We propose a discrete multispecies cooperation and competition predator-prey systems. For general nonautonomous case, sufficient conditions which ensure the permanence and the global stability of the system are obtained; for periodic case, sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system are obtained.

1. Introduction

In this paper, we consider the dynamic behavior of the following non-autonomous discrete n+m-species cooperation and competition predator-prey systemsxi(k+1)=xi(k)exp[r1i(k)(1-xi(k)ai(k)+∑l=1,l≠inbil(k)xl(k)-ci(k)xi(k))-∑l=1mdil(k)yl(k)()],yj(k+1)=yj(k)exp[r2j(k)+∑l=1nejl(k)xl(k)-∑l=1mpjl(k)yl(k)],
where i=1,2,…n;j=1,2,…,m;xi(k) is the density of prey species i at kth generation. yj(k) is the density of predator species j at kth generation.

Dynamic behaviors of population models governed by difference equations had been studied by a number of papers, see [1–4] and the references cited therein. It has been found that the autonomous discrete systems can demonstrate quite rich and complicated dynamics, see [5, 6]. Recently, more and more scholars paid attention to the non-autonomous discrete population models, since such kind of model could be more appropriate.

May [7] suggested the following set of equations to describe a pair of mutualists: u̇=r1u(1-ua1+b1v-c1u),v̇=r2v(1-va2+b2u-c2v),
where u,v are the densities of the species U,V at time t, respectively. ri,ai,bi,ci,i=1,2 are positive constants. He showed that system (1.2) has a globally asymptotically stable equilibrium point in the region u>0,v>0.

Bai et al. [8] argued that the discrete case of cooperative system is more appropriate, and they proposed the following system: x1(k+1)=x1(k)exp{r1(k)[1-x1(k)a1(k)+b1(k)x2(k)-c1(k)x1(k)]},x2(k+1)=x2(k)exp{r2(k)[1-x2(k)a2(k)+b2(k)x2(k)-c2(k)x1(k)]}.

Chen [9] investigated the dynamic behavior of the following discrete n+m-species Lotka-Volterra competition predator-prey systemsxi(k+1)=xi(k)exp[bi(k)-∑l=1nail(k)xl(k)-∑l=1mcil(k)yl(k)],yj(k+1)=yj(k)exp[rj(k)+∑l=1ndjl(k)xl(k)-∑l=1mejl(k)yl(k)],
he investigated the dynamic behavior of the system (1.4).

The aim of this paper is, by further developing the analysis technique of Huo and Li [10] and Chen [9], to obtain a set of sufficient condition which ensure the permanence and the global stability of the system (1.1); for periodic case, sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system (1.1) are obtained.

We say that system (1.1) is permanent if there are positive constants M and m such that for each positive solution (x1(k),…,xn(k),y1(k),…,ym(k)) of system (1.1) satisfiesm≤limk→+∞infxi(k)≤limk→+∞supxi(k)≤M,m≤limk→+∞infyi(k)≤limk→+∞supyi(k)≤M,
for all i=1,2,…,n;j=1,2,…,m.

Throughout this paper, we assume that r1i(k), bil(k), ai(k), ci(k), r2j(k), dil(k), ejl(k), pjl(k) are all bounded nonnegative sequence, and use the following notations for any bounded sequence {a(k)}au=supk∈Na(k),al=infk∈Na(k).

For biological reasons, we only consider solution (x1(k),…,xn(k),y1(k),…,ym(k)) withxi(0)>0;i=1,2,…,n,yj(0)>0,j=1,2,...,m.

Then system (1.1) has a positive solution (x1(k),…,xn(k),y1(k),…,ym(k))k=0∞ passing through (x1(0),…,xn(0),y1(0),…,ym(0)).

The organization of this paper is as follows. In Section 2, we obtain sufficient conditions which guarantee the permanence of the system (1.1). In Section 3, we obtain sufficient conditions which guarantee the global stability of the positive solution of system (1.1). As a consequence, for periodic case, we obtain sufficient conditions which ensure the existence of a globally stable positive solution of system (1.1).

2. Permanence

In this section, we establish permanence results for system (1.1).

Lemma 2.1 (see [<xref ref-type="bibr" rid="B11">11</xref>]).

Let k∈Nk0+={k0,k0+1,…,k0+l,…},r≥0. For any fixed k,g(k,r) is a non-decreasing function with respect to r, and for k≥k0, the following inequalities hold:
y(k+1)≤g(k,y(k)),u(k+1)≥g(k,u(k)).
If y(k0)≤u(k0), then y(k)≤u(k) for all k≥k0.

Now let one consider the following single species discrete model:
N(k+1)=N(k)exp{a(k)-b(k)N(k)},
where {a(k)} and {b(k)} are strictly positive sequences of real numbers defined for k∈N={0,1,2,…} and 0<al≤au,0<bl≤bu.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Any solution of system (2.5) with initial condition N(0)>0 satisfies
m≤limk→+∞infN(k)≤limk→+∞supN(k)≤M,
where
M=1blexp{au-1},m=albuexp{al-buM}.

Proposition 2.3.

Assume that
-r2ju+∑l=1nejllMl>0,j=1,2,…,m
holds, then
limk→+∞supxi(k)≤Mi,i=1,2,…,n,limk→+∞supyi(k)≤Qi,i=1,2,...,m,
where
Mi=1cilr1ilexp{r1iu-1},Qi=1piilexp{∑l=1neiluMl-r2il-1}.

Proof.

Let u(k)=(x1(k),…,xn(k),y1(k),…,ym(k)) be any positive solution of system (1.1), from the ith equation of (1.1) we have
xi(k+1)≤xi(k)exp{r1i(k)[1-ci(k)xi(k)]}=xi(k)exp{r1i(k)-r1i(k)ci(k)xi(k)}.
By applying Lemmas 2.1 and 2.2, it immediately follows that
limk→+∞supxi(k)≤1cilr1ilexp{r1iu-1}:=Mi.
For any positive constant ε small enough, it follows from (2.9) that there exists enough large K0 such that
xi(k)≤Mi+ε,i=1,2,…,n,∀k≥K0.

From the n+jth equation of the system (1.1) and (2.10), we can obtain
yj(k+1)≤yj(k)exp{-r2j(k)+∑l=1nejl(k)(Ml+ε)-pjj(k)yj(k)}.
Condition (2.5) shows that Lemmas 2.1 and 2.2 could be applied to (2.11), and so, by applying Lemmas 2.1 and 2.2, it immediately follows that
limk→+∞supyj(k)≤1pjjlexp{∑l=1nejlu(Ml+ε)-r2jl-1},1≤j≤m.
Setting ε→0 in the above inequality leads to
limk→+∞supyj(k)≤1pjjlexp{∑l=1nejluMl-r2jl-1}:=Qj,1≤j≤m.
This completes the proof of Proposition 2.3.

Now we are in the position of stating the permanence of the system (1.1).

Theorem 2.4.

In addition to (2.5), assume further that
r1il-∑l=1mdiluQl>0,i=1,2,…,n,-r2ju+∑l=1nejllml-∑l=1,l≠jmpjluQl>0,j=1,2,…m,
then system (1.1) is permanent, where
mi=r1il-∑l=1mdiluQlr1iu(1/ail+ciu)exp{r1il-∑l=1mdiluQl-r1iu(1ail+ciu)Mi},Qi=1piilexp{∑l=1neiluMl-r2il-1}.

Proof.

By applying Proposition 2.3, we see that to end the proof of Theorem 2.4, it is enough to show that under the conditions of Theorem 2.4,
limk→+∞infxj(k)≥mi,1≤i≤n,limk→+∞infyj(k)≥qj,1≤j≤m.
From Proposition 2.3, ∀ε>0, there exists a K1>0,K1∈N,∀k>K1,
xi(k)≤Mi+ε,i=1,2,…,n,yj(k)≤Qj+ε,j=1,2,…,m.
From the ith equation of system (1.1) and (2.17), we have
xi(k+1)≥xi(k)exp{r1i(k)-r1i(k)(1ail+ciu)xi(k)-∑l=1mdil(k)(Ql+ε)}=xi(k)exp{r1i(k)-∑l=1mdil(k)(Ql+ε)-r1i(k)(1ail+ciu)xi(k)},
for all k>K1.

By applying Lemmas 2.1 and 2.2 to (2.18), it immediately follows that
limk→+∞infxi(k)≥r1il-∑l=1mdilu(Ql+ε)r1iu(1/ail+ciu)×exp{r1il-∑l=1mdilu(Ql+ε)-r1iu(1ail+ciu)Mi}.
Setting ε→0 in (2.19) leads to
limk→+∞infxi(k)≥r1il-∑l=1mdiluQlr1iu(1/ail+ciu)×exp{r1il-∑l=1mdiluQl-r1iu(1ail+ciu)Mi}:=mi.

Then, for any positive constant ε small enough, from (2.20) we know that there exists an enough large K2>K1 such that
xi(k)≥mi-ε,∀k≥k2.
Equations (2.17), (2.21) combining with the n+jth equation of the system (1.1) leads to
yj(k+1)≥yj(k)exp{-r2j(k)+∑l=1nejl(k)(ml-ε)-∑l=1,l≠jmpjl(k)(Ql+ε)-pjj(k)yj(k)},
under the condition (2.14), by applying Lemmas 2.1 and 2.2 to (2.22), it immediately follows that
limk→+∞infyj(k)≥-r2ju+∑l=1nejll(ml-ε)-∑l=1,l≠jmpjlu(Ql+ε)pjju×exp{-r2ju+∑l=1nejll(ml-ε)-∑l=1,l≠jmpjlu(Ql+ε)-pjjuQj}.
Setting ε→0 in (2.23) leads to
limk→+∞infyj(k)≥-r2ju+∑l=1nejllml-∑l=1,l≠jmpjluQlpjju×exp{-r2ju+∑l=1nejllml-∑l=1,l≠jmpjluQl-pjjuQj}:=qj.
This ends the proof of Theorem 2.4.

It should be noticed that, under the assumption of Theorem 2.4, the set [m1,M1]×⋯[mn,Mn]×[q1,Q1]×⋯[qm,Qm]
is an invariant set of system (1.1).

3. Global Stability

Now we study the stability property of the positive solution of system (1.1).

Theorem 3.1.

Assume that
λi=max{|1-(ciu+1ail)r1iuMi|,|1-(cil+1aiu)r1ilmi|}+r1iuMi(ail)2∑l=1,l≠inbiluMl+∑l=1mdiluQl<1,δj=max{|1-pjjuQj|,|1-pjjlqj|}+∑l=1nejluMl+∑l=1npjluQl<1.
Then for any two positive solution (x1(k),…,xn(k),y1(k),…,ym(k)) and (x̃1(k),…,x̃n(k),ỹ1(k),…,ỹm(k)) of system (1.1), one has
limk→+∞(x̃i(k)-xi(k))=0,limk→+∞(ỹj(k)-yj(k))=0.

Proof.

Let
xi(k)=x̃i(k)exp{ui(k)},yj(k)=ỹj(k)exp{vj(k)}.
Then system (1.1) is equivalent to
ui(k+1)=ui(k)+r1i(k)x̃i(k)ai(k)+∑l=1,l≠inbil(k)x̃l(k)-r1i(k)x̃i(k)exp{ui(k)}ai(k)+∑l=1,l≠inbil(k)x̃l(k)exp{ul(k)}-r1i(k)ci(k)x̃i(k)(exp{ui(k)}-1)-∑l=1mdil(k)ỹl(k)(exp{vl(k)}-1),vj(k+1)=vj(k)-∑l=1nejl(k)x̃l(k)(exp{ul(k)}-1)-∑l=1mpjl(k)ỹl(k)(exp{vl(k)}-1).
So,
|ui(k+1)|≤|1-(ci(k)+1ai(k))r1i(k)x̃i(k)exp{θi(k)ui(k)}||ui(k)|+r1i(k)x̃i(k)ai2(k)∑l=1,l≠inbil(k)x̃l(k)exp{θl(k)ul(k)}|ul(k)|+∑l=1mdil(k)ỹl(k)exp{ξl(k)vl(k)}|vl(k)||vi(k+1)|≤|1-pjj(k)ỹj(k)exp{ξj(k)vj(k)}||vj(k)|+∑l=1nejl(k)x̃l(k)exp{θl(k)ul(k)}|ul(k)|+∑l=1mpil(k)ỹl(k)exp{ξl(k)vl(k)}|vl(k)|,
where θl(k),ξl(k)∈[0,1], to complete the proof, it suffices to show that
limk→+∞ui(k)=0,limk→+∞vj(k)=0.
In view of (3.1), we can choose ε>0 small enough such that
λiε=max{|1-(ciu+1ail)r1iu(Mi+ε)|,|1-(cil+1aiu)r1il(mi-ε)|}+r1iu(Mi+ε)(ail)2∑l=1,l≠inbilu(Ml+ε)+∑l=1mdilu(Ql+ε)<1,δjε=max{|1-pjju(Qj+ε)|,|1-pjjl(qj-ε)|}+∑l=1nejlu(Ml+ε)+∑l=1npjlu(Ql+ε)<1.
For the above ε>0, according to Theorem 2.4 in Section 2, there exists a k*∈N such that
mi-ε≤xi(k),x̃i(k)≤Mi+ε,qj-ε≤yj(k),ỹj(k)≤Qj+ε,
for all k≥k*.

Noticing that θl(k),ξl(k)∈[0,1] implies that x̃l(k)exp{θl(k)ul(k)} lies between x̃l(k) and xl(k),ỹl(k)exp{ξl(k)vl(k)} lies between ỹl(k) and yl(k). From (3.5), we get
|ui(k+1)|≤max{|1-(ciu+1ail)r1iu(Mi+ε)|,|1-(cil+1aiu)r1il(mi-ε)|}|ui(k)|+r1iu(Mi+ε)(ail)2∑l=1,l≠inbilu(Ml+ε)|ul(k)|+∑l=1mdilu(Ql+ε)|vl(k)|,|vj(k+1)|≤max{|1-pjju(Qj+ε)|,|1-pjjl(qj-ε)|}|vj(k)|+∑l=1nejlu(Ml+ε)|ul(k)|+∑l=1npjlu(Ql+ε)|vl(k)|.
Let γ=max{λiε,δjε}, then γ<1. In view of (3.9), for k≥k*, we get
max{|ui(k+1)|,|vj(k+1)|}≤γmax{|ui(k)|,|vj(k)|}.
This implies
max{|ui(k)|,|vj(k)|}≤γk-k*max{|ui(k*)|,|vj(k*)|}.
From (3.11), we have
limk→+∞ui(k)=0,limk→+∞vj(k)=0.
This ends the proof of Theorem 3.1.

4. Existence and Stability of Periodic Solution

In this section, we further assume that the coefficients of the system (1.1) satisfies (4.1).

There exists a positive integer ω such that for k∈N, 0<r1i(k+ω)=r1i(k),0<bil(k+ω)=b1i(k),0<ai(k+ω)=ai(k),0<ci(k+ω)=ci(k),0<r2j(k+ω)=r2j(k),0<dil(k+ω)=dil(k),0<ejl(k+ω)=ejl(k),0<pjl(k+ω)=pjl(k).

Our first result concerned with the existence of a positive periodic solution of system (1.1).

Theorem 4.1.

Assume that (2.5) and (2.14) hold, then system (1.1) admits at least one positive ω-periodic solution which ones denotes by (x̃1(k),…,x̃n(k),ỹ1(k),…,ỹm(k)).

Proof.

As noted at the end of Section 2,
Dn+m:=[m1,M1]×⋯[mn,Mn]×[q1,Q1]×⋯[qm,Qm]
is an invariant set of system (1.1). Thus, we can define a mapping F on Dn+m by
F(x1(0),…,xn(0),y1(0),…,ym(0))=(x1(ω),…,xn(ω),y1(ω),…,ym(ω)),
for (x1(0),…,xn(0),y1(0),…,ym(0))∈Dn+m. Obviously, F depends continuously on (x1(0),…,xn(0),y1(0),…,ym(0)). Thus, F is continuous and maps the compact set Dn+m into itself. Therefore, F has a fixed point. It is easy to see that the solution (x̃1(k),…,x̃n(k),ỹ1(k),…,ỹm(k)) passing through this fixed point is an ω-periodic solution of the system (1.1). This completes the proof of Theorem 4.1.

Theorem 4.2.

Assume that (2.5), (2.14), and (3.1) hold, then system (1.1) has a global stable positive ω-periodic solution.

Proof.

Under the assumption of Theorem 4.2, it follows from Theorem 4.1 that system (1.1) admits at least one positive ω-periodic solution. Also, Theorem 3.1 ensures the positive solution to be globally stable. This ends the proof of Theorem 4.2.

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