Nonautonomous Lotka-Volterra Competitive Systems with Delays and Impulses

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Volume 2012, Article ID 125197,12pages doi:10.1155/2012/125197

Research Article

Permanence in Multispecies

Nonautonomous Lotka-Volterra Competitive Systems with Delays and Impulses

Xiaomei Feng,

^{1, 2}

Fengqin Zhang,

²

Kai Wang,

³

and Xiaoxia Li

²

1College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

2Department of Mathematics, Yuncheng University, Yuncheng 044000, China

3Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830011, China

Correspondence should be addressed to Kai Wang,[email protected] Received 10 November 2011; Accepted 2 February 2012

Academic Editor: Zhen Jin

Copyrightq2012 Xiaomei Feng 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.

This paper studies multispecies nonautonomous Lotka-Volterra competitive systems with delays and fixed-time impulsive eﬀects. The suﬃcient conditions of integrable form on the permanence of species are established.

1. Introduction

In this paper, we consider the nonautonomous n-species Lotka-Volterra type competitive systems with delays and impulses

x_it xit

⎡

⎣ait−bitxit−ⁿ

j1

aijtxj

t−τijt⎤

⎦, t /tk,

xi

t_k

hikxitk, i1,2, . . . , n, k1,2, . . . ,

1.1

wherexitrepresents the population density of theith species at timet, the functionsait, bit, aijt, andτijt i, j 1,2, . . . , nare bounded and continuous functions defined on R 0,∞,aijt≥ 0,bit ≥0,τijt ≥ 0 for allt∈ R, and impulsive coeﬃcientshik for anyi1,2, . . . , nandk1,2, . . .are positive constants.

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In particular, when the delaysτijt ≡ 0 for allt ∈ R andi, j 1,2, . . . , n, then the system 1.1 degenerate into the following nondelayed non-autonomous n-species Lotka- volterra system

x_it xit

⎡

⎣ait−ⁿ

j1

bijtxjt

⎤

⎦, t /tk,

xi

t_k

hikxitk, i1,2, . . . , n, k1,2, . . . ,

1.2

wherebiit bit aiitandbijt aijtfori, j 1,2, . . . , nandi /j. For system1.2, the author establish some new suﬃcient condition on the permanence of species and global attractivity in1 .

As we well know, systems like1.1and 1.2without impulses are very important in the models of multispecies populations dynamics. Many important results on the permanence, extinction, global asymptotical stability for the two species or multi-species nonautonomous Lotka-Volterra systems and their special cases of periodic and almost periodic systems can be found in2–14 and the references therein.

However, owing to many natural and man-made factorse.g., fire, flooding, crop-dust- ing, deforestation, hunting, harvesting, etc., the intrinsic discipline of biological species or ecological environment usually undergoes some discrete changes of relatively short duration at some fixed times. Such sudden changes can often be characterized mathematically in the form of impulses. In the last decade, much work has been done on the ecosystem with impulsivesee1,15–21 and the reference therein. Specially, the following system is considered in22 :

x_it xit

⎡

⎣ait−biitxit− ⁿ

j1,j /i 0

−∞kjsxjtsds

⎤

⎦, t /tk,

xi

t_k

hikxitk, i1,2, . . . , n, k1,2, . . . .

1.3

The author establish some new suﬃcient conditions on the permanence of species and global attractivity for system1.3. However, the eﬀect of discrete delays on the possibility of species survival has been an important subject in population biology. We find that infinite delays are considered in the system1.3. In this paper, it is very meaningful that discrete delays are proposed in the impulsive system1.1.

2. Preliminaries

Letτ sup{τijt, t≥0, i, j 1,2, . . . , n}. We defineCⁿ−τ,0 the Banach space of bounded continuous functionφ:−τ,0 → Rⁿwith the supremum norm defined by:

φ

c sup

−τ≤s≤0

φs, 2.1 whereφ φ1, φ2, . . . , φn, and|φs|_n

i1|φis|.DefineCⁿ−τ,0 {φ φ1, φ2, . . . , φn∈ Cⁿ−τ,0 : φis ≥ 0, and φi0 ≥ 0 for all s ∈ −τ,0 and i 1,2, . . . , n}.

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Motivated by the biological background of system1.1, we always assume that all solutions x1t, x2t, . . . , xntof system1.1satisfy the following initial condition:

xis φis ∀s∈−τ,0 , i1,2, . . . , n, 2.2 whereφ φ1, φ2, . . . , φn∈Cⁿ−τ,0 .

It is obvious that the solution x1t, x2t, . . . , xnt of system 1.1 with initial condition2.2is positive, that is,xit > 0i 1,2, . . . , non the interval of the existence and piecewise continuous with points of discontinuity of the first kindtk k∈Nat which it is left continuous, that is, the following relations are satisfied:

xi

t⁻_k

xitk, xi

t_k

hikxitk, i1,2, . . . , n, k∈N. 2.3 For system1.1, we introduce the following assumptions:

H1functionsait, bit, aijtandτijtare bounded continuous on0,∞ , andbit, aijtandτijt i, j1,2, . . . , nare nonnegative for allt≥0.

H2for each 1≤i≤n, there are positive constantsωi>0 such that

lim inf

t→ ∞

tωi

t

bisds

>0, 2.4

and the functions

hi

t, μ

t≤tk<tμ

lnhik 2.5

are bounded for allt∈Randμ∈0, ωi .

First, we consider the following impulsive logistic system

xt xt

αt−βtxt

, t /tk, x

t_k

hkxtk, k1,2, . . . , 2.6

whereαtand βtare bounded and continuous functions defined onR,βt ≥ 0 for all t∈R, and impulsive coeﬃcientshkfor anyk 1,2, . . .are positive constants. We have the following results.

Lemma 2.1. Suppose that there is a positive constantωsuch that

lim inf

t→ ∞

tw t

βsds

>0,

lim inf

t→ ∞

tw t

αsds

t≤tk<tω

lnhk

>0,

2.7

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and function

h t, μ

t≤tk<tω

lnhk 2.8

is bounded ont∈Randμ∈0, ω . Then we have athere exist positive constantsmandMsuch that

m≤lim inf

t→ ∞ xt≤lim sup

t→ ∞ xt≤M, 2.9

for any positive solutionxtof system2.6;

blim_{t→ ∞}x¹t−x²t 0 for any two positive solutionsx¹tandx²tof system 2.6.

The proof ofLemma 2.1can be found as Lemma 2.1 in1 by Hou et al.

On the assumptionH2, we firstly have the following result.

Lemma 2.2. If assumptionH2holds, then there exist constantsd >0 andD >0 such that for any t2≥t1≥0

t1≤tk<t2

lnhik

≤dt2−t1 D, i1,2, . . . , n. 2.10

The proof ofLemma 2.2is simple, we hence omit it here.

3. Main Results

Letxi0tbe some fixed positive solution of the following impulsive logistic systems as the subsystems of system1.1:

x_itxitait−bitxit , t /tk, xi

t_k

hikxitk, k1,2, . . . . 3.1

On the permanence of all speciesxi i 1,2, . . . , nfor system1.1, we have the following result.

Theorem 3.1. Suppose that assumptionsH1-H2hold. If there exist positive constantsωi such that for each 1≤i≤n:

lim inf

t→ ∞

⎛

⎝ ^tωⁱ

t

⎛

⎝ais−ⁿ

j /i

aijsxj0

s−τijs⎞

⎠ds

t≤tk<tωi

lnhik

⎞

⎠>0, 3.2

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and the functions

hi

t, μ

t≤tk<tμ

lnhik 3.3

are bounded for allt ∈ R andμ ∈ 0, ωi . Then the system1.1is permanent, that is, there are positive constantsγ >0 andM >0 such that

γ≤lim inf

t→ ∞ xit≤lim sup

t→ ∞ xit≤M, i1,2, . . . , n, 3.4 for any positive solutionxt x1t, x2t, . . . , xntof system1.1.

Proof. Letxt x1t, x2t, . . . , xntbe any positive solution of system1.1. We first prove that the componentsxi i 1,2, . . . , nof system1.1are bounded. From assumptionH1 and theith equation of system1.1, we have

x_it≤xitait−bitxit , t /tk, xi

t_k

hikxitk, k∈N. 3.5 by the comparison theorem of impulsive diﬀerential equation, we have

xit≤yit, ∀t≥0, 3.6

whereyitis the solution of3.1with initial valueyi0 xi0. From the condition3.2, we directly have

lim inf

t→ ∞

tωi

t

aisds

t≤tk<tωi

lnhik

>0, i1,2, . . . , n. 3.7

Hence, from conclusionaofLemma 2.1, we can obtain a constantMi1 > 0, and there is a Ti1>0 such thatyit< Mi1for allt≥Ti1. LetMmax_1≤i≤n{Mi1}andT1 max_1≤i≤n{Ti1}, we have

xit≤M, ∀t≥T1, i1,2, . . . , n. 3.8

Hence, we finally have

lim sup

t→ ∞ xt≤M. 3.9

Next, we prove that there is a constantγ >0 such that lim inf

t→ ∞ xt≥γ, i1,2, . . . , n. 3.10

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For anyt1andt2directly from system1.1, we have

xit1 xit2exp

⎛

⎝ ^t¹

t2

⎡

⎣ait−bitxit−ⁿ

j1

aijtxj

t−τijt⎤

⎦dt

t2≤tk≤t1

lnhik

⎞

⎠. 3.11

From condition 3.2, we can choose constants 0 < ε < 1 small enough and T2 > 0 large enough such that

tωi

t

⎛

⎝ais−bis aiis ε−ⁿ

j /i

aijs xj0

s−τijs ε⎞

⎠ds

t≤tk<tωi

lnhik> ε, 3.12

for allt ≥ T2andi 1,2, . . . , n. Considering3.5, by the comparison theorem of impulsive diﬀerential equation and the conclusionbofLemma 2.1., we obtain for the aboveε≥0 that there is aT3> T2such that

xit≤xi0t ε ∀t≥T3, i1,2, . . . , n, 3.13

wherexi0tis a globally uniformly attractive positive solution of system3.1.

Claim 1. There is a constant η > 0 such that lim sup_t_{→ ∞}xit > η i 1,2, . . . , n for any positive solutionxt x1t, x2t, . . . , xntof system1.1. In fact, ifClaim 1is not true, then there is an integerk ∈ {1,2, . . . , n}and a positive solutionxt x1t, x2t, . . . , xnt of system1.1such that

lim sup

t→ ∞ xkt< ε. 3.14

Hence, there is a constantT4> T3such that

xkt< ε ∀t≥T4. 3.15

On the other hand, by3.13there is aT5≥T4such that

xit≤xi0t ε ∀t≥T5, 3.16

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wherei1,2, . . . , nandi /k. By3.11and3.16, we obtain

xkt xkT5τexp

⎛

⎝ ^t

T5τ

⎡

⎣aks−bksxkt−ⁿ

j1

akjsxj

s−τijs⎤

⎦ds

T5τ≤tk≤t

lnhkk

≥xkT5τexp

⎛

⎝ ^t

T5τ

⎡

⎣aks−bks akksε− ⁿ

j1,j /k

aijs xj0

s−τijs ε⎤

⎦ds

T5τ≤tk≤t

lnhkk

,

3.17

for all t ≥ T5 τ. Thus, from 3.12 we finally obtain lim_{t→ ∞}xkt ∞, which lead to a contradiction.

Claim 2. There is a constantγ >0 such that lim inf_{t→ ∞}xit> γ i1,2, . . . , nfor any positive solution of system1.1.

IfClaim 2is not true, then there is an integerk∈ {1,2, . . . , n}and a sequence of initial function{φm} ⊂C−τ,0 such that

lim inf

t→ ∞ xk

t, φm

< η

m² ∀m1,2, . . . , 3.18

where constantηis given inClaim 1. ByClaim 1, for every m there are two time sequences s^mq andt^mq , satisfying:

0< s^m₁ < t^m₁ < s^m₂ < t^m₂ <· · ·< s^mq < t^mq <· · · , lim

q→ ∞s^mq ∞, 3.19

such that

xk

s^mq , φm

≥ η

m, xk

t^mq , φm

≤ η

m², 3.20

η m² ≤xk

t, φm

≤ η

m ∀t∈

s^mq , t^mq

. 3.21

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From the above proof, there is a constantT^m≥T2such thatxit, φm< Mi1,2, . . . , nfor allt ≥T^m. Further, there is an integerK₁^m >0 such thats^mq > T^m for allq > K^m₁ . From 3.11and lemma 2.2., we can obtain

xk

t^mq , φm

≥xk

s^mq , φm

exp

⎛

⎜⎝ ^t

mq

s^mq

⎡

⎣aks−bksM−ⁿ

j1

akjsM

⎤

⎦ds

s^mq ≤tk≤t^mq

lnhkk

⎞

⎟⎠

≥xk

s^mq , φm

exp

−r1d

t^mq −s^mq

−D ,

3.22 wherer1sup_t≥0{|ait|bitM_n

j1aijtM}. Consequently, from3.20we have t^mq −s^mq ≥ lnm−D

r1d ∀q > K^m₁ . 3.23 By3.12, there is a large enoughP >0 such that for allt≥T2,a≥Panda∈lwi,l1wi andi1,2, . . . , n, then, we obtain

ta t

⎛

j /i

aijs xj0

s−τijs ε⎞

⎠ds

t≤tk<ta

lnhik

^tlwⁱ

t

⎛

j /i

aijs xj0

s−τijs ε⎞

⎠ds

t≤tk<tlwi

lnhik

^ta

tlwi

⎛

j /i

aijs xj0

s−τijs ε⎞

⎠ds

tlwi≤tk<ta

lnhik

> lε−r2wi,

3.24 wherer2 sup_t≥0{|ait| bit aiit ε_n

j /iaijsxj0s−τijs ε }. So, we choose L2 r2wi/εsuch that for alll > L, we have

ta t

⎛

j /i

aijs xj0

s−τijs ε⎞

⎠ds

t≤tk<ta

lnhik> ε. 3.25

From3.23, there is an integerN0such that for anym > N0andq > K₁^m, we have η

m< ε, t^mq −s^mq >2Q, 3.26

where constantQ > P τ.

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So, whenm > N0andq > K₁^m, for anyt∈s^mq Qτ, t^mq , from3.11,3.21,3.25, and3.26we can obtain

xk

t^mq , φm

xk

s^mq Qτ, φm

×exp

⎛

⎜⎝ ^t

mq

s^mq Qτ

⎡

⎣akt−bktxk

t, φm

−ⁿ

j1

akjtxj0

t−τkjt , φm

⎤

⎦dt

s^mq Qτ≤tk≤t^mq

lnhkk

⎞

⎟⎠

3.27

Consequently, from3.20and3.25it follows η

m² ≥ η m²

×exp

⎛

⎜⎝ ^t

mq

s^mq Qτ

⎡

⎣akt−bkt akktε− ⁿ

j1,j /k

akjtxj0

t−τkjt ε

⎤

⎦dt

s^mq Qτ≤tk≤t^mq

lnhkk

⎞

⎟⎠

> η m².

3.28

This leads to a contradiction. Therefore,Claim 2is true. This completes the proof.

When system1.1degenerates into the periodic case, then we can assume that there is a constantω >0 and an integerq >0 such thataitω ait,bitω bit,aijt ω aijt, t_kq tk ω andh_ikq hik for all t ∈ R,k 1,2, . . . and i, j 1,2, . . . , n.

From Remarks 2.3 and 2.4 in1 , we can see the fixed positive solutionxj0 of system3.1 can be chosen to be theω-periodic solution of system3.1. Therefore, as a consequence of Theorem 3.1. we have the following result.

Corollary 3.2. Suppose that system1.1isω-periodic and for eachi1,2, . . . , n,

ω 0

bisds >0,

ω 0

⎛

⎝ais−ⁿ

j /i

aijsxj0

s−τijs⎞

⎠ds q k1

lnhik>0.

3.29

Then, system1.1is permanent.

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4. Numerical Example

In this section, we will give an example to demonstrate the eﬀectiveness of our main results.

We consider the following two species competitive system with delays and impulses:

x₁t x1ta1t−b1tx1t−a11tx1t−τ11t−a12tx2t−τ12t , x₂t x2ta2t−b2tx2t−a21tx1t−τ21t−a22tx2t−τ22t , t /tk

x1

t_k

h1kx1tk, x2

t_k

h2kx2tk, k1,2, . . . .

4.1

We takea1t 2,a2t b1t b2t a11t a12t a22t 1,a21 1− |sinπ/2t|, τijt 2,h1k e⁻¹,h2k e,tk k. Obviously, system4.1is periodic with periodω 2.

Forq2, we havetkq tkω,h1kq h1kandh2kq h2kfor allk 1,2, . . .. Consider the following impulsive logistic systems as the subsystems of system4.1:

x₁t x1t2−x1t, x₂t x2t1−x2t, t /k

x1t e⁻¹x1tk,

x2t ex2tk, tk.

4.2

According to the formula in1 , we can obtain that subsystem4.2has a unique globally asymptotically stable positive 2-periodic solutionx10t, x20t, which can be expressed in following form:

x10t 2x10

x10 2−x10e^−2t−k, t∈k, k1, k0,1,2, . . . ,

x20t x20

x20 1−x20e^−t−k, t∈k, k1, k0,1,2, . . . ,

4.3

wherex10 2e^−0.2−e⁻²/1−e⁻²andx20 e−e⁻¹/1−e⁻¹. Since

ω 0

a1t−a12tx20t−τ12tdt q k1

lnh1k

2

1 0

2− x20

x20 1−x20e^−t−2

dt²

k1

lnh1k

≈1.5244,

ω 0

a2t−a21tx10t−τ21tdt^q

k1

lnh2k

2

1 0

1−

1−sinπ

2t 2x10

x10 2−x10e^−2t−2

dt²

k1

lnh2k

≈3.8398,

4.4

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0 10 20 30 40 50 60 0

0.5 1 1.5 2 2.5 3

x1(t) t x2(t)

Figure 1: Time series ofx1tandx2t.

we obtain that all conditions inCorollary 3.2for system1.1holds. Therefore, from Theorem 3.1. we see that system1.1is permanentseeFigure 1.

Acknowledgments

This paper was supported by the National Sciences Foundation of China 11071283, the Sciences Foundation of Shanxi2009011005-3, the Young foundation of Shanxi provinceno.

2011021001-1, research project supported by Shanxi Scholarship Council of China 2011- 093, the Major Subject Foundation of Shanxi, and Doctoral Scientific Research fund of Xinjiang Medical University.

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Nonautonomous Lotka-Volterra Competitive Systems with Delays and Impulses

Research Article

Permanence in Multispecies

Nonautonomous Lotka-Volterra Competitive Systems with Delays and Impulses

Xiaomei Feng,

Fengqin Zhang,

Kai Wang,

and Xiaoxia Li

1. Introduction

2. Preliminaries

3. Main Results

4. Numerical Example

Acknowledgments

References

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

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