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Volume 2009, Article ID 973714,23pages doi:10.1155/2009/973714

Research Article

Boundedness, Attractivity, and Stability of a Rational Difference Equation with Two Periodic Coefficients

G. Papaschinopoulos, G. Stefanidou, and C. J. Schinas

School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece

Correspondence should be addressed to G. Papaschinopoulos,gpapas@env.duth.gr Received 24 August 2008; Accepted 11 January 2009

Recommended by Yong Zhou

We study the boundedness, the attractivity, and the stability of the positive solutions of the rational difference equationxn1 pnxn−2xn−3/qnxn−3,n 0,1, . . ., wherepn, qn,n 0,1, . . .are positive sequences of period 2.

Copyrightq2009 G. Papaschinopoulos 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.

1. Introduction

In1, Camouzis et al. studied the global character of the positive solutions of the difference equation:

xn1 δxn−2xn−3

Axn−3 , n0,1, . . . , 1.1

whereδ, Aare positive parameters and the initial valuesx−3, x−2, x−1, x0are positive real numbers.

The mathematical modeling of a physical, physiological, or economical problem very often leads to difference equations for partial review of the theory of difference equations and their applications see 2–12. Moreover, a lot of difference equations with periodic coefficients have been applied in mathematical models in biologysee13–15. In addition, between others in16–19, we can see some more difference equations with periodic coefficients that have been studied.

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In this paper, we investigate the difference equation xn1 pnxn−2xn−3

qnxn−3 , n0,1, . . . , 1.2

where pn, qn,n 0,1, . . . are positive sequences of period 2 and the initial values xi, i

−3,−2,−1,0 are positive numbers.

Our goal in this paper is to extend some results obtained in1. More precisely, we study the existence of a unique positive periodic solution of1.2of prime period 2. In the sequel, we investigate the boundedness, the persistence, and the convergence of the positive solutions to the unique periodic solution of1.2. Finally, we study the stability of the positive periodic solution and the zero solution of1.2.

If we setynx2n−1,znx2n, it is easy to prove that1.2is equivalent to the following system of difference equations:

yn1 p0zn−1yn−1

q0yn−1 , zn1 p1ynzn−1

q1zn−1 , n0,1, . . . , 1.3 wherepi, qi,i 0,1 are positive constants and the initial valuesyi, zi,i −1,0 are positive numbers. So in order to study1.2we investigate system1.3.

2. Existence of the Unique Positive Equilibrium of System 1.3

In the following proposition, we study the existence of the unique positive equilibrium of system1.3.

Proposition 2.1. Consider system1.3wherepi, qi,i 0,1 are positive constants and the initial valuesyi, zi,i−1,0 are positive numbers. Suppose that

q0−1< p0, q1−1< p1 2.1 are satisfied. Then system1.3possesses a unique positive equilibrium.

Proof. Lety, zbe a positive equilibrium of system1.3then y p0zy

q0y , z p1yz

q1z . 2.2

Equations2.2imply thatzis a solution of the equation fx x32

q1−1 x2

q1−12p1

q0−1 x

q1−1 q0−1

p1p0p210. 2.3 Suppose that

q1≥1. 2.4

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Letλ1, λ2, andλ3be the solutions of2.3. Then from2.1,2.3, and2.4we take λ1λ2λ32

1−q1

≤0, λ1λ2λ3

q1−1 q0−1

p1p0p12>0, 2.5 and so2.3has unique positive solutionz. Then from2.2and2.4we have

z >1−q1, y z2 q1−1

z

p1 >0, 2.6

and so system1.3has a unique positive equilibrium.

Now suppose that

q1<1,

q1−1 q0−1

> p1p0. 2.7

Ifλ1, λ2, andλ3are the solutions of2.3, then from2.3and2.7we take λ1λ2λ32

1−q1

>0, λ1λ2λ3

q1−1 q0−1

p1p0p21<0, 2.8 and so2.3has a negative solution, but also 2.3has a solution in the interval0,1−q1, since

f0

q1−1 q0−1

p1p0p21>0, f

1−q1

−p0p21 <0. 2.9

Moreover,2.3has a solutionzin the interval1−q1,∞, since

xlim→ ∞fx ∞, 2.10

therefore, we get2.6and so system1.3has a unique positive equilibrium.

Finally, suppose that

q1<1,

q1−1 q0−1

< p1p0. 2.11

Ifλ1, λ2, andλ3are the solutions of2.3, then from2.3and2.11, we take λ1λ2λ32

1−q1

>0, λ1λ2λ3

q1−1 q0−1

p1p0p21>0. 2.12

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We have limx→ ∞fx ∞,and sincef1q1<0, it is obvious that2.3has a solutionzin the interval1−q1,∞. From2.3, we get

fx 3x24x q1−1

q1−12

p1 q0−1

. 2.13

If equationfx 0 has complex roots, then it is obvious thatzis the unique solution of2.3. Therefore, we get2.6, and so system1.3has a unique positive equilibrium.

Now, suppose that the roots offx 0 μ1 2

1−q1

−√ D

3 , μ2 21−q1

D

3 , D 1−q12

3p11−q0

, 2.14

are real numbers.

Suppose thatq0<1, then it is obvious that

μ1<1−q1< μ2, 2.15

and so we have that2.3has a unique solutionz∈1−q1,∞.

Ifq0≥1, then it holds that

0< μ1μ2≤1−q1, 2.16

which implies that2.3has a unique solutionz∈1−q1,∞.

Therefore, we can take2.6 and so system1.3has a unique positive equilibrium.

This completes the proof of the proposition.

3. Boundedness and Persistence of the Solutions of System 1.3

In the following propositions we study the boundedness and the persistence of the positive solutions of system1.3. In the sequel we will use the following result which has proved in 20.

Theorem 3.1. Assume that all roots of the polynomial

Pt tNs1tN−1− · · · −sN, 3.1

wheres1, s2, . . . , sN0 have absolute value less than 1, and letynbe a nonnegative solution of the inequality

ynNs1ynN−1· · ·sNynzn. 3.2

Then, the following statements are true.

iIfznis a nonnegative bounded sequence, thenynis also bounded.

iiIf limn→ ∞zn0, then limn→ ∞yn0.

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Proposition 3.2. One considers the system of difference equations1.3where pi, qi,i 0,1 are positive constants and the initial valuesyi, zi,i −1,0 are positive numbers. Then the following statements are true.

iIf

q0q1

p0p1 ≥1, 3.3

then every solution of 1.3is bounded.

iiIf

q0−1< p0q0, q1−1< p1q1, 3.4 then every solution of 1.3is bounded and persists.

Proof. Letyn, znbe an arbitrary solution of1.3.

iFrom3.3, we get that one of the three following conditions holds:

q0 p0

>1, 3.5

q1

p1 >1, 3.6

p0q0p, p1 q1q. 3.7

We assume that3.5is satisfied. We prove that there exists a positive integerNsuch that

yn<1, zn< q0

p0, nN. 3.8

First, we show that if there exists a positive integern0such that zn0< q0

p0

, 3.9

then

zn03p< q0

p0, p0,1, . . . . 3.10

In contradiction, we assume that

zn03 p1yn02zn01

q1zn01q0

p0. 3.11

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Using relations1.3,3.5, and3.11, we get that yn02 p0zn0yn0

q0yn0

> q0q1

p0p1, 3.12

and so relations1.3and3.3imply that zn0 > q02q1

p02p1 > q0

p0, 3.13

which contradicts3.9. Sozn03< q0/p0and working inductively, we get3.10.

Ifz−1< q0/p0,then from the analogous relations3.9and3.10, we get z−13p< q0

p0, p0,1, . . . . 3.14

Now, suppose that

z−1q0

p0

, 3.15

we prove that there exists a positive integerqsuch that z−13q< q0

p0. 3.16

From3.3, there exists a positive integerhsuch that

z−1<

q0q1 p0p1

h

. 3.17

Ifz2< q0/p0, then3.16is true forq1.

Now, suppose that

z2q0

p0. 3.18

Then from1.3,3.5, and3.18, we gety1 > q0q1/p0p1and so from1.3,3.3, and3.5, we have that

z−1> q1q02 p1p02 > q1q0

p1p0. 3.19

Ifz5< q0/p0, then3.16is true forq2.

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Now, suppose that

z5q0

p0. 3.20

Using1.3,3.3,3.5,3.20and arguing as to prove3.19we get

z−1>

q1q0 p1p0

2

. 3.21

Working inductively, we get that

if z−13wq0

p0, w1,2, . . . , thenz−1>

q1q0

p1p0 w

. 3.22

From3.22forwh, we getz−1 >q1q0/p1p0hwhich contradicts3.17. Soz−13h < q0/p0 which means that3.16holds forqh.

Arguing as forz−1, we can prove that there exist positive integersk, lsuch that z03k< q0

p0, z13l< q0

p0. 3.23

From3.16and3.23, we get that there exists a positive integerrsuch that zr < q0

p0, nr. 3.24

Finally, from1.3and3.24, we getyr2<1 and so3.8is true forNr2.

Similarly, we can prove that if3.6holds, then there exists a positive integerNsuch that

zn<1, yn< q1

p1, nN. 3.25

Finally, suppose that3.7hold. From1.3and3.7, we have

yn1−1 p

zn−1−1

pyn−1 , zn1−1 q yn−1

qzn−1 , 3.26

and so,

yn1−1 p pyn−1

q qzn−3

yn−2−1

. 3.27

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From3.27, we get

0≤yn1−1≤yn−2−1, or 0≥yn1−1≥yn−2−1, 3.28 and so the subsequencesy3n, y3n1, y3n2either are bounded from below by 1 and decreasing or bounded from above by 1 and increasing. Hence,yn is bounded and persists. Similarly, we can prove thatzn is bounded and persists. This completes the proof of partiof the proposition.

iiIn statementi, we have already proved that if3.7hold, then every solution of 1.3is bounded and persists. So, from3.4, it remains to show that if either

q0−1< p0< q0, q1−1< p1q1, 3.29 or

q0−1< p0q0, q1−1< p1< q1, 3.30 holds, then the solutionyn, znpersists. From3.3,3.8,3.25,3.29, and3.30, we get that

yn< q1

p1, zn< q0

p0, nN. 3.31

We consider the positive numbermsuch that m <min

yN, zN, yN1, zN1, p01−q0, p11−q1 . 3.32

Moreover, if

fy, z p0zy

q0y , gy, z p1yz

q1z , 3.33

then it is easy to see that for the functions3.33,fis increasing with respect toyfor anyz, z < q0/p0andgis increasing with respect tozfor anyy, y < q1/p1.

Therefore, from1.3,3.31, and3.32we have yN2>

p01 m

q0m > m, zN2>

p11 m

q1m > m, 3.34

and working inductively, we take

yNsm, zNsm, s0,1, . . . . 3.35 Therefore,yn, znpersists and using statementi, thenyn, znis bounded and persists. This completes the proof of the proposition.

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Proposition 3.3. One considers the system of difference equations1.3where pi, qi,i 0,1 are positive constants, and the initial valuesyi, zi,i −1,0 are positive numbers. Then, the following statements are true.

iIf

q0q1

p0p1 <1, 3.36

then every solution of 1.3persists.

iiIf

q0p0q01, q1p1q11, 3.37

then every solution of 1.3is bounded and persists.

Proof. Letyn, znbe an arbitrary solution of1.3.

iFrom3.36, we have

q0

p0

<1, 3.38

or

q1

p1

<1. 3.39

Arguing as in the proof of statementiofProposition 3.2, we can easily prove that if 3.38holds, then there exists a positive integerMsuch that

yn>1, zn> q0

p0, nM, 3.40

and if3.39holds, then there exists a positive integerMsuch that zn>1, yn> q1

p1, nM. 3.41

iiFrom Proposition 3.2, we have that if 3.7 holds, then every solution of 1.3 is bounded and persists. So, from3.37, it remains to show that if either

q0 < p0q01, q1p1q11, 3.42

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or

q0p0q01, q1< p1q11, 3.43

holds, then the solutionyn, znis bounded and persists.

From3.36,3.40,3.41,3.42, and3.43, we get that yn> q1

p1, zn> q0

p0, nM. 3.44

Suppose that

p0/q01 or p1/q11. 3.45

From1.3and3.44, we have

zM1>1, yM3>1. 3.46

We have for the functions3.33thatf is decreasing with respect toyfor anyz,z > q0/p0 andgis decreasing with respect tozfor anyy,y > q1/p1. Therefore, relations1.3,3.44, and3.46imply that

zM3p1yM21

q11 , 3.47

and so from1.3and3.46,

yM5p0p1 q01

q11yM2 p0 q01

q111. 3.48

Working inductively, we can prove that yn5p0p1

q01

q11yn2 p0 q01

q11 1, n≥M. 3.49

Then from3.42,3.43,3.45, andTheorem 3.1,ynis bounded. Similarly, we take thatznis bounded. Therefore, from3.44, the solutionyn, znis bounded and persists.

Now, suppose that

p0q01, p1q11. 3.50

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We claim thatynis bounded. For the sake of contradiction, we assume thatynis not bounded.

Then, there exists a subsequencenisuch that

ilim→ ∞yni1∞, 3.51

yni1>max

yj, j < ni . 3.52

Moreover, from1.3and3.50, we get

yni1< q01 q0

zni−11, 3.53

and so from3.51,

ilim→ ∞zni−1∞. 3.54

Moreover, from1.3and3.50,

zni−1< q11 q1

yni−21, 3.55

and so from3.54,

ilim→ ∞yni−2∞. 3.56

Working inductively, we can prove that

ilim→ ∞yni1−3s∞, lim

i→ ∞zni−1−3s∞, s0,1, . . . . 3.57 We claim that yni−6 is a bounded sequence. Suppose on the contrary that there exists an unbounded subsequence ofyni−6and without loss of generality, we may suppose that

ilim→ ∞yni−6∞. 3.58

Arguing as above, we can easily prove that

ilim→ ∞yni−9 lim

i→ ∞yni−12∞. 3.59

Also, since from1.3, yni−6

q01 zni−8

/ yni−8

1 q0/yni−81 <

q01 zni−8

yni−8 1, 3.60

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from3.58, we have that limi→ ∞zni−8/yni−8 ∞and so eventually,

zni−8> yni−8. 3.61

From1.3,3.50, and3.61, we have

yni1

q01

zni−1yni−1 q0yni−1

< q01

q0 zni−11 q01

q0

q11

yni−2zni−3 q1zni−3

1

<1q01

q0 q01 q0

q11 q1

yni−2

<· · ·< AByni−8

< ABzni−8,

3.62

where

A1q01

q0 q01 q0

q11 q1

q01 q0

2

q11 q1

q01 q0

2 q11

q1 2

q01 q0

3 q11

q1 2

,

B

q01 q0

3 q11

q1 3

.

3.63

Therefore, using1.3and3.50, we get

yni1< AB

q11

yni−9zni−10 q1zni−10

, 3.64

and since from3.57and3.59, we have thatyni−9 → ∞, zni−10 → ∞asi → ∞,we can easily prove that eventually,

yni1< yni−9, 3.65

which contradicts to3.52.

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Therefore,yni−6is a bounded sequence. From1.3,3.50, and3.57, we get

zni−5

q11

yni−6zni−7 q1zni−7

q11

yni−6/zni−7 1

q1/zni−71 −→1, i−→ ∞. 3.66 Similarly, from1.3,3.50and3.57and3.66follows,

yni−3

q01

zni−5yni−5

q0yni−5

q01

zni−5/yni−5 1

q0/yni−51 −→1, i−→ ∞. 3.67

Now, we prove that

lim inf

i→ ∞ yni−1>1. 3.68

Otherwise, and without loss of generality, we may suppose that limi→ ∞yni−1 ≤1.So, relations 1.3,3.50, and3.67imply that

i→ ∞limyni−1

q01

limi→ ∞zni−3limi→ ∞yni−3

q0limi→ ∞yni−3 ≤1, 3.69

and so

ilim→ ∞zni−3q0

q01. 3.70

Moreover, from1.3,3.44, and3.50, we get eventually

zni−3

q11

yni−4zni−5 q1zni−5 >

q11

q1/q11 zni−5

q1zni−5 1, 3.71

and so from3.66, limi→ ∞zni−3≥1 which contradicts to3.70.

Hence,3.68is true.

Similarly, we can prove that

lim inf

i→ ∞ zni−3>1. 3.72

Therefore, from3.68and3.72, we have eventually

yni−1>1k, zni−3>1m, 3.73

wherek, mare positive real numbers.

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Hence, from1.3,3.50, and3.73we have

yni1 q01 q11

yni−2zni−3 /

q1zni−3

yni−1 q0yni−1

<

q01 q11 q11m

q01kyni−2q01 q0 1.

3.74

Then from3.57, we can prove that eventually

yni1< yni−2, 3.75

which contradicts to3.52.

Therefore,ynis a bounded sequence. Moreover, from1.3,3.50, we take thatzn is bounded. Therefore, the solutionyn, znis bounded and persists. This completes the proof of the proposition.

4. Attractivity of the Positive Equilibrium of System 1.3

In the following propositions, we study the convergency of the solutions of system1.3to its positive equilibrium.

Proposition 4.1. One considers the system of difference equations1.3where pi, qi,i 0,1 are positive constants, and the initial values yi, zi,i −1,0 are positive numbers. If either 3.29or 3.30hold, then every solution of 1.3tents to the positive equilibrium of 1.3.

Proof. Letyn, znbe an arbitrary solution of1.3. FromProposition 3.2, there exist L1lim sup

n→ ∞ yn, L2lim sup

n→ ∞ zn, l1lim inf

n→ ∞ yn, l2lim inf

n→ ∞ zn, 0< L1, L2, l1, l2<∞.

4.1

From1.3,3.31, and the monotony of functions3.33, we have

L1p0L2L1

q0L1

, L2p1L1L2

q1L2

, l1p0l2l1

q0l1

, l2p1l1l2

q1l2

, 4.2

and hence

L12L1 q0−1

p0L2≤0, L22L2 q1−1

p1L1≤0, l12l1

q0−1

p0l2≥0, l22l2

q1−1

p1l1≥0. 4.3

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The third inequality of4.3, implies that

l1≥ 1−q0

1−q024p0l2

2 , 4.4

and so from the last inequality of4.3, we have 2l222l2

q1−1

q0−1

p1p1 1−q0

2

4p0l2. 4.5

Hence, we get

2l222l2 q1−1

q0−1

p12

p1

1−q024p0l22

, 4.6

or

l232l22 q1−1

l2

q1−12p1

q0−1 p1

q1−1 q0−1

p0p12≥0. 4.7

The first inequality of4.3, implies that

0< L1 ≤ 1−q0 1−q02

4p0L2

2 , 4.8

and so from second inequality of4.3, we get 2L222L2

q1−1

q0−1

p1p1 1−q02

4p0L2. 4.9

Using4.3, we have

L1l1>1−q0, L2l2>1−q1. 4.10

Therefore, from4.5and4.10, we get 2L222L2

q1−1

q0−1

p12L2

L2q1−1

q0−1 p1

≥2l222l2

q1−1

q0−1 p1

>0.

4.11

Using4.9and4.11, we have L232L22

q1−1 L2

q1−12 p1

q0−1 p1

q1−1 q0−1

p0p12≤0. 4.12

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InProposition 2.1, we proved that2.3has a unique positive solutionz, z∈1−q1,∞. We can write

fx xz

x2axb

, a, b∈R, 4.13

wherefxis defined in2.3andx2axb >0 for anyx >1−q1. Then from4.7,4.12, and4.13, we have

L2z

L22aL2b

≤0, l2z

l22al2b

≥0. 4.14

Therefore, from4.10and4.14,

L2zl2,

which implies that

L2l2z. 4.15

In addition, using4.15, the first and the third inequalities of4.3, we have L21

q0−1

L1l21 q0−1

l1, 4.16

and so4.10implies that

L1l1. 4.17

This completes the proof of the proposition.

Proposition 4.2. One considers the system of difference equations1.3where pi, qi,i 0,1 are positive constants, and the initial values yi, zi,i −1,0 are positive numbers. If either 3.42or 3.43hold, then every solution of 1.3tents to the positive equilibrium of 1.3.

Proof. Let yn, znbe an arbitrary solution of 1.3. From Proposition 3.3, there existLi, li, i1,2 such that4.1are satisfied.

From1.3, the monotony of functions3.33and3.44, we have L1p0L2l1

q0l1 , L2p1L1l2

q1l2 , l1p0l2L1

q0L1 , l2p1l1L2

q1L2 , 4.18

and hence

L1l1L1q0p0L2l1, L1l1l1q0p0l2L1,

L2l2L2q1p1L1l2, L2l2l2q1p1l1L2, 4.19

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which implies that 1q0

L1l1

p0 L2l2

,

1q1 L2l2

p1 L1l1

. 4.20

Therefore,

1q0 1q1

p0p1 L1l1

≤0. 4.21

First suppose that3.45holds. Then from3.42or3.43, and3.45, we getL1−l1≤0,which means that

L1l1. 4.22

Using4.20, it is obvious that

L2l2. 4.23

So if3.45holds, the proof is completed.

Now, suppose that3.50hold. Then from4.20, we have

L2l2L1l1. 4.24

Moreover, from4.24, it follows that q01

l2L1l1q0 q01

L2l1L1q0. 4.25

In addition, from3.50, the first and the second inequalities of4.19, we get q01

l2L1l1q0L1l1q01

L2l1L1q0. 4.26

Therefore, from4.25and4.26, we have L1

q01 L2l1

q0l1 . 4.27

We may assume that there exists a positive integernisuch that

ilim→ ∞yni−jAj, lim

i→ ∞zni−jBj, lim

i→ ∞yni1L1. 4.28

Moreover, from1.3,3.50, and4.28, we get L1

q01 B1A1

q0A1 . 4.29

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Sincefx, y q01xy/q0yis decreasing with respect toy, for anyx >q0/q01, ifB1< L2orl1< A1, then from3.44, and3.50, we get

L1<

q01 L2l1

q0l1 , 4.30

which contradicts to4.27. So,

B1L2, l1A1. 4.31

Using the same argument, we can prove that

A2L1, B3 l2, B3l2, A3L1, B4L2, A4 l1, A4l1, B5L2, B5L2, A5 l1, A5L1, B6 l2,

4.32

and soL1l1A.Also, from4.24, we haveL2 l2B.Therefore,

nlim→ ∞ynA, lim

n→ ∞znB, 4.33

where obviouslyAB2. This completes the proof of the proposition.

Proposition 4.3. One considers the system of difference equations1.3where pi, qi,i 0,1 are positive constants, and the initial valuesyi, zi,i−1,0 are positive numbers. If relations3.7hold, then every solution of 1.3tents to the positive equilibrium1,1of 1.3.

Proof. Let yn, zn be an arbitrary solution of 1.3. From the proof ofProposition 3.2, the subsequencesy3n, y3n1, y3n2, z3n, z3n1, andz3n2 are monotone andyn,zn are bounded and persist. So, there exist positive numbersL1, L2, L3, M1, M2, andM3such that

L1 lim

n→ ∞y3n, L2 lim

n→ ∞y3n1, L3 lim

n→ ∞y3n2, M1 lim

n→ ∞z3n, M2 lim

n→ ∞z3n1, M3 lim

n→ ∞z3n2, 4.34

(19)

and from1.3and3.7, we get

L1 pM2L2 pL2

, M1 qL3M2 qM2

, L2 pM3L3

pL3 , M2 qL1M3

qM3 , L3 pM1L1

pL1 , M3 qL2M1 qM1 .

4.35

Then, we have

L1pL1L2pM2L2, M1qM1M2qL3M2, L2pL2L3pM3L3, M2qM2M3qL1M3, L3pL1L3pM1L1, M3qM3M1qL2M1,

4.36

and hence,

L1M2 pL2

1−L1

,

M1L3

qM2

1−M1 , L2M3

pL3 1−L2

,

M2L1

qM3

1−M2 , L3M1

pL1

1−L3

,

M3L2

qM1

1−M3

.

4.37

Therefore, we take

1 pL2

1−L1

1 qM3

M2−1 , 1

pL3

1−L2

1 qM1

M3−1 , 1

pL1

1−L3

1 qM2

M1−1 .

So,

ifL1 ≥1

resp., L1≤1

, thenM2≤1

resp., M2≥1 , ifL2 ≥1

resp., L2≤1

, thenM3≤1

resp., M3≥1 , ifL3 ≥1

resp., L3≤1

, thenM1≤1

resp., M1≥1 .

4.38

Therefore, if L1 ≥ 1, M2 ≤ 1 resp.,L1 ≤ 1, M2 ≥ 1, we have L1M2 ≥ 0 resp.,L1M2 ≤ 0and so from4.37,L1 ≤ 1resp.,L1 ≥ 1. Hence,L1 1 and from4.37,M2 1.

Similarly, we can prove thatL21, L31, M11, M3 1. This completes the proof of the proposition.

(20)

5. Stability of System 1.3

In this section we find conditions so that the positive equilibrium y, z and the zero equilibrium of1.3are stable.

Proposition 5.1. Consider system1.3wherepi, qi,i 0,1 are positive constants and the initial valuesyi, zi,i−1,0 are positive numbers. Then, the following statements are true.

iIf

q0−1< p0q0, q1−1< p1q1, q0q1p0p1q0q1 <1, 5.1 then the unique positive equilibriumy, zof1.3is globally asymptotically stable.

iiIf

q0q1p0p11< q0q1, 5.2 then the zero equilibrium of 1.3is locally asymptotically stable.

Proof. iSincey, zis the unique positive positive equilibrium of1.3, we have y p0zy

q0y , z p1yz

q1z . 5.3

Then from5.1and5.3, we get

yq0zy

q0y , zq1yz

q1z . 5.4

Without loss of generality we assume thatzy. Then from5.4, it results that yq0yy

q0y , 5.5

which means that

y≤1. 5.6

Moreover, from5.4and5.6, we get

zq1z

q1z 1. 5.7

In addition, from5.3, we have

y > y

q0y, z > z

q1z, 5.8

(21)

and so

y >1−q0, z >1−q1. 5.9

Then the linearized system of1.3about the positive equilibriumy, zis

Zn1AZn, 5.10

where

A

⎜⎜

⎜⎜

⎜⎜

⎜⎜

0 0 1 0

0 0 0 1

q0p0z

q0y2 p0

q0y 0 0

0 q1p1y q1z2 p1

q1z 0

⎟⎟

⎟⎟

⎟⎟

⎟⎟

, Zn

⎜⎜

wn−1

vn−1 wn

vn

⎟⎟

. 5.11

The characteristic equation ofAis

λ4

q0p0z

q0y2 q1p1y q1z2

λ2p1p0

q0y

q1

q0p0z

q1p1y q0y2

q1z2 0. 5.12 According to Remark 1.3.1 of7, all the roots of5.12are of modulus less than 1 if and only if

q0p0z

q0y2 q1p1y q1z2

p1p0

q0y

q1z

q0p0z

q1p1y q0y2

q1z2

<1. 5.13

From5.3, we get

q0p0z 1−y yq0

, q1p1y 1−z zq1

. 5.14

Then from5.6,5.7, and5.14, inequality5.13is equivalent to 1−y

q0y 1−z

q1z p1p0 q0y

q1z 1−y1z q0y

q1z <1. 5.15 Using5.9, inequality5.15holds if5.1are satisfied. Using Propositions4.1and4.3, we have that the unique positive equilibriumy, zof1.3is globally asymptotically stable.

iiArguing as above, we can prove that the linearized system of1.3about the zero equilibrium is

Zn1AZn, 5.16

参照

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