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Volume 2009, Article ID 515706, 10 pagesdoi:10.1155/2009/515706 Research Article Schoener Competition System with Time Delays and Feedback Controls Xuepeng Li and Wensheng Yang School of

Volume 2009, Article ID 515706, 10 pages

doi:10.1155/2009/515706

Research Article

Schoener Competition System with Time

Delays and Feedback Controls

Xuepeng Li and Wensheng Yang

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

Correspondence should be addressed to Wensheng Yang,ywensheng@126.com

Received 4 March 2009; Revised 25 July 2009; Accepted 3 September 2009

Recommended by John Graef

A discrete n-species Schoener competition system with time delays and feedback controls is

proposed By applying the comparison theorem of difference equation, sufficient conditions are obtained for the permanence of the system

Copyrightq 2009 X Li and W Yang 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

1 Introduction

In 1974, Schoener1 proposed the following competition model:

˙x  r1x



I1

x  e1 − r11x − r12y − c1



,

˙y  r2y



I2

y  e2 − r21x − r22y − c2



,

1.1

where r i , I i , e i , r ij , c i i  1, 2; j  1, 2 are all positive constants.

May2 suggested the following set of equations to describe a pair of mutualists:

˙u  r1u



a1 b1v − c1u



,

˙v  r2v



a2 b2u − c2v



,

1.2

where u, v are the densities of the species U, V at time t, respectively r i , a i , b i , c i , 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.

Both of the above-mentioned works are considered the continuous cases However, many authors 3 5 have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations Bai et al.6 argued that the discrete case of cooperative system

is more appropriate, and they proposed the following system:

x1k  1  x1k exp



r1k



a1k  b1k x2k − c1k x1k



,

x2k  1  x2k exp



r2k



a2k  b2k x2k − c2k x1k



.

1.3

On the other hand, as was pointed out by Huo and Li7, ecosystem in the real world is continuously disturbed by unpredictable forces which can result in changes in the biological parameters such as survival rates Practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time In the language of control variables, we call the disturbance functions as control variables During the last decade, many scholars did excellent works on the feedback control ecosystemssee 8 11 and the references cited therein

Chen11 considered the permanence of the following nonautonomous discrete N-species cooperation system with time delays and feedback controls of the form

x i k  1  x i k exp



r i k

1− x i k − τ ii

a i k  n

j 1,j / i b ij k x j

k − τ ij − c i k x i k − τ ii

−d i k μ i k − e i k μ i

k − η i



,

Δμ i k  −α i k μ i k  β i k x i k  γ i k x i k − σ i ,

1.4

where x i k i  1, , n is the density of cooperation species X i , μ i k i  1, , n is the

control variable11 and the references cited therein

Motivated by the above question, we consider the following discrete n-species

Schoener competition system with time delays and feedback controls:

x i k  1  x i k exp

⎧

⎨

⎩

r i k

x i k − τ i  a i k −

n



j1

b ij k x j

k − τ j

− c i k

−d i k μ i k − e i k μ i

k − η i

⎫⎬

⎭,

Δμ i k  −α i k μ i k  β i k x i k  γ i k x i k − σ i ,

1.5

where x i k i  1, 2, , n is the density of competitive species at kth generation; μ i k is the

control variable;Δ is the first-order forward difference operator Δμ i k  μ i k 1 −μ i k , i 

1, 2, , n.

Throughout this paper, we assume the following

H1 α i k , β i k , γ i k , a i k , b ij k , r i k , c i k , d i k , e i k , i  1, 2, , n are all bounded

nonnegative sequence such that

0 < α l i ≤ α u

i < 1, 0 < β l i ≤ β u

i , 0 < γ i l ≤ γ u

i , 0 < a l i ≤ a u

i ,

0 < b ij l ≤ b u

ij , 0 < r i l ≤ r u

i , 0 < c i l ≤ c u

i , 0 < d l i ≤ d u

i , 0 < e i l ≤ e u

Here, for any bounded sequence{ak }, a u supk ∈N a k , a l infk ∈N a k

H2 τ i , η i , σ i , i  1, , n are all nonnegative integers.

Let τ  max{τ i , η i , σ i , i  1, , n}, we consider 1.5 together with the following initial conditions:

x i θ  ϕ i θ , θ ∈ N−τ, 0  {−τ, −τ  1, , 0}, ϕ i 0 > 0,

μ i θ  φ i θ , θ ∈ N−τ, 0  {−τ, −τ  1, , 0}, φ i 0 > 0. 1.7

It is not difficult to see that solutions of 1.5 and 1.7 are well defined for all k ≥ 0 and satisfy

x i k > 0, μ i k > 0 for k ∈ Z, i  1, 2, , n. 1.8

The aim of this paper is, by applying the comparison theorem of difference equation,

to obtain a set of sufficient conditions which guarantee the permanence of the system 1.5

2 Permanence

In this section, we establish a permanence result for system1.5

Definition 2.1 System1.5 is said to be permanent if there exist positive constants M and m

such that

k→ ∞inf x i k ≤ lim

k→ ∞sup x i k ≤ M, i  1, 2, , n,

k→ ∞inf μ i k ≤ lim

k→ ∞sup μ i k ≤ M, i  1, 2, , n 2.1 for any solution xk  x1k , , x n k , μ1k , , μ n k of system 1.5

Now, let us consider the first-order difference equation

where A, B are positive constants Following Lemma 2.1 is a direct corollary of Theorem 6.2

of L Wang and M Q Wang12, page 125

Lemma 2.2 Assuming that |A| < 1, for any initial value y(0), there exists a unique solution y(k) of

2.2 which can be expressed as follow:

y k  A k

y 0 − y∗

where y∗ B/1 − A Thus, for any solution {yk } of system 2.2 , one has

lim

Following comparison theorem of difference equation is Theorem 2.1 of 12, page 241

k0  {k0, k0  1, , k0  l, }, r ≥ 0 For any fixed k, gk, r is a

nondecreasing function with respect to r, and for k ≥ k0, the following inequalities hold:

y k  1 ≤ gk, y k ,

If y k0 ≤ uk0 , then yk ≤ uk for all k ≥ k0.

Now let us consider the following single species discrete model:

where{ak } and {bk } are strictly positive sequences of real numbers defined for k ∈ N  {0, 1, 2, } and 0 < a l ≤ a u , 0 < b l ≤ b u Similarly to the proof of Propositions 1 and 313,

we can obtain the following

k→ ∞inf Nk ≤ lim

where

M 1

b l exp{au − 1}, m  a l

b u exp

a l − b u M

lim

k→ ∞sup x i k ≤ M i , i  1, , n,

lim

b l iiexp

−r i u τ i /a l i exp



r u i

a l i − 1



, Q i

β u

i  γ u i

M i

Proof Let x k  x1k , , x n k , μ1k , , μ n k be any positive solution of system 1.5 ,

from the ith equation of1.5 , we have

x i k  1 ≤ x i k exp



r i k

a l i



Let x i k  exp{N i k }, the inequality above is equivalent to

N i k  1 − N i k ≤ r i k

a l i

Summing both sides of2.12 from k − τ i to k− 1 leads to

k−1



j k−τ i

N i

j 1− N i

j

≤ k−1

j k−τ i

r i

j

a l i ≤ r i u

and so,

N i k − τ i ≥ N i k − r i u τ i

therefore,

x i k − τ i ≥ x i k exp



−r

u

i τ i

a l i



Substituting2.15 to the ith equation of 1.5 leads to

x i k  1 ≤ x i k exp



r i k

a l i − b ii k exp



−r i u τ i

a l i



x i k



By applying Lemmas2.3and2.4, it immediately follows that

lim

k→ ∞sup x i k ≤ 1

b l

iiexp

−r u

i τ i /a l i

 exp



r i u

a l i

− 1



For any positive constant ε small enough, it follows from2.17 that there exists enough large

K0such that

x i k ≤ M i  ε, i  1, , n, ∀ k ≥ K0. 2.18

From the n  ith equation of the system 1.5 and 2.18 , we can obtain

Δμ i k ≤ −α i k μ i k β i k  γ i k M i  ε , 2.19

for all k ≥ K0 max{σ i , i  1, , n.} And so,

μ i k  1 ≤1− α l

i



μ i k β u i  γ u

i

for all k ≥ K0max{σ i , i  1, 2, , n.} Noticing that 0 < 1−α l

i < 1 i  1, 2, , n , by applying

Lemmas2.2and2.3, it follows from2.20 that

lim

k→ ∞sup μ i k ≤

β u i  γ u i

M i  ε

α l i

Setting ε → 0 in the inequality above leads to

lim

k→ ∞sup μ i k ≤

β u i  γ u i

M i

α l i

This completes the proof ofProposition 2.5

Now we are in the position of stating the permanence of system1.5

r l i

M i  a u i

− n

j 1,j / i

b u ij M j − c u

i −d u i  e u

i

Q i > 0, i  1, 2, , n, 2.23

then system1.5 is permanent.

Proof By applyingProposition 2.5, we see that to end the proof ofTheorem 2.6, it is enough

to show that under the conditions ofTheorem 2.6,

lim

k→ ∞inf x i k ≥ m i , i  1, 2, , n,

lim

k→ ∞inf μ i k ≥ q i , i  1, 2, , n. 2.24

FromProposition 2.5, for all ε > 0, there exists a K1> 0, K1 ∈ N, for all k > K1,

x i k ≤ M i  ε; μ i k ≤ Q i  ε, i  1, 2, , n. 2.25

From the ith equation of system1.5 and 2.25 , we have

x i k  1 ≥ x i k exp{A ε k }, ∀ k > K1 τ, 2.26

where

A ε k  r i k

M i  ε  a u

i

−n

j1

b ij k M j  ε− c i k − d i k  e i k Q i  ε 2.27

Let x i k  exp{N i k }, the inequality above is equivalent to

Summing both sides of2.28 from k − τ i to k− 1 leads to

k−1



j k−τ i

N i

j 1− N i

j

and so,

where

A ε l r i l

M i  ε  a u

i

−n

j1

b u ij

M j  ε− c u

i −d u

i  e u i

Therefore,

x i k − τ i ≤ x i k exp−A ε l τ i

Substituting2.32 to the ith equation of 1.5 leads to

x i k  1 ≥ x i k exp

⎧

⎨

⎩

r i k

M i  ε  a u

i

− n

j 1,j / i

b ij k M j  ε− c i k

−b ii k exp−A ε l τ i



x i k − d i k  e i k Q i  ε

⎫

⎬

⎭

 x i k expB ε k − b ii k exp−A ε l

τ i



x i k ,

2.33

for all k > K1 τ, where

B ε k  r i k

M i  ε  a u

i

− n

j 1,j / i

b ij k M j  ε− c i k − d i k  e i k Q i  ε 2.34

Condition2.23 shows thatLemma 2.4could be apply to2.33 , and so, by applying Lemmas

2.3and2.4, it immediately follows that

lim

k→ ∞inf x i k ≥ B ε l

b u iiexp

−A ε l τ i

 expB ε l − b u

iiexp

−A ε l τ i



M i



where

B ε l r i l

M i  ε  a u

i

− n

j 1,j / i

b u ij

M j  ε− c u

i −d u i  e u

i

Setting ε → 0 in 2.35 leads to

lim

k→ ∞inf x i k ≥

B0l

b u

iiexp

−A0 l τ i exp

B0l

− b u

iiexp



−A0l

τ i



M i



where

A0 l r i l

M i  a u i

−n

j1

b u ij M j − c u

i −d u i  e u

i

Q i ,

B0 l r i l

M i  a u i

− n

j 1,j / i

b u ij M j − c u

i −d i u  e u

i

Q i

2.38

For any positive constant ε small enough, it follows from2.37 that there exists enough large

K2such that

x i k ≥ m i − ε, i  1, , n, ∀ k ≥ K2. 2.39

From the n  ith equation of the system 1.5 and 2.39 , we can obtain

Δμ i k ≥ −α i k μ i k β i k  γ i k m i − ε , 2.40

for all k ≥ K2 max{σ i , i  1, , n} And so,

μ i k  1 ≥1− α u

i

μ i k β l i  γ l

i



for all k ≥ K2max{σ i , i  1, 2, , n} Noticing that 0 < 1−α u

i < 1 i  1, 2, , n , by applying

Lemmas2.2and2.3, it follows from2.41 that

lim

k→ ∞inf μ i k ≥



β l i  γ l i



m i − ε

Setting ε → 0 in the inequality above leads to

lim

k→ ∞inf μ i k ≥



β i l  γ l i



m i

This ends the proof ofTheorem 2.6

Now let us consider the following discrete N-species Schoener competition system

with time delays:

x i k  1  x i k exp

⎧

⎨

⎩

r i k

x i k − τ i  a i k −

n



j1

b ij k x j

k − τ j

− c i k

⎫

⎬

where x i k i  1, , n is the density of species X i Obviously, system 2.44 is the generalization of system 1.5 From the previous proof, we can immediately obtain the following theorem

r l i

M i  a u i

− n

j 1,j / i

b u ij M j − c u

i > 0, i  1, 2, , n, 2.45

then system2.44 is permanent.

This work is supported by the Foundation of Education, Department of Fujian Province

JA05204 , and the Foundation of Science and Technology, Department of Fujian Province

2005K027

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1.5

where... Cao, and A P Chen, “Positive periodic solutions of a class of

non-autonomous single species population model with delays and feedback control,” Acta Mathematica

Sinica,... class="text_page_counter">Trang 4

Lemma 2.2 Assuming that |A| < 1, for any initial value y(0), there exists a unique solution y(k) of< /b>

2.2