1. Introduction Throughout this paper, for any bounded sequence {h(n)}, set hu=supn∈N{h(n)} and hl=infn∈N{h(n)}.

Li and Xu [1] studied the following two-species integrodifferential model of mutualism: (1)N1′t=r1tN1tK1t+α1t∫0∞J2sN2t-sds1+∫0∞J2sN2t-sds-N1t-σ1t,N2′t=r2tN2tK2t+α2t∫0∞J1sN1t-sds1+∫0∞J1sN1t-sds-N2t-σ2t.By applying the coincidence degree theory, they showed that system (1) admits at least one positive ω-periodic solution. Chen and You [2] argued that a general nonautonomous nonperiodic system is more appropriate, and for the general nonautonomous case, they showed that the system is permanent. For more background and biological adjustments of system (1), one could refer to [1–6] and the references cited therein. For more work on mutualism model, one could refer to [7–34] and the references cited therein.

Li and Yang [31] and Li [32] argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations; corresponding to system (1), they proposed the following two-species discrete model of mutualism with infinite deviating arguments: (2)x1k+1=x1kexpr1kK1k+α1k∑s=0+∞J2sx2k-s1+∑s=0+∞J2sx2k-s-x1k-δ1k,x2k+1=x2kexpr2kK2k+α2k∑s=0+∞J1sx1k-s1+∑s=0+∞J1sx1k-s-x2k-δ2k,where xi(k), i=1,2, is the density of mutualism species i at the kth generation; rik,Kik,αik,Jik, and {δi(k)}, i=1,2, are bounded nonnegative sequences such that (3)0<ril≤riu,0<ail≤aiu,0<Kil≤Kiu,0<δil≤δiu,∑j=1+∞Jin=1,αi>Ki.They showed that, under the above assumption, system (2) is permanent.

It brings to our attention the fact that the main results of [31, 32] deeply depend on the assumption αi>Ki, i=1,2. Now, an interesting issue is proposed: is it possible for us to investigate the persistent property of system (2) under the assumption Ki>αi, i=1,2?

From the point of view of biology, in the sequel, we shall consider (2) together with the initial conditions: (4)xis=ϕis, s=…,-k,-k+1,…,-2,-1, i=1,2.Then system (2) has a unique positive solution x1k,x2kk=0+∞ satisfying the initial condition (4).

From now on, we assume that the coefficients of system (2) satisfy the following.

(A) rik,{Ki(k)},{αi(k)},{Ji(k)}, and {δi(k)}, i=1,2, are bounded nonnegative sequences such that(5)0<ril≤riu,0<ail≤aiu,0<Kil≤Kiu,0<δil≤δiu,∑j=1+∞Jin=1,Ki>αi.

We mention here that such an assumption implies that the relationship between two species is competition; indeed, under the assumption (A), the first equation in system (1) can be rewritten as follows: (6)x1k+1=x1kexpr1kK1k+α1k∑s=0+∞J2sx2k-s1+∑s=0+∞J2sx2k-s-x1k-δ1k=x1kexpr1kK1k1+∑s=0+∞J2sx2k-s1+∑s=0+∞J2sx2k-s-x1k-δ1k-K1k-α1k∑s=0+∞J2sx2k-s1+∑s=0+∞J2sx2k-s=x1kexpr1kK1k-x1k-δ1k-K1k-α1k∑s=0+∞J2sx2k-s1+∑s=0+∞J2sx2k-s.Similar to the above analysis, the second equation in system (2) can be rewritten as follows: (7)x2k+1=x2kexpr2kK2k-x2k-δ2k-K2k-α2k∑s=0+∞J1sx1k-s1+∑s=0+∞J1sx1k-s.From (6) and (7), one could easily see that both species have negative effect on the other species; that is, the relationship between two species is competition.

Concerned with the persistent property of systems (2) and (4), we have the following result.

Theorem 1. In addition to (A), assume further that (8)r1lK1l>r1uK1u-α1lM21+M2,r2lK2l>r2uK2u-α2lM11+M1holds, where (9)Mi=1rilexpriuKiuδ+1-1, i=1,2;then system (2) is permanent; that is, there exist positive constants mi,Mi, i=1,2 (Mi is defined by (9)), which are independent of the solution of system (2), such that, for any positive solution (N1(t),N2(t)) of system (2) with initial condition (4), one has(10)mi≤liminfk→+∞ xik≤limsupk→+∞ xik≤Mi, i=1,2.

2. Proof of the Main Result Now we state several lemmas which will be useful in proving our main result.

Lemma 2 (see [<xref ref-type="bibr" rid="B11">11</xref>]). Assume that {xk} satisfies xk>0 and (11)xk+1≤xkexpak-bkxkfor k∈N, where a(k) and b(k) are nonnegative sequences bounded above and below by positive constants. Then (12)limsupk→+∞ xk≤1blexpau-1.

Lemma 3 (see [<xref ref-type="bibr" rid="B11">11</xref>]). Assume that {x(k)} satisfies (13)xk+1≥xkexpak-bkxk, k≥N0,lim supk→+∞x(k)≤x∗ and x(N0)>0, where a(k) and b(k) are nonnegative sequences bounded above and below by positive constants and N0∈N. Then (14)liminfk→+∞ xk≥minalbuexpal-bux∗,albu.

Lemma 4 (see [<xref ref-type="bibr" rid="B35">34</xref>]). Let x:Z→R be nonnegative bounded sequences, and let H:N→R be nonnegative sequences such that ∑n=0∞Hn=1. Then (15)liminfn→+∞xn≤liminfn→+∞∑s=-∞nHn-sxs≤limsupn→+∞∑s=-∞nHn-sxs≤limsupn→+∞ xn.

Now we are in the position to prove the main result of this paper.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>. Set(16)δ=maxδ1u,δ2u.Let x1k,x2k be any positive solution of system (2) with initial condition (4). From (6), it follows that (17)x1k+1≤x1kexpr1kK1k≤x1kexpr1uK1u.By using (17), one could easily obtain that (18)x1k-δ1k≥x1kexp-r1uK1uδ.Substituting (18) into (6), it follows that (19)x1k+1≤x1kexpr1uK1u-r1lexp-r1uK1uδx1k.Thus, as a direct corollary of Lemma 2, according to (19), one has (20)limsupk→+∞ x1k≤1r1lexpr1uK1uδ+1-1≝M1.By using (7), similar to the analysis of (17)–(20), we can obtain (21)limsupk→+∞ x2k≤1r2lexpr2uK2uδ+1-1≝M2.From (20), (21), and Lemma 4, we have (22)limsupk→+∞ ∑s=0+∞Jisxik-s=limsupk→+∞ ∑s=-∞kJik-sxis≤limsupk→+∞ xik≤Mi, i=1,2.Condition (6) implies that, for enough small ε>0, inequalities (23)r1lK1l>r1uK1u-α1lM2+ε1+M2+ε,r2lK2l>r2uK2u-α2lM1+ε1+M1+εhold. For above ε>0, from (19)–(22), it follows that there exists N1>0 such that, for all k>N1 and i=1,2, (24)∑s=0+∞Jisxik-s<Mi+ε,xik<Mi+ε.By using the fact that ex≥1+x, x≥0, from (9), one could easily see that (25)M1=1r1lexpr1uK1uδ+1-1≥1r1lr1uK1uδ+1>K1u≥α1l.For k≥N1+δ, from (24) and (6), we have (26)x1k+1=x1kexpr1kK1k-x1k-δ1k-K1k-α1k∑s=0+∞J2sx2k-s1+∑s=0+∞J2sx2k-s≥x1kexpr1kK1k-x1k-δ1k-K1k-α1k∑s=0+∞J2sx2k-s∑s=0+∞J2sx2k-s=x1kexpr1kK1k-x1k-δ1k-K1k-α1k=x1kexpr1kα1k-x1k-δ1k≥x1kexpr1lα1l-r1uM1+ε;we mention here that, from (25), r1lα1l-r1u(M1+ε)<0. By using (26), we obtain (27)x1k-δ1k≤x1kexp-r1lα1l-r1uM1+εδ.Substituting (27) into (4), using (24), for k≥N1+δ, it follows that (28)x1k+1≥x1kexpr1kK1k-x1k-δ1k-K1k-α1k∑s=0+∞J2sx2k-s1+∑s=0+∞J2sx2k-s≥x1kexpr1lK1l-r1uexp-r1lα1l-r1uM1+εδx1k-K1u-α1l∑s=0+∞J2sx2k-s1+∑s=0+∞J2sx2k-s≥x1kexpr1lK1l-r1uexp-r1lα1l-r1uM1+εδx1k-r1uK1u-α1lM2+ε1+M2+ε.Thus, as a direct corollary of Lemma 3, according to (23) and (28), one has (29)liminfk→+∞ x1k≥minA1ε,A2ε,(30)A1ε=Δ1εr1uexpr1lα1l-r1uM1+εδ,A2ε=A1εexpΔ1ε-r1uexp-r1lα1l-r1uM1+εδM1,Δ1ε=r1lK1l-r1uK1u-α1lM2+ε1+M2+ε.Letting ε→0, it follows that (31)liminfk→+∞ x1k≥12minA1,A2≝m1>0,where (32)A1=Δ1r1uexpr1lα1l-r1uM1δ,A2=A1expΔ1-r1uexp-r1lα1l-r1uM1δM1,Δ1=r1lK1l-r1uK1u-α1lM21+M2.Similar to the analysis of (26)–(29), by applying (24), from (7), we also have (33)liminfk→+∞ x2k≥12minB1,B2≝m2>0,where (34)B1=Δ2r2uexpr2lα2l-r2uM2δ,B2=B1expΔ2-r2uexp-r2lα2l-r2uM2δM2,Δ2=r2lK2l-r2uK2u-α2lM11+M1.(20), (21), (31), and (33) show that system (2) is permanent. The proof of the theorem is completed.