# Dynamics in a Lotka-Volterra Predator-Prey Model with Time-Varying Delays

Authors: Changjin Xu; Yusen Wu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A Lotka-Volterra predator-prey model with time-varying delays is investigated. By using the differential inequality theory, some sufficient conditions which ensure the permanence and global asymptotic stability of the system are established. The paper ends with some interesting numerical simulations that illustrate our analytical predictions.


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## Body

## 1. Introduction

In 1992, Berryman [1] pointed out that the dynamical relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Dynamical behavior of predator-prey models has been studied by a lot of papers. It is well known that the investigation on predator-prey models not only focuses on the discussion of stability, periodic oscillatory, bifurcation, and chaos [2–26], but also involves many other dynamical behaviors such as permanence. In many applications, the nature of permanence is of great interest. Recently, Chen [27] investigated the permanence of a discrete n-species food-chain system with delays. Fan and Li [28] gave a theoretical study on permanence of a delayed ratio-dependent predator-prey model with Holling type functional response. Chen [29] focused on the permanence and global attractivity of Lotka-Volterra competition system with feedback control. Zhao and Jiang [30] analyzed the permanence and extinction for nonautonomous Lotka-Volterra system. Teng et al. [31] addressed the permanence criteria for delayed discrete nonautonomous-species Kolmogorov systems. For more research on the permanence behavior of predator-prey models, one can see [32–40].In 2010, Lv et al. [41] investigated the existence and global attractivity of periodic solution to the following Lotka-Volterra predator-prey system:
(1)dx1(t)dt=x1(t)[r1(t)-a11(t)x1(t-τ11(t))-a12(t)x2(t-τ12(t))-a13(t)x3(t-τ13(t))],dx2(t)dt=x2(t)[-r2(t)+a21(t)x1(t-τ21(t))-a22(t)x2(t-τ22(t))-a23(t)x3(t-τ23(t))],dx3(t)dt=x3(t)[-r3(t)+a31(t)x1(t-τ31(t))-a32(t)x2(t-τ32(t))-a33(t)x3(t-τ33(t))],
where x1(t) denotes the density of prey species at time t, x2(t) and x3(t) stand for the density of predator species at time t, ri,aij∈C(ℝ,[0,∞)) and τij∈C(ℝ,ℝ). Using Krasnoselskii’s fixed point theorem and constructing Lyapunov function, Lv et al. obtained a set of easily verifiable sufficient conditions which guarantee the permanence and global attractivity of system (1).For the viewpoint of biology, we shall consider (1) together with the initial conditions xi(0)≥0(i=1,2,3). The principle object of this paper is to explore the dynamics of system (1), applying the differential inequality theory to study the permanence of system (1). Using the method of Lyapunov function, we investigated the globally asymptotically stability of system (1).The remainder of the paper is organized as follows: in Section2, basic definitions and Lemmas are given, and some sufficient conditions for the permanence of the Lotka-Volterra predator-prey model in consideration are established. A series of sufficient conditions for the global stability of the Lotka-Volterra predator-prey model in consideration are included in Section 3. In Section 4, we give an example which shows the feasibility of the main results. Conclusions are presented in Section 5.


## 2. Permanence

For convenience in the following discussing, we always use the notations:(2)fl=inft∈ℝf(t),fu=supt∈ℝf(t),
where f(t) is a continuous function. In order to obtain the main result of this paper, we shall first state the definition of permanence and several lemmas which will be useful in the proving the main result.Definition 1 (see [41]).
We say that system (1) is permanence if there are positive constants M and m such that for each positive solution (x1(t),x2(t),x3(t)) of system (1) satisfies
(3)m≤limt→+∞infxi(t)≤limt→+∞supxi(t)≤M(i=1,2,3).Lemma 2 (see [42]).
Ifa>0,b>0, and x˙≥x(b-ax), when t≥0 and x(0)>0, we have
(4)limt→+∞infx(t)≥ba.
If a>0,b>0, and x˙≤x(b-ax), when t≥0 and x(0)>0, we have
(5)limt→+∞supx(t)≤ba.Now we state our permanence result for system (1).Theorem 3.
LetM1,M2,M3, and m1 be defined by (11), (18), (24), and (30), respectively. Suppose that the following conditions:(H1)
a22uM1>r2l,a31uM1>r3l,(H2)
r1l>a12uM2+a13uM3,r2u>a23uM3,a31lm1>r3u+a32uM2hold, and then system (1) is permanent; that is, there exist positive constants mi,Mi(i=1,2,3) which are independent of the solution of system (1), such that, for any positive solution (x1(t),x2(t),x3(t)) of system (1) with the initial condition xi(0)≥0(i=1,2,3), one has
(6)mi≤limt→+∞infxi(t)≤limt→+∞supxi(t)≤Mi.Proof.
It is easy to see that system (1) with the initial value condition (x1(0),x2(0),x3(0)) has positive solution (x1(t),x2(t),x3(t)) passing through (x1(0),x2(0),x3(0)). Let (x1(t),x2(t),x3(t)) be any positive solution of system (1) with the initial condition (x1(0),x2(0),x3(0)). It follows from the first equation of system (1) that
(7)dx1(t)dt≤r1(t)x1(t)≤r1ux1(t).
Integrating both sides of (7) from t-τ11(t) to t, we get
(8)ln[x1(t)x1(t-τ11(t))]≤∫t-τ11(t)t‍r1uds≤r1uτ11u,
which leads to
(9)x1(t-τ11(t))≥x1(t)exp{-r1uτ11u}.
Substituting (9) into the first equation of system (1), it follows that
(10)dx1(t)dt≤x1(t)[r1u-a11lexp{-r1uτ11u}x1(t)].
It follows from (10) and Lemma 2 that
(11)limt→+∞supx1(t)≤r1ua11lexp{r1uτ11u}:=M1.
For any positive constant ε>0, it follows from (11) that there exists a T1>0 such that, for all t>T1,
(12)x1(t)≤M1+ε.
For t≥T1+τ21u, from (12) and the second equation of system (1), we have
(13)dx2(t)dt≤x1(t)[-r2(t)+a21(t)x1(t-τ21(t))]≤x1(t)[-r2l+a21u(M1+ε)].
Integrating both sides of (13) from t-τ22(t) to t, we get
(14)ln[x2(t)x2(t-τ22(t))]≤∫t-τ22(t)t‍[-r2l+a21u(M1+ε)]ds≤[-r2l+a21u(M1+ε)]τ22u,
which leads to
(15)x2(t-τ22(t))≥x2(t)exp{[r2l-a21u(M1+ε)]τ22u}.
Substituting (15) into the second equation of system (1), it follows that
(16)dx2(t)dt≤x2(t){-r2l+a22u(M1+ε)-a22lexp{[r2l-a21u(M1+ε)]τ22u}x2(t)}.
Thus, as a direct corollary of Lemma 2, according to (16), one has
(17)limt→+∞supx2(t)≤-r2l+a22u(M1+ε)a22lexp{[r2l-a21u(M1+ε)]τ22u}.
Setting ε→0, it follows that
(18)limt→+∞supx2(t)≤-r2l+a22uM1a22lexp{(r2l-a21uM1)τ22u}:=M2.
For t≥T1+τ31u, from (12) and the third equation of system (1), we have
(19)dx3(t)dt≤x3(t)[-r3(t)+a31(t)x1(t-τ31(t))]≤x3(t)[-r3l+a31u(M1+ε)].
Integrating both sides of (19) from t-τ33(t) to t, we get
(20)ln[x3(t)x3(t-τ33(t))]≤∫t-τ33(t)t‍[-r3l+a31u(M1+ε)]ds≤[-r3l+a31u(M1+ε)]τ33u,
which leads to
(21)x3(t-τ33(t))≥x3(t)exp{[r3l-a31u(M1+ε)]τ33u}.
Substituting (21) into the third equation of system (1), it follows that
(22)dx3(t)dt≤x3(t){-r3l+a31u(M1+ε)-a33lexp{[r3l-a31u(M1+ε)]τ33u}x3(t)}.
Thus, as a direct corollary of Lemma 2, according to (22), one has
(23)limt→+∞supx3(t)≤-r3l+a31u(M1+ε)a33lexp{[r3l-a31u(M1+ε)]τ33u}.
Setting ε→0, it follows that
(24)limt→+∞supx3(t)≤-r3l+a31uM1a33lexp{(r3l-a31uM1)τ33u}:=M3.
For t≥T1+max{τ21u,τ31u,τ11u,τ12u,τ13u}, it follows from the first equation of system (1) that
(25)dx1(t)dt≥x1(t)[r1l-a11u(M1+ε)-a12u(M2+ε)-a13u(M3+ε)r1l].
Integrating both sides of (25) from t-τ11(t) to t, one has
(26)ln[x1(t)x1(t-τ11(t))]≥∫t-τ11(t)t‍[r1l-a11u(M1+ε)-a12u(M2+ε)-a13u(M3+ε)]ds≥[r1l-a11u(M1+ε)-a12u(M2+ε)-a13u(M3+ε)]τ22u,
which leads to
(27)x1(t-τ11(t))≤x1(t)exp{τ22u-[r1l-a11u(M1+ε)-a12u(M2+ε)-a13u(M3+ε)r1l]τ22u}.
Substituting (27) into the first equation of system (1), it follows that
(28)dx1(t)dt≥x1(t){r1l-a12u(M2+ε)-a13u(M3+ε)-a11uexp{-[r1l-a11u(M1+ε)-a12u(M2+ε)-a13u(M3+ε)r1l]×τ22u[r1l-a11u(M1+ε)}x1(t)r1l}.
According to Lemma 2, it follows from (28) that
(29)limt→+∞infx1(t)≥(r1l-a12u(M2+ε)-a13u(M3+ε))×(a11uexp{-a12u(M2+ε)-a13u(M3+ε)]τ22u-a12u(M2+ε)-a13u(M3+ε)r1l]τ22u[-r1l-a11u(M1+ε)-a12u(M2+ε)-a13u(M3+ε)r1l]τ22u})-1.
Setting ε→0 in (29), we can get
(30)limt→+∞infx1(t)≥r1l-a12uM2-a13uM3a11uexp{-(r1l-a11uM1-a12uM2-a13uM3)τ22u}:=m1.
For t≥T1+max{τ21u,τ22u,τ23u,τ31u,τ11u,τ12u,τ13u}, from the second equation of system (1), we have
(31)dx2(t)dt≥x2(t)[-r2u+a21l(m1-ε)-a22u(M2+ε)-a23u(M3+ε)].
Integrating both sides of (31) from t-τ22(t) to t leads to
(32)ln[x2(t)x2(t-τ22(t))]≥∫t-τ22(t)t‍[-r2u+a21l(m1-ε)-a22u(M2+ε)-a23u(M3+ε)a21l]ds≥[-r2u+a21l(m1-ε)-a22u(M2+ε)-a23u(M3+ε)]τ22u,
which leads to
(33)x2(t-τ22(t))≤x2(t)exp{[r2u-a21l(m1-ε)+a22u(M2+ε)+a23u(M3+ε)a21l]τ22u}.
Substituting (33) into the second equation of system (1), it follows that
(34)dx2(t)dt≥x2(t){[r2u-a21l(m1-ε)+a22u(M2+ε)r2u-a22u×exp{[r2u-a21l(m1-ε)+a22u(M2+ε)+a23u(M3+ε)a21l]τ22u[r2u-a21l(m1-ε)+a22u(M2+ε)}x2(t)-a23u(M3+ε)[r2u-a21l(m1-ε)+a22u(M2+ε)}.
By Lemma 2 and (34), we can get
(35)limt→+∞infx2(t)≥(r2u-a23u(M3+ε))×(a22uexp{[r2u-a21l(m1-ε)+a22u(M2+ε)+a23u(M3+ε)a21l]τ22u})-1.
Setting ε→0 in the above inequality, it follows that
(36)limt→+∞infx2(t)≥r2u-a23uM3a22uexp{(r2u-a21lm1+a22uM2+a23uM3)τ22u}:=m2.
For t≥T1+max{τ31u,τ32u,τ33u,τ21u,τ22u,τ23u,τ31u,τ11u,τ12u,τ13u}, it follows from the third equation of system (1) that
(37)dx3(t)dt=x3(t)[-r3(t)+a31(t)x1(t-τ31(t))-a32(t)x2(t-τ32(t))-a33(t)x3(t-τ33(t))],≥x3(t)[-r3u+a31l(m1-ε)-a32u(M2+ε)-a33u(M3+ε)a31l].
Integrating both sides of (37) from t-τ33(t) to t, we get
(38)ln[x3(t)x3(t-τ33(t))]≥∫t-τ33(t)t‍[-r3u+a31l(m1-ε)-a32u(M2+ε)-a33u(M3+ε)a31l]ds≥[-r3u+a31l(m1-ε)-a32u(M2+ε)-a33u(M3+ε)]τ33u.
Hence
(39)x3(t-τ33(t))≤x3(t)exp{[r3u-a31l(m1-ε)+a32u(M2+ε)+a33u(M3+ε)r3ua31l]τ33u}.
Substituting (39) into the third equation of system (1), we derive
(40)dx3(t)dt≥x3(t){-r3u+a31l(m1-ε)-a32(M2+ε)-a33exp{[r3u-a31l(m1-ε)+a32u(M2+ε)+a33u(M3+ε)r3ua31l]τ33u}x3(t)}.
In view of Lemma 2 and (40), one has
(41)limt→+∞infx3(t)≥(-r3u+a31l(m1-ε)-a32u(M2+ε))×(a33exp{[r3u-a31l(m1-ε)+a32u(M2+ε)+a33u(M3+ε)a31l]τ33u})-1.
Setting ε→0 in (41) leads to
(42)limt→+∞infx3(t)≥-r3u+a31lm1-a32uM2a33uexp{(r3u-a31lm1+a32uM2+a33uM3)τ33u}:=m3.
Equations (11), (18), (24), (30), (36), and (42) show that system (1) is permanent. The proof of Theorem 3 is complete.


## 3. Global Asymptotically Stability of Positive Solutions

In this section, we formulate the global asymptotically stability of positive solutions of system (1).Definition 4.
A bounded positive solution(x1*(t),x2*(t),x3*(t))T of system (1) is said to be globally asymptotically stable if, for any other positive bounded solution (x1(t),x2(t),x3(t))T of system (1), the following equality holds:
(43)limt→+∞[∑i=13‍|xi(t)-xi*(t)|]=0.Definition 5 (see [24]).
Leth~ be a real number and f be a nonnegative function defined on [h~,+∞) such that f is integrable on [h~,+∞) and is uniformly continuous on [h~,+∞), then limt→+∞f(t)=0.Theorem 6.
In addition to (H1)-(H2), assume further that(H3)
limt→∞infAi(t)>0,where Ai(i=1,2,3) are defined by (48), (49), and (50), respectively. Then system (1) has a unique positive solution (x1*(t),x2*(t),x3*(t))T which is global attractivity.Proof.
According to the conclusion of Theorem3, there exists  T>0 and positive constants mi,Mi(i=1,2,3) such that
(44)mi<xi*(t)≤Mii=1,2,3,t>T.
Define(45)V(t)=∑i=13‍|lnxi*(t)-lnxi(t)|.
Calculating the upper-right derivative of V(t) along the solution of (1), it follows for t≥T that
(46)D+V(t)=∑i=13‍(xi*′(t)xi*(t)-xi′(t)xi(t))sgn(xi*(t)-xi(t))=sgn(x1*(t)-x1(t))×∑i=13‍-a1i(t)[xi*(t-τ1i(t))-xi(t-τ1i(t))]+sgn(x2*(t)-x2(t))×∑i=13‍-a2i(t)[xi*(t-τ2i(t))-xi(t-τ2i(t))]+sgn(x3*(t)-x3(t))×∑i=13‍-a3i(t)[xi*(t-τ3i(t))-xi(t-τ3i(t))]≤-a11(t)|x1*(t-τ11(t))-x1(t-τ11(t))|+a12(t)|x2*(t-τ12(t))-x2(t-τ12(t))|+a13(t)|x3*(t-τ13(t))-x3(t-τ13(t))|+a21(t)|x1*(t-τ21(t))-x1(t-τ21(t))|-a22(t)|x2*(t-τ22(t))-x2(t-τ22(t))|+a23(t)|x3*(t-τ23(t))-x3(t-τ23(t))|+a31(t)|x1*(t-τ31(t))-x1(t-τ31(t))|+a32(t)|x2*(t-τ32(t))-x2(t-τ32(t))|-a33(t)|x3*(t-τ33(t))-x3(t-τ33(t))|.
It follows that
(47)D+V(t)≤-a11(t)|x1*(t-τ11(t))-x1(t-τ11(t))|+a12(t)|x2*(t-τ12(t))-x2(t-τ12(t))|+a13(t)|x3*(t-τ13(t))-x3(t-τ13(t))|+a21(t)|x1*(t-τ21(t))-x1(t-τ21(t))|-a22(t)|x2*(t-τ22(t))-x2(t-τ22(t))|+a23(t)|x3*(t-τ23(t))-x3(t-τ23(t))|+a31(t)|x1*(t-τ31(t))-x1(t-τ31(t))|+a32(t)|x2*(t-τ32(t))-x2(t-τ32(t))|-a33(t)|x3*(t-τ33(t))-x3(t-τ33(t))|≤-a11(t){{-ruτ11u}exp{[r1l-a11u(M1+ε)-a12u(M2+ε)-a13u(M3+ε)r1l]τ22u[r1l-a11u(M1+ε)-a12u(M2+ε)}+exp{-ruτ11u}{[r1l-a11u(M1+ε)-a12u(M2+ε)}×|x1*(t)-x1(t)|+2a12(t)exp{[-r2u+a21l(m1-ε)-a22u(M2+ε)-a23u(M3+ε)a21l]τ22u}×|x2*(t)-x2(t)|+2a13(t)exp{[-r3u+a31l(m1-ε)-a32u(M2+ε)-a33u(M3+ε)r3ua31l]τ33u}×|x3*(t)-x3(t)|+2a21(t)exp{-[r1l-a11u(M1+ε)-a12u(M2+ε)-a13u(M3+ε)r1l]τ22u[r1l-a11u(M1+ε)}×|x1*(t)-x1(t)|-a22(t){exp{[{[r2l-a21u(M1+ε)]τ22u}-r2u+a21l(m1-ε)-a22u(M2+ε)-a23u(M3+ε)r3ua31l]τ22ua21l}+exp{[r2l-a21u(M1+ε)]τ22u}}×|x2*(t)-x2(t)|+2a23(t)exp{[r3u-a31l(m1-ε)+a32u(M2+ε)+a33u(M3+ε)r3ua31l]τ33u}×|x3*(t)-x3(t)|+2a31(t)exp{-[r1l-a11u(M1+ε)-a12u(M2+ε)-a13u(M3+ε)r1l]τ22u[r1l-a11u(M1+ε)}×|x1*(t)-x1(t)|+2a32(t)exp{[-r2u+a21l(m1-ε)-a22u(M2+ε)-a23u(M3+ε)r3ua31l]τ22u}×|x2*(t)-x2(t)|-a33(t){exp{[-a32u(M2+ε)-a33u(M3+ε)τ33u-r3u+a31l(m1-ε)-a32u(M2+ε)-a33u(M3+ε)r3ua31l]τ33u}+exp{[r3l-a31u(M1+ε)]τ33u}-a32u(M2+ε)-a33u(M3+ε)]τ33u}×|x3*(t)-x3(t)|≤[-A1(t)|x1*(t)-x1(t)|+A2(t)|x2*(t)-x2(t)|+A3|x3*(t)-x3(t)|A1(t)|x1*(t)-x1(t)|],
where ε is defined by Theorem 3 and
(48)A1(t)=a11(t){{-ruτ11u}exp{[r1l-a11u(M1+ε)-a12u(M2+ε)-a13u(M3+ε)r1l]τ22u}+exp{-ruτ11u}}-2a21(t)exp{-a13u(M3+ε)r1l]τ22u-[r1l-a11u(M1+ε)-a12u(M2+ε)-a13u(M3+ε)r1l]τ22u}-2a31(t)exp{-a13u(M3+ε)r1l]τ22uτ22u-[r1l-a11u(M1+ε)-a12u(M2+ε)-a13u(M3+ε)r1l]τ22u[r1l-a11u(M1+ε)-a12u(M2+ε)},(49)A2(t)=a22(t){{[r2l-a21u(M1+ε)]τ22u}exp{[-r2u+a21l(m1-ε)-a22u(M2+ε)-a23u(M3+ε)r3ua31l]τ22u}+exp{[r2l-a21u(M1+ε)]τ22u}}-2a12(t)exp{[-r2u+a21l(m1-ε)-a22u(M2+ε)-a23u(M3+ε)a21l]τ22u}-2a32(t)exp{[-r2u+a21l(m1-ε)-a22u(M2+ε)-a23u(M3+ε)a21l]τ22u},(50)A3(t)=a33(t){exp{[-r3u+a31l(m1-ε)+exp{[r3l-a31u(M1+ε)]τ33u}-a32u(M2+ε)-a33u(M3+ε)r3ua31l]τ33u}+exp{[r3l-a31u(M1+ε)]τ33u}-2a13(t)exp{[-r3u+a31l(m1-ε)-a32u(M2+ε)-a33u(M3+ε)r3ua31l]τ33u}-2a23(t)exp{[r3u-a31l(m1-ε)+a32u(M2+ε)+a33u(M3+ε)r3ua31l]τ33u}.
By hypothesis (H3), there exist constants αi(i=1,2,3) and T*>T such that
(51)Ai(t)≥αi>0,(i=1,2,3)fort≥T*.
Integrating both sides of (51) on interval [T*,t] yields
(52)V(t)+∑i=13‍∫T*t‍Ai(t)|xi*(t)-xi(t)|ds≤V(T*).
It follows from (51) and (52) that
(53)∑i=13‍∫T*t‍Ai(t)|xi*(t)-xi(t)|ds≤V(T*)<∞,fort≥T*.
Since xi*(t)(i=1,2,3) are bounded for t≥T*, so |xi*(t)-xi(t)|(i,j=1,2,3) are uniformly continuous on [T*,∞). By Barbalat’s Lemma [24], we have
(54)limt→∞|xi*(t)-xi(t)|=0,(i=1,2,3).
By Theorems 7.4 and 8.2 in [43], we know that the positive solution (x1*(t),x2*(t),x3*(t))T of (1) is uniformly asymptotically stable. The proof of Theorem 6 is complete.


## 4. Numerical Example

To illustrate the theoretical results, we present some numerical simulations. Let us consider the following discrete system:(55)dx1(t)dt=x1(t)[5-cosπt2-(4+cosπt5)x1(t-(1-sinπt4))-(1+sinπt4)x2(t-(0.5-sinπt4))-(1+cosπt3)x3(t-(0.9-cosπt4))],dx2(t)dt=x2(t)[-(48-cosπt12)+(2-cosπt4)x1(t-(0.7-cosπt5))-(4-cosπt12)x2(t-(1+sinπt4))-(1+sinπt4)x3(t-(0.2-sinπt12))],dx3(t)dt=x3(t)[-(1-cosπt4)+(8+sinπt4)x1(t-(0.8-sinπt5))-(0.6-sinπt8)x2(t-(0.6-cosπt12))-(20+sinπt4)x3(t-(0.5+sinπt2))].
Here
(4)r1(t)=5-cosπt2,r2(t)=48-cosπt12,r3(t)=2-cosπt4,a11(t)=4+cosπt5,a12(t)=1+sinπt4,a13(t)=1+cosπt3,a21(t)=2-cosπt4,a22(t)=4-cosπt12,a23(t)=1+sinπt4,a31(t)=8+sinπt4,a32(t)=0.6-sinπt8,a33(t)=20+sinπt4,τ11(t)=1-sinπt4,τ12(t)=0.5-sinπt4,τ13(t)=0.9-cosπt4,τ21(t)=0.7-cosπt5,τ22(t)=1+sinπt4,τ23(t)=0.2-sinπt12,τ31(t)=0.8-sinπt5,τ32(t)=0.6-cosπt12,τ33(t)=0.5+sinπt2.
All the coefficients ri(t)(i=1,2,3),aij(t)(i,j=1,2,3),τij(t)(i,j=1,2,3) are functions with respect to t, and it is easy to see that
(57)a22u=4912,a31u=334,r2l=4712,r3l=14,a12u=12,a13u=23,r2u=4912,a23u=54,a31l=314,r3u=34,a32u=0.2,r1u=5.5,a11l=3.8,τ11u=1.25,a22l=4712,a21u=2.25,τ22u=0.5,a33l=19.75.
Then M1=1.2451,M2=0.7395,M3=2.1093,m1=0.6422. Thus it is easy to see that all the conditions of Theorem 6 are satisfied. Thus system (55) is permanent which is shown in Figures 1, 2, and 3.Figure 1
The dynamical behavior of the first component of the solution(x1(t),x2(t),x3(t)).Figure 2
The dynamical behavior of the second component of the solution(x1(t),x2(t),x3(t)).Figure 3
The dynamical behavior of the third component  of  the solution(x1(t),x2(t),x3(t)).


## 5. Conclusions

In this paper, we have investigated the dynamical behavior of a Lotka-Volterra predator-prey model with time-varying delays. Sufficient conditions which ensure the permanence of the system are derived. Moreover, we also deal with the global stability of the system. It is shown that delay has influence on the permanence and the global stability of system. Thus delay is an important factor to decide the permanence and global stability of the system. Numerical simulations show the feasibility of our main results.



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*Source: 956703-2013-09-22.xml*