Research Article | Open Access
Seasonal Effects on a Beddington-DeAngelis Type Predator-Prey System with Impulsive Perturbations
We study a Beddington-DeAngelis type predator-prey system with impulsive perturbation and seasonal effects. First, we numerically observe the influence of seasonal effects on the system without impulsive perturbations. Next, we find the conditions for the local and global stabilities of prey-free periodic solutions by using Floquet theory for the impulsive equation and small amplitude perturbation skills, and for the permanence of the system via comparison theorem. Finally, we show that seasonal effects and impulsive perturbation can give birth to various kinds of dynamical behavior of the system including chaotic phenomena by numerical simulations.
1. Introduction and Model Formulation
In ecology, one of main goals is to understand the dynamical relationship between predator and prey. Such relationship can be represented by the functional response which refers to the change in the density of prey attached per unit time per predator as the prey density changes. One of well-known functional responses is Beddington-DeAngelis functional response introduced by Beddington  and DeAngelis et al. , independently. It is similar to Holling type II functional response but contains an extra term describing mutual interference by predators. In fact, there are much significant evidences to suggest that functional responses with predator interference occur quite frequently in laboratory and natural systems . Thus, we can establish a predator-prey model with Beddington-DeAngelis functional response as the following form [1, 4, 5]: where , represent the population densities of prey and predator at time , respectively. In this system, the prey grows according to a logistic growth with intrinsic growth rate and is called the carrying capacity of the prey. For parameters setting, is the per-capita rates of predation of the predator, the constants are the conversion rate and the death rate of the predator, respectively, and the term measures the mutual interference between predators.
It is necessary and important to consider models with periodic ecological parameters which might be quite naturally exposed such as those due to seasonal effects of weather or food supply . Thus when the environmental factors that affect various parameters of the ecological model fluctuate periodically, then the corresponding parameters should be taken as periodic functions of time . There are a number of ways to apply periodic perturbation in ecological models. Especially, one of the most popular ways to describe periodic phenomena is to use the sine (or cosine) wave or sinusoid function which describes a wave-like function of time with peak deviation from center and angular frequency [8–15]. Thus we consider the intrinsic growth rate in system (1.1) as periodically varying function of time due to seasonal variation and adopt the sine wave as mentioned above to investigate the seasonality on the system. In fact the seasonality is superimposed as follows:
where the parameter represents the degree of seasonality; is the magnitude of the perturbation in , is the angular frequency of the fluctuation caused by seasonality. Since is assumed to be positive, we have . With this idea of periodic forcing, we consider the following predator-prey system with periodic variation in the intrinsic growth rate of the prey: where and represent the magnitude and frequency of the forcing term, respectively. Of course, a number of researchers [14, 16, 17] have studied that dynamical systems with simple dynamic behavior in the constant parameter case display very complex behavior including chaos when they are periodically perturbed. In this context, in Section 2 we illustrate numerical simulations for system (1.3) to show the existence of limit circles and various kinds of dynamical behaviors including chaos. For this reason, system (1.3) reflects more realistic situation than system (1.1).
There are still some other periodic perturbations such as fire, flood, and mating habits or harvesting seasons which are not suitable to be considered continually. Suppose that with the pest outbreak, for example, there are many ways to beat agricultural pests. One of important ways is biological control leading reduction in pest population from the actions of other living organisms, often called natural enemies or beneficial species. As we know, anther important method for pest control is chemical control. Pesticides can reduce farmer's financial losses by preventing crop losses to insects and other pests. Such control tactics should be used not continuously but impulsively. There are many literatures on systems dealing with impulsive controls [10, 11, 13–15, 18–21]. Thus, we consider the following predator-prey system with adding periodic constant impulsive immigration of the predator regarded as natural enemy of the prey (pest) to system (1.3) and spraying pesticides (harvesting) on all species at the same times where is the period of the impulsive immigration or stock of the predator, , present the fraction of the prey and the predator which die due to the harvesting or pesticides, and so forth, and is the size of immigration or stock of the predator.
If we take , system (1.4) can be expressed as the Holling-type II predator-prey system with impulsive perturbations and seasonal effects as follows: While if , then system (1.4) can be expressed as the ratio-dependent predator-prey system with impulsive perturbations and seasonal effects as follows: We will investigate system (1.4) together with systems (1.5) and (1.6). Impulsive differential equations such as (1.4) are found in almost every domain of applied science and have been studied in many investigations [10, 11, 14, 16, 17]. Especially, Zhang and Chen  considered system (1.4) when , . They investigated abundance of complex dynamics for system (1.4) when , and suggested a more executable way for observing chaos and coexistence of attractors. They also gave a threshold that classifies between the permanence and the stability of prey-free solutions for system (1.4). However, in case , , system (1.4) has not been studied yet. Thus, the purpose of this paper is to find conditions for the stability of prey-free periodic solutions. Also, we show that the system is permanent under some conditions. In addition, using numerical simulations various kinds of dynamical phenomena are discussed in Section 4.
2. Numerical Analysis of System (1.3)
In this section we will numerically study the influence of the seasonality parameter on system (1.3). For this, we fix parameters , , , , , , , and we choose as an initial point. It follows from  that system (1.3) with these parameters has a unique stable limit cycle when . Since the corresponding continuous system (1.3) cannot be solved explicitly and system (1.3) cannot be rewritten as equivalent difference equations, it is difficult to study them analytically. However, the influence of may be documented by stroboscopically sampling one of the variables over a range of values. Thus we numerically integrate system (1.3) and seek the behavior of the solutions. The bifurcation diagram provides a summary of essential dynamical behavior of system. Indeed the points that are plotted will represent either fixed or periodic sinks or other attracting sets including chaos. It shows the birth, evolution, and death of the attracting sets. In Figure 1, we illustrate bifurcation diagrams of system (1.3) to examine significant changes in the set of fixed or periodic points or other sets of interest. As is evident from Figures 1 and 2, the solutions are still periodic for values of in the range and quasiperiodic motions appear when (see Figure 2(b)). Periodic windows are intermittently scattered. Also Figures 3(a) and 3(b) show the route to chaos through the cascade of period doubling. Moreover, although the magnitude of seasonality increases, the solutions are stable and even they become periodic cycles like case after (see Figure 3(c)). We can also catch sight of the existence of occurrences of sudden changes in Figure 1 when , and so forth. They can lead to nonunique attractors. For example, there exist at least three different attractors according to initial values when (see Figure 4). This result shows that the seasonality in just one parameter can give rise to multiple attractors. Thus, these numerical examples show that the dynamical behavior of system (1.3) is more abundant than that of system (1.1).
3. Mathematical Analysis
In this section we give some notations, definitions, and lemmas which will be useful for our main results.
Denote the set of all of nonnegative integers, , , , and the right-hand side of system (1.4). Let , then is said to be in a class if(1) is continuous on , and exists;(2) is locally Lipschitzian in .
Definition 3.1. Let , . The upper right derivatives of with respect to the impulsive differential system (1.4) are defined as
Remark 3.2. (1) The solution of system (1.4) is a piecewise continuous function ; that is, is continuous on , and exists. (2) The smoothness properties of guarantee the global existence and uniqueness of the solutions of system (1.4) (see  for the details).
We will use the following important comparison theorem on an impulsive differential equation .
Lemma 3.4 (see ). Suppose and then is continuous on and for , , exists, is nondecreasing. Let be the maximal solution of the scalar impulsive differential equation existing on . Then implies that , , where is any solution of (3.2).
We now indicate a special case of Lemma 3.4 which provides estimations for the solution of impulsive differential inequalities. For this, we let denote the class of real piecewise continuous (real piecewise continuously differentiable) functions defined on .
Lemma 3.5 (see ). Let the function satisfy the inequalities where and , , and are constants and is a strictly increasing sequence of positive real numbers. Then, for ,
Lemma 3.6. The positive quadrant is an invariant region for system (1.4).
Now, we give the basic properties of the following impulsive differential equation considered the absence of the prey: Solving the first equation of (3.7) between pulses implies Substituting it in the second equation of (3.7), the following difference equation is obtained: Then a periodic solution of (3.7) is given by
Thus we can easy obtain the following results.
Lemma 3.7. ( 1) , , , and is a positive periodic solution of (3.7).
(2) is a general solution of (3.7) with , and .
(3) For every solution and every positive periodic solution of system (3.7), it follows that tends to as . Thus, the complete expression for the prey-free periodic solution of system (1.4) is obtained for .
Now, we discuss the stability of the prey-free periodic solution .
Theorem 3.8. (1) The prey-free periodic solution of system (1.4) is locally asymptotically stable if
(2) Moreover, is globally asymptotically stable if
Proof. To show the local stability of the prey-free periodic solution of system (1.4), consider the following impulsive differential system:
By Lemma 3.4, and , where is a solution of system (1.4). Note that if is locally stable, then so is . It is easy to see that the periodic solution of (3.13) is the same as that of (1.4). That is, . The local stability of the periodic solution may be determined by considering the behavior of small amplitude perturbations of the solution. Define , . Then they may be written as
and , where is the identity matrix. The linearization of the third and fourth equations of system (1.4) becomes
Note that all eigenvalues of
are and .
Since , the condition is equivalent to (3.11). According to Floquet theory , is locally asymptotically stable.
It is easy to see that the solution is locally stable if condition (3.12) holds. Now, to prove the global stability of the pest-free periodic solution, let be a solution of system (1.4). From (3.12), we can select a sufficiently small number satisfying It follows from the first equation in (1.4) that for . From Lemma 3.4, we have , where is a solution of the following impulsive differential equation: Since as if , for any with large enough. For simplicity, we may assume that for all . Since , it follow from Lemma 3.4 that for sufficiently large, where is a solution of the following impulsive differential equation: For simplicity, we may suppose that for all . From system (1.4), we obtain Integrating (3.21) on , we get and hence which implies that as . Further, we obtain, for , which implies that as . Now, take a sufficiently small number satisfying . Since , we may assume that for all . It follows from the second equation in (1.4) that, for , Thus, by Lemma 3.4, we induce that , where is the periodic solution of (3.7) with changed into . By taking sufficiently small and , we obtain that tends to as .
Using the similar method to the proof of Theorem 3.8, we obtain the following theorems.
Theorem 3.9. For system (1.5), the periodic solution is locally asymptotically stable if , and moreover, it is globally asymptotically stable if .
Theorem 3.10. For system (1.6), the periodic solution is locally asymptotically stable if , and moreover, it is globally asymptotically stable if .
Now, we prove the boundedness of system (1.4).
Theorem 3.11. There is an such that for all large enough, where is a solution of system (1.4).
Proof. Let be a solution of system (1.4) and let . Then . If , then we obtain and . Clearly, the right-hand side of (3.25) is bounded by a constant if So we can choose such that From Lemma 3.4, we obtain that for Therefore, is bounded by a constant for sufficiently large . Hence , for a solution with all large enough.
Next, we investigate the permanence of system (1.4).
Theorem 3.12. System (1.4) is permanent if
Proof. Let . Consider the following system:
It follows from Lemma 3.4 that and . From Theorem 3.11, we may assume that , , for all large enough and . Let , . From Lemmas 3.4 and 3.7, we obtain for all large enough. Thus we will show that has a lower bound for all large enough. We will do this in the following two steps.
Step 1. From (3.28), we can choose , small enough such that and , where , and Suppose that for all . Then, from the second equation of system (3.29), we obtain . By Lemmas 3.4 and 3.7, we get and as where is the solution of and , . Then there exists such that for . So, if , , then and if , , then . Let and . Integrating (3.31) on , , we have . Then as which is a contradiction to the boundedness of . Hence there exists a such that Step 2. If for all , then we are done. If not, we may let . Then for and, by the continuity of , we have . Suppose that for some . Select such that and , where . Let . Then we have only to consider two possible cases for .Case 1 ( for ). In this case we will show that there exists such that . Suppose not, that is, , . Then for all . By (3.30) with , we obtain for , . So we get and for . Also we obtain that if and if , for . Similarly to Step 1, we have Since (3.29) and we have, for all , if and if . Integrating it on we obtain that Thus which is a contradiction.
Now, let = . Then for and . So, we have, for , if and if . By the integration of it on for , we can get that .Case 2 (there is a such that ). Let . Then for and . For , if . Integrating the equation on , we can get that .
Thus, in both cases the similar argument can be continued since for some . This completes the proof.
Theorem 3.13. System (1.5) is permanent if .
Theorem 3.14. System (1.6) is permanent if .
4. Numerical Analysis of Seasonal Effect and Impulsive Perturbation
In this section we will study the influence of impulsive perturbation and seasonal effects on system (1.4), and the relationship between seasonal effects and impulsive perturbation. For this, we take the same parameters as those in Section 2, and .
First, we display bifurcation diagrams for system (1.4) as increases from 0 to 20 about and in Figure 5. From Figures 5(a) and 5(b), we see that system (1.4) experiences quasiperiodic oscillation (see Figure 6(a)) when . However, when , we see that there is a cascade of periodic bifurcation (see Figure 6(b)) leading to chaos (see Figure 6(c)), which is followed by a cascade of periodic halving bifurcation from chaos to periodic solutions (see Figure 7). Figures 5(c) and 5(d) clearly show that with increasing from to , system (1.4) experiences process of periodic oscillatingperiodic doublingchaosperiodic halving. Figure 8 displays two different strange attractors. Next, Figure 9 illustrates bifurcation diagrams for different values of the pulse and as a bifurcation parameter. It follows from Figures 9(a) and 9(b) that system (1.4) experiences process of periodic oscillating with different periodsperiodic doublingchaosperiodic windows with periodic halving cascade -periodic solutions. Figure 10 exhibits two different strange attractors. It follows from Figures 9(c) and 9(d) that system (1.4) undergoes chaotic motions when . When , chaotic motions suddenly disappear and appear as -periodic solutions. There are also periodic doubling and halving phenomena. Finally, we investigate the relationship between , , and in a view of controlling the population density of the prey and predator. As seen in Figure 11, we figure out that the longer the period is, the larger the permanence region is and the smaller the stability region is. That means that we should release the predator within a short period, or the impulsive perturbations of the predator should be occurred at short intervals, to eradicate the prey. On the contrary, the impulsive perturbations of the predator should be occurred at long-time intervals for coexistence of the prey and the predator. If we choose , we can see coexistence of the prey and predator as shown in Figure 12.
In this paper, we have investigated the effects of periodic forcing in the intrinsic growth rate of the prey and impulsive perturbations on a predator-prey system with the Beddington-DeAngelis functional response. We have shown that there exists an asymptotically stable prey-free periodic solution if the magnitude of seasonality is less than some critical value and have found parameter regions which system (1.4) is permanent. Numerical results have shown that system (1.4) can give birth to various kinds of dynamical behaviors. Especially, the prey and the predator can coexist even if there are seasonal effects on the prey. In addition, conditions for the stability of prey-free solution and for the permanence of Holling-type II or ration-dependent predator-prey systems have been obtained. Thus we have improved the results of .
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