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Volume 2010, Article ID 249364, 9 pagesdoi:10.1155/2010/249364 Research Article Note on the Persistent Property of a Discrete Lotka-Volterra Competitive System with Delays and Feedback C

Volume 2010, Article ID 249364, 9 pages

doi:10.1155/2010/249364

Research Article

Note on the Persistent Property of a Discrete

Lotka-Volterra Competitive System with Delays and Feedback Controls

Xiangzeng Kong,1, 2 Liping Chen,1, 2 and Wensheng Yang1, 2

1 Key Lab of Network Security and Cryptology, Fujian Normal University, Fuzhou 350007, China

2 School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China

Correspondence should be addressed to Xiangzeng Kong,xzkong@fjnu.edu.cn

Received 26 June 2010; Accepted 12 September 2010

Academic Editor: P J Y Wong

Copyrightq 2010 Xiangzeng Kong et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

A nonautonomous N-species discrete Lotka-Volterra competitive system with delays and feedback

controls is considered in this work Sufficient conditions on the coefficients are given to guarantee that all the species are permanent It is shown that these conditions are weaker than those of Liao

et al 2008

1 Introduction

Traditional Lotka-Volterra competitive systems have been extensively studied by many authors1 7.The autonomous model can be expressed as follows:

ui t  bi u it

⎡

⎣1 −N

j1

a ij u jt

⎤

⎦, i  1, , N, 1.1

where b i > 0, a ii > 0, a ij ≥ 0 i / j, uitdenoting the density of the ith species at time t Montes

de Oca and Zeeman6 investigated the general nonautonomous N-species Lotka-Volterra

competitive system

ui t  uit

⎡

⎣bit −N

j1

c ijtujt

⎤

⎦, cij ≥ 0, i  1, , N, 1.2

and obtained that if the coefficients are continuous and bounded above and below by positive

constants, and if for each i  2, , N, there exists an integer ki < i such that

b i

c ij <

b ki

c k i j , j  1, , i, 1.3

then ui → 0 exponentially for 2 ≤ i ≤ N, and uit → X∗, where X∗is a certain solution of

a logistic equation Teng8 and Ahmad and Stamova 9 also studied the coexistence on a nonautonomous Lotka-Volterra competitive system They obtained the necessary or sufficient conditions for the permanence and the extinction For more works relevant to system1.1, one could refer to1 9 and the references cited therein

However, to the best of the authors’ knowledge, to this day, still less scholars consider the general nonautonomous discrete Lotka-Volterra competitive system with delays and feedback controls Recently, in1 Liao et al considered the following general nonautonomous discrete Lotka-Volterra competitive system with delays and feedback controls:

x in 1  xin exp

⎧

⎨

⎩b in −

N



j1

a ijnxj n − τij− dinuin

⎫

⎬

⎭,

Δuin  rin − einuin cinxin − σi, i  1, 2, , N,

x iθ  φiθ ≥ 0, θ ∈ N−τ, 0 : {−τ, −τ 1, , −1, 0},

1.4

where x in i  1, 2, , N is the density of competitive species; uin is the control variable;

e in : Z → 0, 1; bounded sequences rin, cin, bin, aijn, and din : Z → R ; τ ij and σiare positive integer;Z, R denote the sets of all integers and all positive real numbers, respectively;Δ is the first-order forward difference operator Δuin  uin 1 − uin; τ 

max{max1≤i,j≤Nτ ij , max1≤i≤Nσ i} > 0.

In1, Liao et al obtained sufficient conditions for permanence of the system 1.4 They obtained what follows

Lemma 1.1 Assume that

min

hold, then system1.4 is permanent, where

M iΔi exp



b u i − 1

a l

iiexp

−b u

i τ ii ·a

u

iiexp

τ iiN

j1a u ij M j Wi d u i − b l

i



b l

i−N

j 1,j / i a u

ij M j − d u

i W i ,

W i r i u c u

i M i

e l i

, M i exp b u i − 1

a l

iiexp −b u

i τ ii

1.6

exp

b u i − 1> 0, a l iiexp

−b u

i τ ii

> 0, a u iiexp

⎧

⎨

⎩τ ii

⎛

⎝N

j1

a u ij M j Wi d u i − b l

i

⎞

⎠

⎫

⎬

⎭> 0.

1.7

Hence, the above inequality1.5 implies

b l i− N

j 1,j / i

a u ij M j − d u

i W i > 0. 1.8

That is

b l i >

N



j 1,j / i

a u ij M j d u

i W i

 N

j 1,j / i

a u ij M j d u

i

r u

i c u

i M i

e i l

 N

j 1,j / i

a u ij M j d i u r i u

e l i

d u i c u i M i

e l i

.

1.9

It was shown that in [ 1 ] Liao et al considered system1.4 where all coefficients rin, cin, din,

a ijn, ein, and bin were assumed to satisfy conditions 1.9.

In this work, we shall study system1.4 and get the same results as 1 do under the weaker assumption that

b l

i >

N



j 1,j / i

a u

ij M j d i u r i u

e l i

Our main results are the followingTheorem 1.2

Theorem 1.2 Assume that 1.10 holds, then system 1.4 is permanent.

Remark 1.3 The inequality1.9 implies 1.10, but not conversely, for

N



j 1,j / i

a u ij M j d i u r i u

e l i

≤ N

j 1,j / i

a u ij M j d i u r i u

e l i

d u i c u i M i

e l i

Therefore, we have improved the permanence conditions of1 for system 1.4

Theorem 1.2 will be proved in Section 2 In Section 3, an example will be given to illustrate that1.10 does not imply 1.9; that is, the condition 1.10 is better than 1.9

2 Proof of Theorem 1.2

The following lemma can be found in10

Lemma 2.1 Assume that A > 0 and y0 > 0, and further suppose that

(1)

y n 1 ≤ Ayn Bn, n  1, 2, 2.1

Then for any integer k ≤ n,

y n ≤ A k y n − k k−1

i0

A i B n − i − 1. 2.2

Especially, if A < 1 and B is bounded above with respect to M, then

lim

n→ ∞sup yn ≤ M

2

y n 1 ≥ Ayn Bn, n  1, 2, 2.4

Then for any integer k ≤ n,

y n ≥ A k y n − k k−1

i0

A i B n − i − 1. 2.5

Especially, if A < 1 and B is bounded below with respect to m∗, then

lim

n→ ∞inf yn ≥ m∗

Following comparison theorem of difference equation is Theorem 2.1 of [ 11 , page 241].

Lemma 2.2 Let n ∈ N

n0  {n0, n0 1, , n0 l, }, r ≥ 0 For any fixed n, gn, r is a

nondecreasing function with respect to r, and for n ≥ n0, following inequalities hold: y n 1 ≤

g n, yn, un 1 ≥ gn, un If gn0  ≤ un0, then yn ≤ un for all n ≥ n0.

Now let us consider the following single species discrete model:

N n 1  Nn exp{an − bnNn}, 2.7

where{an} and {bn} are strictly positive sequences of real numbers defined for n ∈ N  {0, 1, 2, } and 0 < a l ≤ a u , 0 < b l ≤ b u Similarly to the proof of Propositions 1 and 3 in12,

we can obtain the following

Lemma 2.3 Any solution of system 2.7 with initial condition N0 > 0 satisfies

m≤ lim

n→ ∞inf Nn ≤ lim

n→ ∞sup Nn ≤ M, 2.8

where

M 1

b lexp{au − 1}, m a l

b uexp

a l − b u M

The following lemma is direct conclusion of1

Lemma 2.4 Let xn  x1n, x2n, , xNn, u1n, u2n, , uN n denote any positive

solution of system1.4.Then there exist positive constants Mi , W ii  1, 2, , N such that

lim

n→ ∞sup xin ≤ Mi , lim

n→ ∞sup uin ≤ Wi , i  1, 2, , N, 2.10

where

M i exp b i u− 1

a l

iiexp −b u

i τ ii , W i r i u c u

i M i

e l i

i  1, 2, , N. 2.11

Proposition 2.5 Suppose assumption 1.10 holds, then there exist positive constant mi and w i such that

lim

n→ ∞inf x in ≥ mi , lim

n→ ∞inf u in ≥ wi 2.12

Proof We first prove lim

n→ ∞inf x in ≥ mi

ByLemma 2.4and by the first equation of system1.4, we have

x in 1  xin exp

⎧

⎨

⎩b in −

N



j1

a ij nxj n − τij− dinuin

⎫

⎬

⎭

≥ xin exp

⎧

⎨

⎩b in −

N



j1

a ij M j ε− dinWi ε

⎫

⎬

⎭

2.13

for n sufficiently large, then

n−1



s n−τ ii

x is 1

x is ≥ exp

⎧

⎨

⎩

n−1



s n−τ ii

⎛

⎝bis −N

j1

a ijs M j ε− disWi ε

⎞

⎠

⎫

⎬

⎭. 2.14

x in − τii ≤ xin exp

 n−1

s n−τ ii

D is

where

D is N

j1

a ijs M j ε disWi ε − bis. 2.16

From the second equation of system1.4, we have

u in  1 − einuin cinxin − σi rin

≤1− e l

i



u in cinxin − σi rin

: Aiu in Bin.

2.17

Then,Lemma 2.1implies that for any k ≤ n − τii,

u in ≤ A k

i u in − k k−1

j0

A j i B i n − j − 1

 A k

i u in − k k−1

j0

A j i

r i n − j − 1 ci n − j − 1x i n − j − 1 − σi

≤ A k

i u in − k k−1

j0

A j i

r i n − j − 1 c u

i exp j 1 σiD u i

x in

≤ A k

i u in − k k−1

j0

A j i r u

i k−1

j0

A j i c u

i c u

i exp j 1 σiD u

i



x in

≤ A k

i W i 1− A k i

1− Ai r i u Hi x in,

2.18

where

H i 

⎡

⎣k−1

j0

A j i c i u c i uexp

j 1 σiD u

i

⎤⎦u

For any small positive constant ε > 0, there exists a K > 0 such that

d u i W i− r i u d u

i

1− Ai A k i < ε ∀k > K. 2.20

From the first equation of system1.4, 2.18, and 2.20, we have

x in 1

≥ xin exp

⎧

⎨

⎩b in −

N



j 1,j / i

a ijnMj − a u

iiexp

τ ii D u i

x in

−d u

i W i A k i −1− A k i

1− Ai r i u d u i − d u

i H i x in

⎫

⎬

⎭

 xin exp

⎧

⎨

⎩b in −

N



j 1,j / i

a ijnMj− r i u d u

i

1− Ai −

d u i W i− r i u d u

i

1− Ai A k i

− a u iiexp

τ ii D u i

d u

i H i

x in

⎫

⎬

⎭

≥ xin exp

⎧

⎨

⎩b in −

N



j 1,j / i

a ijnMj− r i u d u

i

1− Ai − ε − a u iiexp

τ ii D i u

d u

i H i

x in

⎫

⎬

⎭.

2.21

By Lemmas2.2and2.3, we have

lim

n→ ∞inf xin ≥ b

l

i−N

j 1,j / i a u ij M j−r i u d u i /e l i

− ε

a u

iiexp

τ ii D u i



d u

i H i

· exp

⎧

⎨

⎩b i l− N

j 1,j / i

a u ij M j−r i u d u

i

e l i − ε − a u iiexp

τ ii D u i

d u

i H i

M i

⎫

⎬

⎭.

2.22

Setting ε → 0 in 2.22 leads to

lim

n→ ∞inf xin ≥ b

l

i−N

j 1,j / i a u

ij M j−r u

i d u

i /e l i



a u

iiexp

τ ii D u i



d u

i H i

· exp

⎧

⎨

⎩b l i− N

j 1,j / i

a u

ij M j− r i u d i u

e l i

− a u

iiexp

τ ii D u i



d u

i H i

M i

⎫

⎬

⎭.

2.23

Thus,

lim

n→ ∞inf xin ≥ mi , 2.24

m i b

l

i−N

j 1,j / i a u ij M j−r i u d u i /e i l

a u iiexp

τ ii D u i

d u

i H i

· exp

⎧

⎨

⎩b i l− N

j 1,j / i

a u ij M j−r i u d u

i

e l i − a u iiexp

τ ii D u i

d u

i H i

M i

⎫

⎬

⎭.

2.25

Second, we prove limn→ ∞inf u in ≥ wi For enough small ε > 0, from the second equation of

system1.4, we have

u in 1  1 − einuin rin cinxin − σi ≥ r l

i c l

i mi − ε 1− e u

i

u in 2.26

for sufficient large n Hence

u in ≥ 1− e u

i

n

u i0 1− 1− e

u i

e u i



r i l c l

i mi − ε. 2.27 Thus, we obtain

lim

n→ ∞inf uin ≥ wi 2.28 This completes the proof

3 An Example

In this section, we give an example to illustrate that1.10 does not imply 1.9 Consider the

two-species system with delays and feedback controls for t ∈ −∞, ∞

x1n 1  x1n exp

! 1

2 − 2x1n − 1 − 1

2x2n − 3 −1

2u1n

"

,

x2n 1  x2n exp

! 1

2 −1

2x1n − 3 − 2x2n − 1 −1

2u2n

"

,

Δu1n 1  1

8 −1

2u1n x1n − 4,

Δu2n 1  1

8 −1

2u2n x2n − 8.

3.1

We have

b l1 b l

2 1

2, M1 M2 1

2, a

u

12M2 d u

1

r u

1

e l

1

 3

8, a

u

21M1 d u

2

r u

2

e l

2

 3

8. 3.2

b l

1> a u

12M2 d u

1

r1u

e l

1

, b l

2> a u

21M1 d u

2

r2u

e l

2

Therefore1.10 holds

But

1

2  b l

1< a u12M2 d u

1

r1u c u

1M1

e l

1

 7

8,

1

2  b l

2< a u21M1 d u

2

r u

2 c u

2M2

e l

2

 7

8. 3.4 Thus1.9 does not hold

References

1 X Liao, Z Ouyang, and S Zhou, “Permanence of species in nonautonomous discrete

Lotka-Volterra competitive system with delays and feedback controls,” Journal of Computational and Applied

Mathematics, vol 211, no 1, pp 1–10, 2008.

2 S Ahmad, “On the nonautonomous Volterra-Lotka competition equations,” Proceedings of the

American Mathematical Society, vol 117, no 1, pp 199–204, 1993.

3 S Ahmad and A C Lazer, “On the nonautonomous N-competing species problems,” Applicable

Analysis, vol 57, no 3-4, pp 309–323, 1995.

4 K Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol 74 of

Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.

5 M Kaykobad, “Positive solutions of positive linear systems,” Linear Algebra and Its Applications, vol.

64, pp 133–140, 1985

6 F Montes de Oca and M L Zeeman, “Extinction in nonautonomous competitive Lotka-Volterra

systems,” Proceedings of the American Mathematical Society, vol 124, no 12, pp 3677–3687, 1996.

7 M L Zeeman, “Extinction in competitive Lotka-Volterra systems,” Proceedings of the American

Mathematical Society, vol 123, no 1, pp 87–96, 1995.

8 Z D Teng, “Permanence and extinction in nonautonomous Lotka-Volterra competitive systems with

delays,” Acta Mathematica Sinica, vol 44, no 2, pp 293–306, 2001.

9 S Ahmad and I M Stamova, “Almost necessary and sufficient conditions for survival of species,”

Nonlinear Analysis Real World Applications, vol 5, no 1, pp 219–229, 2004.

10 Y.-H Fan and L.-L Wang, “Permanence for a discrete model with feedback control and delay,”

Discrete Dynamics in Nature and Society, vol 2008, Article ID 945109, 8 pages, 2008.

11 L Wang and M Q Wang, Ordinary Difference Equation, Xinjiang University Press, 1991.

12 F Chen, “Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition

predator-prey systems,” Applied Mathematics and Computation, vol 182, no 1, pp 3–12, 2006.