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.
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
p1−p0p210. 2.3 Suppose that
q1≥1. 2.4
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
p1−p0p21>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
We have limx→ ∞fx ∞,and sincef1−q1<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< μ1≤μ2≤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 tN−s1tN−1− · · · −sN, 3.1
wheres1, s2, . . . , sN ≥ 0 have absolute value less than 1, and letynbe a nonnegative solution of the inequality
ynN≤s1ynN−1· · ·sNynzn. 3.2
Then, the following statements are true.
iIfznis a nonnegative bounded sequence, thenynis also bounded.
iiIf limn→ ∞zn0, then limn→ ∞yn0.
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< p0≤q0, q1−1< p1≤q1, 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, n≥N. 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
q1zn01 ≥ q0
p0. 3.11
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−1≥ q0
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
z2≥ q0
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.
Now, suppose that
z5≥ q0
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−13w≥ q0
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, n≥r. 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, n≥N. 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
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< p1≤q1, 3.29 or
q0−1< p0≤q0, 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, n≥N. 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
yNs≥m, zNs≥m, s0,1, . . . . 3.35 Therefore,yn, znpersists and using statementi, thenyn, znis bounded and persists. This completes the proof of the proposition.
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
q0 ≤p0≤q01, q1≤p1≤q11, 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, n≥M, 3.40
and if3.39holds, then there exists a positive integerMsuch that zn>1, yn> q1
p1, n≥M. 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 < p0≤q01, q1≤p1≤q11, 3.42
or
q0 ≤p0≤q01, q1< p1≤q11, 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, n≥M. 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
zM3≤ p1yM21
q11 , 3.47
and so from1.3and3.46,
yM5≤ p0p1 q01
q11yM2 p0 q01
q111. 3.48
Working inductively, we can prove that yn5≤ p0p1
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
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
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.
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−3 ≤ q0
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.
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
L1≤ p0L2L1
q0L1
, L2≤ p1L1L2
q1L2
, l1≥ p0l2l1
q0l1
, l2≥ p1l1l2
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
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
p1≥p1 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
p1≤p1 1−q02
4p0L2. 4.9
Using4.3, we have
L1≥l1>1−q0, L2 ≥l2>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
InProposition 2.1, we proved that2.3has a unique positive solutionz, z∈1−q1,∞. We can write
fx x−z
x2axb
, a, b∈R, 4.13
wherefxis defined in2.3andx2axb >0 for anyx >1−q1. Then from4.7,4.12, and4.13, we have
L2−z
L22aL2b
≤0, l2−z
l22al2b
≥0. 4.14
Therefore, from4.10and4.14,
L2≤z≤l2,
which implies that
L2l2z. 4.15
In addition, using4.15, the first and the third inequalities of4.3, we have L21
q0−1
L1≤l21 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 L1≤ p0L2l1
q0l1 , L2≤ p1L1l2
q1l2 , l1≥ p0l2L1
q0L1 , l2≥ p1l1L2
q1L2 , 4.18
and hence
L1l1L1q0≤p0L2l1, L1l1l1q0≥p0l2L1,
L2l2L2q1≤p1L1l2, L2l2l2q1≥p1l1L2, 4.19
which implies that 1q0
L1−l1
≤p0 L2−l2
,
1q1 L2−l2
≤p1 L1−l1
. 4.20
Therefore,
1q0 1q1
−p0p1 L1−l1
≤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
L2−l2L1−l1. 4.24
Moreover, from4.24, it follows that q01
l2L1−l1q0 q01
L2l1−L1q0. 4.25
In addition, from3.50, the first and the second inequalities of4.19, we get q01
l2L1−l1q0≤L1l1≤ q01
L2l1−L1q0. 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
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
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,
L1−M2 pL2
1−L1
,
M1−L3
qM2
1−M1 , L2−M3
pL3 1−L2
,
M2−L1
qM3
1−M2 , L3−M1
pL1
1−L3
,
M3−L2
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 L1 −M2 ≥ 0 resp.,L1 − M2 ≤ 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.
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< p0≤q0, q1−1< p1≤q1, 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
y≤ q0zy
q0y , z≤ q1yz
q1z . 5.4
Without loss of generality we assume thatz≤y. Then from5.4, it results that y≤ q0yy
q0y , 5.5
which means that
y≤1. 5.6
Moreover, from5.4and5.6, we get
z≤ q1z
q1z 1. 5.7
In addition, from5.3, we have
y > y
q0y, z > z
q1z, 5.8
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
q0−p0z
q0y2 p0
q0y 0 0
0 q1−p1y q1z2 p1
q1z 0
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
, Zn
⎛
⎜⎜
⎝ wn−1
vn−1 wn
vn
⎞
⎟⎟
⎠. 5.11
The characteristic equation ofAis
λ4−
q0−p0z
q0y2 q1−p1y q1z2
λ2− p1p0
q0y
q1zλ
q0−p0z
q1−p1y 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
q0−p0z
q0y2 q1−p1y q1z2
p1p0
q0y
q1z
q0−p0z
q1−p1y q0y2
q1z2
<1. 5.13
From5.3, we get
q0−p0z 1−y yq0
, q1−p1y 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−y1−z 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