Permanence of a discrete multispecies Lotka–Volterra competition predator–prey system with delays

Nonlinear Analysis: Real World Applications 9 (2008) 2185 – 2195 www.elsevier.com/locate/na

Permanence of a discrete multispecies Lotka–Volterra competition predator–prey system with delays Na Fang∗ , Xiao Xing Chen College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, People’s Republic of China Received 5 November 2006; accepted 17 July 2007

Abstract In this paper, we propose a discrete multispecies Lotka–Volterra competition predator–prey system with delays. For general nonautonomous case, sufficient conditions are established for the permanence of the system. 䉷 2007 Elsevier Ltd. All rights reserved. MSC: 34D40; 34D20; 34K20; 92D25 Keywords: Permanence; Delay; Lotka–Volterra competition predator–prey system; Discrete; Nonautonomous

1. Introduction In theoretical ecology, it is important whether or not all species in a multispecies community can be permanent. There have been many studies for the permanence of models governed by differential systems (see [2,7,9,11,13,18,21] and the references cited therein). For example, for two-species Lotka–Volterra differential systems, it is known that time delays are harmless for the permanence of a predator–prey system (see [21]) and a competition system (see [13,18]). On the other hand, in the last two decades, several papers (see, for example, [4,5,8,10,14,15]) have appeared on the permanence of discrete models of Lotka–Volterra type without delay. Recently, some scholars are starting studying the permanence of discrete Lotka–Volterra systems with delays (see [12,17,16,24] and the references cited therein). Saito et al. [17] considered the permanence of the following discrete Lotka–Volterra competition system with delays: x(n + 1) = x(n) · exp{r1 [1 − x(n − k1 ) − 1 y(n − k2 )]}, y(n + 1) = y(n) · exp{r2 [1 − 2 x(n − l1 ) − y(n − l2 )]}.

(1.1)

The initial condition of (1.1) is given as x(−m) 0, m = 0, 1, . . . , max{k1 , l1 },

x(0) > 0,

y(−m)0, m = 0, 1, . . . , max{k2 , l2 },

y(0) > 0.

∗ Corresponding author.

E-mail addresses: [email protected] (N. Fang), [email protected] (X.X. Chen). 1468-1218/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2007.07.005

(1.2)

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Here r1 , r2 , 1 and 2 are constants with r1 > 0, r2 > 0, 1 0 and 2 0, and delays k1 , k2 , l1 and l2 are nonnegative integers. However, the biological population does not all rely on the mutual competition between populations to produce. There is massive population relationship between predators and preys. Now more and more scholars are realizing the importance of such relationship in biosphere. Recently, Saito et al. [16] further studied the permanence of the following discrete Lotka–Volterra predator–prey system with delays: ⎤ ⎡ m m   a1j x(n − k1j ) − b1j y(n − k2j )⎦ , x(n + 1) = x(n) exp ⎣r1 − ⎡ y(n + 1) = y(n) exp ⎣r2 +

j =1

j =1

m 

m 

a2j x(n − l1j ) −

j =1

⎤ b2j y(n − l2j )⎦ .

(1.3)

j =1

The initial condition of (1.3) is given as x(−v) 0, v = 0, 1, . . . , k,

x(0) > 0,

y(−v) 0, v = 0, 1, . . . , k,

y(0) > 0.

(1.4)

Here r1 and r2 are constants, and aij and bij (i = 1, 2; j = 1, 2, . . . , m) are nonnegative constants. Not all of a1j and not all of b2j (j = 1, 2, . . . , m) are zero. Delays kij and lij (i = 1, 2; j = 1, 2, . . . , m) are nonnegative integers. However, there are few papers studying the permanence of discrete multispecies Lotka–Volterra predator–prey systems. Recently, Chen [3] first proposed the following discrete n + m-species Lotka–Volterra competition predator–prey system   n m   ail (k)xl (k) − cil (k)yl (k) , xi (k + 1) = xi (k) exp bi (k) − 

l=1

yj (k + 1) = yj (k) exp −rj (k) +

l=1

n 

dj l (k)xl (k) −

l=1

m 

 ej l (k)yl (k) ,

(1.5)

l=1

sufficient conditions on the permanence and global stability of system (1.5) are obtained. However, the discrete system (1.5) ignores the effect of more than one past generation. And the recent studies of the dynamics of natural populations indicate that the density-dependent population regulation probably takes place over many generations [6,20,19]. Note that Wang et al. [22] studied the following Lotka–Volterra model with delays: ⎧ ⎫ m n  ⎨ ⎬  xi (k + 1) = xi (k) exp ri (k) − aijl (k)xj (k − l) , i = 1, 2, , n. (1.6) ⎩ ⎭ j =1 l=0

They obtained the sufficient conditions which guarantee the global stability of system (1.6). As was pointed out by Berryman [1], the dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. For the above reasons, we will consider the permanence of the following system (1.7) with delays. The model studied by this paper is ⎧ ⎫ h1 h2 n  m  ⎨ ⎬   ails1 (k)xl (k − s1 ) − cils2 (k)yl (k − s2 ) , xi (k + 1) = xi (k) · exp bi (k) − ⎩ ⎭ l=1 s1 =0

l=1 s2 =0

⎧ ⎫ h3 h4 n  m  ⎨ ⎬   s dj 3l (k)xl (k − s3 ) − ejs4l (k)yl (k − s4 ) , yj (k + 1) = yj (k) · exp −rj (k) + ⎩ ⎭ l=1 s3 =0

l=1 s4 =0

(1.7)

N. Fang, X.X. Chen / Nonlinear Analysis: Real World Applications 9 (2008) 2185 – 2195

2187

where i = 1, 2, . . . , n; j = 1, 2, . . . , m. The initial condition of (1.7) is given as xi (−v) 0, v = 0, 1, . . . , max{h1 , h3 },

xi (0) > 0,

yj (−v)0, v = 0, 1, . . . , max{h2 , h4 },

yj (0) > 0.

(1.8)

Here xi (k) is the density of prey species i at kth generation. yj (k) is the density of predator species j at kth generation. ails1 (k) and ejs4l (k) measures the intensity of intraspecific competition or interspecific action of prey species and predator species, respectively. bi (k) representing the intrinsic growth rate of the prey species xi ; rj (k) representing the death rate of the predator species yj . s Throughout this paper, we assume that bi (k), rj (k) are bounded positive sequences, and ails1 (k), cils2 (k), dj 3l (k), ejs4l (k) are bounded nonnegative sequences. We use the following notations for any bounded sequences {f (k)}: f¯ = sup f (k),

f = inf f (k). k∈N

k∈N

And not all of a sii1 and not all of esjj4 are zero, where 0 s1 h1 , 0 s4 h4 . We say that system (1.7) is permanent if there are positive constants mi , mj , Mi and Mj such that for each positive solution (x1 (k), . . . , xn (k), y1 (k), . . . , ym (k)) of system (1.7) satisfies mi  lim inf xi (k) lim sup xi (k)Mi , k→+∞

k→+∞

mj  lim inf yj (k) lim sup yj (k)Mj , k→+∞

k→+∞

for all i = 1, 2, . . . , n; j = 1, 2, . . . , m. The aim of this paper is, by improving system (1.3), the methods used in [16] could not be used directly in this paper. By developing the analysis technique of [17,16,24], using the theory of difference equation, computing the system thoroughly, sufficient conditions to ensure the permanence of system (1.7) are obtained. The methods used in this paper are different from that in Chen [3], and by adding the delays to system (1.5), the former results are extended to more large range. The organization of this paper is the following. In the next section, we state and prove the main results of this paper. 2. Main result In order to present our main result, Theorem 2.1, we need some preparations. Lemma 2.1 (Yang [23, Lemma 2]). Assume that {x(n)} satisfies x(n + 1) x(n) · exp{r(n)(1 − ax(n))},

n N0 ,

lim supn→∞ x(n)K and x(N0 ) > 0, where a is a constant such that aK > 1 and N0 ∈ N . Then lim inf x(n) n→∞

1 · exp{¯r (1 − aK)}. a

 Before proceeding, we make a convention that ni=m F (i) = 1 if m > n. The main result, Theorem 2.1, will follow directly from the following four propositions. Proposition 2.1. For every solution (x1 (k), . . . , xn (k), y1 (k), . . . , ym (k)) of system (1.7) with the initial condition (1.8), we have lim sup xi (k) x¯i ,

(2.1)

k→+∞

where



b¯i

x¯i = min h 1

s1 s1 =0 a ii

exp{(h1 + 1)b¯i }, h

1

s1 1 s1 =0 a ii

 exp{(h1 + 1)b¯i − 1} .

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Proof. Clearly, xi (k) > 0 and yj (k) > 0 (i = 1, . . . , n, j = 1, . . . , m) for k 0. To prove (2.1), we divide the proof into two cases. (i) The case b¯i  1 . We first claim that for any sufficiently small  > 0, there exists a large K0 > 0 such that e

¯

¯

xi (K0 ) hb1i + s1 . If not, we have xi (k) > hb1i + s1 for all large k. Hence, from (1.7) we obtain that s1 =0

a ii

s1 =0

⎧ ⎨

a ii

⎫ ⎬

h1 

xi (k  + 1)xi (k  ) · exp bi (k  ) − aiis1 (k  )xi (k  − s1 ) ⎩ ⎭ s1 =0 ⎧ ⎫ h1 ⎨ ⎬   ¯ xi (k ) · exp bi − a sii1 xi (k  − s1 ) < xi (k  ) exp{−} ⎩ ⎭ s1 =0

¯

for all large k  k + h1 . This implies xi (k  ) → 0 as k  → +∞, which is a contradiction to xi (k) > hb1i + s1 for all large k. Next, we claim that for any m > 0 and v > 0, b¯i +  xi (m + v)  h exp{v b¯i } s1 1 s1 =0 a ii

s1 =0

a ii

(2.2)

¯

¯

if xi (m)  hb1i + s1 . If fact, let xi (m)  hb1i + s1 , then from (1.7) we have s1

=0 a ii

s1 =0

a ii

b¯i +  xi (m + v) xi (m + v − 1) exp{b¯i } xi (m) exp{v b¯i } h exp{v b¯i }. s1 1 a s1 =0 ii Thus, b¯i +  exp{(h1 + 1)b¯i } xi (k) h s1 1 s1 =0 a ii for K0 k K0 + h1 + 1. Now, we will show that for all k > K0 + h1 + 1, b¯i +  xi (k) h exp{(h1 + 1)b¯i }. s1 1 s1 =0 a ii Otherwise, there exists a K1 K0 + h1 + 1 such that b¯i +  xi (k) h exp{(h1 + 1)b¯i } s1 1 s1 =0 a ii for K1 k K0 , and b¯i +  xi (K1 + 1) > h exp{(h1 + 1)b¯i }. s1 1 a s1 =0 ii

(2.3)

On the other hand, from (2.2) and (2.3) we can show that b¯i +  xi (K1 − s1 ) > h s1 1 s1 =0 a ii

(2.4)

N. Fang, X.X. Chen / Nonlinear Analysis: Real World Applications 9 (2008) 2185 – 2195

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for 0 s1 h1 . Hence, from (1.7) and (2.4), ⎧ ⎫ h1 ⎨ ⎬  xi (K1 + 1)xi (K1 ) · exp b¯i − a sii1 xi (K1 − s1 ) < xi (K1 ) exp{−} ⎩ ⎭ s1 =0

b¯i +  < h exp{(h1 + 1)b¯i }. s1 1 a s1 =0 ii This is a contradiction to (2.3). Therefore, it follows from the arbitrariness of  that lim sup xi (k) h

b¯i

1 s1 =0

k→+∞

a sii1

exp{(h1 + 1)b¯i } = x¯i .

(ii) The case b¯i > 1e . By (1.7) we have xi (k − s1 )xi (k) exp{−s1 b¯i } xi (k) exp{−h1 b¯i },

(2.5)

when 0 s1 h1 , and k is large enough. Hence, from (1.7) and (2.5), ⎫ ⎧ h1 ⎬ ⎨  xi (k + 1) xi (k) · exp b¯i − a sii1 xi (k − s1 ) ⎭ ⎩ s1 =0    h1 s1 a ii s =0 1 xi (k) · exp b¯i 1 − exp{−h1 b¯i }xi (k) b¯i

(2.6)

for all large k. Let us define f (x) = x exp{r(1 − ax)}, where r and a are positive constants. It is easy to see that max

0  x<+∞

f (x) =

1 exp{r − 1}. ra

Then, it follows from (2.6) that for all large k, 1

xi (k + 1) h

s1 1 s1 =0 a ii

= h 1

1

s1 s1 =0 a ii

exp{−h1 b¯i }

exp{b¯i − 1}

exp{(h1 + 1)b¯i − 1} = x¯i .

Clearly, lim supk→+∞ xi (k) x¯i . This completes the proof of Proposition 2.1.



Proposition 2.2. Assume that −r j +

h3 n  

s d¯ j3l x¯l > 0

(H1 )

l=1 s3 =0

holds, where x¯l (l = 1, . . . , n) are defined by (2.1). Then for every solution (x1 (k), . . . , xn (k), y1 (k), . . . , ym (k)) of system (1.7) with the initial condition (1.8), we have lim sup yj (k) y¯j , k→+∞

(2.7)

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where

N. Fang, X.X. Chen / Nonlinear Analysis: Real World Applications 9 (2008) 2185 – 2195

⎧ ⎧ ⎫ h3 n  ⎨ −r + n h3 d¯ s3 x¯l ⎨ ⎬  j l=1 s3 =0 j l s ¯ 3 x¯l ) , y¯j = min d · exp (h + 1)(−r + 4  j jl s4 h4 ⎩ ⎩ ⎭ l=1 s =0 s4 =0 ejj ⎫⎫ 3 ⎧ h3 n  ⎬⎬ ⎨  1 ¯ s3 x¯l ) − 1 d + . · exp (h + 1)(−r 4 h4 j j l ⎭⎭ ⎩ es4 l=1 s3 =0

s4 =0 jj

Proof. For sufficiently small 1 > 0, according to Proposition 2.1, there exists a K2 > 0 such that for k K2 , xi (k) x¯i + 1 . Thus, it follows from (1.7) that for k K2 + h3 , ⎧ ⎫ h3 h4 n  m  ⎨ ⎬   s yj (k + 1) yj (k) · exp −r j + d¯ j3l (x¯l + 1 ) − esj4l yl (k − s4 ) . ⎩ ⎭ l=1 s3 =0

(2.8)

l=1 s4 =0

To prove (2.7), we divide the proof into two cases.  h s (i) The case −r j + nl=1 s33=0 d¯ j3l x¯l  1e . By using the same arguments as Proposition 2.1, We first obtain that for any sufficiently small 1 > 0, there exists a large K3 K2 + h3 such that  h s −r j + nl=1 s33=0 d¯j 3l (x¯l + 1 ) + 1 yj (K3 ) . h4 s4 s4 =0 ejj Next, we obtain that for m K2 + h3 and v > 0, yj (m + v) 

−r j +

n

⎧ ⎫ h3 n  ⎨ ⎬ ¯ s3 (x¯l + 1 ) + 1  d s3 =0 j l ¯ s3 (x¯l + 1 )) d exp v(−r + h4 j j l s4 ⎩ ⎭ l=1 s3 =0 s4 =0 ejj

h3

l=1

(2.9)

if yj (m)

−r j +

n

h3

l=1

¯ s3 s3 =0 dj l (x¯ l h4 s4 s4 =0 ejj

+ 1 ) +  1

.

Thus, yj (k)

−r j +

n l=1

h3

¯ s3 s3 =0 d j l (x¯ l h4 s4 s4 =0 ejj

+ 1 ) +  1





exp (h4 + 1) −r j +

n l=1

h3 s3 =0



s d¯ j3l (x¯l + 1 )

for K3 k K3 + h4 + 1. The rest of the proof is similar to the proof of Proposition 2.1 and we omit it. The proof is complete.  Let x¯i and y¯j be the positive constants defined by (2.1) and (2.7). We introduce    n m h2 h1 h1  s1 h1 u1 u1 u2 · exp ( x ¯ a ¯ + a ¯ x ¯ + c ¯ y ¯ − b )s a ¯ i l l 1 i l=1,l = i l=1 ii ii il il s =0 u =0 u =0 u =0 1 1 1 2 (1) i = , m h2 n h1 s2 s1 bi − l=1 s2 =0 c¯il y¯l − l=1,l=i s1 =0 a¯ il x¯l ⎧⎛ ⎫ ⎞ h2 h1 m  n ⎨ ⎬    1 (1) x i = (1) · exp ⎝b¯i − csil2 y¯l − a sil1 x¯l ⎠ (1 − i x¯i ) , ⎩ ⎭ i l=1 s2 =0 l=1,l=i s1 =0

N. Fang, X.X. Chen / Nonlinear Analysis: Real World Applications 9 (2008) 2185 – 2195



   h4  n h3 u3 s4 u4 e¯jj · exp (y¯j hu44 =0 e¯ujj4 + m e ¯ y ¯ + r ¯ − d x )s j l=1,l=j l=1 u4 =0 j l l u3 =0 j l l 4 , n h3 m h4 s3 s4 −¯rj + l=1 s3 =0 d j l x l − l=1,l=j s4 =0 e¯j l y¯l

h4

s4 =0

(2)

j =

yj =

1 (2)

j

2191

⎧⎛ ⎫ ⎞ h3 h4 n  m ⎨ ⎬    s (2) d¯ j3l x l − · exp ⎝−r j + esj4l y¯l ⎠ (1 − j y¯j ) , ⎩ ⎭ l=1 s3 =0

l=1,l=j s4 =0

for all i = 1, 2, . . . , n, j = 1, 2, . . . , m. Proposition 2.3. In addition to (H1 ), assume further that (1)

x¯i i > 1

(H2 )

holds for all i = 1, 2, . . . , n, where x¯i are defined by (2.1). Then for every solution (x1 (k), . . . , xn (k), y1 (k), . . . , ym (k)) of (1.7), we have lim inf xi (k)x i . k→+∞

Proof. Condition (H2 ) implies that for enough small positive number ε > 0, one has (1)

x¯i iε > 1, where (1)

iε =

i = 1, 2, . . . , n,

⎧ h1 ⎨ 

⎡⎛ a¯ iis1 · exp ⎣⎝(x¯i + ε)

s1 =0

⎩

h1  u1 =0

h1 n  

a¯ uii1 +

a¯ uil1 (x¯l + ε)

l=1,l=i u =0

1 ⎞ ⎤⎫ ⎛ ⎞ h2 h1 h m  n m 2 ⎬  u    ⎝bi − + c¯il2 (y¯l + ε) − bi ⎠ s1 ⎦ c¯sil2 (y¯l + ε) − a¯ sil1 (x¯l + ε) ⎠ . ⎭

l=1 u2 =0

l=1 s2 =0

l=1,l=i s1 =0

For above ε > 0, from Propositions 2.1 and 2.2 we know that there exists enough large K5 such that xi (k) x¯i + ε,

yj (k) y¯j + ε, for all k K5 .

Let K6 = K5 + g1 (g1 = max{h1 , h2 }). Thus, by using ith equation in system (1.7), for k K6 , one has ⎧ h2 m  ⎨  cils2 (k)(y¯l + ε) xi (k + 1)xi (k) exp bi (k) − ⎩ l=1 s2 =0 ⎫ h1 h n 1 ⎬   s  − ail1 (k)(x¯l + ε) − aiis1 (k)xi (k − s1 ) . ⎭ l=1,l=i s1 =0

s1 =0

Then, for 0 s1 h1 and k K6 + h1 , ⎛ k−1

k−1

xi (p + 1)  p=k−s1

p=k−s1

−

⎧ h2 m  ⎨  ⎝xi (p) · exp bi (p) − cilu2 (p)(y¯l + ε) ⎩

n 

l=1 u2 =0

h1 

l=1,l=i u1 =0

ailu1 (p)(x¯l + ε) −

h1  u1 =0

⎫⎞ ⎬ aiiu1 (p)xi (k − u1 ) ⎠ , ⎭

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N. Fang, X.X. Chen / Nonlinear Analysis: Real World Applications 9 (2008) 2185 – 2195

or, equivalently, ⎧ ⎛ h1 h1 k−1 n ⎨     u1 ⎝ (x¯i + ε) aii (p) + ailu1 (p)(x¯l + ε) xi (k − s1 )xi (k) · exp ⎩ u1 =0

p=k−s1

l=1,l=i u1 =0

⎞⎫ h2 m  ⎬  + cilu2 (p)(y¯l + ε) − bi (p)⎠ . ⎭ l=1 u2 =0

This, combined with ith equation in system (1.7), gives us ⎧ h1 h2 n m  ⎨    cils2 (k)(y¯l + ε) − ails1 (k)(x¯l + ε) xi (k + 1) xi (k) exp bi (k) − ⎩ l=1 s2 =0

h1 

−

s1 =0

⎧ ⎛ h1 k−1 ⎨   ⎝(x¯i + ε) aiiu1 (p) aiis1 (k)xi (k) · exp ⎩ u1 =0

p=k−s1

h1 n  

+

l=1,l=i s1 =0

ailu1 (p)(x¯l

+ ε) +

h2 m  

cilu2 (p)(y¯l

l=1 u2 =0

l=1,l=i u1 =0

⎞⎫⎫ ⎬⎬ + ε) − bi (p)⎠ ⎭⎭

⎧⎛ ⎞ h2 h1 m  n ⎨    s s xi (k) · exp ⎝bi (k) − cil2 (k)(y¯l + ε) − ail1 (k)(x¯l + ε)⎠ ⎩ l=1 s2 =0

⎫ ⎬ (1) × (1 − iε xi (k)) ⎭

l=1,l=i s1 =0

for k K6 + h1 .

And note that (1)

(1)

lim iε = i ,

ε→0

⎧ ⎨ lim

b¯i −

ε→0 ⎩

= b¯i −

h2 m  

csil2 (y¯l + ε) −

l=1 s2 =0 h2 m  

csil2 y¯l

h1 n   l=1,l=i s1 =0

−

l=1 s2 =0

h1 n  

⎫ ⎬

a sil1 (x¯l + ε)

⎭

a sil1 x¯l .

l=1,l=i s1 =0

It follows from Lemma 2.1 that lim inf xi (k)x i . k→+∞

This completes the proof of Proposition 2.3.



Proposition 2.4. In addition to (H1 ) and (H2 ), assume further that (2)

y¯j j > 1,

(H3 )

N. Fang, X.X. Chen / Nonlinear Analysis: Real World Applications 9 (2008) 2185 – 2195

2193

holds for all j =1, 2, . . . , m, where y¯j are defined by (2.7). Then for every solution (x1 (k), . . . , xn (k), y1 (k), . . . , ym (k)) of (1.7), we have lim inf yj (k)y j . k→+∞

Proof. From (H3 ) one could take ε > 0 enough small, such that (2)

y¯j j ε > 1,

j = 1, 2, . . . , m,

where (2)

j ε =

⎧ h4 ⎨ 

⎡⎛ e¯sjj4 · exp ⎣⎝(y¯j + ε)

s4 =0

−

⎩

h4 

e¯ujj4 +

u4 =0

h3 n  

u d j l3 (x l

l=1 u3 =0

m 

h4 

e¯uj l4 (y¯l + ε) + r¯j

l=1,l=j u4 =0

⎞ ⎤⎫ ⎛ ⎞ h3 h4 n  m ⎬    s s ⎝−¯rj + − ε)⎠ s4 ⎦ d j3l (x l − ε) − e¯j4l (y¯l + ε) ⎠ . ⎭ l=1 s3 =0

l=1,l=j s4 =0

For above ε > 0, according to Propositions 2.1–2.3, there exists a K7 K6 + h1 such that for all k K7 , xi (k) x¯i + ε,

yj (k) y¯j + ε,

xi (k)x i − ε.

Let K8 = K7 + g2 (g2 = max{h3 , h4 }). Thus, by using n + j th equation in system (1.7), for k K8 , one has ⎧ ⎨

yj (k + 1)yj (k) exp −rj (k) + ⎩

−

h4  s4 =0

h3 n  

h4 m  

s

dj 3l (k)(x l − ε) −

l=1 s3 =0

ejs4l (k)(y¯l + ε)

l=1,l=j s4 =0

⎫ ⎬

s4 ejj (k)yj (k − s4 ) . ⎭

Then, for 0 s4 h4 and k K8 + h4 , k−1

⎧ ⎨

k−1

yj (p + 1) p=k−s4

p=k−s4

yj (p) exp −rj (p) + ⎩

m 

−

h4 

h3 n   l=1 u3 =0

ejul4 (p)(y¯l + ε) −

l=1,l=j u4 =0

u

dj l3 (p)(x l − ε)

h4  u4 =0

⎫ ⎬

u4 ejj (p)yj (p − s4 ) , ⎭

or, equivalently, ⎧ ⎛ h4 k−1 m ⎨    u4 ⎝(y¯j + ε) yj (k − s4 )yj (k) · exp ejj (p) + ⎩ p=k−s4

+rj (p) −

h3 n   l=1 u3 =0

u4 =0

⎞⎫ ⎬ u dj l3 (p)(x l − ε)⎠ . ⎭

h4 

l=1,l=j u4 =0

ejul4 (p)(y¯l + ε)

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N. Fang, X.X. Chen / Nonlinear Analysis: Real World Applications 9 (2008) 2185 – 2195

This, combined with n + j th equation in system (1.7), gives us ⎧ h3 h4 n  m ⎨    s dj 3l (k)(x l − ε) − ejs4l (k)(y¯l + ε) yj (k + 1)yj (k) exp −rj (k) + ⎩ l=1,l=j s4 =0

l=1 s3 =0

⎧ ⎛ h4 h4 k−1 ⎨    u4 s4 ⎝ (y¯j + ε) ejj (p) − ejj (k)yj (k) · exp ⎩ s4 =0

m 

+

u4 =0

p=k−s4

h4 

ejul4 (p)(y¯l + ε) + rj (p) −

h3 n   l=1 u3 =0

l=1,l=j u4 =0

⎞⎫⎫ ⎬⎬ u dj l3 (p)(x l − ε)⎠ ⎭⎭

⎧⎛ ⎞ h3 h4 n  m ⎨    s yj (k) · exp ⎝−rj (k) + dj 3l (k)(x l − ε) − ejs4l (k)(y¯l + ε)⎠ ⎩ l=1,l=j s4 =0

l=1 s3 =0

⎫ ⎬ (2) × (1 − j ε yj (k)) , ⎭

for k K8 + h4 .

And note that (2)

(2)

lim j ε = j ,

ε→0

⎧ ⎨

lim

−r j +

h3 n  

ε→0 ⎩

= −r j +

s d¯ j3l (x l − ε) −

l=1 s3 =0 h3 n   l=1 s3 =0

h4 m  

s d¯j 3l x l −

l=1,l=j s4 =0 h4 m  

⎫ ⎬

esj4l (y¯l + ε)

⎭

esj4l y¯l .

l=1,l=j s4 =0

Again, it follows from Lemma 2.1 that lim inf yj (k)y j . k→+∞

This completes the proof of Proposition 2.4.



According to Propositions 2.1–2.4, we have proved the main result of this paper, which is stated below. Theorem 2.1. Assume that (H1 )–(H3 ) hold, then system (1.7) with the initial condition (1.8) is permanent. Remark. When h1 = h2 = h3 = h4 = 0, system (1.7) becomes system (1.6). The results of Proposition 2.1–2.2 are more accurate than the results of Propositions 1–2 of [3], respectively. The results of Propositions 2.3–2.4 are similar to the results of Propositions 3–4 of [3]. The lower bounds of the species xi and yj are greater than or equal to the lower bounds of the species xi and yj of [3], respectively. When b¯i = bi , c¯il = cil , a¯ il = a il and¯rj = r j , d¯j l = d j l , e¯j l = ej l , the lower bounds of the species xi and yj are equal to the lower bounds of the species xi and yj of [3], respectively. So our results are a generalization of the results of [3]. Acknowledgements This work is supported by the Natural Science Foundation of Fujian Province (Z0511014), the Foundation of Developing Science and Technology of Fuzhou University (2005-XQ-18).

N. Fang, X.X. Chen / Nonlinear Analysis: Real World Applications 9 (2008) 2185 – 2195

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