Research Article | Open Access
Impulsive Perturbations of a Three-Species Food Chain System with the Beddington-DeAngelis Functional Response
The dynamics of an impulsively controlled three-species food chain system with the Beddington-DeAngelis functional response are investigated using the Floquet theory and a comparison method. In the system, three species are prey, mid-predator, and top-predator. Under an integrated control strategy in sense of biological and chemical controls, the condition for extinction of the prey and the mid-predator is investigated. In addition, the condition for extinction of only the mid-predator is examined. We provide numerical simulations to substantiate the theoretical results.
Classical two-species continuous time systems such as a Lotka-Volterra system have been used to investigate the interaction between ecological populations. However, in order to understand a complex ecological system, it is necessary to study multispecies systems. For this reason, in this paper, we study three-species food chain system which appears when a top-predator feeds on a mid-predator, which in turn feeds a prey, specially assuming Beddington-DeAngelis functional responses between species .
In recent decades, the effects of impulsive perturbations on population systems have been widely studied and discussed by a number of researchers [2–19]. Thus, in order to control an ecological environment, a discrete impulsive strategy has been suggested. Especially for the three-species food chain system, two impulsive control methods, biological and chemical controls, have been taken into account. Here, a biological control means impulsive and periodic releasing of top-predator to control lower-level populations and a chemical control means that as a result of spreading pesticide the population of all three-species are impulsively lessened.
In this context, the impulsively controlled three-species food chain system with Beddington-DeAngelis functional responses was proposed and studied by Wang et al.  and their system can be described as the following impulsively perturbed system: where , and are the densities of the lowest-level prey, mid-level predator, and top-predator at time , respectively. In this system, the prey grows according to a logistic growth with an intrinsic growth rate and a carrying capacity incorporating the Beddington-DeAngelis functional response. For parameters settings, are the conversion efficiencies, are the mortality rates of the mid-level predator and the top-predator, are the maximum numbers of preys that can be eaten by a predator per unit of time, are the saturation constants, and scale the impact of the predator interference. For an impulsive control strategy, top-predators are impulsively released in the periodic fashion of the period by artificial breeding of species, in a fixed number at each time, and by introducing an impulsive catching or poisoning of the prey populations, fixed proportions of the prey, mid-level predator, and top predator are degraded in an impulsive and periodic fashion, with the same period, but at different moments. Here, all parameters except and are positive, , for , and .
Although the authors in  had introduced the important system (1.1) in a sense of impulsive controlling the food chain system, we find that there are many problems in their theoretical results, where they had showed rich dynamical behaviors in the numerical simulations including a quasiperiodic oscillation, narrow periodic widow, wide periodic window, chaotic band, and period doubling bifurcation, symmetry-breaking pitchfork bifurcation, period-halving bifurcation and crises [20–22].
The authors in  had argued that (1) the prey and mid-predator free periodic solution is always unstable without having any condition and (2) the mid-predator free periodic solution for the system is . But, based on our theoretical computation, the periodic solution can be found only when . It means that under an impulsive control of the population system, the solution is useless and nonmeaningful. In Section 2, we will give a general form of the mid-predator free solution. In the case that the prey is impulsively strong poisoned or caught, there is a possibility that the prey will be eradicated. To examine this possibility, we reinvestigate their system (1.1). Finally, we find out that their theoretical results shown in  are wrong. In this paper we thus may correct and rebuild their theoretical results, in particular, the conditions for stabilities of the periodic solutions and the new mid-predator free periodic solution .
The main purpose of this paper is to reestablish the local and global stability for two periodic solutions and . In addition, we exhibit some numerical examples. To do it, this paper is organized as follows. In Section 2, we first review notations and theorems. Main theorems for two impulsive periodic solutions are given in Section 3. The mathematical proofs for our main results will be provided in Section 4. Conclusions are presented in Section 5.
2. Basic Strategy
In this section we will consider definitions, notations, and auxiliary results for impulsively perturbed dynamical systems.
Let us denote by the set of all nonnegative integers, , and the mapping defined by the right-hand sides of the first three equations in (1.1).
Let , then is said to belong to class if(1) is continuous on for each , and two limits and exist;(2) is locally Lipschitzian in .
Definition 2.1. For , one defines the upper right Dini derivative of with respect to the impulsive differential system (1.1) at by
We suppose that satisfies the following hypotheses: is continuous on and the limits and exist and are finite for and .
Lemma 2.2 (see ). Suppose and where satisfies and are nondecreasing for all . Let be the maximal solution for the impulsive Cauchy problem defined on . Then implies that , where is any solution of (2.2).
A similar result can be obtained when all conditions of the inequalities in the Lemma 2.2 are reversed. Using Lemma 2.2, it is easy to prove that the positive octant is an invariant region for the system (1.1) (see Lemma 2.1 in ).
2.2. Periodic Solutions
In the case in which , that is, mid-predator is eradicated, the system (1.1) is decoupled and led to two impulsive differential equations (2.4) and (2.6). Let us consider the properties of these impulsive differential equations. The following equation or a subsystem of system (1.1) is a periodically forced system: Straightforward computation for getting a positive periodic solution of (2.4) yields the analytic form of : where and .
Next, we consider the impulsive differential equation as follows: The system (2.6) is a periodically forced linear system and its positive periodic solution will be obtained: Moreover, we may get that is a solution of (2.6), where .
Lemma 2.4. For every solution and every positive periodic solution of (2.6), it follows that tends to as .
3. Main Results
In this section we study the local and global stability of the lowest-level prey and mid-level predator free periodic solution and of the mid-level predator free periodic solution . The authors in  had claimed that the solution of the impulsive controlled system (1.1) is always unstable. In the biological point of view, this result is suspected in the sense that the prey and mid-predator will be eradicated under enough impulsive control term, especially, . In Section 3.1 we will reinvestigate the stability of the periodic solution and correct the misleading results shown in , and then the stability of the mid-level predator free periodic solution will be studied in Section 3.2.
3.1. A Stability of a Periodic Solution with Prey and Mid-Predator Eradication
We theoretically and numerically consider the stability of the periodic solution with prey and mid-predator eradications.
Theorem 3.1. The periodic solution is unstable if , and globally asymptotically stable if .
The above Theorem 3.1 says that in case that is sufficiently close to 1 to make negative, the pesticide has a negative effect on the growth of prey in a certain period . To set up a control strategy for impulsive systems, we have to consider the relationship between two important factors, that is, the natural growth rate and (chemical) controlled rate in a controlling period.
The proof of this theorem will be provided in Section 4, and we may numerically consider the dynamical feature related to Theorem 3.1. To do it, we first fix the parameters: . If we choose the parameter , then the value is negative. It implies that trajectories asymptotically approach the periodic orbit as shown in Figure 1. But, for different parameters setting having a positive value , we may expect that a typical trajectory with an initial condition near is repelled from the periodic solution . For instance, we choose the parameters and then . A repelling behavior of a trajectory with a starting point near is shown in Figure 2. It shows the instability of the prey and mid-predator free solution.
As shown in Figures 1 and 2, the dynamical behavior of the periodic solution is depending on the stability condition, the positiveness or negativeness of the value . This stability condition is related to the total growth in a period and the term representing the loss of the prey due to the impulsive control on the prey. It means that species will be eradicated or grow depending on the sum of a natural growth and an artificial loss (impulsive control).
3.2. A Stability of a Periodic Solution with Mid-Predator Eradication
In this section the stability of the periodic solution with mid-predator eradication will be considered. In Theorem 3.2, we will mention the conditions for local and global stability of the periodic orbit . Compared to Theorem 3.1, the positiveness of the value should be added in the condition for being the stable periodic orbit .
Theorem 3.2. Suppose that . Then the periodic solution is locally asymptotically stable if the condition holds. Moreover, the periodic solution is globally asymptotically stable if the condition holds. Here, the values are listed:
The proof of this theorem is provided in Section 4. To illustrate an numerical example related to Theorem 3.2, let , and . These parameters satisfy the condition (3.1). It thus implies that trajectories asymptotically approach the periodic orbit . In Figure 3, we numerically show that the periodic orbit is a sink.
4.1. Proof of Theorem 3.1
Proof. A local stability of the periodic solution of the system (1.1) may be determined by considering the behavior of small amplitude perturbations of the solution. Let be any solution of the system (1.1). Define and . Then they may be written as
and is the identity matrix. Therefore, the fundamental solution matrix is
The resetting impulsive conditions of the system (1.1) become
Note that the eigenvalues of
are , and . Clearly, and . If , then . Therefore, by Floquet theory , the periodic solution is unstable.
To prove a global stability of the periodic solution , first, we assume that . Then . It means that the periodic solution is locally asymptotically stable.
Let be any solution of (1.1). Take a number with and let . Note that . From the first equation in (1.1), we get for and . By Lemma 2.2, we obtain for , where is the solution of (2.4). Using Lemma 2.3, we also get that as . It implies that there exists a number satisfying for . Without loss of generality, we may assume that for all . From the second equation in (1.1), we obtain that for and , Integrating both sides of the inequality (4.7) on , we get It implies that . Therefore as . We also obtain that for , It thus implies that as .
Now, take in order to prove that as . Since , there is a such that for . For the sake of simplicity, we assume that for all . From the third equation in (1.1), we get that, for and , Thus, by Lemma 2.2, we induce that , where is the solution of (2.6) and is also the solution of (2.6) with changed into . Using Lemma 2.4 and letting , and tend to as . We thus prove that as .
4.2. Proof of Theorem 3.2
To determine the stability of the periodic solution , we will use the Floquet theory. First, we construct the monodromy matrix and calculate its eigenvalues: where satisfies and is the identity matrix. Then all eigenvalues of the matrix are Note that where and .
From (4.14) and , we get that and is equivalent to (3.1) in the statement of Theorem 3.2. By the hypothesis of Theorem 3.2, we obtain . Finally, based on the Floquet theory , we get that is locally asymptotically stable.
Suppose that and (3.2) hold.
Let be any solution of (1.1). The condition (3.2) implies Thus the periodic solution is locally asymptotically stable. Further, we can choose such that As the proof of Theorem 3.1, by Lemma 2.2, we obtain for , where is the solution of (2.4). It follows from Lemma 2.3 that there exists a such that for . Without loss of generality, we may assume that for . From the second equation in (1.1), we get that for and , By integrating both sides of (4.17) on , we obtain that It implies that . Finally, we get that as .
From the inequality we get for . Consequently as .
In order to show that as , we take such that . Since , there exists a such that for . For the sake of simplicity, we may suppose that for all . Therefore, for and , we obtain Thus, from Lemma 2.2, we obtain that , where is the solution of (2.4) with changed into . From Lemma 2.3 and taking sufficiently small , it is seen that and tend to as . Thus, we get as .
Note that for and . By using the same process as the proof of Theorem 3.1, we can show that as .
We have considered the impulsively controlled three species food chain system with the Beddington-DeAngelis functional response proposed by the authors in . To control the food chain system with three species, two control terms, biological and chemical controls, are employed. Here, a biological control means an impulsive and periodic releasing of top-predator with a fixed proportion and a chemical control means that, for instance, as a result of pesticide spreading fixed proportions of prey, mid-predator, and top-predator, their population will be impulsively degraded. Under controlling environment, we first show the conditions for extinction and growing of the prey and mid-predator using Floquet theory and comparison method. In addition, a suffcient condition for local and global stability of the mid-predator free solution is established, which means that if the mid-predator is regarded as the pest we can control the pest population under some conditions. These results will correct the misleading results shown in .
The first author was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (no. R01-2008-000-20088-0) and the second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2011–0006087).
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