# Dynamics of a Predator-Prey System with Beddington-DeAngelis Functional Response and Delays

Authors: Nai-Wei Liu; Ting-Ting Kong
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We consider a predator-prey system with Beddington-DeAngelis functional
response and delays, in which not only the stage structure on prey but also the
delay due to digestion is considered. First, we give a sufficient and necessary condition
for the existence of a unique positive equilibrium by analyzing the corresponding
locations of a hyperbolic curve and a line. Then, by constructing an appropriate Lyapunov
function, we prove that the positive equilibrium is locally asymptotically stable
under a sufficient condition. Finally, by using comparison theorem and theω-limit set
theory, we study the global asymptotic stability of the boundary equilibrium and the
positive equilibrium, respectively. Also, we obtain a sufficient condition to assure the
global asymptotic stability.


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## Body

## 1. Introduction

The purpose of this paper is to consider the following predator-prey system with Beddington-DeAngelis functional response and delays:(1)xi′(t)=a~x(t)-dixi(t)-a~e-diτ1x(t-τ1),x′(t)=a~e-diτ1x(t-τ1)-bx2(t)-cx(t)y(t)m1+m2x(t)+m3y(t),y′(t)=-dy(t)+fx(t-τ2)y(t-τ2)m1+m2x(t-τ2)+m3y(t-τ2),
where xi(t) denotes the immature prey population density, x(t) denotes the mature prey population density, and y(t) denotes the mature predator population density, and all the parameters τ1, τ2, a~, di, b, c, d, f, m1, m2, and m3 are positive. More precisely, the parameters τ1 and τ2 represent the time that the prey juveniles and the predator juveniles take to be mature, respectively. a~ is the intrinsic growth rate of the prey and di is the death rate of immature preys. The term a~e-diτ1 denotes the surviving rate of the immature prey during the delay period to be mature. The constant b denotes the intensity of intraspecific action on the prey and d is the death rate of the predator. The function cx(t)y(t)/(m1+m2x(t)+m3y(t)) represents the Beddington-DeAngelis functional response and the function fx(t-τ2)y(t-τ2)/(m1+m2x(t-τ2)+m3y(t-τ2)) represents the predator’s growth rate that comes from the Beddington-DeAngelis functional response during time period [t-τ2,t].Actually, the predator-prey system with Beddington-DeAngelis functional response has been extensively studied in the literature. In the original work, Beddington [1] and DeAngelis et al. [2] proposed the predator-prey system as
(2)x′(t)=ax(t-τ)-bx2(t)-cx(t)y(t)m1+m2x(t)+m3y(t),y′(t)=-dy(t)+fx(t)y(t)m1+m2x(t)+m3y(t).
Shaikhet [3] considered the stability of a positive point of equilibrium of one nonlinear system with aftereffect and stochastic perturbations. Indeed, the author in [3] not only obtained sufficient conditions for asymptotic mean square stability of the considered systems, but also showed that the trivial solution is asymptotically mean square stable and the positive equilibrium is stable in probability by constructing appropriate Lyapunov functions. In particular, Shaikhet expounded a general method of construction of Lyapunov functions for stochastic functional differential equations and obtained the stability conditions for functional differential equations in [4]. See also [5–10] for the corresponding results about the predator’s functional response. Particularly, Huo [8] studied the global attractiveness of delay diffusive prey-predator systems with the Michaelis-Menten functional response.In the real world, many species usually have a life history that takes them through two stages, immature stage and mature stage. Recently, the system with the stage structure on prey has been extensively studied. In particular, She and Li [11] analyzed the following Beddington-DeAngelis model with the stage structure on prey:
(3)xi′(t)=a~x(t)-dixi(t)-a~e-diτx(t-τ),x′(t)=a~e-diτx(t-τ)-bx2(t)-cx(t)y(t)m1+m2x(t)+m3y(t),y′(t)=-dy(t)+fx(t)y(t)m1+m2x(t)+m3y(t),
where the stage structure was introduced by a constant time delay. In fact, it is the case of τ2=0 in system (1). See also [12–15] for predator-prey models with Beddington-DeAngelis functional response and stage structure.More recently, Xiang et al. [16] established a virus dynamics model with Beddington-DeAngelis functional response and delays. In [16], they not only incorporated the delay τ1, which described the time between viral entry into a target cell and the production of new virus particles, but also incorporated the delay τ2, which described the maturation time of the newly produced viruses. Additionally, it is well known that the growth of the predator is affected by the time delay due to digestion. This motivates us to resolve the issue.Motivated by [16], in the present paper, we study the predator-prey system (1) with Beddington-DeAngelis functional response and delays, in which not only the stage structure on prey but also the delay due to digestion is considered.More precisely, leta=a~e-diτ1 and assume that
(4)xi(0)=a~∫-τ10‍e-disx(s)ds
holds. Then, based on similar conclusions for Lemma 3.1 in [10], we obtain
(5)xi(t)=a~∫-τ10‍e-disx(t+s)ds,
which means that xi(t) depends on x(t) completely. Thus, we can obtain the following system, which is separated from (1):
(6)x′(t)=ax(t-τ1)-bx2(t)-cx(t)y(t)m1+m2x(t)+m3y(t),y′(t)=-dy(t)+fx(t-τ2)y(t-τ2)m1+m2x(t-τ2)+m3y(t-τ2).It is obvious that, for studying the dynamics of the predator-prey system (1) with Beddington-DeAngelis functional response and delays, it is sufficient to study the dynamics of (6).The rest of the organization of this paper is as follows. In Section2, we first establish the local stability of the boundary equilibria of (6) by analyzing the characteristic equations, and then, we study the local asymptotic stability of the positive equilibria of (6) by constructing an appropriate Lyapunov function. In Section 3, we consider the global asymptotic stability of the boundary equilibrium and the positive equilibrium, respectively.


## 2. Local Stability of Equilibria

In this section, we study the local stability of the equilibria of (6). By direct calculations, we have that the system (6) has the origin equilibrium E0(0,0) and the boundary equilibrium E1((a/b),0).In order to investigate the existence of the positive equilibrium, we consider the equations(7)(a-bx)(m1+m2x+m3y)-cy=0-d(m1+m2x+m3y)+fx=0
and obtain the following result.Theorem 1.
The system (6) has a unique positive equilibrium E*(x*,y*) if and only if the condition
(8)(f-dm2)ab>dm1
holds, which is equivalent to the condition
(9)(f-dm2)a~e-diτ1b>dm1.Proof.
We prove the existence of the positive equilibrium by analyzing (7). For the first equation of (7), it is easy to see that its corresponding curves go through the points (a/b,0), (-m1/m2,0), and (0,am1/(c-am3)) when c-am3≠0. Furthermore, in case of ((c-am3)/bm3)≠m1/m2, the first equation is a hyperbolic equation and has two asymptotic lines x+((c-am3)/bm3)=0 and y+(m2/m3)x+((bm1m3-cm2)/bm32)=0. If ((c-am3)/bm3)=m1/m2, the first equation can be simplified as (m1+m2x)(am2-bm2x-bm3y)=0, which are two lines.
For the second equation of (7), it implies a line which goes through the points (0,-m1/m3) and (dm1/(f-dm2),0) when dm1/(f-dm2)≠0. Their corresponding figures are as shown in Figures 1, 2, and 3.
Consequently, by analyzing Figures1, 2, and 3 and discussions, we obtain that (6) has a unique positive equilibrium.Figure 1
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0.Condition (9) indicates that the positive equilibrium E* exists for all τ1 in the interval I=(0,τ*), where
(10)τ*=1diln⁡a~(f-dm2)bdm1.
Furthermore, E* will coincide with E1 when τ1=τ* and there exists no positive equilibrium for τ1>τ*.Now, we study the stability of equilibriaE0(0,0),E1((a/b),0) and E*(x*,y*), respectively.At the beginning, we rewrite system (6) as Z-′(t)=F-(Z-(t),Z-(t-τ1),Z-(t-τ2)), where Z-(t)=(x(t),y(t)), Z-(t-τ1)=(x(t-τ1),y(t-τ1)), Z-(t-τ2)=(x(t-τ2),y(t-τ2)).Then, for an arbitrary fixed pointZ-*=(x,y), by letting P=(∂F-/∂Z-(t))Z-*, Q=(∂F-/∂Z-(t-τ1))Z-*, and R=(∂F-/∂Z-(t-τ2))Z-*, we have
(11)P=(-2bx-cqx′-cqy′0-d)X-*,Q=(a000)X-*,R=(00fqx′fqy′)X-*,
where
(12)q(x,y)=xym1+m2x+m3y,qx′=y(m1+m3y)(m1+m2x+m3y)2,qy′=x(m1+m2x)(m1+m2x+m3y)2.
Therefore, the characteristic equation of the system (6) at the point Z-* is as follows:(13)|P+Qe-λτ1+Re-λτ2-λI|=|-2bx-cqx′+ae-λτ1-λ-cqy′fqx′e-λτ2-d+fqy′e-λτ2-λ|=|-2bx-cy(m1+m3y)(m1+m2x+m3y)2+ae-λτ1-λ-cx(m1+m2x)(m1+m2x+m3y)2fy(m1+m3y)(m1+m2x+m3y)2e-λτ2-d+fx(m1+m2x)(m1+m2x+m3y)2e-λτ2-λ|=0.Thus, according to the characteristic equation at the pointE0, we obtain the following result.Theorem 2.
The equilibriumE0(0,0) is unstable.Proof.
The characteristic equation of the system (6) at the point E0 is
(14)|P+Qe-λτ1+Re-λτ2-λI|(0,0)=|ae-λτ1-λ00-d-λ|=(λ+d)(λ-ae-λτ1)=0.
Obviously,λ=-d is a negative eigenvalue and the other eigenvalue depends on the solution of λ-ae-λτ1=0.
Noticing that the liney=λ and the curve y=ae-λτ1 must intersect at a unique point (λ,y), where λ is a positive value, we can obtain that E0 is a saddle. Thus, E0 is unstable.According to the characteristic equation of the pointE1, we have the following theorem.Theorem 3.
The equilibriumE1((a/b),0) is unstable if (f-dm2)(a/b)>dm1 and is locally asymptotically stable if (f-dm2)(a/b)<dm1.Proof.
The characteristic equation of the system (6) at the point E1 is
(15)|P+Qe-λτ1+Re-λτ2-λI|((a/b),0)=|-2a+ae-λτ1-λ-acbm1+am20-d+afbm1+am2e-λτ2-λ|=(λ+2a-ae-λτ1)(λ+d-afbm1+am2e-λτ2)=0.
For the equationλ+2a-ae-λτ1=0, it is clear that the line y=λ+2a and the curve y=ae-λτ1 must intersect at a unique point (λ,y), where λ is a negative value. And for the equation λ+d-(af/(bm1+am2))e-λτ2=0, noticing that the line y=λ+d and the curve y=(af/(bm1+am2))e-λτ2 must intersect at a unique point (λ,y), we have that λ is positive when (f-dm2)(a/b)>dm1 and is negative when (f-dm2)(a/b)<dm1. Consequently, E1 is unstable if (f-dm2)(a/b)>dm1 and locally asymptotically stable if (f-dm2)(a/b)<dm1.In the rest of this section, by constructing an appropriate Lyapunov function, we study the stability of the positive equilibriumE*(x*,y*) under condition (8). Let
(16)x(t)=x*+u(t),y(t)=y*+v(t);
then, the linearization of the system (6) is
(17)u′(t)=Au(t-τ1)-Bu(t)-Cv(t),v′(t)=-Dv(t)+Ev(t-τ2)+Fu(t-τ2),
where
(18)A=a,B=2bx*+cy*(m1+m3y*)(m1+m2x*+m3y*)2,C=cx*(m1+m2x*)(m1+m2x*+m3y*)2,D=d,E=fx*(m1+m2x*)(m1+m2x*+m3y*)2,F=fy*(m1+m3y*)(m1+m2x*+m3y*)2.
Obviously, all the constants defined above are positive, and the system (17) can be recast as
(19)u′(t)=(A-B)u(t)-Cv(t)-A∫t-τ1t‍u′(s)ds=(A-B)u(t)-Cv(t)-A∫t-τ1t‍[Au(s-τ1)-Bu(s)-Cv(s)]dsv′(t)=(E-D)v(t)-E∫t-τ2t‍v′(s)ds+Fu(t)-F∫t-τ2t‍u′(s)ds=(E-D)v(t)+Fu(t)-F∫t-τ2t‍[Au(s-τ1)-Bu(s)-Cv(s)]ds-E∫t-τ2t[-Dv(s)+Ev(s-τ2)+Fu(s-τ2)]ds.
By constructing an appropriate Lyapunov function, we have the following theorem.Theorem 4.
Letτ1∈(0,τ*) with τ* defined in (10), and assume that
(20)τ1,τ2∈(0,τ*),|F-C|<min⁡{2(D-E),2(B-A)},
also holds, where
(21)τ*=min⁡⁡{2(D-E)-|F-C|AC+F(A+B+E)+2(CF+DE+E2),minm2(B-A)-|F-C|A(2A+2B+C)+F(A+B+E)2(D-E)-|F-C|AC+F(A+B+E)+2(CF+DE+E2)}.
Then, the equilibrium (0,0) of the system (17) is locally asymptotically stable, which implies that the positive equilibrium E*(x*,y*) of the system (6) is locally asymptotically stable.Proof.
To illustrate the equilibrium(0,0) of the system (17) is locally asymptotically stable, it is sufficient to study the existence of a strict Lyapunov function. By letting W11(t)=u2(t) and differentiating W11(t) with respect to t, we have
(22)W11′(t)=2(A-B)u2(t)-2Cu(t)v(t)-2Au(t)∫t-τ1t‍[Au(s-τ1)-Bu(s)-Cv(s)]ds≤2(A-B)u2(t)-2Cu(t)v(t)+A(A+B+C)τ1u2(t)+A∫t-τ1t[Au2(s-τ1)+Bu2(s)+Cv2(s)]ds.
Letting W12(t)=A∫t-τ1t‍∫zt‍[Au2(s-τ1)+Bu2(s)+Cv2(s)]dsdz and differentiating W12(t) with respect to t, we have
(23)W11′(t)+W12′(t)≤2(A-B)u2(t)-2Cu(t)v(t)+A(A+B+C)τ1u2(t)+Aτ1[Au2(t-τ1)+Bu2(t)+Cv2(t)].
Furthermore, by letting W13(t)=A2τ1∫t-τ1t‍u2(s)ds and W1(t)=W11(t)+W12(t)+W13(t), we have
(24)W1′(t)≤[2(A-B)+A(2A+2B+C)τ1]u2(t)+ACτ1v2(t)-2Cu(t)v(t).
Next, by lettingW21(t)=v2(t) and differentiating W21(t) with respect to t, we obtain
(25)W21′(t)=2(E-D)v2(t)+2Fu(t)v(t)-2v(t)×∫t-τ2t‍{F[Au(s-τ1)-Bu(s)-Cv(s)]-2v∫t-τ2t‍+E[-Dv(s)+Ev(s-τ2)+Fu(s-τ2)]}ds≤2(E-D)v2(t)+2Fu(t)v(t)+[F(A+B+C)+E(D+E+F)]τ2v2(t)+∫t-τ2t‍{F[Au2(s-τ1)+Bu2(s)+Cv2(s)]+∫t-τ2t‍+E[Dv2(s)+Ev2(s-τ2)+Fu2(s-τ2)]}ds.
Letting W22(t)=∫t-τ2t‍∫zt‍{F[Au2(s-τ1)+Bu2(s)+Cv2(s)]+E[Dv2(s)+Ev2(s-τ2)+Fu2(s-τ2)]}dsdz and differentiating W22(t) with respect to t, we have
(26)W21′(t)+W′22(t)≤2(E-D)v2(t)+2Fu(t)v(t)+[F(A+B+C)+E(D+E+F)]τ2v2(t)+{F[Au2(t-τ1)+Bu2(t)+Cv2(t)]+E[Dv2(t)+Ev2(t-τ2)+Fu2(t-τ2)]}τ2.
By letting W23(t)=AFτ2∫t-τ1t‍u2(s)ds, W24(t)=E2τ2∫t-τ1t‍v2(s)ds, and W25(t)=EFτ2∫t-τ2t‍u2(s)ds, we have
(27)W23′(t)=AFτ2[u2(t)-u2(t-τ1)],W24′(t)=E2τ2[v2(t)-v2(t-τ2)],W25′(t)=EFτ2[u2(t)-u2(t-τ2)].
By letting W2(t)=W21(t)+W22(t)+W23(t)+W24(t)+W25(t), we have
(28)W2′(t)≤{2(E-D)+τ2[F(A+B+E)+2(CF+DE+E2)]}×v2(t)+F(A+B+E)τ2u2(t)+2Fu(t)v(t).
Finally, we let W(t)=W1(t)+W2(t). Obviously, W(t)≥0 and W(t)=0 if and only if u(t)=v(t)=0. Besides, when u(t)=v(t)=0, W′(t)=0. Furthermore,
(29)W′(t)≤{τ2[F(A+B+E)+2(CF+DE+E2)]2(E-D)+ACτ1+τ2[F(A+B+E)+2(CF+DE+E2)]}×v2(t)+2(F-C)u(t)v(t)+[2(A-B)+A(2A+2B+C)τ1+F(A+B+E)τ2]×u2(t)≤{τ2[F(A+B+E)+2(CF+DE+E2)]2(E-D)+ACτ1+τ2[F(A+B+E)+2(CF+DE+E2)]}×v2(t)+2|F-C||u(t)||v(t)|+[2(A-B)+A(2A+2B+C)τ1+F(A+B+E)τ2]×u2(t)≤{τ2[F(A+B+E)+2(CF+DE+E2)]2(E-D)+|F-C|+ACτ1+τ2[F(A+B+E)+2(CF+DE+E2)]}v2(t)+[2(A-B)+|F-C|+A(2A+2B+C)τ1+F(A+B+E)τ2]u2(t).
Based on the conditions in Theorem 4, we have W′(t)<0 if u2(t)+v2(t)>0. Thus, W(t) is a strict Lyapunov function. According to the Lyapunov stability theorem, the equilibrium (0,0) of the system (17) is locally asymptotically stable, which implies that the positive equilibrium E*(x*,y*) of the system (6) is locally asymptotically stable.


## 3. Global Stability of Equilibria

In this section, we study the global asymptotic stability of the boundary equilibriumE1((a/b),0) and the positive equilibrium E*(x*,y*), respectively. We first give two lemmas which will be used later.Lemma 5 (see [10, Lemma 3.2]).
For a given system(30)x′(t)=b1x(t-τ)-a1x(t)-a2x2(t),
where b1>0, a1≥0, a2>0, τ≥0, and x(t)>0 for all -τ≤t≤0, we have (i)
lim⁡t→+∞x(t)=(b1-a1)/a2 if and only if b1>a1;(ii)
lim⁡t→+∞x(t)=0 if and only if b1≤a1.Lemma 6 (see [11, Lemma 3.2]).
LetS={(x,y):x>0,y>0} and S-={(x,y):x≥0,y≥0}. Then, the sets S and S- are both invariant sets.Based on Lemmas5 and 6, we prove that the boundary equilibrium E1 is globally attractive.Theorem 7.
Let(x(t),y(t)) be the solution of (6) with positive initial condition in S.(i)
If(f-dm2)(a/b)<dm1, then E1((a/b),0) is globally attractive.(ii)
If(f-dm2)(a/b)>dm1, then E1((a/b),0) is unstable.Proof.
If(f-dm2)(a/b)<dm1, we first prove that lim⁡t→+∞y(t)=0.
Due to the first equation (6), there is
(31)x′(t)≤ax(t-τ1)-bx2(t).
In view of Lemma 6 and by considering the following comparison equation
(32)p′(t)=ap(t-τ1)-bp2(t),p(0)=x(0)>0,
we have x(t)≤p(t) for all t≥0. In addition, by Lemma 5, we obtain that lim⁡t→+∞p(t)=(a/b). Thus, there exists a sufficiently small positive constant ε with (f-dm2)((a/b)+ε)<dm1 such that, for this ε, there exists a Tε>0 such that x(t)<(a/b)+ε for all t>Tε. Substituting it into the second equation of (6), we have
(33)y′(t)≤-dy(t)+f((a/b)+ε)m1+m2((a/b)+ε)y(t-τ2),
for all t>Tε. Considering the following comparison equation:
(34)q′(t)=-dq(t)+f((a/b)+ε)m1+m2((a/b)+ε)q(t-τ2),q(Tε+τ2)=y(Tε+τ2)>0,
and letting q(t)=eλt, we have
(35)eλt[λ+d-f((a/b)+ε)m1+m2((a/b)+ε)e-λτ2]=0.
Then the line y=λ+d and the curve y=(f((a/b)+ε)/(m1+m2((a/b)+ε)))e-λτ2 must intersect at a unique point (λ,y). Since (f-dm2)((a/b)+ε)<dm1, we can obtain that d>f((a/b)+ε)/(m1+m2((a/b)+ε)). Thus, we deduce that λ<0. Therefore, lim⁡t→+∞q(t)=lim⁡t→+∞eλt=0.
Obviously, based on the comparison theorem, we get thaty(t)≤q(t) for all t≥T+τ2. Thus, lim⁡t→+∞y(t)=0.
Next, we prove thatx(t)→(a/b) as t→+∞.
Sincelim⁡t→+∞y(t)=0, for any given ε1∈(0,(a/b)), there exists a T1>0 such that 0<y(t)<(bm1/2c)ε1 for all t≥T1. Thus, we have
(36)ax(t-τ1)-bx2(t)≥x′(t)=ax(t-τ1)-bx2(t)-cx(t)y(t)m1+m2x(t)+m3y(t)≥ax(t-τ1)-bx2(t)-cx(t)y(t)m1≥ax(t-τ1)-bx2(t)-bε12x(t),
for all t≥T1. That is,
(37)ax(t-τ1)-bx2(t)≥x′(t)≥ax(t-τ1)-bx2(t)-bε12x(t).
Then, we consider the following comparison equation:
(38)p~′(t)=ap~(t-τ1)-bp~2(t)-bε12p~(t),p~(T1)=x(T1)>0.
Since a>bε1, based on Lemma 5, we have that lim⁡t→+∞p~(t)=(a/b)-(ε1/2). Additionally, we have that p~(t)≤x(t)≤p(t) for all t≥T1.
Sincelim⁡t→+∞p(t)=(a/b), for the above ε1, there exists a T2>0 such that p(t)≤(a/b)+ε1 for all t>T2. Similarly, since lim⁡t→+∞p~(t)=(a/b)-(ε1/2) for the above ε1, we also get that there exists a T3>0 such that p~(t)-(a/b)+(ε1/2)>-(ε1/2) for all t>T3. Therefore, if we let T0=max⁡{T1,T2,T3}, there is -ε1<x(t)-(a/b)<ε1 for all t>T0. Thus, we have lim⁡t→+∞x(t)=(a/b).
At last, we prove the second part of this theorem. We can assure that(f-dm2)(a/b)>dm1 and try to derive a contradiction. If (f-dm2)(a/b)>dm1, then (6) has a unique positive equilibrium (x*,y*) by Theorem 1, which is also a solution of (6); that is, (x*,y*)=(x*(t),y*(t)) satisfies (6) and satisfies lim⁡t→+∞(x*(t),y*(t))=((a/b),0) by the above arguments, which contradicts the fact that (x*,y*) is a positive equilibrium. Thus, conclusion (ii) holds. This completes the proofs.Remark 8.
In the above theorem, we derive the global attractiveness of the boundary equilibriumE1((a/b),0). Unfortunately, we just obtain a sufficient condition that (f-dm2)(a/b)<dm1 to ensure that E1 is globally asymptotically stable. If (f-dm2)(a/b)=dm1, since the second equation of (6) is a delay diffusion equation, we cannot derive that lim⁡t→+∞y(t)=0 by using comparison theorem. We leave it for a further study.Furthermore, for the first equation of (1), we can solve explicitly as
(39)lim⁡t→+∞xi(t)=a2(ediτ1-1)bdi.
Thus, by Theorem 7, we can obtain the following corollary directly.Corollary 9.
Let(xi(t),x(t),y(t)) be the solution of (1) with the positive initial condition. If f≤dm2, then
(40)lim⁡t→+∞(xi(t),x(t),y(t))=(a2bdi(ediτ1-1),ab,0).In the rest of this section, we will study the global attractiveness of the positive equilibriumE* and obtain the following result.Theorem 10.
Assume that (8) and the following conditions
(41)am3>c,f(am3-c)bm1m3+m2(am3-c)>d,
hold, and let (x(t),y(t)) be the solution of (6) with the positive initial condition in S such that lim⁡t→+∞(x(t),y(t))=E*. Then, the positive equilibrium E* of (6) is globally attractive.Proof.
Provided that conditions (8) and (41) hold, based on comparison theorem and by similar arguments in [11], we can obtain that, for any solution (x(t),y(t)) of (6) with the positive initial condition in S, there exists a point E-*(x-*,y-*) such that lim⁡t→+∞(x(t),y(t))=(x-*,y-*). Indeed, just by modifying the coefficients of (6), the discussion in [11] also holds. We omit the proof here and refer to [11] for details.
Then, in view of the property of theω-limit set in [17] and by the uniqueness result of the positive equilibrium in Theorem 1, we have E-*=E*. Thus, we derive that E* is globally attractive.


## 4. Conclusion

In the present paper, we considered the stability of the equilibria of a predator-prey system with Beddington-DeAngelis functional response and delays. More precisely, we derived a sufficient and necessary condition for the existence of a unique positive equilibrium and proved that the positive equilibrium is locally asymptotically stable under a sufficient condition. Also, we established the local stability of the boundary equilibria by analyzing the characteristic equations. Finally, by using comparison theorem and theω-limit set theory, we studied the global asymptotic stability for both the boundary equilibrium and the positive equilibrium.In particular, in Theorem7, we derived the global attractiveness of the boundary equilibrium E1((a/b),0). But unfortunately, we just obtained a sufficient condition that (f-dm2)(a/b)<dm1 to ensure that E1 was globally asymptotically stable. If (f-dm2)(a/b)=dm1, since the second equation of (6) was a delay diffusion equation, we could not derive that lim⁡t→+∞y(t)=0 by using comparison theorem.Moreover, as pointed out in [11], there are two interesting problems about system (1), which are related to the phenomena of bifurcations and the existence of periodic orbits of (1). We leave them for a further study.



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*Source: 930762-2014-05-26.xml*