# Global Dynamics of Rational Difference Equationsxn+1=xn+xn-1/q+ynyn-1 and yn+1=yn+yn-1/p+xnxn-1

Authors: Keying Liu; Peng Li; Weizhou Zhong
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

Global dynamics of a system of nonlinear difference equations was investigated, which had five kinds of equilibria including isolated points and a continuum of nonhyperbolic equilibria along the coordinate axes. The local stability of these equilibria was analyzed which led to nine regions in the parameters space. The solution of the system converged to the equilibria or the boundary point(+∞,0) or (0,+∞) in each region depending on nonnegative initial conditions. These results completely described the behavior of the system.


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## Body

## 1. Introduction

In this paper, we focus on the global dynamics of the following system:(1)xn+1=xn+xn-1q+ynyn-1,yn+1=yn+yn-1p+xnxn-1,n=0,1,…,where the parameters p and q are positive and the initial conditions (x-1,y-1) and (x0,y0) are nonnegative.In [1], the stability of (1) was investigated. If p>2 and q>2, the equilibrium (0,0) of (1) is globally asymptotically stable. If p<2 and q<2, the equilibria (0,0) and 2-p,2-q of (1) are locally unstable. The global dynamics of (1) was considered only for the case p>2 and q>2.System (1) can be regarded as a generalization of the equation(2)xn+1=xn+xn-1β+xnxn-1,n=0,1,…,with the parameter β being positive and initial conditions x-1,  x0 being nonnegative, which was studied in [2] on the stability of the equilibria, nonexistence of prime period-two solutions, and global dynamics of the equation. More accurate results were obtained in our forthcoming work that every positive solution {xn} of (2) converged to its equilibria, x¯=0 for β≥2 and x¯+=2-β for 0<β<2.The above equations and systems are also the special cases of a general equation(3)xn+1=Axn2+Bxnxn-1+Cxn-12+Dxn+Exn-1+Faxn2+bxnxn-1+cxn-12+dxn+exn-1+f,n=0,1,…,with nonnegative parameters and proper initial conditions. Several global asymptotic results for some special cases of (3) were obtained in [3–13].As for the definition of stability and the method of linearized stability, see [1–21]. For other types of equations and systems, see [14–19, 22–39]. As for the definition of basin of attraction and the stable manifold and so on, see [35–39].In this article, we try to determine a complete picture of the behavior of (1). First, we part completely the regions of parameters by equilibria. Second, by the theory of linearized stability, we describe the local stability of these equilibria for five cases and derive nine regions in (p,q) plane. At last, we present the main results on global dynamics of (1) in these regions.It is the first time that the parameters spaces are divided into nine regions and complex dynamics of (1) are derived according to the initial conditions for each region. It is also the first time that we determine the details that the equilibrium is nonhyperbolic or a saddle point if it is unstable. It is worth pointing out that the system has a continuum of nonhyperbolic equilibria along a vertical line or/and a horizontal line, which lead to interesting phenomena on the global dynamics.


## 2. Existence of Equilibria

The study of dynamics of difference equations often requires that equilibria be calculated first, followed by a local stability analysis of the equilibria. This is then complemented by other considerations (existence of periodic points, etc.). If the analysis is applied to a class of equations dependent on one or more parameters, the task is complicated by the fact that a formula is not always available for equilibria, and even if it is, determination of stability of parameter-dependent equilibria may be a daunting task.First of all, we obtain the existence of equilibria of (1).As is known, the equilibriumx¯,y¯ of (1) satisfies (4)x¯=2x¯q+y¯2,y¯=2y¯p+x¯2from which it follows that one of five cases holds for the equilibrium points of (1):(1)
E0=(0,0) if one of the following conditions holds:
(i)
p>2 and q>2.(ii)
p>2 and q<2.(iii)
p<2 and q>2.(2)
E0 and E+=(2-p,2-q) if p<2 and q<2.(3)
E0 and Ex¯=(x¯,0) with arbitrary x¯>0 if p≠2 and q=2.(4)
E0 and Ey¯=(0,y¯) with arbitrary y¯>0 if p=2 and q≠2.(5)
E0 and Ex¯=(x¯,0) and Ey¯=(0,y¯) with arbitrary x¯>0 and y¯>0 if p=2 and q=2.


## 3. Local Stability of Equilibria

Now, we consider the local stability of these equilibria of (1).


### 3.1. Local Stability ofE0

Theorem 1.
Suppose thatE0 is the equilibrium of (1). Then one of the following holds: (1)
E0 is locally asymptotically stable for p>2 and q>2.(2)
E0 is nonhyperbolic of the stable type for one of the following three cases:
Case  2.1:p=2,  q>2.Case  2.2:p>2,  q=2.Case  2.3:p=2,  q=2.(3)
E0 is nonhyperbolic of the unstable type for one of the following two cases:
Case  3.1:p=2,  q<2.Case  3.2:p<2,  q=2.(4)
E0 is a saddle point for one of the following three cases:
Case  4.1:p<2 and q<2.Case  4.2:p<2 and q>2.Case  4.3:p>2 and q<2.Proof.
The linearized system of (1) about E0 is(5)xn+1=1qxn+1qxn-1,yn+1=1pyn+1pyn-1.As is shown in [1], we have its characteristic polynomial(6)fλ=f1λf2λ=λ2-1pλ-1pλ2-1qλ-1q.Here, we only focus on one of these two factors. Obviously, we have(7)f10=-1p<0,f11=1-2p,f1-1=1>0. Thus, the distribution of solutions of f1(λ)=0 is one of the following: (i)
Two real roots in(-1,1) for p>2(ii)
One root being1 and the other being -0.5 for p=2(iii)
One root in(-1,0) and the other in (1,+∞) for p<2
By Theorem  1.2.1 in [21], we obtain the conclusions and complete the proof.


### 3.2. Local Stability ofE+

Now, we consider the local stability of the positive equilibriumE+=(2-p,2-q) of (1), which exists only for p<2 and q<2.Theorem 2.
Assume thatp<2 and q<2 and E+=(2-p,2-q) is the positive equilibrium of (1); then E+ is a saddle point.Proof.
The linearized equation of (1) about E+ is(8)xn+1=12xn+12xn-1+αyn+αyn-1,yn+1=αxn+αxn-1+12yn+12yn-1,where α=-2-p2-q/2. It is obvious that 0<α2<1 for p<2 and q<2.
As is shown in [1], its characteristic polynomial is(9)gλ=λ4-λ3-34+α2λ2+12-2α2λ+14-α2.We have g(-1)=1, g(-1/2)=-α2, g(1)=-4α2, and g(5/2)=81/4-49α2/4>0 for 0<α2<1. Thus, g(λ)=0 has one solution in (-1,-1/2) and one in (1,5/2).
Now, we divide it into two cases to show the distribution of the other two solutions ofg(λ)=0.
Case  1 (α2≤1/4). In view of g(0)=1/4-α2≥0, we conclude that g(λ)=0 has three solutions in (-1,1) and one in (1,2.5) and thus E+ is a saddle point by Theorem  1.2.1 in [21].
Case  2 (α2>1/4). In this case, we rewrite (9) as follows:(10)gλ=λ+λ1λ-λ2λ2+sλ+t,where 1/2<λ1<1, 1<λ2<5/2, s and t satisfying(11)s+λ1-λ2=-1,(12)t+λ1-λ2s-λ1λ2=-34+α2,(13)λ1-λ2t-λ1λ2s=12-2α2,(14)-λ1λ2t=14-α2.
Next, we try to prove all roots of the equationλ2+sλ+t=0 to be inside the unit disk for α2>1/4, which is necessary and sufficient to prove s<1+t<2 by Theorem  1.2.2 in [21].
First, we try to show0<t<1 for α2>1/4.
From (14), it is obvious t>0 and thus t<1 is equivalent to α2<1/4+λ1λ2. In view of 0<α2<1 for p<2 and q<2, we try to prove λ1λ2>3/4. To this end, we try to determine the exact range of λ2.
In view ofα2>1/4, from g(3/2)=1/8-25α2/4<0 and g(2.5)>0, we could obtain λ2∈(3/2,5/2). Thus, we have λ1λ2>3/4 and t<1 is proved.
Second, we shows<1+t for α2>1/4. To this end, we only need to show s<1 for t<1.
From (11), we obtain s+1=λ2-λ1 and thus -1/2<s<1 as desired.
In fact, more precisely, in view ofg(2)=25/4-9α2, we have that 3/2<λ2<2 for 1/4<α2<25/36 and 2≤λ2<5/2 for α2≥25/36.
Therefore, forα2>1/4, we have -1/2<s<1/2; for 1/4<α2<25/36 and 0≤s<1 for α2≥25/36. Hence, we obtain s<1+t for α2>1/4 and thus all roots of the equation λ2+sλ+t=0 lie inside the unit disk. By Theorem  1.2.1 in [21], E+ of (1) is a saddle point for α2>1/4.
Thus, we conclude thatE+ of (1) is a saddle point if it exists and we complete the proof.


### 3.3. Local Stability ofEx¯

Now, we consider the local stability of the equilibriaEx¯=x¯,0  x¯>0 of (1), which exists only for p≠2 and q=2.Theorem 3.
Assume thatp≠2 and q=2 and Ex¯=(x¯,0)  (x¯>0) are the equilibria of (1).
Ifp>2 or p<2 and p+(x¯)2≥2, then Ex¯ is nonhyperbolic of the stable type.
Ifp<2 and p+(x¯)2<2, then Ex¯ is nonhyperbolic of the unstable type.Proof.
The linearized equation of (1) about every Ex¯ is(15)xn+1=12xn+12xn-1,yn+1=1p+x¯2yn+1p+x¯2yn-1,from which we have its characteristic polynomial(16)hλ=h1λh2λ=λ2-12λ-12λ2-1p+x¯2λ-1p+x¯2.It is obvious that h1(λ)=0 has two solutions 1 and -0.5.
Similarly, ifp<2 and p+(x¯)2=2, then h2(λ)=0 has two solutions 1 and -0.5.
Ifp>2 or p<2 and p+(x¯)2>2, then h2(λ)=0 has two solutions in (-1,1).
Ifp<2 and p+(x¯)2<2, then h2(λ)=0 has one solution in (-1,0) and the other in (1,+∞).
By Theorem  1.2.1 in [21], we derive the conclusions and complete the proof.


### 3.4. Local Stability ofEy¯

Similar to the proof of Theorem3, we have the following theorem.Theorem 4.
Assume thatp=2 and q≠2 and Ey¯=0,y¯  y¯>0 are the equilibria of (1).
Ifq>2 or q<2 and q+y¯2≥2, then Ey¯ is nonhyperbolic of the stable type.
Ifq<2 and q+(y¯)2<2, then Ey¯ is nonhyperbolic of the unstable type.


### 3.5. Local Stability ofEx¯ and Ey¯

In case ofp=q=2, the equilibria of (1) include E0, Ex¯, and Ey¯  x¯>0,y¯>0.By Theorem1, for p=q=2, E0 of (1) is nonhyperbolic of the stable type.Similar to the proof of Theorem3, the linearized equation of (1) about Ex¯ is (15) with p=2 and its characteristic polynomial is (16) with p=2 which has four roots in (-1,1). It implies that every Ex¯ is nonhyperbolic of the stable type.Similarly, everyEy¯ is nonhyperbolic of the stable type.Theorem 5.
Assume thatp=q=2, E0, Ex¯, and Ey¯  (x¯>0 and y¯>0) are the equilibria of (1); then they are nonhyperbolic of the stable type.There are 9 cases in parametric spacep-q with distinct local stability of distinct equilibria. We list the above results in Table 1. For simplicity, if E0 of (1) is locally asymptotically stable, we denote E0-L.A.S. If E0 is nonhyperbolic, we denote E0-N.H. If E0 is nonhyperbolic of the stable type or the unstable type, we denote E0-N.H.S or N.H.U. If E0 is a saddle point, we denote E0-Saddle.Table 1
Local stability of equilibria of (1).
Region
Parameters
Local stability of equilibria
R
1
p
>
2,
q
>
2
E
0
-
L
.
A
.
S
.
R
2
p
<
2,
q
>
2
E
0
-
S
a
d
d
l
e
R
3
p
<
2,
q
<
2
E
0
-
S
a
d
d
l
e,
E
+
-
S
a
d
d
l
e
R
4
p
>
2,
q
<
2
E
0
-
S
a
d
d
l
e
R
5
p
=
2,
q
>
2
E
0
-
N
.
H
.
S,
Ey--N.H.S
R
6
p
<
2,
q
=
2
E
0
-
N
.
H
.
U,
E
x
-
-
N
.
H
.
R
7
p
=
2,
q
<
2
E
0
-
N
.
H
.
U,
E
y
-
-
N
.
H
.
R
8
p
>
2,
q
=
2
E
0
-
N
.
H
.
S,
E
x
-
-
N
.
H
.
S
R
9
p
=
2,
q
=
2
E
0
-
N
.
H
.
S,
E
x
-
-
N
.
H
.
S,
E
y
-
-
N
.
H
.
S


## 3.1. Local Stability ofE0

Theorem 1.
Suppose thatE0 is the equilibrium of (1). Then one of the following holds: (1)
E0 is locally asymptotically stable for p>2 and q>2.(2)
E0 is nonhyperbolic of the stable type for one of the following three cases:
Case  2.1:p=2,  q>2.Case  2.2:p>2,  q=2.Case  2.3:p=2,  q=2.(3)
E0 is nonhyperbolic of the unstable type for one of the following two cases:
Case  3.1:p=2,  q<2.Case  3.2:p<2,  q=2.(4)
E0 is a saddle point for one of the following three cases:
Case  4.1:p<2 and q<2.Case  4.2:p<2 and q>2.Case  4.3:p>2 and q<2.Proof.
The linearized system of (1) about E0 is(5)xn+1=1qxn+1qxn-1,yn+1=1pyn+1pyn-1.As is shown in [1], we have its characteristic polynomial(6)fλ=f1λf2λ=λ2-1pλ-1pλ2-1qλ-1q.Here, we only focus on one of these two factors. Obviously, we have(7)f10=-1p<0,f11=1-2p,f1-1=1>0. Thus, the distribution of solutions of f1(λ)=0 is one of the following: (i)
Two real roots in(-1,1) for p>2(ii)
One root being1 and the other being -0.5 for p=2(iii)
One root in(-1,0) and the other in (1,+∞) for p<2
By Theorem  1.2.1 in [21], we obtain the conclusions and complete the proof.


## 3.2. Local Stability ofE+

Now, we consider the local stability of the positive equilibriumE+=(2-p,2-q) of (1), which exists only for p<2 and q<2.Theorem 2.
Assume thatp<2 and q<2 and E+=(2-p,2-q) is the positive equilibrium of (1); then E+ is a saddle point.Proof.
The linearized equation of (1) about E+ is(8)xn+1=12xn+12xn-1+αyn+αyn-1,yn+1=αxn+αxn-1+12yn+12yn-1,where α=-2-p2-q/2. It is obvious that 0<α2<1 for p<2 and q<2.
As is shown in [1], its characteristic polynomial is(9)gλ=λ4-λ3-34+α2λ2+12-2α2λ+14-α2.We have g(-1)=1, g(-1/2)=-α2, g(1)=-4α2, and g(5/2)=81/4-49α2/4>0 for 0<α2<1. Thus, g(λ)=0 has one solution in (-1,-1/2) and one in (1,5/2).
Now, we divide it into two cases to show the distribution of the other two solutions ofg(λ)=0.
Case  1 (α2≤1/4). In view of g(0)=1/4-α2≥0, we conclude that g(λ)=0 has three solutions in (-1,1) and one in (1,2.5) and thus E+ is a saddle point by Theorem  1.2.1 in [21].
Case  2 (α2>1/4). In this case, we rewrite (9) as follows:(10)gλ=λ+λ1λ-λ2λ2+sλ+t,where 1/2<λ1<1, 1<λ2<5/2, s and t satisfying(11)s+λ1-λ2=-1,(12)t+λ1-λ2s-λ1λ2=-34+α2,(13)λ1-λ2t-λ1λ2s=12-2α2,(14)-λ1λ2t=14-α2.
Next, we try to prove all roots of the equationλ2+sλ+t=0 to be inside the unit disk for α2>1/4, which is necessary and sufficient to prove s<1+t<2 by Theorem  1.2.2 in [21].
First, we try to show0<t<1 for α2>1/4.
From (14), it is obvious t>0 and thus t<1 is equivalent to α2<1/4+λ1λ2. In view of 0<α2<1 for p<2 and q<2, we try to prove λ1λ2>3/4. To this end, we try to determine the exact range of λ2.
In view ofα2>1/4, from g(3/2)=1/8-25α2/4<0 and g(2.5)>0, we could obtain λ2∈(3/2,5/2). Thus, we have λ1λ2>3/4 and t<1 is proved.
Second, we shows<1+t for α2>1/4. To this end, we only need to show s<1 for t<1.
From (11), we obtain s+1=λ2-λ1 and thus -1/2<s<1 as desired.
In fact, more precisely, in view ofg(2)=25/4-9α2, we have that 3/2<λ2<2 for 1/4<α2<25/36 and 2≤λ2<5/2 for α2≥25/36.
Therefore, forα2>1/4, we have -1/2<s<1/2; for 1/4<α2<25/36 and 0≤s<1 for α2≥25/36. Hence, we obtain s<1+t for α2>1/4 and thus all roots of the equation λ2+sλ+t=0 lie inside the unit disk. By Theorem  1.2.1 in [21], E+ of (1) is a saddle point for α2>1/4.
Thus, we conclude thatE+ of (1) is a saddle point if it exists and we complete the proof.


## 3.3. Local Stability ofEx¯

Now, we consider the local stability of the equilibriaEx¯=x¯,0  x¯>0 of (1), which exists only for p≠2 and q=2.Theorem 3.
Assume thatp≠2 and q=2 and Ex¯=(x¯,0)  (x¯>0) are the equilibria of (1).
Ifp>2 or p<2 and p+(x¯)2≥2, then Ex¯ is nonhyperbolic of the stable type.
Ifp<2 and p+(x¯)2<2, then Ex¯ is nonhyperbolic of the unstable type.Proof.
The linearized equation of (1) about every Ex¯ is(15)xn+1=12xn+12xn-1,yn+1=1p+x¯2yn+1p+x¯2yn-1,from which we have its characteristic polynomial(16)hλ=h1λh2λ=λ2-12λ-12λ2-1p+x¯2λ-1p+x¯2.It is obvious that h1(λ)=0 has two solutions 1 and -0.5.
Similarly, ifp<2 and p+(x¯)2=2, then h2(λ)=0 has two solutions 1 and -0.5.
Ifp>2 or p<2 and p+(x¯)2>2, then h2(λ)=0 has two solutions in (-1,1).
Ifp<2 and p+(x¯)2<2, then h2(λ)=0 has one solution in (-1,0) and the other in (1,+∞).
By Theorem  1.2.1 in [21], we derive the conclusions and complete the proof.


## 3.4. Local Stability ofEy¯

Similar to the proof of Theorem3, we have the following theorem.Theorem 4.
Assume thatp=2 and q≠2 and Ey¯=0,y¯  y¯>0 are the equilibria of (1).
Ifq>2 or q<2 and q+y¯2≥2, then Ey¯ is nonhyperbolic of the stable type.
Ifq<2 and q+(y¯)2<2, then Ey¯ is nonhyperbolic of the unstable type.


## 3.5. Local Stability ofEx¯ and Ey¯

In case ofp=q=2, the equilibria of (1) include E0, Ex¯, and Ey¯  x¯>0,y¯>0.By Theorem1, for p=q=2, E0 of (1) is nonhyperbolic of the stable type.Similar to the proof of Theorem3, the linearized equation of (1) about Ex¯ is (15) with p=2 and its characteristic polynomial is (16) with p=2 which has four roots in (-1,1). It implies that every Ex¯ is nonhyperbolic of the stable type.Similarly, everyEy¯ is nonhyperbolic of the stable type.Theorem 5.
Assume thatp=q=2, E0, Ex¯, and Ey¯  (x¯>0 and y¯>0) are the equilibria of (1); then they are nonhyperbolic of the stable type.There are 9 cases in parametric spacep-q with distinct local stability of distinct equilibria. We list the above results in Table 1. For simplicity, if E0 of (1) is locally asymptotically stable, we denote E0-L.A.S. If E0 is nonhyperbolic, we denote E0-N.H. If E0 is nonhyperbolic of the stable type or the unstable type, we denote E0-N.H.S or N.H.U. If E0 is a saddle point, we denote E0-Saddle.Table 1
Local stability of equilibria of (1).
Region
Parameters
Local stability of equilibria
R
1
p
>
2,
q
>
2
E
0
-
L
.
A
.
S
.
R
2
p
<
2,
q
>
2
E
0
-
S
a
d
d
l
e
R
3
p
<
2,
q
<
2
E
0
-
S
a
d
d
l
e,
E
+
-
S
a
d
d
l
e
R
4
p
>
2,
q
<
2
E
0
-
S
a
d
d
l
e
R
5
p
=
2,
q
>
2
E
0
-
N
.
H
.
S,
Ey--N.H.S
R
6
p
<
2,
q
=
2
E
0
-
N
.
H
.
U,
E
x
-
-
N
.
H
.
R
7
p
=
2,
q
<
2
E
0
-
N
.
H
.
U,
E
y
-
-
N
.
H
.
R
8
p
>
2,
q
=
2
E
0
-
N
.
H
.
S,
E
x
-
-
N
.
H
.
S
R
9
p
=
2,
q
=
2
E
0
-
N
.
H
.
S,
E
x
-
-
N
.
H
.
S,
E
y
-
-
N
.
H
.
S


## 4. Global Dynamics

For nonnegative initial conditionsxi,yi  i=-1,0, we assume that xn,yn is the corresponding solution of (1). For simplicity, we often need to consider the behavior of xn and yn, respectively.In the following, we try to investigate the global dynamics of (1) for these nine cases.Case 1 (R1).
By Theorem  1 in [1], E0 of (1) is globally asymptotically stable for p>2 and q>2; that is, basin of attraction of E0 of (1) is B(E0)=x,y∣x>0,y>0.Case 2 (R2).
In this case,E0 of (1) is a saddle point for p<2 and q>2.
If the initial conditionsxi,yi  i=-1,0 are on y-axis, we have xn=0 for all n, and system (1) is changed into a single equation(17)yn+1=yn+yn-1p,from which we have yn=c1(r1)n+c2(r2)n with r1 and r2 satisfying the characteristic equation r2-r/p-1/p=0. For p<2, one of the modulus of r1 and r2 is smaller than one and the other is greater than one. Therefore, we have limn→∞yn=+∞ with (xi,yi) on y-axis for p<2.
Ifxi,yi are on x-axis, we have yn=0 for all n and thus limn→∞xn=0 for q>2.
We declare that the stable manifold ofE0 is WS(E0)=x,y∣x>0,y=0.
In fact, if(xi,yi)∉WS(E0) in the first quadrant, then from (1), we obtain(18)xn+1≤xn+xn-1q.
We ascertainlimn→∞xn=0 for q>2. In fact, we could deduce that by comparison and the theory of linear difference equations.
Settingu-1=x-1,  u0=x0 and(19)un+1=un+un-1q,we obtain xn≤un for all n≥1 by induction.
From (19), we have un=c1(r1)n+c2(r2)n  n≥1 with r1 and r2 satisfying the characteristic equation r2-r/q-1/q=0 from which we have r1<1 and r2<1 for q>2 and thus un goes to zero as n tends to ∞. Therefore, we have limn→∞xn=0 for q>2.
Next, we consider the behavior of the componentyn.
From the fact oflimn→∞xn=0, there is a positive constant M satisfying p+M2<2 such that xn≤M for n≥n∗ with n∗ being some positive integer. From (1), for n≥n∗+1, we obtain(20)yn+1>yn+yn-1p+M2.By comparison and the theory of linear difference equations, we get limn→∞yn=+∞ for p<2.
Hence, we obtain the following theorem.Theorem 6.
Ifp<2 and q>2, then the global stable manifold E0 of (1) is WS(E0)=x,y∣x>0,y=0. Whenever xi,yi∉WS(E0)i=-1,0, then limn→∞xn,yn=0,+∞.Case 3 (R3).
In this case, bothE0 and E+ of (1) are saddle points for p<2 and q<2.
We claim that sets of the form(21)Δ+=2-p+ϵ,+∞×0,2-q-ϵare invariant for sufficiently small ϵ>0: that is, xn,yn∈Δ+ for all n if xi,yi∈Δ+  i=-1,0.
Suppose(x-1,y-1) and (x0,y0)∈Δ+, from (1); then we have(22)x1=x0+x-1q+y0y-1≥x0+x-12-ϵ≥22-ϵ2-p+ϵ,y1=y0+y-1p+x0x-1≤y0+y-12+ϵ≤22+ϵ2-q-ϵfrom which we have (x1,y1)∈Δ+. By induction, we have (xn,yn)∈Δ+ for all n and(23)xn+1=xn+xn-1q+ynyn-1≥xn+xn-12-ϵ,yn+1=yn+yn-1p+xnxn-1≤yn+yn-12+ϵfrom which it follows that limn→∞(xn,yn)=(+∞,0).
Similarly, sets of the form(24)Δ-=0,2-p-ϵ×2-q+ϵ,+∞are invariant for sufficiently small ϵ>0. For (xi,yi)∈Δ-, then we have limn→∞(xn,yn)=(0,+∞).Theorem 7.
Ifp<2 and q<2, then sets of the form Δ+ and Δ- (defined by (21) and (24)) are invariant of (1) for sufficiently small ϵ>0.
Ifxi,yi∈Δ-  (i=-1,0), then limn→∞xn,yn=0,+∞.
Ifxi,yi∈Δ+  (i=-1,0), then limn→∞xn,yn=+∞,0.Case 4 (R4).
In this case,E0 of (1) is a saddle point for p>2 and q<2.
Similar to that of Case2, we obtain the following theorem.Theorem 8.
Ifp>2 and q<2, then the stable manifold of E0 of (1) is WSE0=x,y∣x=0,y>0. Whenever xi,yi∉WSE0i=-1,0, then limn→∞xn,yn=+∞,0.Case 5 (R5).
In this case,E0 and Ey¯ of (1) are nonhyperbolic of the stable type for p=2 and q>2.
For(xi,yi) on y-axis, we have that xn=0 for all n and yn satisfies (17) with p=2, from which it follows that limn→∞yn exists depending on yi  i=-1,0.
For(xi,yi) on x-axis, we have that yn=0 for all n and xn satisfies(25)xn+1=xn+xn-1q,from which we have limn→∞xn=0 for q>2.
For positive initial conditions, similar to Case2, we also know limn→∞xn=0 for q>2. Specially, it follows that xn=0 for all n≥n0 for some positive integer n0. From (1), for p=2, we have(26)yn+1=yn+yn-1p+xnxn-1=yn+yn-12for n≥n0 and thus limn→∞yn exists. Thus, basin of attraction of E0 is B(E0)=x,y∣x>0,y=0. If xi,yi∉B(E0)  i=-1,0 then limn→∞xn=0 and limn→∞yn exists.Theorem 9.
Ifp=2 and q>2, then basin of attraction of E0 of (1) is BE0=x,y∣x>0,y=0. Whenever xi,yi∉BE0  i=-1,0, then limn→∞xn,yn=Ey¯=0,y¯.Case 6 (R6).
In this case,E0 and Ex¯ of (1) are nonhyperbolic for p<2 and q=2.
For(xi,yi) on y-axis, we have that xn=0 for all n and yn satisfies (17) with p<2, from which it follows that limn→∞yn=+∞.
For(xi,yi) on x-axis, we have that yn=0 for all n and xn satisfies (25) with q=2, from which it follows that limn→∞xn exists depending on xi(i=-1,0).
There is a curveC0(x) such that the first quadrant is divided into two connected parts and(27)C0x=x,y∣x>0,y=x,(28)W+0x=x,y∣x>0,y<x,(29)W-0x=x,y∣x>0,y>x.
If(xi,yi)∈W-0(x), then we have yn>xn for all n. Thus, from (1), we obtain(30)xn+1=xn+xn-1q+ynyn-1<xn+xn-1q+xnxn-1.By comparison and the results of (2), we have limn→∞xn=0 for q=2.
Hence, similar to Case2, we have limn→∞yn=+∞ for p<2.
Ifxi,yi∈C0(x), then we also obtain the above conclusion.
Ifxi,yi∈W+0x, that is, yi<xi  i=-1,0, then we choose such a x¯>0 that p+(x¯)2=2: that is, x¯∗=2-p. There is a curve Cx¯∗(x),(31)Cx¯∗x=x,y∣y>0,y=x-2-p,which is below the curve C0(x) such that W+0(x) is divided into two connected parts(32)W+,10x=x,y∣y>0,Cx¯∗x<y<x,W+,20x=x,y∣y>0,y<Cx¯∗x.
If(xi,yi)∈W+,20(x), that is, yi<Cx¯∗xi  i=-1,0, then we have yn<Cx¯∗(xn) for all n by induction. Thus, from (1), we obtain(33)yn+1=yn+yn-1p+xnxn-1<yn+yn-1p+yn+2-pyn-1+2-p<yn+yn-12+ϵfor arbitrarily small ϵ>0. Thus, we conclude that limn→∞yn=0. Specially, we have yn=0 for n≥n^ for some positive integer n^. Hence, from (1) we also obtain (25) with q=2 for n≥n^+1, from which it follows that limn→∞xn exists depending on xi(i=-1,0).
If(xi,yi) on the curve Cx¯∗(x), then we also derive the above conclusion.
Thus, we obtain the following theorem.Theorem 10.
Ifp<2 and q=2, then E0 of (1) is nonhyperbolic of the unstable type and every Ex¯  x¯>0 of (1) is nonhyperbolic. More precisely, Ex¯ is nonhyperbolic of the unstable type for p+x¯2<2 and is nonhyperbolic of the stable type for p+x¯2≥2.
There is a curveC0(x) defined by (27) such that limn→∞(xn,yn)=(0,+∞) for xi,yii=-1,0 on and above the curve C0(x).
There is a curveCx¯∗(x) defined by (31) with x¯∗=2-p such that limn→∞(xn,yn)=Ex¯=x¯,0 with x¯>0 for xi,yii=-1,0 on and below the curve Cx¯∗(x).Case 7 (R7).
In this case,E0 and Ey¯ of (1) are nonhyperbolic for p=2 and q<2.
Similar to that of Case6, we obtain the following theorem.Theorem 11.
Ifp=2 and q<2, then E0 of (1) is nonhyperbolic of the unstable type and every Ey¯  y¯>0 of (1) is nonhyperbolic. More precisely, Ey¯ is nonhyperbolic of the unstable type for p+y¯2<2 and is nonhyperbolic of the stable type for p+y¯2≥2.
There is a curveC0(x) defined by (27) such that limn→∞(xn,yn)=+∞,0 for (xi,yi)  (i=-1,0) on and below the curve C0(x).
There is a curveCy¯∗(x) defined by Cy¯∗(x)=x,y∣x>0,y=x+2-q such that limn→∞xn,yn=Ey¯=0,y¯ with y¯>0 for (xi,yi)  i=-1,0 on and above the curve Cy¯∗(x) with y¯∗=2-q.Case 8 (R8).
In this case, bothE0 and Ex¯ of (1) are nonhyperbolic of the stable type for p>2 and q=2.
Similar to that of Case5, we obtain the following theorem.Theorem 12.
Ifp>2 and q=2, then basin of attraction of E0 of (1) is B(E0)=x,y∣x=0,y>0. Whenever xi,yi∉BE0  i=-1,0, then limn→∞xn,yn=Ex¯=x¯,0.Case 9 (R9).
Here,E0, Ex¯, and Ey¯  (x¯>0 and y¯>0) of (1) are nonhyperbolic of the stable type for p=q=2.
Now, we focus onE0. There is a curve C0x (defined by (27)) passing through E0 such that the first quadrant is divided into two connected parts and(34)W+0x=x,y∣x>0,y<x,(35)W-0x=x,y∣x>0,y>x.
If(xi,yi)  (i=-1,0) are on the curve C0(x), system (1) is reduced to a single equation and every positive solution of (1) converges to E0.
If(xi,yi)∈W+0(x), we have xi>yi  i=-1,0. By induction, from (1), we have xn>yn for all n. For p=2, from (1), we have(36)yn+1=yn+yn-12+xnxn-1<yn+yn-12+ynyn-1and thus limn→∞yn=0 by comparison and the results of (2). Specially, it follows that yn=0 for all n≥n∗ for some positive integer n∗. Thus, from (1), we have (25) for q=2 and hence limn→∞xn exists depending on initial conditions. Therefore, we have limn→∞xn,yn=Ex¯  x¯>0 for xi,yi∈W+0x.
Similarly, we obtainlimn→∞(xn,yn)=Ey¯  y¯>0 for xi,yi∈W-0x.Theorem 13.
Ifp=2 and q=2, then basin of attraction of E0 of (1) is (37)BE0=C0x=x,y∣x>0,y=x.
If(xi,yi)∈W+0x  i=-1,0, then limn→∞xn,yn=Ex¯.
If(xi,yi)∈W-0x  i=-1,0, then limn→∞xn,yn=Ey¯.
Here,W+0x and W-0x are defined by (34) and (35), respectively.These above theorems completely describe the global dynamics of (1) and are listed in Table 2. The solution of system (1) converged to either the equilibria or the boundary point +∞,0 or (0,+∞) depending on nonnegative initial conditions and parameters.Table 2
Global dynamics of (1).
Region
Theorem
Global dynamics of (1)
R
1 (p>2,q>2)
Theorem  1 [1]
B
(
E
0
)
=
x
,
y
∣
x
>
0
,
y
>
0
R
2 p<2,q>2
Theorem6
W
S
E
0
=
x
,
y
∣
x
>
0
,
y
=
0 limn→∞⁡xn,yn=0,+∞forxi,yi∉WSE0
R
3 (p<2,q<2)
Theorem7
lim
n
→
∞
⁡
x
n
,
y
n
=
(
0
,
+
∞
)
f
o
r
(
x
i
,
y
i
)
∈
Δ
- limn→∞⁡xn,yn=(+∞,0)for(xi,yi)∈Δ+
R
4 (p>2,q<2)
Theorem8
W
S
(
E
0
)
=
x
,
y
∣
x
=
0
,
y
>
0 limn→∞⁡xn,yn=(+∞,0)for(xi,yi)∉WSE0
R
5 (p=2,q>2)
Theorem9
B
(
E
0
)
=
x
,
y
∣
x
>
0
,
y
=
0 limn→∞⁡xn,yn=Ey-y->0for(xi,yi)∉B(E0)
R
6 (p<2,q=2)
Theorem10
lim
n
→
∞
⁡
x
n
,
y
n
=
0
,
+
∞
f
o
r
x
i
,
y
i
o
n
a
n
d
a
b
o
v
e
C
0
(
x
)
. limn→∞⁡xn,yn=Ex-x->0forxi,yionandbelowCx-∗x
R
7 (p=2,q<2)
Theorem11
lim
n
→
∞
⁡
x
n
,
y
n
=
(
+
∞
,
0
)
f
o
r
(
x
i
,
y
i
)
o
n
a
n
d
b
e
l
o
w
C
0
(
x
)
. limn→∞⁡xn,yn=Ey-(y->0)for(xi,yi)onandaboveCy-∗x
R
8 (p>2,q=2)
Theorem12
B
(
E
0
)
=
x
,
y
∣
x
=
0
,
y
>
0 limn→∞⁡xn,yn=Ex-x->0forxi,yi∉BE0
R
9 (p=2,q=2)
Theorem13
B
E
0
=
C
0
x
=
x
,
y
∣
x
>
0
,
y
=
x limn→∞⁡xn,yn=Ex-x->0for(xi,yi)belowC0x limn→∞⁡xn,yn=Ey-y->0for(xi,yi)aboveC0x


## 5. Conclusion

It is known that the techniques in the investigation of the behavior of difference equations can be used in investigating equations arising in mathematical models describing real life situations in biology, economics, physics, sociology, control theory, and vice versa.In this paper, we investigate a system of nonlinear difference equations, which has complex dynamics. We use the results of linearized stability to analyze the local stability of five kinds of equilibria of (1) with parameters p,q in nine regions in the first quadrant. Specially, for particular parameters p and q, dynamics of the system is very interesting with a continuum of nonhyperbolic equilibria along a line. Generally speaking, the solution of (1) converges to its equilibria if the equilibrium is locally asymptotically stable or nonhyperbolic of the stable type, depending on initial conditions. Otherwise, it may converge to the boundary point +∞,0 or 0,+∞ or exhibit somewhat chaos.



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*Source: 1295089-2017-05-03.xml*