Discrete Dynamics in Nature and Society
Volume 2019 (2019), Article ID 4592054, 12 pages
https://doi.org/10.1155/2019/4592054

Research Article

Dynamic Behaviors of a Competitive System with Beddington-DeAngelis Functional Response

Shengbin Yu¹ and Fengde Chen²

¹Department of Basic Teaching and Research, Yango University, Fuzhou, Fujian 350015, China
²College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China

Correspondence should be addressed to Shengbin Yu; yushengbin.8@163.com

Received 30 October 2018; Revised 25 December 2018; Accepted 26 December 2018; Published 5 February 2019

Academic Editor: Yukihiko Nakata

Copyright © 2019 Shengbin Yu and Fengde Chen. 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.

Abstract.

This article studies a competitive system with Beddington-DeAngelis functional response and establishes sufficient conditions on permanence, partial extinction, and the existence of a unique almost periodic solution for the system. The results supplement and generalize the main conclusions in recent literature. Numerical simulations have been presented to validate the analytical results.

1. Introduction

For a continuous bounded function , we define

In the paper, we investigate the dynamic behaviors of the following competitive system:

where and are the biomass of species and at time , respectively. For , are the intrinsic growth rates of species ; are the rates of intraspecific competition of the first and second species; the interspecific competition between two species takes the Beddington-DeAngelis functional response type and , respectively; are all continuous and bounded functions with upper and lower positive bounds. For the biological meaning, we will consider system (2) with the following initial conditions:

It is not difficult to obtain that the corresponding solution satisfies for all .

Motivated by Gopalsamy [1], Wang, Liu and Li [2] introduced the following Lotka-Volterra competitive system:

which is a special case of system (2) with and . Wang et al. [2] showed the existence and stability of positive almost periodic solutions of system (4). We [3] obtained partial extinction of system (4) by constructing some suitable Lyapunov type extinction functions. Liu and Wang [4] incorporated the impulsive perturbations to the system (4) and investigated the uniqueness of positive almost periodic solutions. Liu, Wu and Cheke [5] also investigated dynamic behaviour of (4) with delay, impulsive harvesting and stocking controls. Xie et al. [6] further considered the partial extinction of system (4) with one toxin producing species.

Qin et al.[7] and Wang et al.[8] both studied the discrete time version of system (4)

and obtained the permanence, stability, and almost periodic solutions of the system. Yue [9] considered the partial extinction of system (5) with one toxin producing species.

Considering the interference of unpredictable forces for ecosystems in nature, Wang et al. [10] further incorporated feedback controls to system (5)

and established some results on almost periodic solutions of the system. We [11, 12] investigated the effect of feedback control variables on permanence and extinction of (6).

Motivated by Gopalsamy [1], Ma, Gao, and Xie [13] investigated the following discrete two-species competitive system:

and obtained the almost periodic solutions of the system. However, to the best of our knowledge, there are no researches on the dynamic behaviors of continuous analogue of system (7) which is a special case of system (2) under and .

Based on the above papers, Chen, Chen and Huang [14] proposed system (2) with the effect of toxic substances and obtained the partial extinction of system. However, authors in [14] did not study some important topics such as permanence, stability, and almost periodic solutions of the system. Hence, the goal of this paper is to obtain results on permanence, partial extinction, and the existence of a unique almost periodic solution of system (2) and (3). Our results supplement the main results of [13, 14] and generalize [2, 3]. Many important results concerned this direction; one could see [15–18] and so on.

This paper is distributed as follows: Section 2 is devoted to the results on permanence and extinction for system (2). In Section 3, we discuss the global attractivity of the system (2) and of one species under the other one is extinct. In Section 4, the uniqueness of positive almost periodic solutions of system (2) is obtained. Numerical simulations are presented to validate the analytical results in Section 5. Finally, we conclude in Section 6.

2. Permanence and Extinction

First, let us introduce the following lemma which is useful for our main result.

Lemma 1 (see [19]). If , , and , when and , we have

If , , and , when and , we have

Theorem 2. Suppose

holds, where , then system (2) with initial condition (3) is permanent. That is, each positive solution of system (2) satisfies

where , .

Proof. For any small enough , it follows from condition that

The first equation of system (2) yields

By applying Lemma 1, we obtain

Analogously,

Above two inequalities imply that there exists a such that, for ,

Hence, (15) and the second equation of (2) show that, for ,

From (16), according to (11) and Lemma 1, one can get

Similarly, for , we can easily obtain

Setting , one can get

Equations (12), (13), and (19) show that system (2) is permanent.

Theorem 3. Assume

holds, where is defined in Theorem 2, then for each positive solution of system (2), .

Proof. implies that we can choose small enough and two positive constants , such that

and

Thus,

For above , Theorem 2 shows there exists and for ,

It follows from system (2) that

Consider the following Lyapunov type extinction function

For , from (22)-(24), we have

Integrating the above inequality from to , one has

Choose , then (23) shows that , for . Hence, (27) implies that

and consequently, .

Theorem 4. Assume

holds, where is defined in Theorem 2, then for each positive solution of system (2), .

Proof. Due to , there exist three positive constants , , and such that

and

Thus,

Consider the following Lyapunov type extinction function:

For , it follows from (23)-(24) and (31) that

Similarly to the analysis in Theorem 3, one can get .

3. Stability

We will derive the global attractivity of the system and of one species under the other one is extinct in this section. Firstly, we introduce some useful lemmas firstly.

Lemma 5 (see [20], fluctuation lemma). Let be a bounded differentiable function on . Then there exist sequences and such that

(i) and as ;(ii) and as .

According to Lemma 2.1 in Zhao et al.[21], one can get

Lemma 6. Suppose that and are continuous functions bounded above and below by positive constants, then any positive solutions of the following equation

are defined on , bounded above and below by positive constants and globally attractive.

Theorem 7. Assume and

hold, where and are defined in Theorem 2 and

Let be any two positive solutions of system (2), then

Proof. According to , , and Theorem 2 that there exist and , when ,

and

Set , where Calculation of the upper right derivatives for and along the solution of system (2) yields

and

Thus, by (37), one can obtain that, for ,

Consequently, is nonincreasing on . Integrating inequality (41) from to , we get

Then

and so Moreover, Theorem 2 and system (2) show that both and their derivatives are bounded on , and therefor and are uniformly continuous on Using Barblat’s Lemma [22], one has

Theorem 8. Assume holds, then

where and are any two positive solutions of system (2) and (34), respectively.

Proof. We first show that if holds, then

where and .
By (13) and Theorem 3, one can obtain

Since , one can choose a small enough such that

For above , from (47), there exists a such that for ,

and so

In view of (48) and (50), it derives from Lemma 1 that . Setting in the above inequality, we have and (46) holds. It follows from (46) and Lemma 6 that there exist three positive constants , , and such that and , for . Set , , and , then , , and

Hence,

or

So is bounded and differentiable. By Lemma 5, there exist sequences and satisfying , ; , as . Noting

and

according to and (53)

Hence that is Moreover, since , are bounded functions and , we have

Theorem 9. Suppose holds, then

where and are any two positive solutions of system (2) and equation , respectively.

Proof. Similar arguments as the proof of Theorem 8 can show Theorem 9. We omit the details here.

4. Existence of a Unique Almost Periodic Solution

Now, we are in a position to show the existence of a unique almost periodic solution of system (2) under the assumption that are all continuous almost periodic functions with upper and lower positive bounds. For information on almost periodic functions, one can refer to [23, 24].

Let be the set of all solutions of system (2) with , for

Lemma 10. .

Proof. As are almost periodic functions, there exists a sequence satisfying and

uniformly on as Set to be a solution of system (2) with , for Then and are clearly uniformly bounded and equicontinuous on each bounded subset of According to Ascoli-Arzela theorem, one can get a subsequence of (still denote as ), such that uniformly on each bounded subset of as . Choose any such that for all ; hence for , we have

and

Employing Lebesgue’s dominated convergence theorem and taking limits in above equations lead to

Due to the arbitrariness of , is a solution of (2) on . Obviously, , for , so and Lemma 10 is tenable.

Theorem 11. Assume all conditions in Theorem 7 hold, then system (2) with (3) has a unique positive almost periodic solution.

Proof. By Lemma 10, system (2) with (3) has a bounded positive solution and so one can get a sequence satisfying as and be a solution of the following system

Theorem 2 with the fact that all coefficients in system (2) are continuous, positive almost periodic functions leads to that and are uniformly bounded; therefor are uniformly bounded and equicontinuous. Using Ascoli-Arzela theorem, one can get a uniformly convergent subsequence with the property that, for any , there exists a satisfying that when ,

That is to say are asymptotically almost periodic functions, so one can find almost periodic functions and continuous functions satisfying

where

and are also almost periodic functions.
Thus,

Moreover,

So exists.
By definition of almost periodic function, one can get a sequence with and

uniformly on as .
Obviously, as , hence

and

These show that satisfies system (2) and is a positive almost periodic solution of (2). Combining with Theorem 7, we further obtain that there exists a unique almost periodic solution of system (2).

5. Numeric Simulations

Example 1. Let us choose the following system:

In this case, according to Theorem 2, we have

Considering , , and , we can get

Equations (73)-(76) mean that , , and hold, so system (71) has a unique positive almost periodic solution by Theorem 11 which is also verified in Figure 1.

Figure 1: Numeric simulations of system (71) with initial values , , , and , respectively.

Example 2. Let us choose the following system:

In this case, according to Theorem 3, we can get , and

Equations (78)-(79) show that holds and according to Theorem 8, for system (77), species is driven to extinction while species is asymptotic to any positive solution of

Figure 2 also supports this conclusion.

Figure 2: Numeric simulations of system (77) with initial values , , , and , respectively.

Example 3. Let us choose the following system:

In this case, according to Theorem 4, we can get , and

Equations (82)-(83) show that holds and according to Theorem 9, for system (81), species is driven to extinction while species is asymptotic to any positive solution of

Figure 3 also supports this conclusion.

Figure 3: Numeric simulations of system (81) with initial values , , , and , respectively.

6. Conclusion

We study a competitive system with Beddington-DeAngelis functional response in this paper and obtain sufficient conditions on permanence, partial extinction, and the existence of a unique almost periodic solution for the system. When and , (2) we discussed becomes (4) which was investigated by Wang et al. [2] and We [3]. Our main results Theorems 2 and 11 are in accordance with Lemma 2.2 and Theorem 3.1 in [2]; Theorems 3, 4, 8, and 9 are also consistent with Theorem 1-4 in [3], so our results generalize [2, 3]. Moreover, when and , system (2) reduces to continuous analogue of system (7) investigated by Ma, Gao, and Xie [13]. Since we consider the permanence, global attractivity, and existence of unique almost periodic solution for the system which Ma et al. [13] and Chen et al. [14] have not studied, our results also supplement the main results of [13, 14]. Numeric simulations validate our analytical results.

Data Availability

No data were used to support this study.

Conflicts of Interest

We declare that there are no conflicts of interest to this paper.

Acknowledgments

The research was supported by the Natural Science Foundation of Fujian Province (2019J01089), Program for New Century Excellent Talents in Fujian Province University (2018), and Program for Outstanding Youth Scientific Research Talents Cultivation in Fujian Province University (2016).

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