# Almost Periodic Solution of a Diffusive

Authors: Qiong Liu
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

A delayed predator-prey model with diffusion and competition is
proposed. Some sufficient conditions on uniform persistence of the model have
been obtained. By applying Liapunov-Razumikhin technique, we will point
out, under almost periodic circumstances, a set of sufficient conditions that
assure the existence and uniqueness of the positive almost periodic solution
which is globally asymptotically stable.


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## Body

## 1. Introduction

In the nature world, diffusion often occurs in an
ecological environment; that is, species can diffuse between patches. The works
about autonomous systems in this field were pioneered by Levin, after Levin
[1], Kishimoto
[2], and Takeuchi
[3] studied this kind
of model. But all the coefficients in the system they studied are constants.
Since biological and environmental parameters are naturally subject to
fluctuation in time, the effects of a varying environment are considered as
important selective forces on systems in a fluctuating environment. More
realistic and interesting models should take into account both the seasonality
of the changing
environment and the effects of time delays [4–7]. This motivated
Chen et al.
[8–11], and others to consider
nonautonomous predator-prey models with almost periodic coefficients and
diffusion. In this paper, we study the almost periodic solution of the delayed
predator-prey model with diffusion and competition so as to obtain some
conditions under which three species are uniformly persistent. In addition, we
obtain that for the almost periodic system there exists a unique almost
positive periodic solution which is globally asymptotically stable.The
organization of this paper is as follows. In the next section, we develop our
model, establish its important properties, and give several lemmas, which will
be a key for our proofs and discussions. In Section3, sufficient conditions
are given for uniform persistence of three species. In Section 4, by applying
Liapunov-Razumikhin technique, we prove the existence and uniqueness of the
positive almost periodic solution which is globally asymptotically stable.
Finally, we give a discussion of our results.


## 2. Model and Preliminaries

It is assumed
that the ecosystem is composed of two isolated patches, and the  prey
population can disperse among the patches instantaneously. The state variables
of the models,xi=xi(t)(i=1,2), describe the densities of the prey population
in Patch 1 and Patch 2,
respectively. We suppose that the net exchange of the prey population from
Patch j to Patch i is proportional to the difference of the
concentration between
xj−xi   with Di(t),i,j=1,2.
The state variables of the models, xi=xi(t)(i=3,4), describe the densities of the predator
population in Patch 1 with competition.Let us consider
the following delayed diffusive predator-prey system with competition and
functional response:(2.1)x1′=x1(a10(t)−a11(t)x1)−α1(t)x12x31+β1(t)x12−α2(t)x12x41+β2(t)x12+D1(t)(x2−x1),x2′=x2(a20(t)−a21(t)x2)+D2(t)(x1−x2),x3′=x3(−a30(t)+a31(t)α1(t)x12(t−τ1)1+β1(t)x12(t−τ1)−a32(t)x3−a34(t)x4),x4′=x4(−a40(t)+a41(t)α2(t)x12(t−τ2)1+β2(t)x12(t−τ2)−a42(t)x4−a43(t)x3),with the initial
condition
(2.2)x1(s)=ϕ1(s)∈C([−τ,0],R+),s∈[−τ,0],ϕ1(0)≥0,⁢xi(0)=ϕi≥0⁢(constants),i=2,3,4.
Here, ai0(t) and ai1(t)(i=1,2) represent the intrinsic growth rate and the
intraspecific interference coefficient of the prey population xi⁢(i=1,2),
respectively. We then assume that the death rate of the predator population xi⁢(i=3,4) in Patch 1 is proportional to both the
existing predator population with the proportional functions a30(t) and, respectively, a40(t) and to its square with the proportional
functions a32(t) and, respectively, a42(t).
The predator consumes the prey according to Holling type III functional
response [12, 13], that is, α1(t)x12x3/(1+β1(t)x12) and α2(t)x12x4/(1+β2(t)x12). τi⁢(i=1,2) is the time to digest food in the predator
organism. τ=max⁡{τ1,τ2}. R+≐{z:z≥0}.We introduce some notations
and definitions, and state some preliminary lemmas which will be useful for
establishing our main results.LetR+4={X∈R4:X=(x1,x2,x3,x4),xi>0,i=1,2,3,4}. C=C([−τ,∞)×R+×R+×R+,R+4).
Assume Ω is a subset of R+×C([−τ,0],R+)×R+×R+×R+.
Denote by f=(f1,f2,f3,f4)T:Ω→R+4 the map defined by the right-hand side of
system (2.1). If V:R+×C→R+ is a continuous function, then the upper right
derivative of V(t,x) with respect to system (2.1) is defined
as(2.3)D+V(t,X)=lim⁡h→0+sup⁡1h[V(t+h,X+hf(t,X))−V(t,X)].Obviously, the global existence
and uniqueness of solutions of system (2.1) are guaranteed by the smoothness
properties of f (see [14, 15] for details on fundamental properties of retarded
functional differential equations).For
convenience, we introduce the following notations:(2.4)ψ¯=sup⁡t≥0{ψ(t)},ψ¯=inf⁡t≥0{ψ(t)}.In this paper, we need all the
coefficients to satisfy(2.5)min⁡i=1,2,3,4,j=0,1,2,3,{aij¯,αi¯,βi¯,Di¯}>0,max⁡i=1,2,3,4,j=0,1,2,3.{aij¯,αi¯,βi¯,Di¯}<∞.Definition 2.1.
System (2.1) is said to be uniformly persistent if
there exists a compact region D⊂R+4 such that every solution (x1,x2,x3,x4) of system (2.1) with initial conditions
(2.2)
eventually enters and remains in the region D.For
convenience, the setCIP={u:[0,∞)→[0,∞)∣u(s)is positive and nondecreasing for  s>0,u(0)=0}.Lemma 2.2 (See
[16, 17]).
Consider the following
almost periodic equation:(2.6)x′(t)=g(t,xt).Let CH*={xt∈C:∥xt∥=sup⁡θ∈[−τ,0]|xt(θ)|<H*}, SH*={x∈Rn:|x|<H*}, H*∈R+ or H*=+∞, g:R×CH*→Rn,
and g is uniformly almost periodic with respect to t.
Let V:R+×SH*×SH*→R+.
Assume that the following conditions hold:
(i)
a(∥x−y∥)≤V(t,x,y)≤b(∥x−y∥),a(⋅),b(⋅)∈CIP,b(0)>0;(ii)
|V(t,x1,y1)−V(t,x2,y2)|≤L(∥x1−x2∥+∥y1−y2∥),
where L is a positive constant;(iii)
there exists a continuous nondecreasing functionP(S) such that(2.7)P(S)>S⁢ifS>0,D+(V(t,x1(t),x2(t)))≤−𝒞V(t,x1(t),x2(t)),𝒞∈R+,ifP(V(t,x1(t),x2(t)))≥V(t+θ,x1(t+θ),x2(t+θ)),θ∈[−τ,0].
If system (2.6) has a solution ξ(t):∥ξt∥≤H<H*,⁢t≥t0,
then system (2.6) has a unique positive almost periodic solution η(t) which is uniformly asymptotically stable, and
mod⁡(η)⊂mod(g).
Furthermore, if g is ω-periodic with respect to t,
then system (2.6) has a positive ω-periodic solution which is globally
asymptotically stable.Here,mod(ϕ) denotes the module of ϕ(t) which is the set consisting of all real
numbers which are finite linear combinations of elements of the set(2.8)Λ={α∈R∣lim⁡T→∞1T∫0Tϕ(t)exp⁡(−iαt)dt≠0}with integer
coefficients.Lemma 2.3.
R+4={(x1,x2,x3,x4)∣xi>0,i=1,2,3,4} is a positive invariant set of system (2.1).Proof.
Let(x1,x2,x3,x4) be a solution of system (2.1) with initial
conditions (2.2). Hence, for t∈R and (x1,x2,x3,x4)∈R+4,
we can derive(2.9)x1′|x1=0=D1(t)x2>0forx2>0,x2′|x2=0=D2(t)x1>0forx1>0,x3>x3(0)exp⁡(∫0t(−a30(s)−a32(s)x3(s)−a34(s)x4(s))ds)>0,x4>x4(0)exp⁡(∫0t(−a40(s)−a42(s)x4(s)−a43(s)x3(s))ds)>0.Therefore, we obtain the positive
invariance of R+4.
This completes the proof.We will focus our discussion onR+4 with respect to a biological meaning. This
also ensures the solution with positive initial value to be positive all the
time.


## 3. Uniform Persistence

In what
follows, we want to construct an ultimately bounded region of system (2.1).Theorem 3.1.
There exist three constantsMi>Mi*⁢(i=1,2,3) such that xj(t)≤M1⁢(j=1,2), x3(t)≤M2,
and x4(t)≤M3 for each positive solution (x1(t),x2(t),x3(t),x4(t)) of system (2.1) with t large enough,
where(3.1)M1*=max⁡{a10¯a11¯,a20¯a21¯},M2*=Aa32¯,M3*=Ba42¯,(3.2)A≐a31¯(α1β1)¯−a30¯>0,B≐a41¯(α2β2)¯−a40¯>0.Proof.
Suppose that(x1(t),x2(t),x3(t),x4(t)) is a solution of system (2.1) with initial
conditions (2.2). According to the first two equations of (2.1), we have(3.3)x1′≤a10¯x1−a11¯x12+D1(t)[x2−x1],x2′≤a20¯x2−a21¯x22+D2(t)[x1−x2].We define the following lines in x1-x2  plane: (3.4)LineL1:x1=M1,0≤x2≤M1,LineL2:x2=M1,0≤x1≤M1.Then, we have(3.5)x1′|L1<0,x2′|L2<0.Hence, it follows from(3.6)max⁡{x1(0),x2(0)}≤M1that(3.7)max⁡{x1(t),x2(t)}≤M1fort≥0.If(3.8)max⁡{x1(0),x2(0)}>M1,we only consider what follows.
If xi>M1,i=1,2,
from the given condition we get(3.9)ai0¯xi−ai1¯xi2<M1(ai0¯−ai1¯M1)<0,i=1,2.Let(3.10)−α≐max⁡i=1,2{M1(ai0¯−ai1¯M1)},g(t)=max⁡{x1(t),x2(t)}.Next, we consider the following
three cases.
Case 1.
x1(0)>x2(0), g(0)=x1(0)>M1.
Then, there exists ε>0 such that g(t)=x1(t)>M1 for t∈[0,ε).
We also derive that(3.11)x1′≤a10¯x1−a11¯x12<−α<0.Hence, if t2>t1 and t1,t2∈[0,ε),
we get(3.12)g(t2)−g(t1)<−α(t2−t1).
Case 2.
x2(0)>x1(0),g(0)=x2(0)>M1.
Similarly, we could obtain that there exists [0,ε).
If t2>t1 and t1,t2∈[0,ε),
we get(3.13)g(t2)−g(t1)<−α(t2−t1).
Case 3.
x2(0)=x1(0)=g(0)>M1.
We can also find an interval [0,ε) such that g(t)=x1(t)>M1 or g(t)=x2(t)>M1.
In the same way, if t2>t1 and t1,t2∈[0,ε),
we can obtain(3.14)g(t2)−g(t1)<−α(t2−t1).Now, we can know that if g(0)>M1,
g(t) will monotonously decrease by speed α.
So, there exists T1>0.
If t≥T1,
we have(3.15)g(t)<M1.
According to the third equation of (2.1), we have(3.16)x3′≤x3(−a30¯+a31¯α1(t)x12(t−τ1)1+β1(t)x12(t−τ1)−a32¯x3)<x3(−a30¯+a31¯(α1β1)¯−a32¯x3),x3′|x3=M2<x3(−a30¯+a31¯(α1β1)¯−a32¯x3).Hence, it follows from x3(0)≤M2 that x3(t)≤M2 for t≥0.
If(3.17)x3(0)>M2,we only consider what follows.
If x3>M2,
from the given condition we obtain(3.18)x3(−a30¯+a31¯(α1β1)¯−a32¯x3)<M2[−a30¯+a31¯(α1β1)¯−a32¯M2]<0.Let(3.19)−β=M2[−a30¯+a31¯(α1β1)¯−a32¯M2].We also derive that(3.20)x3′<M2[−a30¯+a31¯(α1β1)¯−a32¯M2]=−β<0.Hence, if t2>t1 and t1,t2∈[0,ε),
we get(3.21)x3(t2)−x3(t1)<−β(t2−t1).Now, we can know that if x3(0)>M2, x3(t) will monotonously decrease by speed β.
So, there exists T2 such that x3(t)<M2 for t≥T2.
Similarly, we also get(3.22)x4′≤x4(−a40¯+a41¯α2(t)x12(t−τ2)1+β2(t)x12(t−τ2)−a42¯x4)<x4(−a40¯+a41¯(α2β2)¯−a42¯x4).We can also choose the same M3.
There exists T3>0 such that x4(t)<M3 for t>T3.
This completes the proof.Theorem 3.2.
Suppose that system (2.1) satisfies the following
conditions:(3.23)a10¯−D1¯>0,a20¯−D2¯>0,E=(a10¯−D1¯)2−4a11¯(α1β1+α2β2)¯Mx>0,a31¯α1¯m121+β1¯m12−a30¯−a34¯Mx>0,a41¯α2¯m121+β2¯m12−a40¯−a43¯Mx>0in which(3.24)m1=a10¯−D1¯+E2a11¯.Then, system (2.1) is uniformly
persistent.Proof.
Suppose(x1,x2,x3,x4) is a solution of system (2.1) with the initial
condition (2.2). According to the first equation of
(2.1), we get(3.25)x1′(t)≥x1(t)((a10(t)−D1(t))−α1(t)x12(t)x3(t)1+β1(t)x12(t)−α2(t)x12(t)x4(t)1+β2(t)x12(t))≥−a11(t)x12(t)+(a10(t)−D1(t))x1(t)−α1(t)Mxβ1(t)−α2(t)Mxβ2(t).So,(3.26)lim⁡inf⁡t→∞⁢x1(t)≥m1>0.Then, there exists a T5>0 such that(3.27)x1(t)≥m1fort≥T5.Similarly,(3.28)lim⁡inf⁡t→∞⁢x2(t)≥m2≐a20¯−D2¯a21¯>0.Then, there exists a T6>0 such that(3.29)x2(t)≥m2fort≥T6.From the third equation of
(2.1), we obtain(3.30)x3′(t)≥x3(t)(−a30(t)+a31(t)α1(t)m121+β1(t)m12−a32(t)x3(t)−a34(t)Mx).So,(3.31)lim⁡inf⁡t→∞⁢x3(t)≥m3≐a31¯(α1¯m12/(1+β1¯m12))−a30¯−a34¯Mxa32¯>0.Then, there exists a T7>0 such that(3.32)x3(t)≥m3fort≥T7.Similarly, we also get(3.33)lim⁡inf⁡t→∞⁢x4(t)≥m4≐a41¯(α2¯m12/(1+β2¯m12))−a40¯−a43¯Mxa42¯>0.Then, there exists a T8>0 such that(3.34)x4(t)≥m4fort≥T8.
Finally, let(3.35)D={(x1,x2,x3,x4)∣mx<xi<Mx,⁢i=1,2,3,4},where mx=min⁡i=1,2,3,4{mi} and Mx=max⁡{M1*,M2*,M3*}; Mi*⁢(i=1,2,3) is given in Theorem 3.1.
From Theorem 3.1 and
the above analysis, we see that D is a bounded compact region in R+4 which has positive distance from coordinate
hyperplanes. Let T^=max⁡{Ti,⁢i=1,…,8},
then we obtain that if t>T^,
then every positive solution of system (2.1) with initial conditions (2.2)
eventually enters and remains in the region D.
This completes the proof.


## 4. Almost Periodic Solution

In this
section, we derive sufficient conditions which guarantee that the periodic
solution of periodic system (2.2) is globally attractive.Theorem 4.1.
In addition to (2.5), (3.2), and (3.23), assume further
that all the coefficients of system (2.1) are continuous and positive almost
periodic functions and(4.1)(a11¯+D1¯m2M12+α1¯m31+β1¯M12+α2¯m41+β2¯M12)m1>(2α1¯β1¯M12M3(1+β1¯m12)2+2α2¯β2¯M12M4(1+β2¯m12)2+D2¯m2)M1+Mx2mx(2α1¯a31¯M1(1+β1¯m12)2+2α2¯a41¯M1(1+β2¯m12)2),(a21¯+D2¯m1M22)m2>D1¯m1M2+Mx2mx(2α1¯a31¯M1(1+β1¯m12)2+2α2¯a41¯M1(1+β2¯m12)2),a32¯m3>(α1¯M11+β1¯m12+a43¯)M3+Mx2mx(2α1¯a31¯M1(1+β1¯m12)2+2α2¯a41¯M1(1+β2¯m12)2),a42¯m4>(α2¯M11+β2¯m12+a34¯)M3+Mx2mx(2α1¯a31¯M1(1+β1¯m12)2+2α2¯a41¯M1(1+β2¯m12)2).Then, system (2.1) has a unique
positive almost periodic solution which is globally asymptotically stable.
Furthermore, if system (2.1) is an ω-periodic system, then system (2.1) has a
positive ω-periodic solution which is globally
asymptotically stable.Proof.
Consider
the product system of (2.1):(4.2)x1′=x1(a10(t)−a11(t)x1)−α1(t)x12x31+β1(t)x12−α2(t)x12x41+β2(t)x12+D1(t)(x2−x1),x2′=x2(a20(t)−a21(t)x2)+D2(t)(x1−x2),x3′=x3(−a30(t)+a31(t)α1(t)x12(t−τ1)1+β1(t)x12(t−τ1)−a32(t)x3−a34(t)x4),x4′=x4(−a40(t)+a41(t)α2(t)x12(t−τ2)1+β2(t)x12(t−τ2)−a42(t)x4−a43(t)x3),y1′=y1(a10(t)−a11(t)y1)−α1(t)y12y31+β1(t)y12−α2(t)y12y41+β2(t)y12+D1(t)(y2−y1),y2′=y2(a20(t)−a21(t)y2)+D2(t)(y1−y2),y3′=y3(−a30(t)+a31(t)α1(t)y12(t−τ1)1+β1(t)y12(t−τ1)−a32(t)y3−a34(t)y4),y4′=y4(−a40(t)+a41(t)α2(t)y12(t−τ2)1+β2(t)y12(t−τ2)−a42(t)y4−a43(t)y3).It is easily noted that the
existence and uniqueness of the positive almost periodic solution of system
(2.1) are equivalent to the existence and uniqueness of the positive almost
periodic solution of system (4.2). Then, choose the
following function:(4.3)V(t)=V(t,xi,yi)=∑i=14|ln⁡xi(t)−ln⁡yi(t)|.Obviously, V(t) satisfies conditions (i) and (ii) of Lemma 2.2. Next, we will prove that V(t) satisfies condition (iii) of Lemma 2.2. It follows that(4.4)x1′x1−y1′y1=−a11(x1−y1)−D1(x2x1−y2y1)−(α1x1x31+β1x12−α1y1y31+β1y12)−(α2x1x41+β2x12−α2y1y41+β2y12)in which(4.5)α1x1x31+β1x12−α1y1y31+β1y12=(α1x31+β1x12−α1β1y1y3(x1+y1)(1+β1x12)(1+β1y12))(x1−y1)+α1y11+β1x12(x3−y3);also,(4.6)x2′x2−y2′y2=(−a21−D2y1x2y2)(x2−y2)+D2x1(x1−y1),x3′x3−y3′y3=a31α1[x1(t−τ1)+y1(t−τ1)][1+β1x12(t−τ1)][1+β1y12(t−τ1)][x1(t−τ1)−y1(t−τ1)]−a32(x3−y3)−a34(x4−y4),x4′x4−y4′y4=a41α2[x1(t−τ2)+y1(t−τ2)][1+β1x12(t−τ2)][1+β1y12(t−τ2)][x1(t−τ2)−y1(t−τ2)]−a42(x4−y4)−a43(x3−y3).In this regard, after few
computations, it is noted that (4.7)D+V(t,xi,yi)=∑i=14sgn(xi(t)−yi(t))(xi′(t)xi(t)−yi′(t)yi(t))=[−a11−D1y2x1y1−α1x3(1+β1x12)−α2x4(1+β2x12)+α1β1y1y3(x1+y1)(1+β1x12)(1+β1y12)+α2β1y1y4(x1+y1)(1+β2x12)(1+β2y12)]|x1−y1|+[−a21−D2y1x2y2]|x2−y2|−a32|x3−y3|−a42|x4−y4|+sgn(x1−y1)D1(x2−y2)x1−sgn(x1−y1)α1y1(1+β1x12)(x3−y3)−sgn(x1−y1)α2y1(1+β2x12)(x4−y4)+sgn(x2−y2)D2(x1−y1)x2+sgn(x3−y3)a31α1[x1(t−τ1)+y1(t−τ1)][1+β1x12(t−τ1)][1+β1y12(t−τ1)][x1(t−τ1)−y1(t−τ1)]−a34sgn(x3−y3)(x4−y4)+sgn(x4−y4)a41α2[x1(t−τ2)+y1(t−τ2)][1+β1x12(t−τ2)][1+β1y12(t−τ2)][x1(t−τ2)−y1(t−τ2)]−a43sgn(x4−y4)(x3−y3)≤−(a11¯+D1¯m2M12+α1¯m31+β1¯M12+α2¯m41+β2¯M12−2α1¯β1¯M12M3(1+β1¯m12)2−2α2¯β2¯M12M4(1+β2¯m12)2−D2¯m2)|x1−y1|+(−a21¯−D2¯m1M22+D1¯m1)|x2−y2|+(−a32¯+α1¯M11+β1¯m12+a43¯)|x3−y3|+(−a42¯+α2¯M11+β2¯m12+a34¯)|x4−y4|+2α1¯a31¯M1(1+β1¯m12)2|x1(t−τ1)−y1(t−τ1)|+2α2¯a41¯M1(1+β2¯m12)2|x1(t−τ2)−y1(t−τ2)|.
It follows from (4.1) that
(4.8)1Mx∑i=14∥xi−yi∥≤V(t,xi,yi)≤1mx∑i=14∥xi−yi∥.
Choose
P(s)=(Mx/mx)s>s>0,a(s)=(1/Mx)s>0,b(s)=(1/mx)s>0.
When(4.9)P(V(t,xi(t),yi(t)))≥V(t+θ,xi(t+θ),yi(t+θ)),θ∈[−τ,0],i=1,2,3,4,|x1(t−τ)−y1(t−τ)|≤Mx|ln⁡x1(t−τ)−ln⁡y1(t−τ)|≤MxV(t−τ,xi(t−τ),yi(t−τ))≤MxMxmxV(t,xi(t),yi(t));then(4.10)2α1¯a31¯M1(1+β1¯m12)2|x1(t−τ1)−y1(t−τ1)|≤Mx2mx2α1¯a31¯M1(1+β1¯m12)2V(t,xi(t),yi(t)),2α2¯a41¯M1(1+β2¯m12)2|x1(t−τ2)−y1(t−τ2)|≤Mx2mx2α2¯a41¯M1(1+β2¯m12)2V(t,xi(t),yi(t)).Hence,(4.11)D+V(t,xi,yi)≤[−(a11¯+D1¯m2M12+α1¯m31+β1¯M12+α2¯m41+β2¯M12)m1+(2α1¯β1¯M12M3(1+β1¯m12)2+2α2¯β2¯M12M4(1+β2¯m12)2+D2¯m2)Mx]|ln⁡x1−ln⁡y1|+[−(a21¯+D2¯m1M22)m2+D1¯m1M2]|ln⁡x2−ln⁡y2|+[−a32¯m3+(α1¯M11+β1¯m12+a43¯)M3]|ln⁡x3−ln⁡y3|+[−a42¯m4+(α2¯M11+β2¯m12+a34¯)M4]|ln⁡x4−ln⁡y4|+Mx2mx(2α1¯a31¯M1(1+β1¯m12)2+2α2¯a41¯M1(1+β2¯m12)2)V(t,xi(t),yi(t))≤−𝒞V(t,xi(t),yi(t)),where(4.12)−𝒞=max⁡{−(a11¯+D1¯m2M12+α1¯m31+β1¯M12+α2¯m41+β2¯M12)m1+(2α1¯β1¯M12M3(1+β1¯m12)22α2¯β2¯M12M4(1+β2¯m12)2+D2¯m2)Mx+Mx2mx(2α1¯a31¯M1(1+β1¯m12)2+2α2¯a41¯M1(1+β2¯m12)2)−(a21¯+D2¯m1M22)m2+D1¯m1M2+Mx2mx(2α1¯a31¯M1(1+β1¯m12)2+2α2¯a41¯M1(1+β2¯m12)2)−a32¯m3+(α1¯M11+β1¯m12+a43¯)M3+Mx2mx(2α1¯a31¯M1(1+β1¯m12)2+2α2¯a41¯M1(1+β2¯m12)2)−a42¯m4+(α2¯M11+β2¯m12+a34¯)M4+Mx2mx(2α1¯a31¯M1(1+β1¯m12)2+2α2¯a41¯M1(1+β2¯m12)2)}.This completes the proof.


## 5. Discussion

In this work,
we consider a nonautonomous delayed predator-prey model with competition and
diffusion. Some sufficient conditions on uniform persistence of the model have
been given. By means of the Liapunov-Razumikhin technique, it is also seen
that, under almost periodic circumstances, the existence and uniqueness of the
positive almost periodic solution which is globally asymptotically stable are
governed by several inequalities.



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*Source: 706154-2008-07-02.xml*