# Dynamic Behaviors of a General Discrete Nonautonomous System of Plankton  Allelopathy with Delays

Authors: Yaoping Chen; Fengde Chen; Zhong Li
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

We study the dynamic behaviors
of a general discrete nonautonomous system of plankton
allelopathy with delays. We first show that under some suitable
assumption, the system is permanent. Next, by constructing a
suitable Lyapunov functional, we obtain a set of sufficient
conditions which guarantee the global attractivity of the two
species. After that, by constructing an extinction-type Lyapunov
functional, we show that under some suitable assumptions, one
species will be driven to extinction. Finally, two examples
together with their numerical simulations show the feasibility of
the main results.


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## Body

## 1. Introduction

The aim of this
paper is to investigate the dynamic behaviors of the following general discrete
nonautonomous system of plankton allelopathy with delay:(1.1)N1(k+1)=N1(k)exp[r1(k)−∑l=0ma1l(k)N1(k−l)−∑l=0mb1l(k)N2(k−l)−∑l=0mc1l(k)N1(k)N2(k−l)],N2(k+1)=N2(k)exp[r2(k)−∑l=0ma2l(k)N2(k−l)−∑l=0mb2l(k)N1(k−l)−∑l=0mc2l(k)N2(k)N1(k−l)]together with the initial
condition(1.2)Ni(−l)≥0,Ni(0)>0,i=1,2;l=0,1,…,m,where m is a positive integer, Ni(k) represent the densities of population i at the kth generation, ri(k) are the intrinsic growth rate of population i at the kth generation, ail(k) measure the intraspecific influence of the (k−l)th generation of population i on the density of own population, bil(k) stand for the interspecific influence of the (k−l)th generation of population i on the density of own population, and cil(k) stand for the effect of toxic inhibition of
population i by population j at the (k−l)th generation, i,j=1,2 and i≠j. Also, {ri(k)},{ail(k)},{bil(k)} and {cil(k)} are all bounded nonnegative sequences defined
for k∈N,
denoted by the set of all nonnegative integers, and l∈{0,1,…,m} such that(1.3)0<riL≤ri(k)≤riM,0<ailL≤ail(k)≤ailM,0<bilL≤bil(k)≤bilM,0<cilL≤cil(k)≤cilM,here, for any bounded sequence {f(k)},
define(1.4)fM=supk∈Nf(k),fL=infk∈Nf(k).As was pointed out by Chattopadhyay [1] the effects of toxic substances
on ecological communities are an important problem from an environmental point of
view. Chattopadhyay [1] and Maynard-Smith [2] proposed the following two
species Lotka-Volterra competition system, which describes the changes of size
and density of phytoplankton:(1.5)dx1(t)dt=x1(t)[r1−a11x1(t)−a12x2(t)−b1x1(t)x2(t)],dx2(t)dt=x2(t)[r2−a21x1(t)−a22x2(t)−b2x1(t)x2(t)],where x1(t) and x2(t) denote the population density of two competing
species at time t for a common pool of resources. The terms b1x1(t)x2(t) and b2x1(t)x2(t) denote the effect of toxic substances. Here,
they made the assumption that each species produces a substance toxic to the
other, only when the other is present. Noticing that the production of the
toxic substance allelopathic to the competing species will not be instantaneous,
but delayed by different discrete time lags required for the maturity of both
species, thus, Mukhopadhyay et al. [3] also incorporated the discrete time
delay into the above system. Tapaswi and Mukhopadhyay [4] also studied a
two-dimensional system that arises in plankton allelopathy involving discrete
time delays and environmental fluctuations. They assumed that the environmental
parameters are assumed to be perturbed by white noise characterized by a
Gaussian distribution with mean zero and unit spectral density. They focus on
the dynamic behavior of the stochastic system and the fluctuations in
population. For more works on system (1.5), one could refer to [1–3, 5–24] and the references cited therein.Since the
discrete time models governed by difference equations are more appropriate than
the continuous ones when the populations have nonoverlapping generations, and
discrete time models can also provide efficient computational models of
continuous models for numerical simulations, corresponding to system (1.5), Huo and Li [25] argued that it is necessary to study the following
discrete two species competition system:(1.6)x1(k+1)=x1(k)exp[r1(k)−a11(k)x1(k)−a12(k)x2(k)−b1(k)x1(k)x2(k)],x2(k+1)=x2(k)exp[r2(k)−a21(k)x1(k)−a22(k)x2(k)−b2(k)x1(k)x2(k)],where x1(k) and x2(k) are the population sizes of the two
competitors at generation k,b1(k) and b2(k) have respectively, shown that each species produces
a toxic substance to the other but the other only is present. In [25],
sufficient conditions were obtained to guarantee the permanence of the above
system, they also investigated the existence and stability property of the
positive periodic solution of system (1.6). Recently, Li and Chen [26]
further investigated the dynamic behaviors of the system (1.6). For general
nonatonomous case, they obtain a set of sufficient conditions which guarantee
the extinction of species x2 and the global stability of species x1 when species x2 is eventually extinct. For periodic case, the
other set of sufficient conditions, which concerned with the average condition
of the coefficients of he system, were obtained to ensure the eventual
extinction of species x2 and the global stability of positive periodic
solution of species x1 when species x2 is eventually extinct. For more works on
discrete population dynamics, one could refer to [7, 10, 25–45].Liu and Chen [32] argued that for a more realistic model, both seasonality of the
changing environment and some of the past states, that is, the effects of time
delays, should be taken into account in a model of multiple species growth.
They proposed and studied the system (1.1), which is more general than system
(1.6). By applying the coincidence degree theory, they obtained a set of
sufficient conditions for the existence of at least one positive periodic
solution of system (1.1)-(1.2). Zhang and Fang [46] also investigated the
periodic solution of the system (1.1), they showed that under some suitable
assumption, system (1.1) could admit at least two positive periodic solution.
As we can see, the works [32, 46] are all concerned with the positive periodic
solution of the system. However, since few things in the nature are really
periodic, it is nature to study the general nonautonomous system (1.1), in this
case, it is impossible to study the periodic solution of the system, however,
such topics as permanence, extinction, and stability become the most important
things. In this paper, we will further investigate the dynamics behaviors of
the system (1.1). More precisely, by developing the analysis technique of Liu [31] and Muroya [35, 36], we study the permanence, global
attractivity and extinction of system (1.1)-(1.2).The organization
of this paper is as follows. We study the persistence property of the system in
Section2 and the stability property in Section 3. Then in Section 4, by
constructing a suitable Lyapunov functional, sufficient conditions which ensure
the extinction of species N2 of system (1.1)-(1.2) are studied. In Section 5, two examples together with their numeric simulations show the feasibility of
main results. For more relevant works, one could refer to
[2, 3, 5–9, 12, 13, 27–30, 33, 34, 37–45] and the references cited therein.


## 2. Permanence

In this section, we study the persistent
property of system (1.1)-(1.2).Lemma 2.1.
For any positive solution{(N1(k),N2(k))} of system (1.1)-(1.2),(2.1)limsupk→∞Ni(k)≤Bi,i=1,2,where(2.2)Bi=defexp(riM−1)ai0L,i=1,2.Proof.
Let{(N1(k),N2(k))} be any positive solution of system
(1.1)-(1.2), in view of the system (1.1) for all k∈N,
we have(2.3)Ni(k+1)≤Ni(k)exp[ri(k)−ai0(k)Ni(k)],i=1,2.Applying Lemma 2.1 of Yang [44] to (2.3), we can obtain(2.4)limsupk→∞Ni(k)≤exp(riM−1)ai0L=defBi,i=1,2.This completes the proof of
Lemma 2.1.Lemma 2.2.
Assume that(2.5)Δ11=defr1L−∑l=1ma1lMB1−∑l=0m[b1lM+c1lMB1]B2>0,Δ21=defr2L−∑l=1ma2lMB2−∑l=0m[b2lM+c2lMB2]B1>0hold, where B1 and B2 are defined in (2.2). Then for any positive
solution {(N1(k),N2(k))} of system (1.1)-(1.2),(2.6)liminfk→∞Ni(k)≥Ai,i=1,2,where(2.7)Ai=Δi1ai0Mexp[Δi2],i=1,2,Δ12=r1L−∑l=0ma1lMB1−∑l=0m[b1lM+c1lMB1]B2,Δ22=r2L−∑l=0ma2lMB2−∑l=0m[b2lM+c2lMB2]B1.Proof.
In view of (2.5), we can choose a constant ε>0 small enough such that(2.8)r1L−∑l=1ma1lM(B1+ε)−∑l=0m[b1lM+c1lM(B1+ε)](B2+ε)>0,(2.9)r2L−∑l=1ma2lM(B2+ε)−∑l=0m[b2lM+c2lM(B2+ε)](B1+ε)>0.In view of (2.1), for above ε>0,
there exists an integer k0∈N such that(2.10)Ni(k)≤Bi+ε∀k≥k0,i=1,2.We consider the following two cases.
Case (i).
We
assume that there exists an integerl0≥k0+m such that N1(l0+1)≤N1(l0).
Note that(2.11)N1(l0+1)=N1(l0)exp[r1(l0)−∑l=0ma1l(l0)N1(l0−l)−∑l=0mb1l(l0)N2(l0−l)−∑l=0mc1l(l0)N1(l0)N2(l0−l)]≥N1(l0)exp{r1L−∑l=1ma1lM(B1+ε)−∑l=0m[b1lM+c1lM(B1+ε)](B2+ε)−a10MN1(l0)}.So we can obtain(2.12)r1L−∑l=1ma1lM(B1+ε)−∑l=0m[b1lM+c1lM(B1+ε)](B2+ε)−a10MN1(l0)≤0.It follows from (2.8) that(2.13)N1(l0)≥r1L−∑l=1ma1lM(B1+ε)−∑l=0m[b1lM+c1lM(B1+ε)](B2+ε)a10M>0.Then we have(2.14)N1(l0+1)≥r1L−∑l=1ma1lM(B1+ε)−∑l=0m[b1lM+c1lM(B1+ε)](B2+ε)a10M×exp{r1L−∑l=0ma1lM(B1+ε)−∑l=0m[b1lM+c1lM(B1+ε)](B2+ε)}.Let(2.15)N1ε=r1L−∑l=1ma1lM(B1+ε)−∑l=0m[b1lM+c1lM(B1+ε)](B2+ε)a10M×exp{r1L−∑l=0ma1lM(B1+ε)−∑l=0m[b1lM+c1lM(B1+ε)](B2+ε)}.Note that(2.16)B1=exp(r1M−1)a10L≥r1Ma10L≥r1La10M,thus r1L−a10MB1≤0,
and so, for above ε>0,(2.17)r1L−∑l=0ma1lM(B1+ε)−∑l=0m[b1lM+c1lM(B1+ε)](B2+ε)<r1L−a10M(B1+ε)<r1L−a10MB1≤0or(2.18)r1L−∑l=1ma1lM(B1+ε)−∑l=0m[b1lM+c1lM(B1+ε)](B2+ε)a10M≥N1ε.
We can claim that(2.19)N1(k)≥N1ε∀k≥l0.By way of contradiction, assume
that there exists an integer p0≥l0 such that N1(p0)<N1ε. Then p0≥l0+2. Let p˜0≥l0+2 be the smallest integer such that N1(p˜0)<N1ε. Then N1(p˜0−1)>N1(p˜0). The above argument produces that N1(p˜0)≥N1ε, a contradiction. Thus (2.19) proved.Case (ii).
We
assume thatN1(k+1)>N1(k) for all k≥k0+m,
then limk→∞N1(k) exists, denoted by N¯1.
We can claim that(2.20)N¯1≥r1L−∑l=1ma1lM(B1+ε)−∑l=0m[b1lM+c1lM(B1+ε)](B2+ε)a10M.By the  way of contradiction, assume
that(2.21)N¯1<r1L−∑l=1ma1lM(B1+ε)−∑l=0m[b1lM+c1lM(B1+ε)](B2+ε)a10M.Taking limit in the first
equation of (1.1) gives(2.22)limk→∞[r1(k)−∑l=0ma1l(k)N1(k−l)−∑l=0mb1l(k)N2(k−l)−∑l=0mc1l(k)N1(k)N2(k−l)]=0,which is a contradiction since(2.23)limk→∞[r1(k)−∑l=0ma1l(k)N1(k−l)−∑l=0mb1l(k)N2(k−l)−∑l=0mc1l(k)N1(k)N2(k−l)]≥r1L−∑l=1ma1lM(B1+ε)−a10MN¯1−∑l=0m[b1lM+c1lM(B1+ε)](B2+ε)>0.The claim is thus proved.
From (2.20), we
see that(2.24)N¯1≥N1ε.Combining Cases (i) and (ii), we see that(2.25)liminfk→∞N1(k)≥N1ε.Setting ε→0, it follows that(2.26)limε→0N1ε=Δ11a10Mexp{Δ12}=defA1.
So we can easily see that(2.27)liminfk→∞N1(k)≥A1.From the second equation of
(1.1), similar to above analysis, we have(2.28)liminfk→∞N2(k)≥A2,where A2 is defined in (2.6). This completes the proof
of Lemma 2.2.It immediately follows from Lemmas2.1 and 2.2 that the following theorem holds.Theorem 2.3.
Assume that (2.5) hold, then
system (1.1)-(1.2) is permanent.


## 3. Global attractivity

This section devotes to study the stability
property of the positive solution of system (1.1)-(1.2).Theorem 3.1.
Assume that there exists a constantη>0 such thatH0min{ai0L,2Bi−ai0M}−∑j=1,j≠i2[∑l=1mailM+(m+1)(bjM+2BjcM)]>η,i=1,2,where, for i,j=1,2,i≠j,Bi and Bj are defined in (2.2),(3.1)bjM=max{bjlM:l=0,1,…,m},ciM=max{cilM:l=0,1,…,m},cM=max{ciM:i=1,2},then for any two positive
solutions {(N1(k),N2(k))} and {(N1*(k),N2*(k))} of system (1.1)-(1.2),(3.2)limk→∞(Ni(k)−Ni*(k))=0,i=1,2.Proof.
First, let(3.3)V11(k)=|lnN1(k)−lnN1*(k)|.Then from the first equation of
(1.1), we have(3.4)V11(k+1)=|lnN1(k+1)−lnN1*(k+1)|≤|lnN1(k)−lnN1*(k)−a10(k)[N1(k)−N1*(k)]|+∑l=1ma1l(k)|N1(k−l)−N1*(k−l)|+∑l=0mb1l(k)|N2(k−l)−N2*(k−l)|+∑l=0mc1l(k)|N1(k)N2(k−l)−N1*(k)N2*(k−l)|.Noticing that by mean-value
theory(3.5)|lnN1(k)−lnN1*(k)|=1θ1(k)|N1(k)−N1*(k)|,where 0<θ1(k)≤max{N1(k),N1*(k)}.
Then(3.6)|lnN1(k)−lnN1*(k)−a10(k)[N1(k)−N1*(k)]|=|lnN1(k)−lnN1*(k)|−|lnN1(k)−lnN1*(k)|+|lnN1(k)−lnN1*(k)−a10(k)[N1(k)−N1*(k)]|=|lnN1(k)−lnN1*(k)|−(1θ1(k)−|1θ1(k)−a10(k)|)|N1(k)−N1*(k)|.Substituting (3.6) into (3.4)
leads to(3.7)V11(k+1)≤|lnN1(k)−lnN1*(k)|−(1θ1(k)−|1θ1(k)−a10(k)|)|N1(k)−N1*(k)|+∑l=1ma1l(k)|N1(k−l)−N1*(k−l)|+∑l=0mb1l(k)|N2(k−l)−N2*(k−l)|+∑l=0mc1l(k)|N1(k)N2(k−l)−N1*(k)N2*(k−l)|.So it follows that(3.8)ΔV11≤−(1θ1(k)−|1θ1(k)−a10(k)|)|N1(k)−N1*(k)|+∑l=1ma1l(k)|N1(k−l)−N1*(k−l)|+∑l=0mb1l(k)|N2(k−l)−N2*(k−l)|+∑l=0mc1l(k)|N1(k)N2(k−l)−N1*(k)N2*(k−l)|.According to (2.1), for any
constant ε>0,
there exists an integer k0∈N such that(3.9)N1(k)≤B1+ε,N2(k)≤B2+ε∀k≥k0.So for all k≥k0+m,l=0,1,…,m,
it follows that(3.10)|N1(k)N2(k−l)−N1*(k)N2*(k−l)|=|N1(k)N2(k−l)−N1(k)N2*(k−l)+N1(k)N2*(k−l)−N1*(k)N2*(k−l)|=|N1(k)[N2(k−l)−N2*(k−l)]+N2*(k−l)[N1(k)−N1*(k)]|≤(B1+ε)|N2(k−l)−N2*(k−l)|+(B2+ε)|N1(k)−N1*(k)|.So for all k≥k0+m,
it follows from (3.4) that(3.11)ΔV11≤−(1θ1(k)−|1θ1(k)−a10(k)|−∑l=0m(B2+ε)c1l(k))|N1(k)−N1*(k)|+∑l=1ma1l(k)|N1(k−l)−N1*(k−l)|+∑l=0m[b1l(k)+(B1+ε)c1l(k)]|N2(k−l)−N2*(k−l)|.Next, let(3.12)V12(k)=∑l=1m∑s=k−lk−1a1l(s+l)|N1(s)−N1*(s)|+∑l=0m∑s=k−lk−1[b1l(s+l)+(B1+ε)c1l(s+l)]|N2(s)−N2*(s)|,and we can obtain(3.13)ΔV12=V12(k+1)−V12(k)=∑l=1m∑s=k+1−lka1l(s+l)|N1(s)−N1*(s)|−∑l=1m∑s=k−lk−1a1l(s+l)|N1(s)−N1*(s)|+∑l=0m∑s=k+1−lk[b1l(s+l)+(B1+ε)c1l(s+l)]|N2(s)−N2*(s)|−∑l=0m∑s=k−lk−1[b1l(s+l)+(B1+ε)c1l(s+l)]|N2(s)−N2*(s)|=∑l=1ma1l(k+l)|N1(k)−N*(k)|−∑l=1ma1l(k)|N1(k−l)−N1*(k−l)|+∑l=0m[b1l(k+l)+(B1+ε)c1l(k+l)]|N2(k)−N2*(k)|−∑l=0m[b1l(k)+(B1+ε)c1l(k)]|N2(k−l)−N2*(k−l)|.Now, we define V1 by(3.14)V1(k)=V11(k)+V12(k).So for all k≥k0+m,
it follows from (3.6) and (3.9) that(3.15)ΔV1=ΔV11+ΔV12≤−(1θ1(k)−|1θ1(k)−a10(k)|−∑l=0m(B2+ε)c1l(k)−∑l=1ma1l(k+l))|N1(k)−N1*(k)|+∑l=0m[b1l(k+l)+(B1+ε)c1l(k+l)]|N2(k)−N2*(k)|.Similar to above arguments, we
can define(3.16)V2(k)=V21(k)+V22(k),where(3.17)V21(k)=|lnN2(k)−lnN2*(k)|,V22(k)=∑l=1m∑s=k−lk−1a2l(s+l)|N2(s)−N2*(s)|+∑l=0m∑s=k−lk−1[b2l(s+l)+(B2+ε)c2l(s+l)]|N1(s)−N1*(s)|.Then for all k≥k0+m, we can obtain(3.18)ΔV2=ΔV21+ΔV22≤−(1θ2(k)−|1θ2(k)−a20(k)|−∑l=0m(B1+ε)c2l(k)−∑l=1ma2l(k+l))|N2(k)−N2*(k)|+∑l=0m[b2l(k+l)+(B2+ε)c2l(k+l)]|N1(k)−N1*(k)|,where θ2(k) lies between N2(k) and N2*(k).
Now, we defineV by(3.19)V(k)=V1(k)+V2(k).It is easy to see that V(k)≥0 for all k∈Z and V(k0+m)<+∞. For the arbitrariness of ε>0 and by (H0),
we can choose ε>0 small enough such that for i=1,2,(3.20)min{ai0L,2Bi+ε−ai0M}−∑j=1,j≠i2[∑l=1mail(k+l)+(m+1)(bjM+2(Bj+ε)cM)]>η.So for all k≥k0+m,
it follows from (3.15) and (3.18) that(3.21)ΔV≤−∑i=12{1θi(k)−|1θi(k)−ai0(k)|−∑j=1,j≠i2[∑l=1mail(k+l)+∑l=0m[bjl(k+l)+(Bj+ε)(cil(k)+cjl(k+l))]]}×|Ni(k)−Ni*(k)|≤−∑i=12{min{ai0L,2Bi+ε−ai0M}−∑j=1,j≠i2[∑l=1mail(k+l)+(m+1)(bjM+2(Bj+ε)cM)]}×|Ni(k)−Ni*(k)|≤−η∑i=12|Ni(k)−Ni*(k)|,So we have(3.22)∑p=k0+mk[V(p+1)−V(p)]≤−η∑p=k0+mk∑i=12|Ni(p)−Ni*(p)|,which implies(3.23)V(k+1)+η∑p=k0+mk∑i=12|Ni(p)−Ni*(p)|≤V(k0+m).It follows that(3.24)∑p=k0+mk∑i=12|Ni(p)−Ni*(p)|≤V(k0+m)η.Then(3.25)∑k=k0+m∞∑i=12|Ni(k)−Ni*(k)|≤V(k0+m)η<+∞,which implies that limk→∞∑i=12|Ni(k)−Ni*(k)|=0, that is, (3.26)limk→∞(Ni(k)−Ni*(k))=0,i=1,2.This completes the proof of Theorem 3.1.


## 4. Extinction of speciesN2

This section
devotes to study the extinction of the speciesN2.Lemma 4.1.
For any positive solution{(N1(k),N2(k))} of system (1.1)-(1.2), there exists a constant σ>0 such that(4.1)liminfk→+∞[N1(k)+N2(k)]>σ.Proof.
By (2.1), there exists a constant B>0 such that(4.2)Ni(k)<B∀k>k0,i=1,2.In view of (1.1) for all k>k0+m,i=1,2, it follows that(4.3)Ni(k)≤Ni(k+1)exp{−riL+∑l=0m(ailM+bilM+cilMB)B}.So we have(4.4)Ni(k−l)≤Ni(k)exp{l[−riL+∑l=0m(ailM+bilM+cilMB)B]},l=0,1,…,m.Let(4.5)C1=max{exp{l[−riL+∑l=0m(ailM+bilM+cilMB)B]}:i=1,2,l=0,1,…,m},C2=max{dilMC1:i=1,2,l=0,1,…,m},where dilM=max{ailM,bilM+cilMB},i=1,2,l=0,1,…,m. For all k>k0+m,i,j=1,2,i≠j, it follows from (1.1) and (4.4) that(4.6)Ni(k+1)≥Ni(k)exp{riL−∑l=0mailMC1Ni(k)−∑l=0m(bilM+cilMB)C1Nj(k)}≥Ni(k)exp{riL−∑l=0mdilMC1[Ni(k)+Nj(k)]}≥Ni(k)exp{min{r1L,r2L}−(m+1)C2[N1(k)+N2(k)]},so we have(4.7)N1(k+1)+N2(k+1)≥[N1(k)+N2(k)]exp{min{r1L,r2L}−(m+1)C2[N1(k)+N2(k)]}.Let x(k)=N1(k)+N2(k), then we have(4.8)x(k+1)≥x(k)exp{min{r1L,r2L}−(m+1)C2x(k)}=x(k)exp{min{r1L,r2L}[1−(m+1)C2min{r1L,r2L}x(k)]}=defx(k)exp{r[1−ax(k)]}.Note that for all k>k0,x(k)=N1(k)+N2(k)<2B, so similar to the proof of Lemma 2.2 of Chen [27], we have(4.9)liminfk→+∞x(k)≥1aexp{r(1−2aB)}>0.Then, there is a positive
constant σ>0 such that(4.10)liminfk→+∞[N1(k)+N2(k)]=liminfk→+∞x(k)≥1aexp{r(1−2aB)}>σ.This completes the proof of
Lemma 4.1.Theorem 4.2.
Assume thatH1∑l=0m[r1Lb2lL−r2Ma1lM]>0,∑l=0m[r1La2lL−r2M(b1lM+c1lMB1)]>0,where B1 is defined in (2.2). Let {(N1(k),N2(k))} be any positive solution of system
(1.1)-(1.2), then N2(k)→0 as k→+∞.Corollary 4.3.
Assume that for alll=0,1,…,m,
the following inequalities
H1*r2Mr1L−min{b2lLa1lM,a2lLb1lM+c1lMB1}<0hold, where B1 is defined in (2.2). Let {(N1(k),N2(k))} be any positive solution of system
(1.1)-(1.2), then N2(k)→0 as k→+∞.Proof of Corollary4.3.
Obviously, if
condition (H1*) holds, one could easily see that condition (H1) holds, thus, the conclusion of Corollary 4.3
follows from Theorem 4.2. The proof is complete.Proof of Theorem4.2.
It follows from (H1) that we can choose a constant ε>0 small enough such that(4.11)∑l=0m[r1Lb2lL−r2Ma1lM]>0,∑l=0m[r1La2lL−r2M(b1lM+c1lM(B1+ε))]>0.Set(4.12)Δε=min{∑l=0m[r1Lb2lL−r2Ma1lM],∑l=0m[r1La2lL−r2M(b1lM+c1lM(B1+ε))]}>0.For above ε>0
from (2.1), there is an integer K∈N such that for i=1,2,(4.13)Ni(k)≤Bi+ε∀k>K.Lemma 4.1 also implies that
there exists K1>K such that(4.14)N1(k)+N2(k)≥σ2∀k≥K1.Set(4.15)u(k)=N2r1L(k)N1r2M(k)exp{−∑l=0m∑s=k−lk−1r1L[a2lLN2(s)+b2lLN1(s)]+∑l=0m∑s=k−lk−1r2M[a1lMN1(s)+b1lMN2(s)+c1lM(B1+ε)N2(s)]}.So for all k>K2>K1+m,
it follows from (1.1), (4.13), and (4.14) that(4.16)u(k+1)u(k)=exp{r1Lr2(k)−∑l=0mr1L[a2l(k)N2(k−l)+b2l(k)N1(k−l)+c2l(k)N2(k)N1(k−l)]−r2Mr1(k)+∑l=0mr2M[a1l(k)N1(k−l)+b1l(k)N2(k−l)+c1l(k)N1(k)N2(k−l)]−∑l=0m∑s=k+1−lkr1L[a2lLN2(s)+b2lLN1(s)]+∑l=0m∑s=k−lk−1r1L[a2lLN2(s)+b2lLN1(s)]+∑l=0m∑s=k+1−lkr2M[a1lMN1(s)+b1lMN2(s)+c1lM(B1+ε)N2(s)]−∑l=0m∑s=k−lk−1r2M[a1lMN1(s)+b1lMN2(s)+c1lM(B1+ε)N2(s)]}=exp{[r1Lr2(k)−r2Mr1(k)]−∑l=0mr1L[(a2l(k)−a2lL)N2(k−l)+(b2l(k)−b2lL)N1(k−l)]−∑l=0mr1Lc2l(k)N2(k)N1(k−l)−∑l=0mr2M[(a1lM−a1l(k))N1(k−l)+(b1lM−b1l(k))N2(k−l)+(c1lM(B1+ε)−c1l(k)N1(k))N2(k−l)]−∑l=0m[r1Lb2lL−r2Ma1lM]N1(k)−∑l=0m[r1La2lL−r2Mb1lM−r2Mc1lM(B1+ε)]N2(k)}≤exp{−∑l=0m[r1Lb2lL−r2Ma1lM]N1(k)−∑l=0m[r1La2lL−r2Mb1lM−r2Mc1lM(B1+ε)]N2(k)}.≤exp{−Δε(N1(k)+N2(k))}≤exp{−Δεσ2}.
That is, for allk>K2,(4.17)u(k)≤u(k2)exp{−Δεσ2(k−K2)}.So from the definition of u(k) it follows that(4.18)N2r1L(k)≤N1r2M(k)exp{∑l=0m∑s=k−lk−1r1L[a2lLN2(s)+b2lLN1(s)]−∑l=0m∑s=k−lk−1r2M[a1lMN1(s)+b1lMN2(s)+c1lM(B1+ε)N2(s)]}×exp{−Δεσ2(k−K2)}≤(2B1)r2Mexp{∑l=0m∑s=k−lk−12r1L[a2lLB2+b2lLB1]+∑l=0m∑s=k−lk−12r2M[a1lMB1+b1lMB2+c1lM(B1+ε)B2]}×exp{−Δεσ2(k−K2)}→0ask→+∞.The above analysis shows that(4.19)limk→∞N2(k)=0.This completes the proof of
Theorem 4.2.


## 5. Examples

The following two examples show
the feasibility of our results.Example 5.1.
Consider the following system(5.1)N1(k+1)=N1(k)exp[1.4−(2.52+0.02sin(k))N1(k)−0.5N1(k−1)−0.55N2(k)−0.3N2(k−1)−0.1N1(k)N2(k)−0.09N1(k)N2(k−1)],N2(k+1)=N2(k)exp[0.7−(2.62+0.02sin(k))N2(k)−1.2N2(k−1)−0.01N1(k)−0.01N1(k−1)−0.09N1(k)N2(k)−0.1N2(k)N1(k−1)].One could easily see that(5.2)Δ11=r1L−a11MB1−∑l=01[b1lM+c1lMB1]B2≈0.8271394917>0,Δ21=r2L−a21MB2−∑l=01[b2lM+c2lMB2]B1≈0.3138443044>0.Clearly, conditions (2.5) are satisfied. From Theorem 2.3, it follows
that system (5.1) is permanent. Also, by simple computation, we have(5.3)min{a10L,2B1−a10M}−[a11M+2(b2M+2B2cM)]≈0.1776281960,min{a20L,2B2−a20M}−[a21M+2(b1M+2B1cM)]≈0.613080483.The above inequality shows that (H0) is fulfilled. From Theorem 3.1, it follows
that(5.4)limk→∞(Ni(k)−Ni*(k))=0,i=1,2.Figures 1 and 2 are the
numeric simulations of the solution of system (5.1) with initial condition (N1(k),N2(k))=(0.42,0.175),(0.41,0.178), and (0.4,0.18),k=−1,0.Figure 1
Dynamic
behaviors of the speciesN1 of system (5.1) with initial conditions (N1(p),N2(p))=(0.42,0.175),(0.41,0.178), and (0.4,0.18),p=−1,0.Figure 2
Dynamic
behaviors of the speciesN2 of system (5.1) with initial conditions (N1(k),N2(k))=(0.42,0.175),(0.41,0.178), and (0.4,0.18),k=−1,0.Example 5.2.
Consider the following system:(5.5)N1(k+1)=N1(k)exp[1.4−(1.5+0.2sin(k))N1(k)−0.9N1(k−1)−(0.3+0.2sin(k))N2(k)−0.5N2(k−1)−(0.4+0.1cos(k))N1(k)N2(k)−0.4N1(k)N2(k−1)],N2(k+1)=N2(k)exp[0.7−(1.1+0.5sin(k))N2(k)−0.5N2(k−1)−(1.1+0.2cos(k))N1(k)−0.7N1(k−1)−2.3N1(k)N2(k)−0.4N2(k)N1(k−1)].One could easily see that(5.6)r2Mr1L=0.71.4=0.5,b20La10M=0.91.7≈0.5294,a20Lb10M+c10MB1≈0.60.5+0.5×1.1476≈0.5588,b21La11M=0.70.9≈0.7778,a21Lb11M+c11MB1≈0.50.5+0.4×1.1476≈0.5214.Then, for l=0,1,(5.7)r2Mr1L−min{b2lLa1lM,a2lLb1lM+c1lMB1}<0.The above inequality shows that (H1*) is fulfilled. From Theorem 4.2, it follows
that limk→∞N2(k)=0.
Numeric simulation of the dynamic behaviors of system (5.5) with the initial
conditions (N1(k),N2(k))=(0.42,0.6),(k=−1,0) is presented in Figure 3.Figure 3
Dynamic
behaviors of the speciesN2 of system (5.5) with initial conditions (N1(k),N2(k))=(0.42,0.6),(k=−1,0),
respectively.Remark 5.3.
In the above two examples, we can takesin(k),cos(k) as the perturbation terms. Our numeric
simulations show that if the perturbation terms are large enough, then those
terms will greatly influence the dynamic behaviors of the system, and in some cases,
may lead to the extinction of the species.



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*Source: 310425-2009-02-26.xml*