A discrete nonlinear prey-competition system with m-preys and
(n-m)-predators and delays is considered. Two sets of sufficient conditions on the
permanence of the system are obtained. One set is delay independent, while
the other set is delay dependent.

1. Introduction

In this paper, we investigate the following discrete nonlinear prey-competition system with delays:

xi(k+1)=xi(k)exp[ri(k)-∑j=1naij(k)xjαij(k)-∑j=1nbij(k)xjβij(k-τij(k))],i=1,2,…,m,xi(k+1)=xi(k)exp[-ri(k)+∑j=1maij(k)xjαij(k)+∑j=1mbij(k)xjβij(k-τij(k))-∑j=m+1naij(k)xjαij(k)-∑j=m+1nbij(k)xjβij(k-τij(k))],i=m+1,2,…,n,
where xi(k)(i=1,2,…,m) is the density of prey species i at kth generation, xi(k)(i=m+1,…,n) is the density of predator species i at kth generation. In this system, the competition among predator species and among prey species is simultaneously considered. For more background and biological adjustments of system (1.1), we can see [1–5] and the references cited therein.

Throughout this paper, we always assume that for all i,j=1,2,…,n,

(H1)ri(k),aij(k),bij(k) are all bounded nonnegative sequences and aiil≥0,biil≥0,aiil+biil>0. Here, for any bounded sequence fu=supk∈Nf(k),fl=infk∈Nf(k);

(H2)τij(k) are bounded nonnegative integer sequences, and αij,βij are all positive constants.

By a solution of system (1.1), we mean a sequence {x1(k),…,xn(k)} which defined for N={0,1,…} and which satisfies system (1.1) forN={0,1,…}. Motivated by application of system (1.1) in population dynamics, we assume that solutions of system (1.1) satisfy the following initial conditions:

xi(θ)=ϕi(θ),θ∈N[-τ,0]=[-τ,-τ+1,…,0],ϕi(0)>0,
where τ=max{τij(k),i,j=1,2,…,n}. The exponential forms of system (1.1) assure that the solution of system (1.1) with initial conditions (1.2) remains positive.

Recently, Chen et al. in [1] proposed the following nonlinear prey-competition system with delays:

By using Gaines and Mawhins continuation theorem of coincidence degree theory and by constructing an appropriate Lyapunov functional, they obtained a set of sufficient conditions which guarantee the existence and global attractivity of positive periodic solutions of the system (1.3). In addition, sufficient conditions are obtained for the permanence of the system (1.3) in [2].

On the other hand, though most population dynamics are based on continuous models governed by differential equations, the discrete time models are more appropriate than the continuous ones when the size of the population is rarely small or the population has nonoverlapping generations [3–15]. Therefore, it is reasonable to study discrete time prey-competition models governed by difference equations.

As we know, a more important theme that interested mathematicians as well as biologists is whether all species in a multispecies community would survive in the long run, that is, whether the ecosystems are permanent. In fact, no such work has been done for system (1.1).

The main purpose of this paper is, by developing the analytical technique of [4, 8, 16], to obtain two sets of sufficient conditions which guarantee the permanence of system (1.1).

2. Main Results

Firstly, we introduce a definition and some lemmas which will be useful in the proof of the main results of this section.

Definition 2.1.

System (1.1) is said to be permanent, if there are positive constants m and M, such that each positive solution (x1(k),…,xn(k)) of system (1.1) satisfies
m≤lim infk→+∞xi(k)≤lim supk→+∞xi(k)≤M,i=1,2,…,n.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B8">8</xref>]).

Assume that {x(k)} satisfies x(k)>0 and
x(k+1)≤x(k)exp{r(k)(1-ax(k))}
for k∈[k1,+∞), whereais a positive constant. Then
lim supk→+∞x(k)≤1aruexp(ru-1).

Lemma 2.3 (see [<xref ref-type="bibr" rid="B8">8</xref>]).

Assume that {x(k)} satisfies
x(k+1)≥x(k)exp{r(k)(1-ax(k))},k≥K0,lim supk→+∞x(k)≤x* and x(K0)>0, where a is a constant such that ax*>1andK0∈N. Then
lim infk→+∞x(k)≥1aexp(ru(1-ax*)).

For system (1.1), we will consider two cases, aiil>0,biil≥0 and aiil≥0,biil>0 respectively, and then we obtain Lemmas 2.4–2.6.

Lemma 2.4.

Assume thataiil>0. Then for every positive solution (x1(k),…,xn(k)) of system (1.1) with initial condition (1.2), one has
lim supk→+∞xi(k)≤Mi,i=1,2,…,n,
where
Mi=(1αiiaiil)1/αiiexp[riu-1αii],i=1,2,…,m,Mi=(1αiiaiil)1/αiiexp[-ril+∑j=1naijuMjαij+∑j=1nbijuMjβij-1αii],i=m+1,…,n.

Proof.

Let x(k)=(x1(k),…,xn(k)) be any positive solution of system (1.1) with initial condition (1.2), for i=1,2,…,m, it follows from system (1.1) that
xi(k+1)≤xi(k)exp[ri(k)-aii(k)xiαii(k)],
thus
xiαii(k+1)≤xiαii(k)exp[αii(ri(k)-aii(k)xiαii(k))].
Letui(k)=xiαii(k), we can have
ui(k+1)≤ui(k)exp[αii(ri(k)-aii(k)ui(k))]≤ui(k)exp[αiiri(k)(1-aiilriuui(k))].
By applying Lemma 2.2 to (2.10), we obtain
lim supk→+∞ui(k)≤(1αiiaiil)exp[αiiriu-1]=̇Li;
so, we immediately get
lim supk→+∞xi(k)≤Mi,i=1,2,…,m.
For anyε>0 small enough, it follows from (2.12) that there exists enough largeK1 such that for alli=1,2,…,m and k≥K1xi(k)≤Mi+ε.
For i=m+1,…,n and k≥K1+τ, (2.13) combining with the i-th equation of system (1.1) leads to
xi(k+1)≤xi(k)exp[-ri(k)+∑j=1maij(k)(Mj+ε)αij+∑j=1mbij(k)(Mj+ε)βij-aii(k)xiαii(k)],
thus
xiαii(k+1)≤xiαii(k)exp[αii(-ri(k)+∑j=1maij(k)(Mj+ε)αij+∑j=1mbij(k)(Mj+ε)βij-aii(k)xiαii(k))].
Similarly, let ui(k)=xiαii(k), we get
ui(k+1)≤ui(k)exp[αii(-ri(k)+∑j=1maij(k)(Mj+ε)αij+∑j=1mbij(k)(Mj+ε)βij-aii(k)ui(k))]≤ui(k)exp[αii(-ri(k)+∑j=1maij(k)(Mj+ε)αij+∑j=1mbij(k)(Mj+ε)βij)×(1-aiil-ril+∑j=1maiju(Mj+ε)αij+∑j=1mbiju(Mj+ε)βijui(k))].
By using (2.16), for i=m+1,…,n, according to Lemma 2.2, it follows that
lim supk→+∞ui(k)≤(1αiiaiil)exp[αii(-ril+∑j=1naiju(Mj+ε)αij+∑j=1nbiju(Mj+ε)βij)-1];
settingε→0 in above inequality, we have
lim supk→+∞ui(k)≤(1αiiaiil)exp[αii(-ril+∑j=1naijuMjαij+∑j=1nbijuMjβij)-1]=̇Li,
then
lim supk→+∞xi(k)≤Mi,i=m+1,…,n.

This completes the proof.

For convenience, we introduce the following notation.

For i=1,2,…,m

Ai=aiiuril-∑j=1,j≠inaijuMjαij-∑j=1nbijuMjβij,Riu=riu-∑j=1,j≠inaijlMjαij-∑j=1nbijlMjβij.
For i=m+1,…,n

Assume thataiil>0 and
min1≤i≤nLiAi>1,
hold. Then for any positive solution (x1(k),…,xn(k)) of system (1.1) with initial condition (1.2), one has
lim infk→+∞xi(k)≥mi,
where
mi=(1Ai)1/αiiexp[Riu(1-AiLi)],i=1,2,…,n.

Proof.

Let x(k)=(x1(k),…,xn(k)) be any positive solution of system (1.1) with initial condition (1.2). From Lemma 2.4, we know that there exists K2>K1+τ, such that for i=1,2,…,n and k≥K2xi(k)≤Mi+ε.
For i=1,…,m and k≥K2+τ, (2.25) combining with the i-th equation of system (1.1) lead to
xi(k+1)≥xi(k)exp[ri(k)-∑j=1,j≠inaij(k)(Mj+ε)αij-∑j=1nbij(k)(Mj+ε)βij-aii(k)xiαii(k)],
thus
xiαii(k+1)≥xiαii(k)exp[αii(ri(k)-∑j=1,j≠inaij(k)(Mj+ε)αij-∑j=1nbij(k)(Mj+ε)βij-aii(k)xiαii(k))];
let ui(k)=xiαii(k), we can have
ui(k+1)≥ui(k)exp[αii(ri(k)-∑j=1,j≠inaij(k)(Mj+ε)αij-∑j=1nbij(k)(Mj+ε)βij-aii(k)ui(k))]≥ui(k)exp[αiiRiε(k)(1-Aiεui(k))],
where
Riε(k)=ri(k)-∑j=1,j≠inaij(k)(Mj+ε)αij-∑j=1nbij(k)(Mj+ε)βij,Aiε=aiiuril-∑j=1,j≠inaiju(Mj+ε)αij-∑j=1nbiju(Mj+ε)βij.
According to Lemma 2.3, we obtain
lim infk→+∞ui(k)≥1Aiεexp[αiiRiεu(1-AiεLi)],
where
Riεu=riu-∑j=1,j≠inaijl(Mj+ε)αij-∑j=1nbijl(Mj+ε)βij.
Settingε→0 in (2.30) leads to
lim infk→+∞ui(k)≥1Aiexp[αiiRiu(1-AiLi)],
therefore
lim infk→+∞xi(k)≥mi,i=1,2,…,m.
For any ε>0 small enough, it follows from (2.33) that there exists enough large K3>K2+τ such that for alli=1,…,m andk≥K3xi(k)≥mi-ε,
and so, fori=m+1,…,nand k≥K3+τ, it follows from system (1.1) that
xi(k+1)≥xi(k)exp[-ri(k)+∑j=1maij(k)(mj-ε)αij+∑j=1mbij(k)(mj-ε)βij-∑j=m+1,j≠inaij(k)(Mj+ε)αij-∑j=m+1nbij(k)(Mj+ε)βij-aii(k)xiαii(k)]≥xi(k)exp[Riε(k)(1-Aiεxiαii(k))],
where
Riε(k)=-ri(k)+∑j=1maij(k)(mj-ε)αij+∑j=1mbij(k)(mj-ε)βij-∑j=m+1,j≠inaij(k)(Mj+ε)αij-∑j=m+1nbij(k)(Mj+ε)βij,Aiε=aiiu/{-riu+∑j=1maijl(mj-ε)αij+∑j=1nbijl(mj-ε)βij-∑j=m+1,j≠inaiju(Mj+ε)αij-∑j=m+1nbiju(Mj+ε)βij},
by using (2.35), similarly to the analysis of (2.33), fori=m+1,…,nlim infk→+∞ui(k)≥1Aiexp[αiiRiu(1-AiLi)],
and therefore, we easily get
lim infk→+∞xi(k)≥mi,i=m+1,…,n.
This ends the proof of Lemma 2.5.

Denote for i=1,2,…,m

L̅i=1βiibiilexp[βiiriu(τ+1)-1];Γi=ril-∑j=1naijuMjαij-∑j=1nbijuMjβij;A̅i=biiuexp[-βiiΓiτ]ril-∑j=1naijuMjαij-∑j=1,j≠inbijuMjβij;R̅iu=riu-∑j=1naijlMjαij-∑j=1,j≠inbijlMjβij.
For i=m+1,…,n

Assume that biil>0 and
(H̅3)min1≤i≤nL̅iA̅i>1,
hold. Then for any positive solution (x1(k),…,xn(k)) of system (1.1) with initial condition (1.2), one has
m̅i≤lim infk→+∞xi(k)≤lim supk→+∞xi(k)≤M̅i,i=1,2,…,n.
where
M̅i=L̅i1/βii,m̅i=(1A̅i)1/βiiexp[R̅iu(1-A̅iL̅i)].

Proof.

Let x(k)=(x1(k),…,xn(k)) be any positive solution of system (1.1) with initial condition (1.2), for i=1,2,…,m, it follows from system (1.1) that
xi(k+1)≤xi(k)exp[ri(k)]≤xi(k)exp[riu],xi(k+1)≤xi(k)exp[ri(k)-bii(k)xiβii(k-τii(k))],
It follows from (2.44)that
∏j=k-τii(k)k-1xi(j+1)xi(j)≤∏j=k-τii(k)k-1exp[riu]≤exp[riuτ],
and hence
xi(k-τii(k))≥xi(k)exp[-riuτ],
which, together with (2.45), produces
xi(k+1)≤xi(k)exp[ri(k)-bii(k)exp[-βiiriuτ]xiβii(k)]≤xi(k)exp[ri(k)(1-biilexp[-βiiriuτ]riuxiβii(k))],
similar to the analysis of (2.11) and (2.12), for i=1,2,…,mlim supk→+∞ui(k)≤L̅i,
and thus, we immediately get
lim supk→+∞xi(k)≤M̅i,i=1,2,…,m.
For any ε>0 small enough, it follows from (2.50) that there exists enough large K̅1 such that for all i=1,2,…,m and k≥K̅1xi(k)≤M̅i+ε.
For i=m+1,…,n and k≥K̅1+τ, (2.51) combining with the i-th equation of system (1.1) lead to
xi(k+1)≤xi(k)exp[-ril+∑j=1maiju(Mj+ε)αij+∑j=1mbiju(Mj+ε)βij]=xi(k)exp[Υiε],xi(k+1)≤xi(k)exp[-ri(k)+∑j=1maij(k)(Mj+ε)αij+∑j=1mbij(k)(Mj+ε)βij-bii(k)xiβii(k-τii(k))],
from (2.53), similar to the argument of (2.44) and (2.47), for k≥K̅1+τ, we have
xi(k-τii(k))≥xi(k)exp[-Υiετ],
substituting (2.54) into (2.53), we get
xi(k+1)≤xi(k)exp[-ri(k)+∑j=1maij(k)(Mj+ε)αij+∑j=1mbij(k)(Mj+ε)βij-bii(k)exp[-βiiΥiετ]xiβii(k)],
similar to the analysis of (2.18) and (2.19), fori=m+1,…,nlim supk→+∞ui(k)≤L̅i,
then,
lim supk→+∞xi(k)≤M̅i,i=m+1,…,n.
For any ε>0 small enough, it follows from (2.51) and (2.57) that there exists enough largeK̅2>K̅1+τ such that for all i=1,2,…,n and k≥K̅2xi(k)≤M̅i+ε.
Hence, fori=1,2,…,m, and k≥K̅2+τ, it follows from system (1.1) that
xi(k+1)≥xi(k)exp[ril-∑j=1naiju(Mj+ε)αij-∑j=1nbiju(Mj+ε)βij]=xi(k)exp[Γiε],xi(k+1)≥xi(k)exp[ri(k)-∑j=1naij(k)(Mj+ε)αij-∑j=1,j≠inbij(k)(Mj+ε)βij-bii(k)xiβii(k-τii(k))],
from(2.59), similar to the argument of (2.44) and (2.47), for k≥K̅2+τ, we have
xi(k-τii(k))≤xi(k)exp[-Γiετ],
and this combined with (2.60) gives
xi(k+1)≥xi(k)exp[ri(k)-∑j=1naij(k)(Mj+ε)αij-∑j=1,j≠inbij(k)(Mj+ε)βij-bii(k)exp[-βiiΓiετ]xiβii(k)].
Similar to the argument of (2.32) and (2.33), for k≥K̅2+τ, we obtain
lim infk→+∞ui(k)≥1A̅iexp[βiiR̅iu(1-A̅iL̅i)],
then
lim infk→+∞xi(k)≥m̅i,i=1,2,…,m.
For anyε>0 small enough, it follows from (2.63) that there exists enough large K̅3>K̅2+τ such that for alli=1,…,m and k≥K̅3xi(k)≥m̅i-ε,
and so, fori=m+1,…,n and k≥K3+τ, it follows from system (1.1) that
xi(k+1)≥xi(k)exp[-riu+∑j=1maijl(mj-ε)αij+∑j=1mbijl(mj-ε)βij-∑j=m+1naiju(Mj+ε)αij-∑j=m+1nbiju(Mj+ε)βij]=xi(k)exp[Γiε],xi(k+1)≥xi(k)exp[-ri(k)+∑j=1maij(k)(mj-ε)αij+∑j=1mbij(k)(mj-ε)βij-∑j=m+1naij(k)(Mj+ε)αij-∑j=m+1,j≠inbij(k)(Mj+ε)βij-bii(k)xiβii(k-τii(k))].

Similar to the argument of (2.61) and (2.62), for k≥K̅3+τ, we have

lim infk→+∞ui(k)≥1A̅iexp[βiiR̅iu(1-A̅iL̅i)],
then
lim infk→+∞xi(k)≥m̅i,i=m+1,…,n.
This ends the proof of Lemma 2.6.

Denote (H3)

aiil>0,min1≤i≤nLiAi>1,
or

biil>0,min1≤i≤nL̅iA̅i>1,

Our main result in this paper is the following theorem about the permanence of system (1.1).

Theorem 2.7.

Assume that (H1),(H2), and (H3) hold, then system (1.1) is permanent.

Proof.

Let x(k)=(x1(k),…,xn(k)) be any positive solution of system (1.1) with initial condition (1.2). Suppose M=maxi=1,…,n{Mi,M̅i},m=mini=1,…,n{mi,m̅i}. By Lemmas 2.4–2.6, if system (1.1) satisfies (H1),(H2), and (H3), then we have
m≤lim infk→+∞xi(k)≤lim supk→+∞xi(k)≤M,i=1,2,…,n.
The proof is completed.

In this paper, we study a discrete nonlinear predator-prey system with m-preys and (n-m)-predators and delays, which can be seen as the modification of the traditional Lotka-Volterra prey-competition model. From our main results, Theorem 2.7 gives two sets of sufficient conditions on the permanence of the system (1.1). One set is delay independent, while the other set is delay dependent.

Acknowledgments

The author is grateful to the Associate Editor, Professor Xue-Zhong He, and two referees for a number of helpful suggestions that have greatly improved her original submission. This research is supported by the Nation Natural Science Foundation of China (no. 10171010), Key Project on Science and Technology of Education Ministry of China (no. 01061), and Innovation Group Program of Liaoning Educational Committee No. 2007T050.

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