Note on the Persistent Property of a Discrete Lotka-Volterra Competitive System with Delays and Feedback Controls

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,

Liping Chen,

and Wensheng Yang

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

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

Correspondence should be addressed to Xiangzeng Kong,[email protected] Received 26 June 2010; Accepted 12 September 2010

Academic Editor: P. J. Y. Wong

Copyrightq2010 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 nonautonomousN-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:

u_it b_iu_it

⎡

⎣1−^N

j1

a_iju_jt

⎤

⎦, i1, . . . , N, 1.1

whereb_i>0,a_ii>0,a_ij≥0i /j,u_itdenoting the density of the ith species at timet. Montes de Oca and Zeeman6investigated the general nonautonomousN-species Lotka-Volterra competitive system

u_it u_it

⎡

⎣b_it−^N

j1

c_ijtujt

⎤

⎦, c_ij≥0, i1, . . . , N, 1.2

and obtained that if the coefficients are continuous and bounded above and below by positive constants, and if for eachi2, . . . , N,there exists an integerk_i< isuch that

bi

c_ij < b_ki

c_kij, j1, . . . , i, 1.3

thenui → 0 exponentially for 2 ≤i≤N,anduit → X^∗,whereX^∗is a certain solution of a logistic equation. Teng8and Ahmad and Stamova9also 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 to1–9and 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, in1Liao et al. considered the following general nonautonomous discrete Lotka-Volterra competitive system with delays and feedback controls:

xin 1 xinexp

⎧⎨

⎩bin−^N

j1

aijnxj

n−τij

−dinuin

⎫⎬

⎭, Δuin r_in−e_inuin c_inxin−σ_i, i1,2, . . . , N,

x_iθ φ_iθ≥0, θ∈N−τ,0:{−τ,−τ 1, . . . ,−1,0},

1.4

wherex_in i1,2, . . . , Nis the density of competitive species;u_inis the control variable;

e_in : Z → 0,1; bounded sequencesr_in,c_in,b_in,a_ijn, andd_in : Z → R ;τ_ij andσiare positive integer;Z,R denote the sets of all integers and all positive real numbers, respectively;Δis the first-order forward difference operatorΔuin u_in 1−u_in;τ max{max1≤i,j≤Nτ_ij,max_1≤i≤Nσ_i}>0.

In1, Liao et al. obtained sufficient conditions for permanence of the system1.4.

They obtained what follows.

Lemma 1.1. Assume that

1≤i≤NminM_iΔi>1 1.5

hold, then system1.4is permanent, where

MiΔi exp b^u_i −1 a^l_iiexp

−b^u_iτ_ii·a^u_iiexp τii_N

j1a^u_ijMj Wid^u_i −b^l_i b^l_i−_N

j1,j /ia^u_ijM_j−d^u_iW_i ,

Wi r_i^u c^u_iM_i

e^l_i , Mi exp b^u_i −1 a^l_iiexp

−b^u_iτ_ii.

1.6

Since

exp b^u_i −1

>0, a^l_iiexp

−b_i^uτ_ii

>0, a^u_iiexp

⎧⎨

⎩τ_ii

⎛

⎝^N

j1

a^u_ijM_j W_id^u_i −b^l_i

⎞

⎠

⎫⎬

⎭>0.

1.7 Hence, the above inequality1.5implies

b^l_i− ^N

j1,j /i

a^u_ijMj−d^u_iWi>0. 1.8

That is

b^l_i>

N j1,j /i

a^u_ijMj d^u_iWi

^N

j1,j /i

a^u_ijM_j d^u_i r_i^u c^u_iM_i e_i^l

^N

j1,j /i

a^u_ijMj

d_i^ur_i^u e^l_i

d^u_ic^u_iMi

e^l_i .

1.9

It was shown that in [1] Liao et al. considered system1.4where all coefficientsr_in,c_in,d_in, aijn,ein, andbinwere assumed to satisfy conditions1.9.

In this work, we shall study system1.4and get the same results as1do under the weaker assumption that

b^l_i>

N j1,j /i

a^u_ijM_j d_i^ur_i^u

e^l_i . 1.10

Our main results are the followingTheorem 1.2.

Theorem 1.2. Assume that1.10holds, then system1.4is permanent.

Remark 1.3. The inequality1.9implies1.10, but not conversely, for N

j1,j /i

a^u_ijMj

d_i^ur_i^u e^l_i ≤ ^N

j1,j /i

a^u_ijMj

d_i^ur_i^u e_i^l

d^u_ic^u_iMi

e^l_i . 1.11

Therefore, we have improved the permanence conditions of1for system1.4.

Theorem 1.2 will be proved in Section 2. In Section 3, an example will be given to illustrate that1.10does not imply1.9; that is, the condition1.10is better than1.9.

2. Proof of Theorem 1.2

The following lemma can be found in10.

Lemma 2.1. Assume thatA >0 andy0>0, and further suppose that (1)

yn 1≤Ayn Bn, n1,2, . . . . 2.1

Then for any integerk≤n,

yn≤A^kyn−k ^k−1

i0

AⁱBn−i−1. 2.2

Especially, ifA <1 andBis bounded above with respect toM, then

nlim→ ∞supyn≤ M

1−A. 2.3

2

yn 1≥Ayn Bn, n1,2, . . . . 2.4

Then for any integerk≤n,

yn≥A^kyn−k ^k−1

i0

AⁱBn−i−1. 2.5

Especially, ifA <1 andBis bounded below with respect tom^∗, then

nlim→ ∞infyn≥ m^∗

1−A. 2.6

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:yn 1 ≤ gn, yn,un 1≥gn, un.Ifgn₀ ≤un0, thenyn≤unfor alln≥n0.

Now let us consider the following single species discrete model:

Nn 1 Nnexp{an−bnNn}, 2.7

where{an}and{bn}are strictly positive sequences of real numbers defined forn∈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 system2.7with initial conditionN0>0 satisfies m≤ lim

n→ ∞infNn≤ lim

n→ ∞supNn≤M, 2.8

where

M 1

b^lexp{a^u−1}, m a^l b^uexp

a^l−b^uM

. 2.9

The following lemma is direct conclusion of1.

Lemma 2.4. Let xn x1n, x2n, . . . , xNn, u1n, u2n, . . . , uNn denote any positive solution of system1.4.Then there exist positive constantsM_i, W_ii1,2, . . . , Nsuch that

nlim→ ∞supxin≤Mi, lim

n→ ∞supuin≤Wi, i1,2, . . . , N, 2.10 where

Mi exp b_i^u−1 a^l_iiexp

−b^u_iτ_ii, Wi r_i^u c^u_iMi

e^l_i i1,2, . . . , N. 2.11 Proposition 2.5. Suppose assumption1.10holds, then there exist positive constantmiandwisuch that

nlim→ ∞infx_in≥m_i, lim

n→ ∞infu_in≥w_i. 2.12

Proof. We first prove lim

n→ ∞infx_in≥m_i.

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

xin 1 xinexp

⎧⎨

⎩bin−^N

j1

aijnxj

n−τij

−dinuin

⎫⎬

⎭

≥xinexp

⎧⎨

⎩bin−^N

j1

a_ij

Mj ε

−dinWi ε

⎫⎬

⎭

2.13

fornsufficiently large, then n−1

sn−τii

x_is 1 xis ≥exp

⎧⎨

⎩

n−1

sn−τii

⎛

⎝b_is−^N

j1

a_ijs

M_j ε

−d_isWi ε

⎞

⎠

⎫⎬

⎭. 2.14

Thus

x_in−τ_ii≤x_inexp _n−1

sn−τii

D_is

, 2.15

where

D_is ^N

j1

a_ijs

M_j ε

d_isWi ε−b_is. 2.16

From the second equation of system1.4, we have

uin 1−einuin cinxin−σi rin

≤ 1−e_i^l

u_in c_inxin−σ_i r_in :Aiuin Bin.

2.17

Then,Lemma 2.1implies that for anyk≤n−τ_ii,

uin≤A^k_iuin−k ^k−1

j0

A^j_iBi

n−j−1

A^k_iu_in−k ^k−1

j0

A^j_i r_i

n−j−1 c_i

n−j−1 x_i

n−j−1−σ_i

≤A^k_iuin−k ^k−1

j0

A^j_i ri

n−j−1

c^u_i exp

j 1 σi

D^u_i xin

≤A^k_iu_in−k ^k−1

j0

A^j_ir_i^u k−1

j0

A^j_ic^u_ic^u_i exp

j 1 σ_i D^u_i

x_in

≤A^k_iW_i 1−A^k_i 1−Ai

r_i^u H_ix_in,

2.18

where

H_i

⎡

⎣^k−1

j0

A^j_ic_i^uc_i^uexp

j 1 σ_iD_i^u⎤

⎦

u

. 2.19

For any small positive constantε >0, there exists aK >0 such that

d^u_iW_i− r_i^ud^u_i

1−A_i A^k_i < ε ∀k > K. 2.20

From the first equation of system1.4,2.18, and2.20, we have

xin 1

≥xinexp

⎧⎨

⎩bin− ^N

j1,j /i

aijnMj−a^u_iiexp τiiD^u_i

xin

−d^u_iWiA^k_i −1−A^k_i

1−Air_i^ud^u_i −d^u_iHixin

⎫⎬

⎭ xinexp

⎧⎨

⎩bin− ^N

j1,j /i

aijnMj− r_i^ud^u_i 1−A_i −

d^u_iWi− r_i^ud^u_i 1−A_i A^k_i

− a^u_iiexp

τiiD^u_i d^u_iHi

xin

⎫⎬

⎭

≥xinexp

⎧⎨

⎩bin− ^N

j1,j /i

aijnMj− r_i^ud^u_i

1−A_i −ε− a^u_iiexp

τiiD_i^u d^u_iHi

xin

⎫⎬

⎭. 2.21

By Lemmas2.2and2.3, we have

nlim→ ∞infxin≥ b^l_i−_N

j1,j /ia^u_ijMj−

r_i^ud^u_i/e^l_i

−ε a^u_iiexp

τ_iiD_i^u d^u_iH_i

·exp

⎧⎨

⎩b_i^l− ^N

j1,j /i

a^u_ijM_j−r_i^ud^u_i e^l_i −ε−

a^u_iiexp τ_iiD^u_i

d_i^uH_i M_i

⎫⎬

⎭.

2.22

Settingε → 0 in2.22leads to

n→ ∞lim infxin≥ b^l_i−_N

j1,j /ia^u_ijM_j−

r_i^ud_i^u/e^l_i a^u_iiexp

τ_iiD_i^u d^u_iH_i

·exp

⎧⎨

⎩b^l_i− ^N

j1,j /i

a^u_ijM_j− r_i^ud_i^u e^l_i −

a^u_iiexp τ_iiD^u_i

d^u_iH_i M_i

⎫⎬

⎭.

2.23

Thus,

nlim→ ∞infxin≥mi, 2.24

where

m_i b^l_i−_N

j1,j /ia^u_ijMj−

r_i^ud^u_i/e_i^l a^u_iiexp

τiiD^u_i d^u_iHi

·exp

⎧⎨

⎩b_i^l− ^N

j1,j /i

a^u_ijM_j−r_i^ud^u_i e^l_i −

a^u_iiexp τ_iiD^u_i

d_i^uH_i M_i

⎫⎬

⎭.

2.25

Second, we prove lim_{n→ ∞}infu_in≥w_i. For enough smallε >0, from the second equation of system1.4, we have

uin 1 1−einuin rin cinxin−σi≥r^l_i c^l_imi−ε 1−e^u_i

uin 2.26 for sufficient largen. Hence

uin≥

1−e^u_in

ui0 1− 1−e^u_i e^u_i

r_i^l c_i^lmi−ε

. 2.27

Thus, we obtain

nlim→ ∞infuin≥wi. 2.28

This completes the proof.

3. An Example

In this section, we give an example to illustrate that1.10does not imply1.9. Consider the two-species system with delays and feedback controls fort∈−∞, ∞

x1n 1 x1nexp

!1

2 −2x1n−1−1

2x2n−3−1 2u1n

"

,

x₂n 1 x₂nexp

!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^l₁b^l₂ 1

2, M1M2 1

2, a^u₁₂M2 d^u₁r₁^u e^l₁ 3

8, a^u₂₁M1 d^u₂r₂^u e^l₂ 3

8. 3.2

So

b^l₁> a^u₁₂M₂ d^u₁r₁^u

e^l₁, b^l₂> a^u₂₁M₁ d^u₂r₂^u

e^l₂. 3.3

Therefore1.10holds.

But 1

2 b^l₁< a^u₁₂M2 d^u₁r₁^u c^u₁M1

e^l₁ 7

8, 1

2 b^l₂< a^u₂₁M1 d^u₂r₂^u c^u₂M₂ e^l₂ 7

8. 3.4

Thus1.9does not hold.

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