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

Authors: Shengbin Yu; Fengde Chen
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## 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.


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## Body

## 1. Introduction

For a continuous bounded functionf(t), we define(1)fl=inft∈Rft,fu=supt∈Rft.In the paper, we investigate the dynamic behaviors of the following competitive system:(2)x˙1t=x1tr1t-a1tx1t-b1tx2tα1t+β1tx1t+γ1tx2t,x˙2t=x2tr2t-a2tx2t-b2tx1tα2t+β2tx1t+γ2tx2t,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:(3)x0>0,y0>0.It is not difficult to obtain that the corresponding solution(x1(t),x2(t))T satisfies x1(t)>0,x2(t)>0 for all t≥0.Motivated by Gopalsamy [1], Wang, Liu and Li [2] introduced the following Lotka-Volterra competitive system:(4)x˙1t=x1tr1t-a1tx1t-b1tx2t1+x2t,x˙2t=x2tr2t-a2tx2t-b2tx1t1+x1t,which is a special case of system (2) with αi(t)=γi(t)=1 and βi(t)=0(i=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)(5)x1n+1=x1nexp⁡r1n-a1nx1n-b1nx2n1+x2n,x2n+1=x2nexp⁡r2n-a2nx2n-b2nx1n1+x1nand 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)(6)x1n+1=x1nexp⁡r1n-a1nx1n-b1nx2n1+x2n-e1nu1n,x2n+1=x2nexp⁡r2n-a2nx2n-b2nx1n1+x1n-e2nu2n,Δu1n=-f1nu1n+d1nx1n,Δu2n=-f2nu2n+d2nx2nand 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:(7)x1n+1=x1nexpr1n-a1nx1n-b1nx2n1+β1nx1n,x2n+1=x2nexp⁡r2n-a2nx2n-b2nx1n1+β2nx2nand 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 γi(t)=0 and αi(t)=1(i=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 [15–18] and so on.This paper is distributed as follows: Section2 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]).
Ifa>0, b>0, and x˙≥x(b-ax), when t≥0 and x(0)>0, we have (8)liminft→+∞xt≥ba.If a>0, b>0, and x˙≤x(b-ax), when t≥0 and x(0)>0, we have(9)limsupt→+∞xt≤ba.Theorem 2.
SupposeH1r1lα1l>b1u-r1lγ1lM2,r2lα2l>b2u-r2lγ2lM1holds, where Mi=riu/ail,(i=1,2), then system (2) with initial condition (3) is permanent. That is, each positive solution (x1(t),x2(t))T of system (2) satisfies(10)0<m1≤liminft→+∞x1t≤limsupt→+∞x1t≤M1,0<m2≤liminft→+∞x2t≤limsupt→+∞x2t≤M2,where m1=r1lα1l-(b1u-r1lγ1l)M2/a1u(α1l+γ1lM2), m2=r2lα2l-(b2u-r1lγ2l)M1/a2u(α2l+γ2lM1).Proof.
For any small enoughε>0, it follows from condition H1 that(11)r1lα1l>b1u-r1lγ1lM2+ε,r2lα2l>b2u-r2lγ2lM1+ε.The first equation of system (2) yields(12)x˙1t≤x1tr1u-a1lx1t.By applying Lemma 1, we obtain(13)limsupt→+∞x1t≤r1ua1l=ΔM1.Analogously,(14)limsupt→+∞x2t≤r2ua2l=ΔM2.Above two inequalities imply that there exists a T1>0 such that, for t≥T1,(15)x1t≤M1+ε≜M1ε,x2t≤M2+ε≜M2ε.Hence, (15) and the second equation of (2) show that, for t≥T1,(16)x˙1t≥x1tr1l-a1ux1t-b1uM2εα1l+γ1lM2ε.From (16), according to (11) and Lemma 1, one can get(17)liminft→+∞x1t≥r1lα1l-b1u-r1lγ1lM2εa1uα1l+γ1lM2ε.Similarly, for t≥T1, we can easily obtain(18)liminft→+∞x2t≥r2lα2l-b2u-r1lγ2lM1εa2uα2l+γ2lM1ε.Setting ε→0, one can get(19)liminft→+∞x1t≥r1lα1l-b1u-r1lγ1lM2a1uα1l+γ1lM2≜m1,liminft→+∞x2t≥r2lα2l-b2u-r1lγ2lM1a2uα2l+γ2lM1≜m2.Equations (12), (13), and (19) show that system (2) is permanent.Theorem 3.
AssumeH2r2ur1l<min⁡b2la1uα2u+β2uM1+γ2uM2,a2lα1lb1u,holds, where Mi is defined in Theorem 2, then for each positive solution (x1(t),x2(t))T of system (2), limt→+∞x2(t)=0.Proof.
H 2 implies that we can choose ε1>0 small enough and two positive constants p, q such that(20)r2ur1l<min⁡b2la1uα2u+β2uM1+ε1+γ2uM2+ε1,a2lα1lb1u,and(21)r2ur1l<pq<min⁡b2la1uα2u+β2uM1+ε1+γ2uM2+ε1,a2lα1lb1u.Thus,(22)-pr1l+qr2u=def-d1<0,pa1u-qb2lα2u+β2uM1+ε1+γ2uM2+ε1<0,pb1uα1l-qa2l<0.For above ε1, Theorem 2 shows there exists T2>0 and for t≥T2,(23)xit<Mi+ε1,i=1,2.It follows from system (2) that(24)x˙1tx1t=r1t-a1tx1t-b1tx2tα1t+β1tx1t+γ1tx2t,x˙2tx2t=r2t-a2tx2t-b2tx1tα2t+β2tx1t+γ2tx2t.Consider the following Lyapunov type extinction function(25)Vt=x1-ptx2qt.For t≥T2, from (22)-(24), we have(26)V˙t=Vt-pr1t+qr2t+pa1t-qb2tα2t+β2tx1t+γ2tx2tx1t+pb1tα1t+β1tx1t+γ1tx2t-qa2tx2t≤Vt-pr1l+qr2u+pa1u-qb2lα2u+β2uM1+ε1+γ2uM2+ε1x1t+pb1uα1l-qa2lx2t≤-d1Vt<0.Integrating the above inequality from T2 to t(t≥T2), one has(27)Vt≤VT2exp⁡-d1t-T2.Choose M=2max⁡M1,M2, then (23) shows that xi(t)<M, for t≥T2. Hence, (27) implies that(28)x2t<Mp/qx1T2-p/qx2T2exp⁡-d1qt-T2,and consequently, limt→+∞x2(t)=0.Theorem 4.
AssumeH3r2lr1u>max⁡b2uα2la1l,a2uα1u+β1uM1+γ1uM2b1lholds, where Mi is defined in Theorem 2, then for each positive solution (x1(t),x2(t))T of system (2), limt→+∞x1(t)=0.Proof.
Due toH3, there exist three positive constants ε2, p, and q such that(29)r2lr1u>max⁡b2uα2la1l,a2uα1u+β1uM1+ε2+γ1uM2+ε2b1l.and(30)r2lr1u>pq>max⁡b2uα2la1l,a2uα1u+β1uM1+ε2+γ1uM2+ε2b1l.Thus,(31)pr1u-qr2l=def-d2<0,-pa1l+qb2uα2l<0,-pb1lα1u+β1uM1+ε2+γ1uM2+ε2+qa2u<0.Consider the following Lyapunov type extinction function:(32)Vt=x1ptx2-qt.For t≥T2, it follows from (23)-(24) and (31) that(33)V˙t=Vtpr1t-qr2t+-pa1t+qb2tα2t+β2tx1t+γ2tx2tx1t+-pb1tα1t+β1tx1t+γ1tx2t+βa2tx2t≤Vtpr1u-βr2l+-pa1l+qb2uα2lx1t+-pb1lα1u+β1uM1+ε2+γ1uM2+ε2+qa2ux2t≤-d2Vt<0.Similarly to the analysis in Theorem 3, one can get limt→+∞x1(t)=0.


## 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).
Letx(t) be a bounded differentiable function on [α,∞). Then there exist sequences τn→∞ and σn→∞ such that (i)
x˙(τn)→0 and x(τn)→limsupt→∞x(t)=x¯ as n→∞;(ii)
x˙(σn)→0 and x(σn)→liminft→∞x(t)=x_ as n→∞.According to Lemma 2.1 in Zhao et al.[21], one can getLemma 6.
Suppose thatr1(t) and a1(t) are continuous functions bounded above and below by positive constants, then any positive solutions of the following equation(34)x˙t=xtr1t-a1txtare defined on [0,+∞), bounded above and below by positive constants and globally attractive.Theorem 7.
AssumeH1 and H4a1t-b1tβ1tM2Δ12t,m1,m2-b2tα2tΔ22t,m1,m2-b2tγ2tM2Δ22t,m1,m2l>0,H5a2t-b1tβ1tM1Δ12t,m1,m2-b1tα1tΔ12t,m1,m2-b2tγ2tM1Δ22t,m1,m2l>0hold, where mi and Mi(i=1,2) are defined in Theorem 2 and(35)Δit,xt,yt=αit+βitxt+γityt,i=1,2.Let z1(t)=(x1(t),x2(t))T,z2(t)=(y1(t),y2(t))T be any two positive solutions of system (2), then (36)limt→+∞x1t-y1t=0,limt→+∞x2t-y2t=0.Proof.
According toH4, H5, and Theorem 2 that there exist ε3>0 and T3>T1, when t≥T3,(37)a1t-b1tβ1tM2+ε3Δ12t,m1-ε3,m2-ε3-b2tα2tΔ22t,m1-ε3,m2-ε3-b2tγ2tM2+ε3Δ22t,m1-ε,m2-ε3l>ε3,a2t-b1tβ1tM1+ε3Δ12t,m1-ε3,m2-ε3-b1tα1tΔ12t,m1-ε3,m2-ε3-b2tγ2tM1+ε3Δ22t,m1-ε3,m2-ε3l>ε3and(38)m1-ε3≤x1t≤M1+ε3,m2-ε3≤x2t≤M2+ε3,m1-ε3≤y1t≤M1+ε3,m2-ε3≤y2t≤M2+ε3.Set V(t)=V1(t)+V2(t), where Vi(t)=|ln⁡xit-ln⁡yi(t)|,i=1,2. Calculation of the upper right derivatives for V1(t) and V2(t) along the solution of system (2) yields(39)D+V1t=sgnx1t-y1t-a1tx1t-y1t-b1tx2tΔ1t,x1t,x2t-y2tΔ1t,y1t,y2t=-a1tx1t-y1t-sgn⁡x1t-y1tb1tα1tx2t-y2t+β1ty1tx2t-β1tx1ty2tΔ1t,x1t,x2t·Δ1t,y1t,y2t=-a1tx1t-y1t-sgn⁡x1t-y1tb1tα1tx2t-y2tΔ1t,x1t,x2t·Δ1t,y1t,y2t+β1ty1tx2t-y2t+β1ty2ty1t-x1tΔ1t,x1t,x2t·Δ1t,y1t,y2t≤-a1tx1t-y1t+b1tα1tx2t-y2tΔ1t,x1t,x2t·Δ1t,y1t,y2t+β1ty1tx2t-y2t+β1ty2tx1t-y1tΔ1t,y1t,y2t·Δ1t,x1t,x2t,and(40)D+V2t≤-a2tx2t-y2t+b2tα2tx1t-y1tΔ2t,x1t,x2t·Δ2t,y1t,y2t+γ2ty2tx1t-y1t+γ2ty1tx2t-y2tΔ2t,y1t,y2t·Δ2t,x1t,x2t.Thus, by (37), one can obtain that, for t≥T3,(41)
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.Consequently, V(t) is nonincreasing on [T3,+∞). Integrating inequality (41) from T3 to t, we get(42)Vt+ε3∫T3tx1s-y1sds+ε3∫T3tx2s-y2sds<VT3<+∞,t≥T3.Then(43)∫T3tx1s-y1sds<VT3ε3<+∞,∫T3tx2s-y2sds<VT3ε<+∞,and so |x1(t)-y1(t)|,|x2(t)-y2(t)|∈L1[T3,+∞). Moreover, Theorem 2 and system (2) show that both x1(t),y1(t),x2(t),y2(t) and their derivatives are bounded on [T3,+∞), and therefor |x1(t)-y1(t)| and |x2(t)-y2(t)| are uniformly continuous on [T3,+∞). Using Barbaˇlat’s Lemma [22], one has(44)limt→+∞x1t-y1t=0,limt→+∞x2t-y2t=0.Theorem 8.
AssumeH2 holds, then(45)limt→+∞x2t=0andlimt→+∞x1t-x1∗t=0,where (x1(t),x2(t))T and x1∗(t) are any two positive solutions of system (2) and (34), respectively.Proof.
We first show that ifH2 holds, then(46)A1≤liminft→+∞x1t≤limsupt→+∞x1t≤M1,where M1=r1u/a1l and A1=r1l/a1u.
By (13) and Theorem 3, one can obtain(47)limsupt→+∞x1t≤M1,limt→+∞x2t=0.Since r1l>0, one can choose a small enough ε4>0 such that(48)Aε=defr1l-b1uα1lε4>0.For above ε4, from (47), there exists a T4>T3 such that for t≥T4,(49)x1t≤M1+ε4,x2t≤ε4,and so(50)x˙1t≥x1tr1l-a1ux1t-b1uα1lε4.In view of (48) and (50), it derives from Lemma 1 that liminft→+∞x1(t)≥Aε/a1u. Setting ε4→0 in the above inequality, we have liminft→+∞x1(t)≥r1l/a1u and (46) holds. It follows from (46) and Lemma 6 that there exist three positive constants T5>T4, η1, and η2 such that A1/2<x1(t)<3M1/2 and η1<x1∗(t)<η2, for t≥T5. Set v(t)=1/x1(t), v∗(t)=1/x1∗(t), and z(t)=v(t)-v∗(t), then 2/3M1<v(t)<2/A1, 1/η2<v∗(t)<1/η1, and (51)v˙t=-r1tvt+a1t+b1tx2tv2tα1tvt+β1t+γ1tx2tvt,v˙∗t=-r1tv∗t+a1t.Hence, (52)z˙t=-r1tzt+b1tx2tv2tα1tvt+β1t+γ1tx2tvt,or(53)zt=-z˙tr1t+b1tx2tv2tr1tα1tvt+β1t+γ1tx2tvt.So z(t) is bounded and differentiable. By Lemma 5, there exist sequences τn→∞ and σn→∞ satisfying z˙(τn)→0, z(τn)→z¯; z˙(σn)→0, z(σn)→z_ as n→∞. Noting (54)0<b1tx2tw2tr1tα1twt+β1t+γ1tx2twt<4b1ux2tr1l2α1lA1+β1lA12+2γ1lA1x2t,and(55)0<1r1t<1r1l,according to limt→+∞x2(t)=0 and (53)(56)limn→∞zτn=-limn→∞z˙τnr1τn+limn→∞b1τnx2τnw2τnr1τnα1τnwτn+β1τn+γ1τnx2τnwτn;limn→∞zσn=-limn→∞z˙σnr1σn+limn→∞b1σnx2σnw2σnr1σnα1σnwσn+β1σn+γ1σnx2σnwσnHence z¯=z_=0 that is limt→+∞z(t)=0. Moreover, since x1(t), x1∗(t) are bounded functions and |x1(t)-x1∗(t)|=|v(t)-v∗(t)|x1(t)x1∗(t)=|z(t)|x1(t)x1∗(t), we have limt→+∞(x1(t)-x1∗(t))=0.Theorem 9.
SupposeH3 holds, then(57)limt→+∞⁡x1t=0andlimt→+∞⁡x2t-x2∗t=0,where (x1(t),x2(t))T and x1∗(t) are any two positive solutions of system (2) and equation x˙(t)=(r2(t)-a2(t)x(t)), respectively.Proof.
Similar arguments as the proof of Theorem8 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 ri(t),ai(t),bi(t),αi(t),βi(t),γi(t)(i=1,2) are all continuous almost periodic functions with upper and lower positive bounds. For information on almost periodic functions, one can refer to [23, 24].LetS(E) be the set of all solutions (x1(t),x2(t))T of system (2) with m1≤x1(t)≤M1, m2≤x2(t)≤M2 for t∈R.Lemma 10.
S ( E ) ≠ Ø.Proof.
Asri(t),ai(t),bi(t),αi(t),βi(t),γi(t)(i=1,2) are almost periodic functions, there exists a sequence {tn} satisfying tn→∞ and (58)rit+tn→rit,ait+tn→ait,bit+tn→bit,αit+tn→αit,βit+tn→βit,γit+tn→γiti=1,2,uniformly on R as n→∞. Set (x1(t),x2(t))T to be a solution of system (2) with m1≤x1(t)≤M1, m2≤x2(t)≤M2 for t>T. Then x1(t+tn) and x2(t+tn) are clearly uniformly bounded and equicontinuous on each bounded subset of R. According to Ascoli-Arzela theorem, one can get a subsequence of {tn} (still denote as {tn}), such that x1(t+tn)→p1(t),x2(t+tn)→p2(t) uniformly on each bounded subset of R as n→∞. Choose any T6∈R such that tn+T6≥T for all n; hence for t≥0, we have (59)x1t+tn+T6-x1tn+T6=∫T6t+T6x1s+tnr1s+tn-a1s+tnx1s+tn-b1s+tnx2s+tnα1s+tn+β1s+tnx1s+tn+γ1s+tnx2s+tndsand(60)x2t+tn+T6-x2tn+T6=∫T6t+T6x2s+tnr2s+tn-a2s+tnx1s+tn-b2s+tnx1s+tnα2s+tn+β2s+tnx1s+tn+γ2s+tnx2s+tndsEmploying Lebesgue’s dominated convergence theorem and taking limits in above equations lead to (61)p1t+T6-p1T6=∫T6t+T6p1sr1s-a1sx1s-b1sx2sα1s+β1sx1s+γ1sx2sds,p2t+T6-p2T6=∫T6t+T6p2sr2s-a2sx2s-b2sx1sα2s+β2sx1s+γ2sx2sds.Due to the arbitrariness of T1, (p1(t),p2(t))T is a solution of (2) on R. Obviously, m1≤p1(t)≤M1, m2≤p2(t)≤M2 for t∈R, so (p1(t),p2(t))T∈S(E) and Lemma 10 is tenable.Theorem 11.
Assume all conditions in Theorem7 hold, then system (2) with (3) has a unique positive almost periodic solution.Proof.
By Lemma10, system (2) with (3) has a bounded positive solution w(t)=(w1(t),w2(t))T and so one can get a sequence {tk′} satisfying {tk′}→∞ as k→∞ and (w1(t+tk′),w2(t+tk′))T be a solution of the following system (62)x˙1t=x1tr1t+tk′-a1t+tk′x1t-b1t+tk′x2tα1t+tk′+β1t+tk′x1t+γ1t+tk′x2t,x˙2t=x2tr2t+tk′-a2t+tk′x2t-b2t+tk′x1tα2t+tk′+β2t+tk′x1t+γ2t+tk′x2t.Theorem 2 with the fact that all coefficients in system (2) are continuous, positive almost periodic functions leads to that {wi(t+tk′)} and {u˙(t+tk′)}(i=1,2) are uniformly bounded; therefor {wi(t+tk′)}(i=1,2) are uniformly bounded and equicontinuous. Using Ascoli-Arzela theorem, one can get a uniformly convergent subsequence {wi(t+tk)}⊆{wi(t+tk′)} with the property that, for any ε5>0, there exists a K(ε5)>0 satisfying that when m,k≥K(ε5), (63)wit+tm-wit+tk<ε5,i=1,2.That is to say wi(t)(i=1,2) are asymptotically almost periodic functions, so one can find almost periodic functions ui(t+tk) and continuous functions vi(t+tk)(i=1,2) satisfying (64)wit+tk=uit+tk+vit+tk,i=1,2,where (65)limk→+∞⁡uit+tk=uit,limk→+∞⁡vit+tk=0,i=1,2,and ui(t)(i=1,2) are also almost periodic functions.
Thus,(66)limk→+∞⁡wit+tk=uit,i=1,2.Moreover, (67)limk→+∞w˙it+tk=limk→+∞limh→0wit+tk+h-wit+tkh=limh→0limk→+∞wit+tk+h-wit+tkh=limh→0uit+h-uith,i=1,2.So u˙i(t)(i=1,2) exists.
By definition of almost periodic function, one can get a sequence{tn} with {tn}→∞ and (68)rit+tn→rit,ait+tn→ait,bit+tn→bit,αit+tn→αit,βit+tn→βit,γit+tn→γiti=1,2,uniformly on R as n→∞.
Obviously,wi(t+tn)→ui(t)(i=1,2) as n→∞, hence (69)u˙1t=limn→+∞w˙1t+tn=limn→+∞w1t+tnr1t+tn-a1t+tnw1t+tn-b1t+tnw2t+tnα1t+tn+β1t+tnw1t+tn+γ1t+tnw2t+tn=u1tr1t-a1tu1t-b1tu2tα1t+β1tu1t+γ1tu2tand(70)u˙2t=limn→+∞w˙2t+tn=limn→+∞w2t+tnr2t+tn-a2t+tnw2t+tn-b2t+tnw2t+tnα2t+tn+β2t+tnu1t+tn+γ2t+tnu2t+tn=u2tr2t-a2tu2t-b2tu2tα2t+β2tu1t+γ2tu2t.These show that (u1(t),u2(t))T 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:(71)x˙1t=x1t10+2cos⁡7t-5+sin⁡5tx1t-0.5+0.3sin⁡3tx2t0.5+8+sin⁡11tx1t+0.6x2t,x˙2t=x2t17+3sin⁡5t-5+cos⁡2tx2t-0.7+0.3cos⁡7tx2t0.4+10+cos⁡11tx1t+0.5x2t.In this case, according to Theorem 2, we have (72)M1=3,M2=5,m1≈1.1429,m2≈1.2807.Considering H1, H4, and H5, we can get(73)r1lα1l-b1u-r1lγ1lM2=24>0,(74)r2lα2l-b2u-r2lγ2lM1=23.6>0,(75)a1t-b1tβ1tM2Δ12t,m1,m2-b2tα2tΔ22t,m1,m2-b2tγ2tM2Δ22t,m1,m2l≥a1l-b1uβ1uM2α1l+β1lm1+γ1lm22-b2uα2uα2l+β2lm1+γ2lm22-b2uγ2uM2α2l+β2lm1+γ2lm22≈3.5583>0,(76)a2t-b1tβ1tM1Δ12t,m1,m2-b1tα1tΔ12t,m1,m2+b2tγ2tM1)Δ22t,m1,m2l≥a2l-b1uβ1uM1α1l+β1lm1+γ1lm22-b1uα1uα1l+β1lm1+γ1lm22-b2uγ2uM1α2l+β2lm1+γ2lm22≈3.7322>0,Equations (73)-(76) mean that H1, H4, and H5 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 (x(0),y(0))=(3,3)T, (4,2)T, (2.5,4.3)T, and (1.5,2.5)T, respectively.Example 2.
Let us choose the following system:(77)x˙1t=x1t6+cos⁡7t-0.5+0.3sin⁡5tx1t-0.5+0.3sin⁡3tx2t8+8+sin⁡11tx1t+0.6x2t,x˙2t=x2t0.2+0.1sin⁡5t-3x2t-7+cos⁡7tx2t0.4+0.6+0.1cos⁡11tx1t+0.1x2t.In this case, according to Theorem 3, we can get M1=35,M2=0.1, and(78)r2ur1l-b2la1uα2u+β2uM1+γ2uM2≈-0.2411<0,(79)r2ur1l-a2lα1lb1u=-29.94<0,Equations (78)-(79) show that H2 holds and according to Theorem 8, for system (77), species x2 is driven to extinction while species x1 is asymptotic to any positive solution of (80)x˙t=xt6+cos⁡7t-0.5+0.3sin⁡5txt.Figure 2 also supports this conclusion.Figure 2
Numeric simulations of system (77) with initial values (x(0),y(0))=(20,10)T, (5,5)T, (35,2)T, and (10,7)T, respectively.Example 3.
Let us choose the following system:(81)x˙1t=x1t0.6+0.1cos⁡7t-6+sin⁡5tx1t-9+3sin⁡3tx2t8+8+sin⁡11tx1t+0.6x2t,x˙2t=x2t5+sin⁡5t-0.3x2t-0.5+0.3sin⁡3tx2t4+0.6+0.1cos⁡11tx1t+0.1x2t.In this case, according to Theorem 4, we can get M1=0.14,M2=20, and(82)r2lr1u-a2uα1u+β1uM1+γ1uM2b1l≈4.6513>0,(83)r2lr1u-b2uα2la1l=5.6743>0,Equations (82)-(83) show that H3 holds and according to Theorem 9, for system (81), species x1 is driven to extinction while species x2 is asymptotic to any positive solution of(84)x˙t=xt5+sin⁡5t-0.3xt.Figure 3 also supports this conclusion.Figure 3
Numeric simulations of system (81) with initial values (x(0),y(0))=(3,14)T, (8,5)T, (9,2)T, and (5,6)T, 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. Whenβi(t)=0 and αi(t)=γi(t)=1(i=1,2), (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 αi(t)=1 and γi(t)=0(i=1,2), 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.



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*Source: 4592054-2019-02-05.xml*