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

where x1(t) and x2(t) are the biomass of species x1 and x2 at time t, respectively. For i = 1, 2, ri(t) are the intrinsic growth rates of species xi; ai(t) 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 b1(t)x2(t)/(α1(t) + β1(t)x1(t) + γ1(t)x2(t)) and b2(t)x1(t)/(α2(t) + β2(t)x1(t) + γ2(t)x2(t)), respectively; ri(t), ai(t), bi(t), αi(t), βi(t), γi(t) 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: x (0) > 0, y (0) > 0. (3)

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   () =   () = 1 and   () = 0 ( = 1, 2).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   () = 0 and   () = 1 ( = 1, 2).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][16][17][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.

eorem . Assume
holds, where   is defined in Theorem 2, then for each positive solution Proof.Due to ( 3 ), there exist three positive constants  2 , , and  such that and Consider the following Lyapunov type extinction function: For  ≥  2 , it follows from ( 23)-( 24) and (31) that Similarly to the analysis in Theorem 3, one can get lim →+∞  1 () = 0.

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.
According to Lemma 2.1 in Zhao et al. [21], one can get Lemma .Suppose that  1 () and  1 () are continuous functions bounded above and below by positive constants, then any positive solutions of the following equation are defined on [0, +∞), bounded above and below by positive constants and globally attractive.
Proof.Similar arguments as the proof of Theorem 8 can show Theorem 9. We omit the details here.
eorem .Assume all conditions in Theorem 7 hold, then system (2) with (3) has a unique positive almost periodic solution.
Example 2. Let us choose the following system: In this case, according to Theorem 3, we can get  1 = 35,  2 = 0.1, and Equations ( 78)-(79) show that ( 2 ) holds and according to Theorem 8, for system (77), species  2 is driven to extinction while species  1 is asymptotic to any positive solution of