# Dynamics of a Nonautonomous Leslie-Gower Type Food Chain Model with Delays

Authors: Hongying Lu; Weiguo Wang
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

A nonautonomous Leslie-Gower type food chain model with time delays is
investigated. It is proved the general nonautonomous system is permanent and
globally asymptotically stable under some appropriate conditions. Furthermore,
if the system is periodic one, some sufficient conditions are established, which
guarantee the existence, uniqueness, and global asymptotic stability of a positive
periodic solution of the system. The conditions for the permanence, global stability of system, and the existence, uniqueness of positive periodic solution depend
on delays; so, time delays are profitless.


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## Body

## 1. Introduction

Among the relationships between the species living in the same outer environment, the predator-prey theory plays an important and fundamental role. The dynamic relationship between predators and their preys 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 [1]. Food-chain predator-prey system, as one of the most important predator-prey system, has been extensively studied by many scholars, many excellent results concerned with the permanent property and positive periodic solution of the system, see [2–4] and the references cited therein. Recently, Nindjin and Aziz-Alaoui [5] proposed the following autonomous delayed predator-prey model with modified Leslie-Gower functional response (1)ẋ1(t)=x1(t)(a1-a11x1(t-τ11)-a12x2(t)x1(t)+d1),ẋ2(t)=x2(t)(-a2+a21x1(t-τ21)x1(t-τ21)+d1-a22x2(t-τ22)-a23x3(t)x2(t)+d2),ẋ3(t)=x3(t)(a3-a33x3(t-τ33)x2(t-τ32)+d3),
system (1) represents an ecological situation where a prey population x1 is the only food for a predator x2. This specialist predator x2, in turns, serves as the prey of a top-predator x3. The interaction between species x2 and its prey x1 has been modeled by the Volterra scheme. But, the interaction between species x3 and its prey x2 has been modeled by a modified version of Leslie-Gower scheme. About Leslie-Gower scheme one could refer to [6–13] and the references cited therein.In [5], the authors showed that the system is uniformly persistent under some appropriate conditions and obtained sufficient conditions for global stability of the positive equilibrium of system (1).We note that any biological or environment parameters are naturally subject to fluctuation in time, and if a model is to take into account such fluctuation then the model must be nonautonomous. On the other hand, time delays occur so often in nature, a number of models can be formulated as systems of differential equations with time delays (see, e.g., [2, 14, 15] and the references cited therein). Motivated by above considerations, in this paper, we consider the following general nonautonomous Leslie-Gower type food chain model with time delays of the form (1.2)ẋ1(t)=x1(t)(a1(t)-a11(t)x1(t-τ11(t))-a12(t)x2(t)x1(t)+d1(t)),ẋ2(t)=x2(t)(-a2(t)+a21(t)x1(t-τ21(t))x1(t-τ21(t))+d1(t)-a22(t)x2(t-τ22(t))-a23(t)x3(t)x2(t)+d2(t)),ẋ3(t)=x3(t)(a3(t)-a33(t)x3(t-τ33(t))x2(t-τ32(t))+d3(t)),
where τii(t), i=1,2,3 denote the time delays due to negative feedbacks of the prey, specialist predator and top-predator, respectively. τ21(t) is a time delay due to gestation, that is, mature adult predators can only contribute to the reproduction of predator biomass. τ32(t) can be regarded as a gestation period.In this paper, for system (1.2) we always assume that for all i,j=1,2,3(H1)
ai(t),aij(t),di(t) are continuous and bounded above and below by positive constants on [0,+∞), and(H2)
τij(t) are continuous and differentiable bounded functions on [0,+∞), and τ̇ij(t) is uniformly continuous with respect to t on [0,+∞) and inft∈[0,+∞){1-τ̇ij(t)}>0.Letτ=sup{τij(t):t∈[0,+∞),i,j=1,2,3}, then we have 0≤τ<+∞. Let σij(t)=t-τij(t), then the function σij-1(t) is the inverse function of the function σij(t), i,j=1,2,3. Motivated by the application of system (1.2) to population dynamics, we assume that solutions of system (1.2) satisfies the following initial conditions (1.3)xi(θ)=ϕi(θ)≥0,θ∈[-τ,0],ϕi(0)>0,ϕi∈C([-τ,0],R+),i=1,2,3.It is well known that by the fundamental theory of functional differential equation [16], one can prove the solution of system (1.2) with initial conditions (1.3) exists and remains positive for t≥0, we call such a solution the positive solution of system (1.2).The organization of this paper is as follows. In Section2, by using comparison theorem and further developing the analytical technique of [2, 14], we obtain a set of sufficient conditions, which ensure the permanence of the system (1.2). In Section 3, by constructing a suitable Lyapunov function, we establish a set of sufficient conditions, which ensure the global stability of the system (1.2). In Section 4, we will explore the existence and stability of the solutions of the periodic system (1.2). At last, the conclusion ends with brief remarks.


## 2. Permanence

In this section, we establish a permanent result for system (1.2).Here, for any bounded function{f(t)}(2.1)fu=lim⁡t→+∞⁡sup⁡{f(t)},fl=lim⁡t→+∞⁡inf⁡{f(t)}.Definition 2.1.
System (1.2) is said to be permanent, if there are positive constants mi and Mi, such that each positive solution (x1(t),x2(t),x3(t)) of system (1.2) satisfies
(2.2)mi≤lim⁡t→+∞inf⁡xi(t)≤lim⁡t→+∞sup⁡xi(t)≤Mi,i=1,2,3.Theorem 2.2.
Assume that(H1) and (H2) hold assume further that(H3)
a1l>a12uM2/d1l, a21lm1/(m1+d1u)>a2u+a23uM3/d2lhold. Then system (1.2) is permanent.Proof.
From the first equation of the system (1.2) it follows that
(2.3)ẋ1(t)≤x1(t)(a1(t)-a11(t)x1(t-τ11(t)))≤x1(t)(a1u-a11lx1(t-τ11(t))).
Letτ=sup{τij(t):t∈[0,+∞),i,j=1,2,3}, by (H2), then we have 0≤τ<+∞. Taking M̃1=(a1u/a11l)(1+h1), where 0<h1<exp{a1uτ}-1. Firstly, suppose x1(t) is not oscillatory about M̃1. That is, there exists a T1>0, for t>T1 such that(2.4)x1(t)<M̃1,
or
(2.5)x1(t)>M̃1.
If (2.4) holds, then our aim is obtained. Suppose (2.5) holds, then for t≥T1+τ, we obtain(2.6)ẋ1(t)<-h1a1ux1(t),
thus x1(t)<x1(0)exp[-h1a1ut]→0, as t→+∞, which is contradiction with (2.5). Hence there must exist T1*>T1+τ such that x1(t)<M̃1 for t>T1*. Secondly now assume that x1(t) is oscillatory about M̃1 for t≥T1, that is, there exists a time sequence {tn} such that τ<t1<t2<⋯<tn<⋯  is a sequence of zeros of x1(tn)-M̃1 with limn→∞tn=+∞ and x1(tn)=M̃1. Set t̃n a point where x1(t) attends its maximum in (tn,tn+1). Thus we get x1(t̃n)≥x1(tn)=M̃1. Then it follows from (2.3) that
(2.7)0=ẋ1(t)|t=t̃n≤x1(t̃n)(a1u-a11lx1(t̃n-τ11(t̃n))),
which leads to
(2.8)x1(t̃n-τ11(t̃n))≤a1ua11l.
Integrating the both sides of (2.3) from t̃n-τ11(t̃n) to t̃n, it follows that
(2.9)ln⁡x1(t̃n)x1(t̃n-τ11(t̃n))≤∫t̃n-τ11(t̃n)t̃n(a1u-a11lx1(t-τ11(t)))dt≤a1uτ11(t̃n).
From (2.8) and (2.9) we have
(2.10)x1(t̃n)≤a1ua11lexp⁡{a1uτ}.
Sincex1(t̃n) is an arbitrary local maximum of x1(t), we can see that there exists a T2>T1 such that for all t≥T2(2.11)x1(t)≤a1ua11lexp⁡{a1uτ}:=M1.
Thus
(2.12)lim⁡t→+∞sup⁡x1(t)≤M1.
For t≥T2+τ, from (2.11) and the second equation of the system (1.2) it follows that
(2.13)ẋ2(t)≤x2(t)(-a2l+a21uM1M1+d1l-a22lx2(t-τ22(t))).
Similar to the argument above, from (2.13) we obtain
(2.14)lim⁡t→+∞sup⁡x2(t)≤-a2l+a21uM1/(M1+d1l)a22lexp⁡{(-a2l+a21uM1M1+d1l)τ}:=M2.
Similarly, from the third equation of the system (1.2), we have
(2.15)ẋ3(t)≤x3(t)(a3u-a33lx3(t-τ33(t))M2+d3u),
and so
(2.16)lim⁡t→+∞sup⁡x3(t)≤a3u(M2+d3u)a33lexp⁡{a3uτ}:=M3.
Condition (H3) of Theorem 2.2 also implies that we could choose ε>0 small enough such that
(2.17)a1l>a12u(M2+ε)d1l,a21l(m1-ε)(m1-ε)+d1u>a2u+a23u(M3+ε)d2l,
hence, for ε>0 satisfies (2.17), from (2.12), (2.14), and (2.16), we know that there exists T3>T2+τ such that for i=1,2,3 and t≥T3(2.18)xi(t)≤Mi+ε.
From the first equation and (2.18) it follows that for t≥T3+τ,
(2.19)ẋ1(t)=x1(t)(a1(t)-a11(t)x1(t-τ11(t))-a12(t)x2(t)x1(t)+d1(t))≥x1(t)(a1l-a11u(M1+ε)-a12u(M2+ε)d1l).
Note that a1u/a11l≤M1 implies that
(2.20)a1l-a11u(M1+ε)-a12u(M2+ε)d1l≤a1l-a11u(M1+ε)≤a1u-a11l(M1+ε)≤0.
Now we consider the following two cases.Case 1.
Supposea1l-a11u(M1+ε)-a12u(M2+ε)/d1l=0, then for t≥T3+τ, from the positivity of the solution and (2.19) it follows that
(2.21)ẋ1(t)=0.
Then from (2.12) and (2.21) it follows that limt→+∞x1(t)=:p1<a1l/a11u, then there exists T4>T3+τ such that for t≥T4(2.22)x1(t)<p1+a1l/a11u-p12<a1la11u<M1.
From the positivity of the solution, (2.19) and (2.22) it follows that
(2.23)ẋ1(t)≥x1(t)(a1l-a11ua1l/a11u+p12-a12u(M2+ε)d1l)>x1(t)(a1l-a11u(M1+ε)-a12u(M2+ε)d1l)=0,t≥T4+τ,x1(t)≥x1(T4+τ)exp⁡{(a1l-a11ua1l/a11u+p12-a12u(M2+ε)d1l)(t-(T4+τ))},
then we can see that x1(t)→+∞ as t→+∞, which is contradiction with (2.12). Hence we have limt→+∞x1(t)≥a1l/a11u, which implies that there exists T4*>T4+τ such that x1(t)≥a1l/2a11u for t≥T4*.Case 2.
Supposea1l-a11u(M1+ε)-a12u(M2+ε)/d1l<0, from (2.19), for t≥T3+τ, it follows that
(2.24)ẋ1(t)≥x1(t)(a1l-a12u(M2+ε)d1l-a11ux1(t-τ11(t))).
Let(2.25)m̃1=a1l-a12u(M2+ε)/d1la11u(1-σ1),
where 0<σ1<1-exp{(a1l-a11u(M1+ε)-a12u(M2+ε)/d1l)τ}.Firstly, suppose x1(t) is not oscillatory about m̃1. That is, there exists a T5>0, for t>T5 such that
(2.26)x1(t)>m̃1,
or
(2.27)x1(t)<m̃1.
If (2.26) holds, then our aim is obtained. Suppose (2.27) holds, then for t≥T5+τ, we obtain(2.28)ẋ1(t)≥σ1(a1l-a12u(M2+ε)d1l)x1(t).
thus there must exist T5*>T5+τ such that x1(t)>m̃1 for t>T5*, which is a contradiction. Hence, (2.27) could not hold. Secondly now assume that x1(t) is oscillatory about m̃1 for t≥T3+τ, that is, there exists a time sequence {tn} such that τ<t1<t2<⋯<tn<⋯ is a sequence of zeros of x1(tn)-m̃1 with limn→∞tn=+∞ and x1(tn)=m̃1. Set t̂n be a point where x1(t) attends its minimum in (tn,tn+1). Thus, we get x1(t̂n)≤x1(tn)=m̃1. Then it follows from (2.24) that
(2.29)0=ẋ1(t)|t=t̂n≥x1(t̂n)(a1l-a12u(M2+ε)d1l-a11ux1(t̂n-τ11(t̂n))),
which implies that
(2.30)x1(t̂n-τ11(t̂n))≥a1l-a12u(M2+ε)/d1la11u.
Integrating (2.24) on the interval [t̂n-τ11(t̂n),t̂n], we have
(2.31)ln⁡x1(t̂n)x1(t̂n-τ11(t̂n))≥∫t̂n-τ11(t̂n)t̂n(a1l-a12u(M2+ε)d1l-a11ux1(t-τ11(t)))dt≥∫t̂n-τ11(t̂n)t̂n(a1l-a12u(M2+ε)d1l-a11u(M1+ε))dt=(a1l-a11u(M1+ε)-a12u(M2+ε)d1l)τ11(t̂n).
From (2.30) and (2.31), we get that
(2.32)x1(t̂n)≥a1l-a12u(M2+ε)/d1la11uexp⁡{(a1l-a11u(M1+ε)-a12u(M2+ε)d1l)τ}.
Sincex1(t̂n) is an arbitrary local minimum of x1(t), we can have that there exists a q1≤m̃1 such that for all t≥T6*(2.33)x1(t)≥q1,
where
(2.34)0<q1≤a1l-a12uM2/d1la11uexp⁡{(a1l-a11uM1-a12uM2d1l)τ}.
Taken m1=min{a1l/2a11u,q1}, thus, we have
(2.35)lim⁡t→+∞inf⁡x1(t)≥m1.
Thus, for ε>0 satisfies (2.17), it follows from (2.35) that there exists large enough T6 such that for t≥T6(2.36)x1(t)≥m1-ε.
Fort≥T6+τ, by using (2.36), from the second equation of system (1.2) it follows that(2.37)ẋ2(t)=x2(t)(-a2(t)+a21(t)x1(t-τ21(t))x1(t-τ21(t))+d1(t)-a22(t)x2(t-τ22(t))-a23(t)x3(t)x2(t)+d2(t))≥x2(t)(-a2u+a21l(m1-ε)(m1-ε)+d1u-a22u(M2+ε)-a23u(M3+ε)d2l),ẋ2(t)≥x2(t)(-a2u+a21l(m1-ε)(m1-ε)+d1u-a23u(M3+ε)d2l-a22(t)x2(t-τ22(t)));
from (2.37), by a procedure similar to the discussion above, we can verify that
(2.38)lim⁡t→+∞inf⁡x2(t)≥m2,
where
(2.39)m2=min⁡{-a2u+a21lm1/(m1+d1u)2a22u,q2},0<q2≤-a2u+a21lm1/(m1+d1u)-a23uM3/d2la22uexp⁡{(-a2u+a21lm1m1+d1u-a22uM2-a23uM3d2l)τ}.
From (2.38), we see that there exists large enough T7 such that for t≥T7(2.40)x2(t)≥m2-ε.
Substituting (2.18) to the last equation of system (1.2), it follows that
(2.41)ẋ3(t)≥x3(t)(a3l-a33u(M3+ε)d3l),ẋ3(t)≥x3(t)(a3l-a33ux3(t-τ33(t))d3l);
from (2.41), similar to the argument of (2.35), we also have
(2.42)lim⁡t→+∞inf⁡x3(t)≥m3,
where
(2.43)m3=min⁡{a3ld3l2a33u,q3},0<q3≤a3ld3la33uexp⁡{(a3l-a33uM3d3l)τ}.
Consequently, (2.12), (2.14), (2.16), (2.35), (2.38), and (2.42) show that under the assumption (H1)–(H3), for any positive solution (x1(t),x2(t),x3(t)) of system (1.2), one has
(2.44)mi≤lim⁡t→+∞inf⁡xi(t)≤lim⁡t→+∞sup⁡xi(t)≤Mi,i=1,2,3,
where mi and Mi, i=1,2,3 are independent of the solution of system (1.2), thus system (1.2) is permanent. This completes the proof of Theorem 2.2.


## 3. Global Stability

Now we study the global stability of the positive solution of system (1.2). We say a positive solution of system (1.2) is globally asymptotically stable if it attracts all other positive solution of the system.The following lemma is from [17], and will be employed in establishing the global stability of positive solution of system (1.2).Lemma 3.1.
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 3.2.
In addition to(H1)–(H3), assume further that(H4)
there exist constantsλi>0 such that(3.1)lim⁡inf⁡t→+∞⁡{Ai(t)}>0,i=1,2,3,
where
(3.2)A1(t)=λ1(a11(t)-a12(t)M2d12(t)-a11(σ11-1(t))M11-τ̇11(σ11-1(t))∫σ11-1(t)σ11-1(σ11-1(t))a11(s)ds-[a1(t)+a11(t)M1+a12(t)M2d1(t)]∫tσ11-1(t)a11(s)ds)-λ2(a21(σ21-1(t))d1(σ21-1(t))(1-τ̇21(σ21-1(t)))+a21(σ21-1(t))M2d1(σ21-1(t))(1-τ̇21(σ21-1(t)))∫σ21-1(t)σ22-1(σ21-1(t))a22(s)ds),A2(t)=λ2(a22(t)-a23(t)M3d22(t)-a22(σ22-1(t))M21-τ̇22(σ22-1(t))∫σ22-1(t)σ22-1(σ22-1(t))a22(s)ds-[a2(t)+a22(t)M2+a21(t)+a23(t)d2(t)M3]∫tσ22-1(t)a22(s)ds)-λ1a12(t)(1d1(t)+∫tσ11-1(t)a11(s)ds)-λ3(a33(σ32-1(t))M3d32(σ32-1(t))(1-τ̇32(σ32-1(t)))+a33(σ32-1(t))M32d32(σ32-1(t))(1-τ̇32(σ32-1(t)))∫σ32-1(t)σ33-1(σ32-1(t))a33(s)d3(s)ds),A3(t)=λ3(a33(t)M2+d3(t)-[a3(t)+a33(t)M3d3(t)]∫tσ33-1(t)a33(s)d3(s)ds-a33(σ33-1(t))M3d3(σ33-1(t))(1-τ̇33(σ33-1(t)))∫σ33-1(t)σ33-1(σ33-1(t))a33(s)d3(s)ds)-λ2a23(t)(1d2(t)+∫tσ22-1(t)a22(s)ds).
Then for any positive solutions (x1(t),x2(t),x3(t)) and (x1*(t),x2*(t),x3*(t)) of system (1.2), one has
(3.3)lim⁡t→+∞[|xi(t)-xi*(t)|]=0,i=1,2,3.Proof.
For two arbitrary nontrivial solutions(x1(t),x2(t),x3(t)) and (x1*(t),x2*(t),x3*(t)) of system (1.2), we have from Theorem 2.2 that there exist positive constants T>T7 and Mi, mi(i=1,2,3) such that for all t≥T and i=1,2,3(3.4)mi≤xi(t)≤Mi.
We define
(3.5)V11(t)=|ln⁡x1(t)-ln⁡x1*(t)|.
Calculating the upper right derivative of V11(t) along the solution of system (1.2), for t≥T+τ, it follow that
(3.6)D+V11(t)=(ẋ1(t)x1(t)-ẋ1*(t)x1*(t))sgn⁡(x1(t)-x1*(t))=sgn⁡(x1(t)-x1*(t))×{-a11(t)(x1(t-τ11(t))-x1*(t-τ11(t)))-a12(t)x2(t)x1(t)+d1(t)+a12(t)x2*(t)x1*(t)+d1(t)}=sgn⁡(x1(t)-x1*(t))×{-a11(t)(x1(t)-x1*(t))-a12(t)x1(t)+d1(t)(x2(t)-x2*(t))+a12(t)x2*(t)(x1(t)+d1(t))(x1*(t)+d1(t))(x1(t)-x1*(t))+a11(t)∫t-τ11(t)t(ẋ1(u)-ẋ1*(u))du}≤-a11(t)|x1(t)-x1*(t)|+a12(t)d1(t)|x2(t)-x2*(t)|+a12(t)M2d12(t)|x1(t)-x1*(t)|+a11(t)|∫t-τ11(t)t(ẋ1(u)-ẋ1*(u))du|.
On substituting (1.2) into (3.6), we derive that
(3.7)D+V11(t)≤-a11(t)|x1(t)-x1*(t)|+a12(t)d1(t)|x2(t)-x2*(t)|+a12(t)M2d12(t)|x1(t)-x1*(t)|+a11(t)|∫t-τ11(t)t{a12(u)x1(u)x1(u)+d1(u)a1(u)-a11(u)x1*(u-τ11(u))-a12(u)d1(u)x2*(u)(x1(u)+d1(u))(x1*(u)+d1(u))(x1(u)-x1*(u))-a11(u)x1(u)(x1(u-τ11(u))-x1*(u-τ11(u)))-a12(u)x1(u)x1(u)+d1(u)(x2(u)-x2*(u))}du|.
It follows (3.4) and (3.7) that for t>T+τ(3.8)D+V11(t)≤-a11(t)|x1(t)-x1*(t)|+a12(t)d1(t)|x2(t)-x2*(t)|+a12(t)M2d12(t)|x1(t)-x1*(t)|+a11(t)∫t-τ11(t)t{[a1(u)+a11(u)M1+a12(u)M2d1(u)]|x1(u)-x1*(u)|+a11(u)M1|x1(u-τ11(u))-x1*(u-τ11(u))|+a12(u)|x2(u)-x2*(u)|a12(u)M2d1(u)}du.
Letσ11(t)=t-τ11(t), by (H2), we can obtain the inverse function of the function σ11(t) denoted by σ11-1(t).
Define(3.9)V12(t)=∫tσ11-1(t)∫σ11(s)ta11(s)×{[a1(u)+a11(u)M1+a12(u)M2d1(u)]|x1(u)-x1*(u)|+a11(u)M1|x1(u-τ11(u))-x1*(u-τ11(u))|+a12(u)|x2(u)-x2*(u)|a12(u)M2d1(u)}duds.
We obtain from (3.8) and (3.9) for t>T+τ(3.10)D+V11(t)+V̇12(t)≤-(a11(t)-a12(t)M2d12(t))|x1(t)-x1*(t)|+a12(t)d1(t)|x2(t)-x2*(t)|+∫tσ11-1(t)a11(s)ds×{[a1(t)+a11(t)M1+a12(t)M2d1(t)]|x1(t)-x1*(t)|+a11(t)M1|x1(t-τ11(t))-x1*(t-τ11(t))|+a12(t)|x2(t)-x2*(t)|a12(t)M2d1(t)}.
We now define
(3.11)V1(t)=V11(t)+V12(t)+V13(t),
where
(3.12)V13(t)=M1∫t-τ11(t)t∫σ11-1(l)σ11-1(σ11-1(l))a11(s)a11(σ11-1(l))1-τ̇11(σ11-1(l))|x1(l)-x1*(l)|dsdl.
It then follows from (3.10) that for t>T+τ(3.13)D+V1(t)≤-(a11(t)-a12(t)M2d12(t))|x1(t)-x1*(t)|+a12(t)×[1d1(t)+∫tσ11-1(t)a11(s)ds]|x2(t)-x2*(t)|+[a1(t)+a11(t)M1+a12(t)M2d1(t)]∫tσ11-1(t)a11(s)ds|x1(t)-x1*(t)|+a11(σ11-1(t))M11-τ̇11(σ11-1(t))∫σ11-1(t)σ11-1(σ11-1(t))a11(s)ds|x1(t)-x1*(t)|.
Similarly, we define
(3.14)V2(t)=V21(t)+V22(t)+V23(t),
where
(3.15)V21(t)=|ln⁡x2(t)-ln⁡x2*(t)|,V22(t)=∫t-τ21(t)ta21(σ21-1(s))d1(σ21-1(s))(1-τ̇21(σ21-1(s)))|x1(s)-x1*(s)|ds+∫tσ22-1(t)∫σ22(s)ta22(s)×{[a2(u)+a22(u)M2+a21(u)+a23(u)M3d2(u)]|x2(u)-x2*(u)|+a23(u)|x3(u)-x3*(u)|+a22(u)M2|x2(u-τ22(u))-x2*(u-τ22(u))|+a21(u)M2d1(u)|x1(u-τ21(u))-x1*(u-τ21(u))|}duds,V23(t)=M2∫t-τ22(t)t∫σ22-1(l)σ22-1(σ22-1(l))a22(s)a22(σ22-1(l))1-τ̇22(σ22-1(l))|x2(l)-x2*(l)|dsdl+M2∫t-τ21(t)t∫σ21-1(l)σ22-1(σ21-1(l))a22(s)a21(σ21-1(l))d1(σ21-1(l))(1-τ̇21(σ21-1(l)))|x1(l)-x1*(l)|dsdl.
Calculating the upper right derivative of V2(t) along solutions of (1.2), we derive that  for t>T+τ(3.16)D+V2(t)≤-(a22(t)-a23(t)M3d22(t))|x2(t)-x2*(t)|+a23(t)[1d2(t)+∫tσ22-1(t)a22(s)ds]|x3(t)-x3*(t)|+a21(σ21-1(t))d1(σ21-1(t))(1-τ̇21(σ21-1(t)))|x1(t)-x1*(t)|+[a2(t)+a22(t)M2+a21(t)+a23(t)M3d2(t)]×∫tσ22-1(t)a22(s)ds|x2(t)-x2*(t)|+a22(σ22-1(t))M21-τ̇22(σ22-1(t))∫σ22-1(t)σ22-1(σ22-1(t))a22(s)ds|x2(t)-x2*(t)|+a21(σ21-1(t))M2d1(σ21-1(t))(1-τ̇21(σ21-1(t)))∫σ21-1(t)σ22-1(σ21-1(t))a22(s)ds|x1(t)-x1*(t)|.
Similarly, we define
(3.17)V3(t)=V31(t)+V32(t)+V33(t),
where
(3.18)V31(t)=|ln⁡x3(t)-ln⁡x3*(t)|,V32(t)=M3∫t-τ32(t)ta33(σ32-1(s))d32(σ32-1(s))(1-τ̇32(σ32-1(s)))|x2(s)-x2*(s)|ds+∫tσ33-1(t)∫σ33(s)ta33(s)d3(s)×{[a3(u)+a33(u)M3d3(u)]|x3(u)-x3*(u)|+a33(u)M3d3(u)|x3(u-τ33(u))-x3*(u-τ33(u))|+a33(u)M32d32(u)|x2(u-τ32(u))-x2*(u-τ32(u))|}duds,V33(t)=M3∫t-τ33(t)t∫σ33-1(l)σ33-1(σ33-1(l))a33(s)a33(σ33-1(l))d3(s)d3(σ33-1(l))(1-τ̇33(σ33-1(l)))|x3(l)-x3*(l)|dsdl+M32∫t-τ32(t)t∫σ32-1(l)σ33-1(σ32-1(l))a33(s)a33(σ32-1(l))d3(s)d32(σ32-1(l))(1-τ̇32(σ32-1(l)))|x2(l)-x2*(l)|dsdl.
Calculating the upper right derivative of V3(t) along solutions of (1.2), we derive that  for t>T+τ(3.19)D+V3(t)≤-a33(t)M2+d3(t)|x3(t)-x3*(t)|+M3a33(σ32-1(t))d32(σ32-1(t))(1-τ̇32(σ32-1(t)))|x2(t)-x2*(t)|+[a3(t)+a33(t)M3d3(t)]∫tσ33-1(t)a33(s)d3(s)ds|x3(t)-x3*(t)|+a33(σ33-1(t))M3d3(σ33-1(t))(1-τ̇33(σ33-1(t)))∫σ33-1(t)σ33-1(σ33-1(t))a33(s)d3(s)ds|x3(t)-x3*(t)|+a33(σ32-1(t))M32d32(σ32-1(t))(1-τ̇32(σ32-1(t)))∫σ32-1(t)σ33-1(σ32-1(t))a33(s)d3(s)ds|x2(t)-x2*(t)|.
We now define a Lyapunov functional as
(3.20)V(t)=∑i=13λiVi(t).
It then follows from (3.13), (3.16), (3.19), and (3.20) that for t>T+τ(3.21)D+V(t)≤-∑i=13Ai(t)|xi(t)-xi*(t)|.
By the hypothes is (H4), there exist enough small positive constants αi, i=1,2,3 and a large enough constant T*≥T+τ, such that for all i=1,2,3 and t≥T*(3.22)Ai(t)≥αi>0.
Integrating both sides of (3.21) on interval [T*,t](3.23)V(t)+∑i=13∫T*tAi(s)|xi(s)-xi*(s)|ds≤V(T*).
It follows from (3.22) and (3.23) that
(3.24)V(t)+∑i=13αi∫T*t|xi(s)-xi*(s)|ds≤V(T*).
Therefore, V(t) is bounded on [T*,+∞) and also
(3.25)∫T*t|xi(s)-xi*(s)|ds<∞,i=1,2,3.
By Theorem 2.2, we know that |xi(t)-xi*(t)|, i=1,2,3 are bounded on [T*,+∞). On the other hand, it is easy to see that ẋi(t), i=1,2,3 are bounded for t≥T*. Therefore, |xi(t)-xi*(t)|, i=1,2,3 are uniformly continuous on [T*,+∞). By Lemma 3.1, one can conclude that
(3.26)lim⁡t→+∞|xi(t)-xi*(t)|=0,i=1,2,3.
This completes the proof of Theorem 3.2.Remark 3.3.
In the proof of the stability of system (1.2), we construct the Lyapunov functional, which need the condition (H2). Similar method and conditon can be found in the [2, 14].


## 4. Existence and Stability of the Positive Periodic Solutions

In this section, we suppose that all the coefficients in system (1.2) are continuous and positive ω-periodic functions, then the system (1.2) is an ω-periodic system for this case.We let the following denote the unique solution of periodic system (1.2) for initial value Z0={x10,x20,x30}: (4.1)Z(t,Z0)={x1(t,Z0),x2(t,Z0),x3(t,Z0)},fort>0,Z(0,Z0)=Z0.Now define Poincâre transformationA:R+3→R+3 is (4.2)A(Z0)=Z(ω,Z0).
In this way, the existence of periodic solution of system (1.2) will be equal to the existence of the fixed point A.Theorem 4.1.
Assume that the conditions of(H1)–(H3) hold, then system (1.2) with initial condition (1.3) has at least one positive ω-periodic solution.Proof.
If assumption(H1)–(H3) are satisfied, then from Theorem 2.2 we have that there exist positive constants T>T7 and Mi,mi(i=1,2,3) such that for all t≥T and i=1,2,3(4.3)mi≤xi(t)≤Mi.
Let
(4.4)K={x1(t),x2(t),x3(t)∣mi≤xi(t)≤Mi,i=1,2,3},
then the compact region K⊂R+3 is a positive invariant set of system (1.2), and K is also a close bounded convex set. So we have
(4.5)Z0∈K⟹Z(t,Z0)∈K,
also Z(ω,Z0)∈K, thus AK⊂K. The operator A is continuous because the solution is continuous about the initial value. Using the fixed point theorem of Brower, we can obtain that A has at least one fixed point in K, then there exists at least one strictly positive ω-periodic solution of system (1.2). This ends the proof of Theorem 4.1.By constructing similar Lyapunov functional to those of Theorem3.2, and using Theorem 4.1, we have the following theorem.Theorem 4.2.
Assume that the conditions of(H1)–(H4) hold, then system (1.2) has a unique positive ω-periodic solution which is globally asymptotically stable.


## 5. Concluding Remarks

In this paper, a nonautonomous Leslie-Gower type food chain model with time delays is investigated, which is based on the Holling type II and a Leslie-Gower modified functional response. By using comparison theorem, we prove the system is permanent under some appropriate conditions. Further, by constructing the suitable Lyapunov functional, we show that the system is globally asymptotically stable under some appropriate conditions. If the system is periodic one, some sufficient conditions are established, which guarantee the existence, uniqueness and global asymptotic stability of a positive periodic solution of the system. Our results have showed that the permanence, global stability of system and the existence, uniqueness of positive periodic solution depend on delays, and so time delays are profitless.



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*Source: 380279-2011-03-27.xml*