doi:10.1155/2009/132802
Research Article
Global Dynamics of a Competitive System of
Rational Difference Equations in the Plane
S. Kalabu ˇsi ´c,
1M. R. S. Kulenovi ´c,
2and E. Pilav
11Department of Mathematics, University of Sarajevo, 71 000 Sarajevo, Bosnia and Herzegovina 2Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USA
Correspondence should be addressed to M. R. S. Kulenovi´c,[email protected]
Received 26 August 2009; Accepted 8 December 2009
Recommended by Panayiotis Siafarikas
We investigate global dynamics of the following systems of difference equationsxn1 α1
β1xn/yn,yn1 α2γ2yn/A2xn,n 0,1,2, . . ., where the parametersα1,β1,α2,γ2, and
A2are positive numbers and initial conditionsx0andy0are arbitrary nonnegative numbers such thaty0>0. We show that this system has rich dynamics which depend on the part of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points are separated by the global stable manifolds of either saddle points or of nonhyperbolic equilibrium points.
Copyrightq2009 S. Kalabuˇsi´c 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 and Preliminaries
In this paper, we study the global dynamics of the following rational system of difference equations:
xn1 α1yβ1xn
n
,
yn1
α2γ2yn
A2xn
,
n0,1,2, . . . , 1.1
where the parametersα1, β1, α2, γ2,andA2are positive numbers and initial conditionsx0≥
0 andy0 > 0 are arbitrary numbers. System1.1was mentioned in 1as a part of Open
1.1has a variety of dynamics that depend on the value of parameters. We show that system
1.1may have between zero and two equilibrium points, which may have different local character. If system1.1 has one equilibrium point, then this point is either locally saddle point or non-hyperbolic. If system1.1has two equilibrium points, then the pair of points is the pair of a saddle point and a sink. The major problem is determining the basins of attraction of different equilibrium points. System1.1gives an example of semistable non-hyperbolic equilibrium point. The typical results are Theorems4.1and4.5below.
System 1.1 is a competitive system, and our results are based on recent results developed for competitive systems in the plane; see 2,3. In the next section, we present some general results about competitive systems in the plane. The third section deals with some basic facts such as the non-existence of period-two solution of system1.1. The fourth section analyzes local stability which is fairly complicated for this system. Finally, the fifth section gives global dynamics for all values of parameters.
LetIandJbe intervals of real numbers. Consider a first-order system of difference equations of the form
xn1f
xn, yn
,
yn1g
xn, yn
, n0,1,2, . . . , 1.2
wheref:I × J → I, g :I × J → J, and x0, y0∈ I × J.
When the functionfx, y is increasing in x and decreasing in y and the function gx, yis decreasing inxand increasing iny, the system1.2is calledcompetitive. When the functionfx, yis increasing inxand increasing inyand the functiongx, yis increasing in xand increasing in y, the system1.2is called cooperative. A mapT that corresponds to the system 1.2 is defined as Tx, y fx, y, gx, y. Competitive and cooperative maps, which are called monotone maps, are defined similarly.Strongly competitivesystems of difference equations or maps are those for which the functionsf andgare coordinate-wise strictly monotone.
Ifv u, v ∈ R2, we denote with Q
v, ∈ {1,2,3,4}, the four quadrants inR2
relative tov, that is,Q1v {x, y∈R2 :x≥u, y ≥v}, Q2v {x, y∈R2 :x≤u, y ≥ v}, and so on. Define theSouth-East partial orderse onR2 byx, yses, tif and only if x≤ sandy≥ t. Similarly, we define theNorth-Eastpartial orderneonR2byx, ynes, t
if and only ifx ≤ sandy ≤ t. For A ⊂ R2 and x ∈ R2, define the distance fromxtoA as
distx,A:inf{x−y : y∈ A}. By intA,we denote the interior of a setA.
It is easy to show that a mapFis competitive if it is nondecreasing with respect to the South-East partial order, that is if the following holds:
x1
y1
se
x2
y2
⇒F
x1
y1
seF
x2
y2
. 1.3
Competitive systems were studied by many authors; see4–19, and others. All known results, with the exception of4,6,10, deal with hyperbolic dynamics. The results presented here are results that hold in both the hyperbolic and the non-hyperbolic cases.
Definition 1.1. LetR be a nonempty subset ofR2. A competitive mapT : R → R is said
to satisfy conditionOif for everyx,yinR,TxneTyimpliesxney, andT is said to
satisfy conditionO−if for everyx,yinR,TxneTyimpliesynex.
The following theorem was proved by de Mottoni and Schiaffino 20 for the Poincar´e map of a periodic competitive Lotka-Volterra system of differential equations. Smith generalized the proof to competitive and cooperative maps15,16.
Theorem 1.2. LetRbe a nonempty subset ofR2. IfTis a competitive map for which (O) holds, then
for allx∈R,{Tnx}is eventually componentwise monotone. If the orbit ofxhas compact closure, then it converges to a fixed point ofT. If instead (O−) holds, then for allx∈R,{T2n}is eventually componentwise monotone. If the orbit ofxhas compact closure inR, then its omega limit set is either a period-two orbit or a fixed point.
The following result is from18, with the domain of the map specialized to be the Cartesian product of intervals of real numbers. It gives a sufficient condition for conditions
OandO−.
Theorem 1.3Smith18. LetR⊂R2be the Cartesian product of two intervals inR. LetT :R → Rbe aC1competitive map. IfTis injective anddetJ
Tx>0for allx∈R,thenTsatisfies (O). IfT is injective anddetJTx<0for allx∈R,thenT satisfies (O−).
Theorem 1.4. LetTbe a monotone map on a closed and bounded rectangular regionR ⊂R2.Suppose
thatThas a unique fixed pointeinR.Theneis a global attractor ofTonR.
The following theorems were proved by Kulenovi´c and Merino3for competitive systems in the plane, when one of the eigenvalues of the linearized system at an equilibrium
hyperbolic or non-hyperbolicis by absolute value smaller than 1 while the other has an arbitrary value. These results are useful for determining basins of attraction of fixed points of competitive maps.
Our first result gives conditions for the existence of a global invariant curve through a fixed pointhyperbolic or notof a competitive map that is differentiable in a neighborhood of the fixed point, when at least one of two nonzero eigenvalues of the Jacobian matrix of the map at the fixed point has absolute value less than one. A regionR ⊂R2isrectangularif it is
the Cartesian product of two intervals inR.
Theorem 1.5. LetT be a competitive map on a rectangular regionR ⊂R2. Let x∈ Rbe a fixed point
ofTsuch thatΔ:R ∩intQ1x∪ Q3xis nonempty (i.e., x is not the NW or SE vertex ofR, and
Tis strongly competitive onΔ. Suppose that the following statements are true.
aThe mapThas aC1extension to a neighborhood of x.
bThe Jacobian matrix ofT atxhas real eigenvaluesλ,μsuch that0<|λ|< μ, where|λ|<1, and the eigenspaceEλassociated withλis not a coordinate axis.
Corollary 1.6. If T has no fixed point nor periodic points of minimal period-two in Δ, then the endpoints ofCbelong to∂R.
For maps that are strongly competitive near the fixed point, hypothesis b. of
Theorem 1.5reduces just to|λ|<1. This follows from a change of variables18that allows the Perron-Frobenius Theorem to be applied to give that, at any point, the Jacobian matrix of a strongly competitive map has two real and distinct eigenvalues, the larger one in absolute value being positive, and that corresponding eigenvectors may be chosen to point in the direction of the second and first quadrants, respectively. Also, one can show that in such case no associated eigenvector is aligned with a coordinate axis.
The following result gives a description of the global stable and unstable manifolds of a saddle point of a competitive map. The result is the modification ofTheorem 1.7from12.
Theorem 1.7. In addition to the hypotheses of Theorem 1.5, suppose that μ > 1 and that the eigenspaceEμassociated withμis not a coordinate axis. If the curveCofTheorem 1.5has endpoints in∂R, thenCis the global stable manifoldWsxof x, and the global unstable manifoldWuxis a curve inRthat is tangential toEμat x and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints ofWuxinRare fixed points ofT.
The next result is useful for determining basins of attraction of fixed points of competitive maps.
Theorem 1.8. Assume the hypotheses of Theorem 1.5, and let C be the curve whose existence is guaranteed byTheorem 1.5. If the endpoints ofCbelong to∂R, thenCseparatesRinto two connected components, namely
W−:{x∈ R \ C:∃y∈ Cwith xsey}, W:{x∈ R \ C:∃y∈ Cwith ysex}, 1.4
such that the following statements are true.
iW−is invariant, anddistTnx,Q2x → 0asn → ∞for every x∈ W −. iiWis invariant, anddistTnx,Q4x → 0asn → ∞for every x∈ W
.
If, in addition, x is an interior point ofRand T isC2 and strongly competitive in a neighborhood
of x, thenT has no periodic points in the boundary ofQ1x∪ Q3xexcept for x, and the following statements are true.
iiiFor every x∈ W−,there existsn0∈Nsuch thatTnx∈intQ2xforn≥n0. ivFor every x∈ W,there existsn0∈Nsuch thatTnx∈intQ4xforn≥n0.
2. Some Basic Facts
2.1. Equilibrium Points
The equilibrium pointsx, yof system1.1satisfy
x α1β1x
y ,
y α2γ2y A2x
.
2.1
First equation of System2.1gives
y α1
x β1. 2.2
Second equation of System2.1gives
yA2x α2γ2y. 2.3
Now, using2.2, we obtain
α1β1x
A2x
x α2γ2
α1β1x
x . 2.4
This implies
α1β1x
A2x α2xγ2α1β1γ2x, 2.5
which is equivalent to
β1x2x
α1−α2 β1
A2−γ2
α1
A2−γ2
0. 2.6
Solutions of2.6are
x1
−α1−α2 β1
A2−γ2
α1−α2 β1
A2−γ2 2
−4α1β1
A2−γ2
2β1
,
x2
−α1−α2 β1
A2−γ2
−α1−α2 β1
A2−γ2 2−
4α1β1
A2−γ2
2β1
.
Table1
E1 A2< γ2,
E1, E2 A2> γ2, α1−α2 β1A2−γ22−4α1β1A2−γ2>0, β1A2−γ2< α2−α1
E1≡E2 A2> γ2, α1−α2 β1A2−γ22−4α1β1A2−γ2 0, β1A2−γ2< α2−α1
E A2γ2, α2> α1
No equilibrium A2> γ2, α1−α2 β1A2−γ22−4α1β1A2−γ2<0
No equilibrium A2> γ2, α1−α2 β1A2−γ22−4α1β1A2−γ2≥0, β1A2−γ2> α2−α1 No equilibrium A2γ2, α2≤α1
Now,2.2gives
y1 α1−α2−β1
A2−γ2
α1−α2 β1
A2−γ2 2−
4α1β1
A2−γ2
−2A2−γ2
,
y2 α1−α2−β1
A2−γ2
−α1−α2 β1
A2−γ2 2−
4α1β1
A2−γ2
−2A2−γ2
.
2.8
The equilibrium points are:
E1
x1, y1
,
E1
x2, y2
,
2.9
wherex1, y1, x2, y2are given by the above relations.
Note that
xy /β1γ2. 2.10
The discriminant of2.6is given by
Dα1−α2 β1
A2−γ2 2−
4α1β1
A2−γ2
. 2.11
The criteria for the existence of equilibrium points are summarized inTable 1where
E α2−α1 β1
, β1α2 α2−α1
. 2.12
2.2. Condition
O
and Period-Two Solution
In this section we prove three lemmas.
Proof. The mapTassociated to system1.1is given by
Equations2.15and2.16are equivalent, respectively, to
α1
Lemma 2.2. System1.1has no minimal period-two solution.
Period-two solution satisfies
xA2α1
β1
α1xβ1
/y
α2yγ2 −
x0, 2.22
yα2
γ2
α2yγ2
/xA2 yA2α1xβ1 −
y0. 2.23
We show that this system has no other positive solutions except equilibrium points. Equations2.22and2.23are equivalent, respectively, to
xyα1yA2α1−xyα2xα1β1A2α1β1x2β12xA2β12−xy2γ2 yα2yγ2
0, 2.24
−xy2A
2−y2A22−xyα1−yA2α1xyα2yA2α2−x2yβ1−xyA2β1yα2γ2y2γ22 xA2yA2α1xβ1
0.
2.25
Equation2.24implies
yA2α1xyα1−α2 A2α1β1x2β12xβ1
α1A2β1
−xy2γ2 0. 2.26
Equation2.25implies
−xy2A
2−x2yβ1xy
−α1α2−A2β1
yA2−α1α2 α2γ2
y2−A22γ22
0. 2.27
Using2.26, we have
x2 −yA2α1−xyα1−α2−A2α1β1−xβ1
α1A2β1
xy2γ 2
β12
. 2.28
Putting2.28into2.27, we have
yyxA2α1y
−A2xA2β1−xyγ2β1γ22
α2
−xyβ1
xA2γ2
0. 2.29
This is equivalent to
x −yA2 2β
1A2
yα1α2β1
β1γ2
α2yγ2
−yα1α2
y−β1
yA2β1yγ2
Putting2.30into2.24, we obtain Discriminant of2.32is given by
Δ:A2γ2
which is equivalent to
This is true since
−α1β1
A2γ2
<−α1α2β1
A2γ2
≤0. 2.38
2Assume−α1α2β1 A2−γ2<0.Then it is obvious thatx2<0. Now, assume
−α1α2β1
A2−γ2
≥0. 2.39
Thenx2<0 if and only if
Δ−−α1α2β1
A2−γ2 2
A2γ2 2
>0. 2.40
This is equivalent to
4β1γ2
A2γ2
A2
−α1α2A2β1
α2γ2−β1γ22
>0. 2.41
Using2.39, we have
A2
−α1α2A2β1
α2γ2−β1γ22
≥A2γ2β1α2γ2−β1γ22≥α1γ2>0, 2.42
which implies that the inequality2.41is true.
Now, the proof of theLemma 2.2follows from theClaim 2.2.
Lemma 2.3. The mapT associated to System1.1satisfies the following:
Tx, yx, y only forx, yx, y. 2.43
Proof. By using2.1, we have
Tx, yx, y,
⇐⇒α1β1x
y
α1β1x
y ,
α2γ2y A2x
α2γ2y A2x
.
2.44
First equation implies
α1
y−yβ1
xy−xy0. 2.45
Second equation implies
α2x−x γ2A2
y−yγ2
Note the following
xy−xy x−xyxy−y. 2.47
Using2.47, Equations2.45and2.46, respectively, become
β1yx−x
α1β1x
y−y0,
α2γ2y
x−x γ2A2x
y−y0. 2.48
Note that System 2.48 is linear homogeneous system in x− x and y − y. The determinant of System2.48is given by
β1y α1β1y
α2γ2y γ2A2x
. 2.49
Using2.1, the determinant of System2.48becomes
β1y xy
yA2x γ2A2x
yA2x
β1γ2−xy
/
0. 2.50
This implies that System2.48has only trivial solution, that is
xx, yy. 2.51
3. Linearized Stability Analysis
The Jacobian matrix of the mapT has the following form:
JT
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
β1
y −
α1β1x y2
− α2γ2y A2x2
γ2 A2x
⎞ ⎟ ⎟ ⎟ ⎟
⎠. 3.1
The value of the Jacobian matrix ofTat the equilibrium point is
JTx, y ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
β1
y −
x y
− y A2x
γ2 A2x
⎞ ⎟ ⎟ ⎟ ⎟
The determinant of3.2is given by
det JTx, y β1γ2−xy yA2x
. 3.3
The trace of3.2is
Tr JT
x, y β1 y
γ2 A2x
. 3.4
The characteristic equation has the form
λ2−λ β1 y
γ2 A2x
β1γ2−xy
yA2x 0. 3.5
Theorem 3.1. Assume thatA2 < γ2.Then there exists a unique positive equilibriumE1 which is a
saddle point, and the following statements hold.
aIfβ1γ2A2< α1α2,thenλ1∈1,∞andλ2∈−1,0.
bIfβ1γ2A2> α1α2, β1A2 < α2 < β1γ2andα1β1γ2−α2< β1γ2α2−β1A2,then λ1∈1,∞andλ2∈0,1.
cIfβ1γ2A2> α1α2, α2> β1γ2andα1α2−β1γ2> β1γ2α2−β1A2,thenλ1∈1,∞
andλ2∈0,1.
dIfβ1γ2A2> α1α2, α2> β1γ2andα1α2−β1γ2< β1γ2α2−β1A2,thenλ1∈1,∞
andλ2∈−1,0.
Proof. The equilibrium is a saddle point if and only if the following conditions are satisfied:
Tr JTx, y>1det JTx, y, Tr2JTx, y−4 det JTx, y>0. 3.6
The first condition is equivalent to
β1 y
γ2 A2x
>1 β1γ2−xy yA2x
. 3.7
This implies the following:
β1A2x γ2y > yA2x β1γ2−xy
⇐⇒A2x
β1−y
γ2
y−β1
>−xy
⇐⇒y−β1
A2−γ2x
< xy.
This is equivalent to
x2y2< x1y1. 3.13
The last condition is equivalent to
x2y2−x1y1<0⇐⇒ −
y2x1−x2 x1
y1−y2<0 3.14
which is true sincex1 > x2andy1> y2.
The second condition is equivalent to
β1 y
γ2 A2x
2
−4 β1γ2 yA2x
4 xy yA2x
>0. 3.15
This is equivalent to
β1 y −
γ2 A2x
2
4 xy
yA2x
>0, 3.16
establishing the proof ofTheorem 3.1.
Since the mapT is strongly competitive, the Jacobian matrix3.2has two real and distinct eigenvalues, with the larger one in absolute value being positive.
From3.5atE1,we have
λ1λ2 β1 y1
γ2 A2x1
,
λ1λ2
β1γ2−x1y1 y1A2x1
.
3.17
The first equation implies that either both eigenvalues are positive or the smaller one is negative.
Consider the numerator of the right-hand side of the second equation. We have
β1γ2−x1y1β1γ2−
−α1−α2−β1
A2−γ2
√D 2β1
α1−α2−β1
A2−γ2
√D −2A2−γ2
β1γ2− √
D α1α2−β1
A2−γ2
2
β1
γ2A2
−α1α2− √
D
2 ,
3.18
aIfβ1γ2A2< α1α2,then the smaller root is negative, that is,λ2∈−1,0.
We now perform a similar analysis for the other cases inTable 1.
Theorem 3.2. Assume
The equilibrium is a sink if the following condition is satisfied:
becomes
x1 β1 A2−γ2
y1A2−γ2 β1
> x2y2, 3.28
that is,
x1y1> x2y2, 3.29
which is true.seeTheorem 3.1. Condition
1det JT
x, y<2 3.30
is equivalent to
β1γ2 yA2x−
xy yA2x
<1. 3.31
This implies
β1γ2−xy < yA2x⇐⇒β1γ1−yA2<2xy. 3.32
We have to prove that
β1γ2−y2A2<2x2y2. 3.33
Using2.2, we have
β1γ2− α1 x2
β1
A2<2x2 α1 x2
β1
. 3.34
This is equivalent to
β1
γ2−A2
<2α12x2β1 α1A2
x2
, 3.35
which is always true sinceA2> γ2and the left side is always negative, while the right side is
Notice that conditions
x1y1> x2y2,
β1 y −
γ2 A2x
2
4 xy
yA2x >0,
3.36
imply thatE1is a saddle point.
From3.5atE1,we have
λ1λ2 β1 y1
γ2 A2x1
,
λ1λ2
β1γ2−x1y1 y1A2x1
.
3.37
The first equation implies that either both eigenvalues are positive or the smaller one is negative.
Consider the numerator of the right-hand side of the second equation. We have
β1γ2−x1y1β1γ2−
−α1−α2−β1
A2−γ2
√D 2β1
α1−α2−β1
A2−γ2
√D −2A2−γ2
β1γ2− √
D α1α2−β1
A2−γ2
2
β1
γ2A2
−α1α2− √
D
2 .
3.38
We have
β1γ2−xy >0⇐⇒ β1
γ2A2
−α1α2− √
D
2 >0. 3.39
Inequality
β1
γ2A2
−α1α2− √
D >0 3.40
is equivalent to
β1γ2
β1A2−α2
α1
α2−β1γ2
>0, 3.41
which is obvious ifβ1γ2 < α2 < β1A2. Then inequality3.41holds. This confirmsa.The
Theorem 3.3. Assume
Then there exists a unique positive equilibrium point
E1E2
which is non-hyperbolic. The following holds.
aIfβ1A2γ2> α1α2,thenλ11andλ2∈0,1.
The characteristic equation of JTx, yis
detJTx, y−λI
which is simplified to
Solutions of3.46are
Note thatλ2can be written in the following form:
λ2
The corresponding eigenvectors, respectively, are
−A2−γ2
Note that the denominator of3.48is always positive. Consider numerator of3.48
−α1−α22β21
Now, we consider the special case of System1.1whenA2γ2.
In this case system1.1becomes
xn1
α1β1xn
yn
,
yn1 αA2A2xn 2yn
,
n0,1,2, . . . . 3.55
Equilibrium points are solutions of the following system:
x α1β1x
y ,
y α2A2y A2x
.
3.56
The second equation implies
x α2−α1 β1
, α2> α1. 3.57
Now, the first equation implies
y β1α2
α2−α1, α2> α1.
3.58
The mapTassociated to System3.55is given by
Tx, y α1β1x
y ,
α2A2y A2x
. 3.59
The Jacobian matrix of the mapT has the following form:
JT
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
β1
y −
α1β1x y2
−α2A2y A2x2
γ2 A2x
⎞ ⎟ ⎟ ⎟ ⎟
⎠. 3.60
The value of the Jacobian matrix ofTat the equilibrium point is
JTx, y ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
β1
y −
x y
− y A2x
A2 A2x
⎞ ⎟ ⎟ ⎟ ⎟
The determinant of3.61is given by
det JTx, y β1A2−xy yA2x
. 3.62
The trace of3.61is
Tr JTx, y β1 y
A2 A2x
. 3.63
Theorem 3.4. Assume
A2γ2, α2> α1. 3.64
Then there exists a unique positive equilibrium point
E α2−α1 β1
, β1α2 α2−α1
3.65
of system1.1, which is a saddle point. The following statements hold.
aIfβ1A2−α2 >0,thenλ1 ∈1,∞andλ2 ∈0,1. bIfβ1A2−α2 <0,thenλ1 ∈1,∞andλ2 ∈−1,0.
Proof. We prove thatEis a saddle point. We check the conditions
Tr JT
x, y>1det JT
x, y, Tr2JT
x, y−4 det JT
x, y>0. 3.66
Condition|Tr JTx, y|>1det JTx, yis equivalent to
β1 y
A2 A2x
>1 β1A2
A2xy− x A2x
. 3.67
This implies
β1A2x A2y > yA2xyβ1A2−xy⇐⇒β1x >0. 3.68
Condition
Tr2JT
x, y−4 det JT
is equivalent to
β1 y −
γ2 A2x
2
4 xy
yA2x
>0. 3.70
HenceEis a saddle point. Now,
λ1λ2 β1
y,
λ1λ2
β1A2−α2 yA2x .
3.71
The first equation implies that either both eigenvalues are positive or the smaller one is less then zero. The second equation implies that
Ifβ1A2 > α2, thenλ1 ∈1,∞, λ2∈0,−1;
Ifβ1A2 < α2, thenλ1 ∈1,∞, λ2∈−1,0,
3.72
establishing the proof of theorem.
4. Global Behavior
Theorem 4.1. Assume
A2< γ2. 4.1
Then system 1.1 has a unique equilibrium point E1 which is a saddle point. Furthermore, there
exists the global stable manifoldWsE1that separates the positive quadrant so that all orbits below this manifold are asymptotic to∞,0,and all orbits above this manifold are asymptotic to0,∞.All orbits that start onWsE1are attracted toE
1.The global unstable manifoldWuE1is the graph of
a continuous, unbounded, strictly decreasing function.
Proof. The existence of the global stable manifoldWsE1with the stated properties follows
from Theorems1.5,1.7, and1.8and Lemmas2.1and2.2.
Theorem 4.2. Assume
A2> γ2, β1
A2−γ2
< α2−α1,
α1−α2 β1
A2−γ2 2−
4α1β1
A2−γ2
>0. 4.2
Then system1.1has two equilibrium points:E1 which is a saddle point andE2 which is a sink.
manifoldWuE1is the graph of a continuous, unbounded, strictly decreasing function with end point
E2.
Proof. The existence of the global stable manifoldWsE2with the stated properties follows
from Theorems1.5,1.7, and1.8and Lemmas2.1and2.2.
Theorem 4.3. Assume
A2> γ2, β1
A2−γ2
< α2−α1,
α1−α2 β1
A2−γ2 2
−4α1β1
A2−γ2
0. 4.3
Then system 1.1 has a unique equilibrium E2 E3 which is non-hyperbolic. The sequences {x2n}, {x2n1}, {y2n}, and{y2n1}are eventually monotonic. Every solution that starts inQ4E2
is asymptotic to∞,0,and every solution that starts inQ2E2is asymptotic to the equilibriumE2.
Furthermore, there exists the global stable manifoldWsE2that separates the positive quadrant into three invariant regions, so that all orbits below this manifold are asymptotic to∞,0,and all orbits that start above this manifold are attracted to the equilibriumE2.All orbits that start onWsE2are
attracted toE2.
Proof. The existence of the global stable manifoldWsE2with the stated properties follows
from Theorems1.5,1.7, and1.8and Lemmas2.1and2.2.
First we prove that for all pointsMx x,α1/x β1, x /0,the following holds:
MxseTMx. 4.4
Observe that Mx is actually an arbitrary point on the curve x α1 β1x/y, which
represents one of two equilibrium curves for system1.1. Indeed,
Tx,α1/x β1
x,α2γ2
α1/x β1
A2x
x,α2xγ2
α1β1x
A2xx
.
4.5
Now we have
x,α1
x β1
se
x,α2xγ2
α1β1x
xA2x
⇐⇒x≤x, α1β1x
x ≥
α2xγ2
α1β1x
xA2x .
4.6
The last inequality is equivalent to
α1β1x
This is equivalent to
β1x2x
α1−α2β1
A2−γ2
α1
A2−γ2
≥0, 4.8
which always holds since the discriminant of the quadratic polynomial on the left-hand side is zero.
Note thatMxseE2,andMx E2forx α2−α1−β1A2−γ2/2β1.
Monotonicity of the mapTimplies
TnMxseTn1Mx. 4.9
Set TnMx {x
n, yn}. Then the sequence {xn} is increasing and bounded by
x-coordinate of the equilibrium, and the sequence {yn} is decreasing and bounded by
y-coordinate of the equilibrium. This implies that {xn, yn} converges to the equilibrium
asn → ∞.
Now, take any point Ax, y ∈ Q2E2. Then there exists point Mx∗ such that Mx∗seAx, yseE2.By using monotonicity of the mapT,we obtain
TnMx∗seTn
Ax, yseE2. 4.10
Lettingn → ∞in4.10, we have
lim
n→ ∞T
nAx, yE
2. 4.11
Now, we considerQ4E2.By choosingMxsuch thatE2seMx, we note that
E2seMxseTMx. 4.12
By using monotonicity of the mapT,we have
TnMxseTn1Mx. 4.13
SetTnMx {x
n, yn}.Then the sequence{xn} is increasing, and the sequence
{yn} is decreasing and bounded by y-coordinate of equilibrium and has to converge. If
{xn}converges, then{xn, yn}has to converge to the equilibrium, which is impossible. This
implies thatxn → ∞, n → ∞.Sinceyn1 α2γ2yn/A2xn,then limn→ ∞yn 0.
Now, take any point Bx, y in Q4E2. Then there is point Mx∗∗ such that E2seMx∗∗seBx, y.Using monotonicity of the mapT,we have
E2seTnMx∗∗seTn
Bx, y. 4.14
Since,TnMx∗∗is asymptotic to∞,0,then lim
Theorem 4.4. Assume
A2γ2, α2> α1. 4.15
Then system1.1has a unique equilibriumEwhich is a saddle point. Furthermore, there exists the global stable manifoldWsEthat separates the positive quadrant so that all orbits below this manifold are asymptotic to∞,0,and all orbits above this manifold are asymptotic to0,∞.All orbits that start onWsEare attracted toE.The global stable manifoldWuEis the graph of a continuous, unbounded, strictly increasing function.
Proof. The existence of the global stable manifoldWsEwith the stated properties follows
from Theorems1.5,1.7, and1.8and Lemmas2.1and2.2.
Theorem 4.5. Assume
A2> γ2,
α1−α2 β1
A2−γ2 2−
4α1β1
A2−γ2
<0, 4.16
A2> γ2,
α1−α2 β1
A2−γ2 2
−4α1β1
A2−γ2
≥0, β1
A2−γ2
> α2−α1 4.17
or
A2γ2, α2≤α1. 4.18
Then system1.1does not possess an equilibrium point. Its global behavior is described as follows:
xn−→ ∞, yn−→0, n−→ ∞. 4.19
Proof. If the conditions of this theorem are satisfied, then2.6implies that there is no realif the first condition of this theorem is satisfiedor positive equilibrium pointsif the second condition of this theorem is satisfied.
Consider the second equation of system1.1. That is,
yn1
α2γ2yn
A2xn
. 4.20
Note the following
yn1≤ Aα2 2
γ2
A2yn. 4.21
Now, consider equation
un1−
γ2 A2
un
α2 A2
Its solution is given by
unc
γ2 A2
n
α2 A2−γ2
. 4.23
SinceA2> γ2,then lettingn → ∞we obtain thatun → 0.Now,4.21implies
yn ≤
α2 A2−γ2
ε, n−→ ∞. 4.24
This means that sequence{yn}is bounded forA2> γ2.
In order to prove the global behavior in this case, we decompose System1.1into the system of even-indexed and odd-indexed terms as
x2n1 α1yβ1x2n 2n ,
x2n
α1β1x2n−1 y2n−1
,
y2n1 αA2γ2y2n 2x2n ,
y2n
α2γ2y2n−1 A2x2n−1
,
4.25
for n1,2, . . ..
Lemma 2.1implies that subsequences{x2n1}, {x2n}, {y2n1},and {y2n}are
eventu-ally monotone.
Since sequence {yn} is bounded, then the subsequences {y2n1} and {y2n} must
converge. If the sequences {x2n1} and {x2n} would converge to finite numbers, then
the solution of 1.1 would converge to the period-two solution, which is impossible by
Lemma 2.2. Thus at least one of the subsequences{x2n1}and{x2n} tends to∞, n → ∞.
Assume thatx2n → ∞asn → ∞. In view of third equation of4.25,y2n1 → 0,and in
view of first equation of4.25,x2n1 → ∞which by fourth equation of4.25implies that y2n1 → 0 asn → ∞.
Now, we prove the case whenA2γ2andα2α1.
In this case System1.1becomes
xn1 α1yβ1xn
n
,
yn1
α1A2yn
A2xn
.
4.26
The mapTassociated to System4.26is given by
Tx, y α1β1x
y ,
α1A2y A2x
Equilibrium curvesC1andC2can be given explicitly as the following functions ofx:
C1:y1 α1 x β1,
C2:y2 α1 x.
4.28
It is obvious that these two curves do not intersect, which means that System4.26
does not possess an equilibrium point.
Similarly, as in the proof ofTheorem 4.3, for all pointsC1x x,α1/x β1, x /0
the following holds:
C1xTC1x. 4.29
Indeed,
Tx,α1 x β1
⎛ ⎜ ⎝x,
α1A2 α1
x β1
A2x
⎞ ⎟ ⎠
x,α1xA2
α1β1x
A2xx
.
4.30
Now, we have
x,α1
x β1
se
x,α1xA2
α1β1x
A2xx
⇐⇒x≤xand α1 x β1 ≥
α1xA2
α1β1x
A2xx .
4.31
The last inequality is equivalent to
α1β1x
A2x≥α1xA2α1β1A2x, 4.32
which always holds.
Monotonicity ofTimplies
TnC1xseTn1C1x. 4.33
SetTnC1x {x
n, yn}.Then the sequence{xn} is increasing and the sequence
{y
n}is decreasing. Since{yn}is decreasing andyn>0, n1,2, . . . ,then it has to converge.
If{xn}converges, then{xn, yn}has to converge to the equilibrium, which is impossible. This implies thatxn → ∞, n → ∞. The second equation of System4.26implies that
Now, take any pointx, y∈R2
.Then there exists pointC1x x, α1/x such that
C1xse
x, y. 4.34
Monotonicity ofTimplies
TnC1x seTn
x, y. 4.35
Set
Tnx, yxn, yn
, TnC1x xn, yn. 4.36
Then, we have
xn≤xn,
yn≥yn.
4.37
Since
xn−→ ∞, yn−→0, n−→ ∞. 4.38
we conclude, using the inequalities4.37, that
xn−→ ∞, yn−→0, n−→ ∞. 4.39
Similarly, we can prove the caseA2γ2, α2< α1.
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