Research Article | Open Access
Dynamic Behaviors of a Competitive System with Beddington-DeAngelis Functional Response
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.
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 , Wang, Liu and Li  introduced the following Lotka-Volterra competitive system:which is a special case of system (2) with and . Wang et al.  showed the existence and stability of positive almost periodic solutions of system (4). We  obtained partial extinction of system (4) by constructing some suitable Lyapunov type extinction functions. Liu and Wang  incorporated the impulsive perturbations to the system (4) and investigated the uniqueness of positive almost periodic solutions. Liu, Wu and Cheke  also investigated dynamic behaviour of (4) with delay, impulsive harvesting and stocking controls. Xie et al.  further considered the partial extinction of system (4) with one toxin producing species.
Qin et al. and Wang et al. both studied the discrete time version of system (4)and obtained the permanence, stability, and almost periodic solutions of the system. Yue  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.  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 , Ma, Gao, and Xie  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  proposed system (2) with the effect of toxic substances and obtained the partial extinction of system. However, authors in  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 ). If , , and , when and , we have If , , and , when and , we have
Proof. For any small enough , it follows from condition thatThe first equation of system (2) yieldsBy applying Lemma 1, we obtainAnalogously,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 getSimilarly, for , we can easily obtainSetting , one can getEquations (12), (13), and (19) show that system (2) is permanent.
Proof. implies that we can choose small enough and two positive constants , such thatandThus,For above , Theorem 2 shows there exists and for ,It follows from system (2) thatConsider the following Lyapunov type extinction functionFor , from (22)-(24), we haveIntegrating the above inequality from to , one hasChoose , then (23) shows that , for . Hence, (27) implies thatand consequently, .
Proof. Due to , there exist three positive constants , , and such thatandThus,Consider the following Lyapunov type extinction function:For , it follows from (23)-(24) and (31) thatSimilarly to the analysis in Theorem 3, one can get .
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 , 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., 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 equationare defined on , bounded above and below by positive constants and globally attractive.
Proof. According to , , and Theorem 2 that there exist and , when ,andSet , where Calculation of the upper right derivatives for and along the solution of system (2) yieldsandThus, by (37), one can obtain that, for ,Consequently, is nonincreasing on . Integrating inequality (41) from to , we getThenand 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 , one has
Proof. We first show that if holds, thenwhere and .
By (13) and Theorem 3, one can obtainSince , one can choose a small enough such thatFor above , from (47), there exists a such that for ,and soIn 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, orSo is bounded and differentiable. By Lemma 5, there exist sequences and satisfying , ; , as . Noting andaccording to and (53)Hence that is Moreover, since , are bounded functions and , we have
Theorem 9. Suppose holds, thenwhere and are any two positive solutions of system (2) and equation , respectively.
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