relationship with the population density or number; when the biological populations reach the threshold, the number of natural enemies released during the pulse is constant and regardless of density. However, in actual situations, in order to better control the biological population and obtain benefits, it is necessary to make appropriate control in combination with factors such as biological population density and fishing rate. Therefore, in this paper, for the proposed herbivore-plankton impulsive semidynamic system, the amount of plankton released is a function of its density, and the number of herbivores caught is related to the maximum fishing rate. This allows both plankton and herbivores to be effectively controlled, more in line with actual conditions.

This article is arranged in the following sections: in Section 2, the useful definition of the pulsed semidynamic system and related lemmas are presented. In Section 3, we introduce the plankton-herbivore model and give its qualitative properties. The definition of Poincaré map and its main properties are studied in Section 4. In Section 5, the stability conditions of the boundary periodic solution and the order-1 periodic solution are given, and the existence of the order-k () periodic solution is further proved. Finally, the numerical simulation is performed to verify the conclusion.

2. Preliminaries

In practical biological problems, state feedback control strategies can be broadly defined, usually modeled by impulsive semidynamic systems. Control strategies (harvesting, applying pesticides, treatment, etc.) are only implemented when a particular species attains the previously defined threshold. In many papers, a common assumption is that all solutions from a phase set must undergo an unlimited number of pulses. In order to better research the paper, we propose the following useful definitions and lemma:

Definition 1. A generalized plane pulse semidynamic system with the state-dependent feedback control can be depicted as the following:where are continuous funtions and represents the impulsive set. , , and . For any , indicates the impulsive point; thus, the impulsive function is defined as . We define as the phase set; then, for all , while .
Define or as the semidynamical system, then represents a metric space and represents all nonnegative reals. When , the function is clearly continuous such that and for all and . The set is described as the positive orbit of . For any , we have , and for all set , then .

Definition 2. A trajectory in is called the periodic of period and order k if there are integers and which make k the smallest integer of and .
More details on the concept and nature of the impulsive semidynamical system can be found in [37–41].

Lemma 1. The order-k periodic solution of systemis orbitally asymptotically stable and enjoys the property of the asymptotic phase if the Floquet multiplier satisfies the condition , wherewithwhere are calculated at the point , , and . And is continuously differentiable with respect to . is equivalent to .

3. Herbivore-Plankton Interaction Model and Its Properties

Ntr et al. [42] proposed the following herbivore-plankton interaction model:where and are the density of plankton and herbivores, respectively. is the constant carrying capacity without herbivores, is the rate of biological conversion, indicates the function response of the herbivore to change in the density of plankton, is the semisaturation constant, and represents the maximum consumption of herbivores. For more biological definitions, see [42].

We nondimensionalize system (5) and get the following state feedback control model [43]:where and denote the density of plankton and herbivore at time , respectively, and indicates the effect of density dependence on herbivores. and indicate the quantity of plankton and herbivore reduced by catching or exhaustion, respectively. System (6) shows that when the quantity of herbivores reaches certain threshold , without considering the quantity of plankton, harvesting both herbivores and plankton causes excessive depletion of plankton and leads to a lack of food for herbivores, which in turn affects their production. And at this time, the control of plankton and herbivores is only related to the density, but in actual situations, we need to control the biological population based on the fishing rate and actual factors.

Therefore, we establish the state feedback control model as follows:

system (7) shows that whenever the quantity of plankton reaches certain threshold , people will catch herbivores and increase the amount of plankton to and , respectively, where represents the maximum fishing rate, indicates the maximum amount of plankton added, and represents the morphological parameter. For more definition of biological parameters, see [44]. Define , and by calculating, we get the function is increasing in .

The qualitative analysis of system (8) is discussed as follows when system (7) has no pulse:

Theorem 1. The extinction equilibrium point of system (8) is unstable, the boundary equilibrium point is the saddle point, and is the internal equilibrium point [43] (Figure 1).(i)If , then internal equilibrium point is stable(ii)If , then internal equilibrium point is unstable and has a limit cycleIn addition, branch occurs in the case of , where meets the following condition:

(a)

(b)

(c)

To facilitate the study of the dynamics of system (7), we define the following two isoclines:

4. Poincaré Map and Its Main Properties

We first construct the Poincaré map to study the dynamics of system (7). According to the biological significance, this article only needs to be discussed in , where

Therefore, we first define the following two sets:

It is easy to know set and represents two parallel lines and in . We represent the intersection of and as and the intersection of and as , where and . And there exists a curve that starts at point and intersects with at point .

According to the above definition, we construct the impulsive semidynamical system of system (7).

Define the impulsive set aswhile the continuous function uses the expression as follows:and the impulsive function is increasing in . So, we represent the phase set as follows (Figure 2):

Unless otherwise specified, in this article, we assume that the initial point belongs to . The trajectory through the point is expressed as follows:

Next, we define the Poincaré map of system (6) in the case .

Since the internal equilibrium point is the stable point, we assume that . Then, for any point , the trajectory passing through point will reach the impulse set M through time , sowhere

This indicates that the ordinate of the point at which the initial point on the phase set reaches the pulse set is determined by . Then, we define .

Therefore, the Poincaré map can be expressed as follows:

Then, system (8) is represented as the following scalar differential equation:

Obviously, is continuous and differentiable in . We make and with for which .

Define

According to model (20), function with initial value of can be expressed as

Therefore, the form of the Poincaré map is

Theorem 2. We assume that system (8) has a stable internal equilibrium point which meets and ; then, the Poincaré map of system (8) contains the following properties (Figure 3):(i)The domain of is (ii) is increasing on and decreasing on (iii)Poincaré map is continuously differentiable(iv) has a unique fixed point(v) takes the minimum value at and has no maximum value

(a)

(b)

Proof. (i)According to Theorem 1 and , for any point , where and the trajectory through point always intersects the impulse set . Therefore, the function is meaningful when . So, the domain of is .(ii)We first prove the monotonicity of the function on the interval . Choose any two points with , , where . Then, after time , the trajectory where . From the properties of the solution of the differential equation, we get that ; then, from system (19), the Poincaré map . Therefore, is increasing on .If with , the trajectory across the isocline and reach at points and . From the Cauchy–Lipschitz theorem, we have and . According to the above proof, ; therefore, is decreasing on .(iii)From system (20), it is clear that is continuous and differentiable in the first quadrant. According to the Cauchy–Lipschitz theorem and system (19), the function is continuously differentiable in .(iv)First, we consider the value of the Poincaré map of and prove the existence of the fixed point. If , then the Poincaré map possess a fixed point.If , let . From property (ii), we get when , then . In the light of property (iii) and the theorem of zero existence, there exists at least a point , which meets . From property (ii), has at least a fixed point on .
If , then . When , from model (19), we obtain and . Then, . From the monotonicity and continuity of , has at least a fixed point in which . Thus, possess the fixed point on .
Next, the uniqueness of the fixed point is proved as follows. We assume that system (7) admits two fixed points and , where . Define . Sincewheretherefore, when , we have . Then, which means that is a monotonically decreasing function and . Then,which leads to a contradiction. Thus, the fixed point of system (7) is unique when .(v)From property (ii), we obtain that is decreasing on and increasing on . Thus, for any , there is . So, when , the function reaches the minimal value which is also the minimum value. And the minimum value is .Since is increasing on , then has only a minimum value on and no maximum value.

5. Periodic Solution of the System

5.1. Boundary Periodic Solution

In the absence of herbivores and , system (7) possess the boundary periodic solution. Thus, we represent the system aswhere the initial condition is , and by calculating the first equation, we have

Supposing the solution starting from reaches the line at time , one has

Solving the above equation on , then

Thus, the boundary periodic solution of model (7) with period is

Theorem 3. When condition is satisfied, then the boundary periodic solution is orbitally asymptotically stable, andwhere

Proof. Let , , , , and .
Then,Furthermore,Therefore, from Lemma 1, we obtain the conclusion of Theorem 3 when .

5.2. Order-k Periodic Solution and Its Stability Analysis

According to Theorem 2 (iv), we conclude that system (7) has a unique fixed point, which implies that the system possesses an order-1 periodic solution. In this section, we prove the stability of periodic solution and the existence of order-k () periodic solution. So, we make the following provisions.

For , the trajectory of the initial point will reach the impulse set M, which is applied to the point by the impulse. In the light of the definition of the Poincaré map, we have , and repeating the process above, then

Further promotion,

The sequence is available, k = 0, 1, 2, ….

Theorem 4. The order-1 period solution is orbitally asymptotically stable ifwhere

Proof. We define the starting and the ending point of the order-1 periodic solution as and , respectively.
Thus,The multiplier is obtained:From (38), we obtain ; thus, the order-1 periodic solution is locally stable.

Theorem 5. When conditions and are established, then the order-1 periodic solution of system (7) is globally asymptotic stable.

Proof. According to Theorem 2, when , system (7) possesses unique fixed point , which means system (7) has an order-1 periodic solution, where . Also, we get the Poincaré map which increases on .
For any trajectory in which the initial point is , when , since is increasing on and there exists a unique fixed point, then and . It can be inferred from this that . According to the monotonically bounded sequence, we get .
When , by using a similar approach, we have and . Furthermore, we get . Then, . Therefore, the order-1 periodic solution of system (7) is globally asymptotic stable.

Theorem 6. If conditions , , and are established, then model (7) has a stable order-1 or order-2 periodic solution.

Proof. For any point , where , we discuss the following two cases of function which are defined by point .(i)When , since the function monotonically increases and has no fixed point when , therefore, there must be a positive integer so that and ; then, and (ii)When , since the function which decreases in and and , there must exist which satisfies , where Through the above discussion, when , there always exist a positive integer m which satisfies . Since is decreasing in , thus,So, we only need to discuss the case of function in interval . is the trajectory of the initial point, where . After the action of the impulse, we get the sequence , where .
According to the magnitude of , there exist four situations:

Situation 1. (Figure 4(a)); then, , and thus, we have and , so . After an infinite number of pulses, we get

(a)

(b)

Situation 2. (Figure 4(b)); furthermore, we have and . After an infinite number of pulses, we get

Situation 3. ; from the above similar methods, we obtain

Situation 4. ; from the above similar methods, we getFor situations 2 and 3, sequences and are monotonically bounded. Then, we have or and , where . This indicates that, in situations 2 and 3, system (7) has stable order-1 or order-2 periodic solution.
From situations 1 and 4, and . Therefore, in situations 1 and 4, system (7) has stable order-2 periodic solution.
Theorem 6 shows that system (7) has stable order-1 or order-2 periodic solutions under certain conditions. However, this theorem does not give sufficient and necessary conditions for the global stability of order-1 periodic solution. Therefore, Theorem 7 is given as follows:

Theorem 7. When , then the sufficient and necessary conditions for the global asymptotic stability of the order-1 periodic solution of system (7) are for any , where .

Proof. From Theorem 2, a stable order-1 periodic solution is available in system (7) when . Then, system (7) has in which . For any , make , and according to the properties of Poincaré map , we have ; furthermore, we get . From the monotone boundedness of the sequence, we know . Therefore, the order-1 periodic solution of system (7) is globally asymptotically stable.
We assume the order-1 periodic solution is globally stable, and the following proves that for any satisfies . If this condition is not met, then there exists minimum which makes . From Theorem 6, we get that, for any small , there exists which makes . From the continuity of , there exists which makes . Therefore, system (7) has order-2 periodic solution, which is contradictory to the global stability of order-1 periodic solution.

Theorem 8. When and , where , , then system (7) has the order-3 periodic solution.

Proof. Since , from Theorem 2, we get that system (7) has unique , which makes . Since Poincaré map is continuous in closed intervals , , and , from the intermediate value theorem, there is which makes , and then . Thus, we have .
On the contrary, according to the expression of Poincaré map, we obtain that . So, which makes . This indicates that system (7) has order-3 period solution with as the initial point.
In the same way, we can prove that if , Z, system (7) possesses the order-k periodic solution.

6. Numerical Simulations

An impulsive semidynamic system of plankton and herbivore is established in this paper. The numerical simulation illustrates the correctness of our arguments.

We set the parameter value as , and obtain that system (7) possesses a globally asymptotically stable order-1 periodic solution if (Figure 5(a)). Furthermore, let . We get that model (7) has unique order-1 periodic solution when , and it is globally asymptotically stable (Figure 6(a)). Figures 5(b) and 5(c) are time series diagram when , and Figures 6(b) and 6(c) are time series diagram when . In addition, the red trajectories represent the phase portrait and time series of system (7) without pulses. By comparison, we get that plankton and herbivore can survive stably under effective control.

(a)

(b)

(c)

(a)

(b)

(c)

From Figure 7, we get that although the trajectories start from different initial points, they all converge to the same order-1 periodic solution. And the trajectory eventually stabilizes, which implies that the order-1 periodic solution is globally asymptotically stable.

When herbivores are absent and , we get the expression of the boundary period solution. Through analysis, the boundary periodic solution is asymptotically stable under certain conditions. Thus, the rationality of the plankton release method can be obtained. After no longer harvesting herbivores, that is, when , we get that the number of herbivores increases and eventually approaches a stable value, and the plankton population will oscillate periodically from Figures 8(a)–8(c)). Although herbivores will increase temporarily, they will be in equilibrium after a period of time, and the number will no longer increase. However, pulse control allows herbivores to grow periodically, resulting in sustainable harvests and thus greater gains. Thus, we get the rationality and necessity of taking pulse control of this paper.

(a)

(b)

(c)

7. Conclusion

In this paper, an impulsive semidynamic system for the relationship between plankton and herbivores is proposed. We adopt the method of Poincaré map to comprehensively analyze the dynamic properties of system (7). Moreover, state feedback control is adopted to enable plankton and herbivores to survive stably.

Firstly, the model is introduced, and the conditions for the existence and stability of the equilibrium point of the model are analyzed. Secondly, we define the Poincaré map of the model and study its main properties such as monotonicity, continuity, fixed point, and extremum. Then, we discuss the existence and local and global stability of order-1 periodic solution and give sufficient conditions for the local stability of the boundary periodic solution. In addition, the existence of the order-k periodic solution is obtained through the proof of the existence of the order-3 periodic solution. Finally, our theoretical results are verified by numerical simulation.

From biological perspective, periodic solutions and state feedback control have important roles in ecology and fisheries. When the plankton is reduced to a certain amount, a state feedback control strategy of increasing plankton and hunting herbivores is adopted. In the end, a periodic solution can be formed, and the plankton and herbivore are stabilized. The analysis of the boundary periodic solution verifies the rationality and necessity of the pulse form used in this paper. In particular, if not controlled, the number of plankton will continue to decrease, which will reduce the food for herbivores and affect yield.

We summarize some of the innovations in this paper: (1) the impulse semidynamic system studied in this paper uses the new form of pulse, and the control parameters are changed according to the actual situation. This makes the system have more complex dynamic properties, which is worthy of our in-depth study. (2) In this paper, the Poincaré map is defined and used as a tool to carry out more comprehensive dynamic analysis of the system. For example, by analyzing the impulse function, the precise range of the pulse and phase set is obtained, and a more comprehensive and detailed study of periodic solutions is carried out. Compared with [43], we not only prove the existence and stability of the order-1 periodic solution but also give the necessary and sufficient condition for the global stability of the order-1 periodic solution and the existence conditions of the order-k periodic solution. (3) Through the analysis and numerical simulation of the model, the practical biological significance of its complex dynamics is revealed. When , the system has more complicated dynamic properties. In future research work, we will further discuss the complex dynamics of the system and optimize to obtain maximum benefits.

Data Availability

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

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11371230), the SDUST Research Fund (2014TDJH102), the Shandong Provincial Natural Science Foundation, China (ZR2015AQ001), and Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, SDUST Innovation Fund for Graduate Students (SDKDYC170225).

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Copyright © 2020 Yufei Wang 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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