Advances in Difference Equations

, 2009:132802 | Cite as

Global Dynamics of a Competitive System of Rational Difference Equations in the Plane

Open Access
Research Article

Abstract

We investigate global dynamics of the following systems of difference equations Open image in new window , Open image in new window , Open image in new window , where the parameters Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window are positive numbers and initial conditions Open image in new window and Open image in new window are arbitrary nonnegative numbers such that Open image in new window . 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.

Keywords

Equilibrium Point Saddle Point Jacobian Matrix Unstable Manifold Global Behavior 

1. Introduction and Preliminaries

In this paper, we study the global dynamics of the following rational system of difference equations:

where the parameters Open image in new window   and Open image in new window are positive numbers and initial conditions Open image in new window and Open image in new window are arbitrary numbers. System (1.1) was mentioned in [1] as a part of Open Problem 3 which asked for a description of global dynamics of three specific competitive systems. According to the labeling in [1], system (1.1) is called Open image in new window . In this paper, we provide the precise description of global dynamics of system (1.1). We show that system (1.1) has a variety of dynamics that depend on the value of parameters. We show that system (1.1) may have between zero and two equilibrium points, which may have different local character. If system (1.1) has one equilibrium point, then this point is either locally saddle point or non-hyperbolic. If system (1.1) has 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. System (1.1) gives an example of semistable non-hyperbolic equilibrium point. The typical results are Theorems 4.1 and 4.5 below.

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 system (1.1). 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.

Let Open image in new window and Open image in new window be intervals of real numbers. Consider a first-order system of difference equations of the form

where Open image in new window

When the function Open image in new window is increasing in Open image in new window and decreasing in Open image in new window and the function Open image in new window is decreasing in Open image in new window and increasing in Open image in new window , the system (1.2) is called competitive. When the function Open image in new window is increasing in Open image in new window and increasing in Open image in new window and the function Open image in new window is increasing in Open image in new window and increasing in Open image in new window the system (1.2) is called cooperative. A map Open image in new window that corresponds to the system (1.2) is defined as Open image in new window . Competitive and cooperative maps, which are called monotone maps, are defined similarly. Strongly competitive systems of difference equations or maps are those for which the functions Open image in new window and Open image in new window are coordinate-wise strictly monotone.

If Open image in new window , we denote with Open image in new window , the four quadrants in Open image in new window relative to Open image in new window , that is, Open image in new window , and so on. Define the South-East partial order Open image in new window on Open image in new window by Open image in new window if and only if Open image in new window and Open image in new window . Similarly, we define the North-East partial order Open image in new window on Open image in new window by Open image in new window if and only if Open image in new window and Open image in new window . For Open image in new window and Open image in new window , define the distance from Open image in new window to Open image in new window as Open image in new window . By Open image in new window we denote the interior of a set Open image in new window .

It is easy to show that a map Open image in new window is competitive if it is nondecreasing with respect to the South-East partial order, that is if the following holds:

Competitive systems were studied by many authors; see [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], and others. All known results, with the exception of [4, 6, 10], deal with hyperbolic dynamics. The results presented here are results that hold in both the hyperbolic and the non-hyperbolic cases.

We now state three results for competitive maps in the plane. The following definition is from [18].

Definition 1.1.

Let Open image in new window be a nonempty subset of Open image in new window . A competitive map Open image in new window is said to satisfy condition ( Open image in new window ) if for every Open image in new window , Open image in new window in Open image in new window , Open image in new window implies Open image in new window , and Open image in new window is said to satisfy condition ( Open image in new window ) if for every Open image in new window , Open image in new window in Open image in new window , Open image in new window implies Open image in new window .

The following theorem was proved by de Mottoni and Schiaffino [20] for the Poincaré map of a periodic competitive Lotka-Volterra system of differential equations. Smith generalized the proof to competitive and cooperative maps [15, 16].

Theorem 1.2.

Let Open image in new window be a nonempty subset of Open image in new window . If Open image in new window is a competitive map for which ( Open image in new window ) holds, then for all Open image in new window , Open image in new window is eventually componentwise monotone. If the orbit of Open image in new window has compact closure, then it converges to a fixed point of Open image in new window . If instead ( Open image in new window ) holds, then for all Open image in new window , Open image in new window is eventually componentwise monotone. If the orbit of Open image in new window has compact closure in Open image in new window , then its omega limit set is either a period-two orbit or a fixed point.

The following result is from [18], with the domain of the map specialized to be the Cartesian product of intervals of real numbers. It gives a sufficient condition for conditions ( Open image in new window ) and ( Open image in new window ).

Theorem 1.3 (Smith [18]).

Let Open image in new window be the Cartesian product of two intervals in Open image in new window . Let Open image in new window be a Open image in new window competitive map. If Open image in new window is injective and Open image in new window for all Open image in new window then Open image in new window satisfies ( Open image in new window ). If Open image in new window is injective and Open image in new window for all Open image in new window then Open image in new window satisfies ( Open image in new window ).

Theorem 1.4.

Let Open image in new window be a monotone map on a closed and bounded rectangular region Open image in new window Suppose that Open image in new window has a unique fixed point Open image in new window in Open image in new window Then Open image in new window is a global attractor of Open image in new window on Open image in new window

The following theorems were proved by Kulenović and Merino [3] for competitive systems in the plane, when one of the eigenvalues of the linearized system at an equilibrium (hyperbolic or non-hyperbolic) is by absolute value smaller than Open image in new window 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 point (hyperbolic or not) of 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 region Open image in new window is rectangular if it is the Cartesian product of two intervals in Open image in new window .

Theorem 1.5.

Let Open image in new window be a competitive map on a rectangular region Open image in new window . Let Open image in new window be a fixed point of Open image in new window such that Open image in new window is nonempty (i.e., Open image in new window is not the NW or SE vertex of Open image in new window , and Open image in new window is strongly competitive on Open image in new window . Suppose that the following statements are true.

  1. (a)

    The map Open image in new window has a Open image in new window extension to a neighborhood of Open image in new window .

     
  2. (b)

    The Jacobian matrix of Open image in new window at Open image in new window has real eigenvalues Open image in new window , Open image in new window such that Open image in new window , where Open image in new window , and the eigenspace Open image in new window associated with Open image in new window is not a coordinate axis.

     

Then there exists a curve Open image in new window through Open image in new window that is invariant and a subset of the basin of attraction of Open image in new window , such that Open image in new window is tangential to the eigenspace Open image in new window at Open image in new window , and Open image in new window is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of Open image in new window in the interior of Open image in new window are either fixed points or minimal period-two points. In the latter case, the set of endpoints of Open image in new window is a minimal period-two orbit of Open image in new window .

Corollary 1.6.

If Open image in new window has no fixed point nor periodic points of minimal period-two in Open image in new window , then the endpoints of Open image in new window belong to Open image in new window .

For maps that are strongly competitive near the fixed point, hypothesis b. of Theorem 1.5 reduces just to Open image in new window . This follows from a change of variables [18] that 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 of Theorem 1.7 from [12].

Theorem 1.7.

In addition to the hypotheses of Theorem 1.5, suppose that Open image in new window and that the eigenspace Open image in new window associated with Open image in new window is not a coordinate axis. If the curve Open image in new window of Theorem 1.5 has endpoints in Open image in new window , then Open image in new window is the global stable manifold Open image in new window of Open image in new window , and the global unstable manifold Open image in new window is a curve in Open image in new window that is tangential to Open image in new window at Open image in new window and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of Open image in new window in Open image in new window are fixed points of Open image in new window .

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 Open image in new window be the curve whose existence is guaranteed by Theorem 1.5. If the endpoints of Open image in new window belong to Open image in new window , then Open image in new window separates Open image in new window into two connected components, namely

such that the following statements are true.

If, in addition, Open image in new window is an interior point of Open image in new window and Open image in new window is Open image in new window and strongly competitive in a neighborhood of Open image in new window , then Open image in new window has no periodic points in the boundary of Open image in new window except for Open image in new window , and the following statements are true.

2. Some Basic Facts

In this section we give some basic facts about the nonexistence of period-two solutions, local injectivity of map Open image in new window at the equilibrium point and Open image in new window condition.

2.1. Equilibrium Points

The equilibrium points Open image in new window of system (1.1) satisfy

First equation of System (2.1) gives

Second equation of System (2.1) gives

Now, using (2.2), we obtain

This implies

which is equivalent to

Solutions of (2.6) are

Now, (2.2) gives

The equilibrium points are:

where Open image in new window are given by the above relations.

Note that

The discriminant of (2.6) is given by

2.2. Condition Open image in new window and Period-Two Solution

In this section we prove three lemmas.

Lemma 2.1.

System (1.1) satisfies either Open image in new window or Open image in new window Consequently, the second iterate of every solution is eventually monotone.

Proof.

The map Open image in new window associated to system (1.1) is given by
then we have
Equations (2.15) and (2.16) are equivalent, respectively, to
Now, using (2.17) and (2.18), we have the following:

Lemma 2.2.

System (1.1) has no minimal period-two solution.

Proof.

Period-two solution satisfies

We show that this system has no other positive solutions except equilibrium points.

Equations (2.22) and (2.23) are equivalent, respectively, to
Equation (2.24) implies
Equation (2.25) implies
Using (2.26), we have
Putting (2.28) into (2.27), we have
This is equivalent to
Putting (2.30) into (2.24), we obtain

From (2.31), we obtain fixed points. In the sequel, we consider (2.32).

Discriminant of (2.32) is given by
Real solutions of (2.32) exist if and only if Open image in new window The solutions are given by
Using (2.30), we have

Claim.

Assume Open image in new window Then

  1. (i)

    for all values of parameters, Open image in new window

     
  2. (ii)

    for all values of parameters, Open image in new window

     

Proof.

( Open image in new window ) Assume Open image in new window Then it is obvious that the claim Open image in new window is true. Now, assume Open image in new window Then Open image in new window if and only if
which is equivalent to
This is true since
This is equivalent to
Using (2.39), we have

which implies that the inequality (2.41) is true.

Now, the proof of the Lemma 2.2 follows from the Claim .

Lemma 2.3.

The map Open image in new window associated to System (1.1) satisfies the following:

Proof.

By using (2.1), we have
First equation implies
Second equation implies
Note the following
Using (2.47), Equations (2.45) and (2.46), respectively, become
Note that System (2.48) is linear homogeneous system in Open image in new window and Open image in new window The determinant of System (2.48) is given by
Using (2.1), the determinant of System (2.48) becomes
This implies that System (2.48) has only trivial solution, that is

3. Linearized Stability Analysis

The Jacobian matrix of the map Open image in new window has the following form:

The value of the Jacobian matrix of Open image in new window at the equilibrium point is

The determinant of (3.2) is given by

The trace of (3.2) is

The characteristic equation has the form

Theorem 3.1.

Assume that Open image in new window Then there exists a unique positive equilibrium Open image in new window which is a saddle point, and the following statements hold.

Proof.

The equilibrium is a saddle point if and only if the following conditions are satisfied:
The first condition is equivalent to
This implies the following:
Notice the following:
That is,
Similarly,
Now, we have
This is equivalent to
The last condition is equivalent to

which is true since Open image in new window and Open image in new window

The second condition is equivalent to
This is equivalent to

establishing the proof of Theorem 3.1.

Since the map Open image in new window is strongly competitive, the Jacobian matrix (3.2) has two real and distinct eigenvalues, with the larger one in absolute value being positive.

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

where Open image in new window

(a) If Open image in new window then the smaller root is negative, that is, Open image in new window

From the last inequality statements Open image in new window and Open image in new window follow.

We now perform a similar analysis for the other cases in Table 1.

Theorem 3.2.

Then Open image in new window exist. Open image in new window is a saddle point; Open image in new window is a sink. For the eigenvalues of Open image in new window the following holds.

Proof.

Note that if Open image in new window and Open image in new window then Open image in new window and Open image in new window which implies Open image in new window , which is a contradiction.

The equilibrium is a sink if the following condition is satisfied:
The condition Open image in new window is equivalent to
This implies

Now, we prove that Open image in new window is a sink.

We have to prove that
Notice the following:
Similarly,
Now, condition
that is,

which is true. (see Theorem 3.1.)

Condition
is equivalent to
This implies
We have to prove that
Using (2.2), we have
This is equivalent to

which is always true since Open image in new window and the left side is always negative, while the right side is always positive.

Notice that conditions

imply that Open image in new window is a saddle point.

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
Inequality
is equivalent to

which is obvious if Open image in new window . Then inequality (3.41) holds. This confirms Open image in new window The other cases follow from (3.41).

Theorem 3.3.

Then there exists a unique positive equilibrium point

which is non-hyperbolic. The following holds.

Proof.

Evaluating the Jacobian matrix (3.2) at equilibrium Open image in new window we have
The characteristic equation of Open image in new window is
which is simplified to
Solutions of (3.46) are
Note that Open image in new window can be written in the following form:

Note that Open image in new window

The corresponding eigenvectors, respectively, are

Note that the denominator of (3.48) is always positive.

Consider numerator of (3.48)
Substituting Open image in new window from (3.52) in (3.50), we obtain
Now, (3.48) becomes

establishing the proof of the theorem.

Now, we consider the special case of System (1.1) when Open image in new window

In this case system (1.1) becomes

Equilibrium points are solutions of the following system:

The second equation implies

Now, the first equation implies

The map Open image in new window associated to System (3.55) is given by

The Jacobian matrix of the map Open image in new window has the following form:

The value of the Jacobian matrix of Open image in new window at the equilibrium point is

The determinant of (3.61) is given by

The trace of (3.61) is

Theorem 3.4.

Then there exists a unique positive equilibrium point

of system (1.1), which is a saddle point. The following statements hold.

Proof.

We prove that Open image in new window is a saddle point.

We check the conditions
Condition Open image in new window is equivalent to
This implies
Condition
is equivalent to

Hence Open image in new window is a saddle point.

The first equation implies that either both eigenvalues are positive or the smaller one is less then zero. The second equation implies that

establishing the proof of theorem.

4. Global Behavior

Theorem 4.1.

Then system (1.1) has a unique equilibrium point Open image in new window which is a saddle point. Furthermore, there exists the global stable manifold Open image in new window that separates the positive quadrant so that all orbits below this manifold are asymptotic to Open image in new window and all orbits above this manifold are asymptotic to Open image in new window All orbits that start on Open image in new window are attracted to Open image in new window The global unstable manifold Open image in new window is the graph of a continuous, unbounded, strictly decreasing function.

Proof.

The existence of the global stable manifold Open image in new window with the stated properties follows from Theorems 1.5, 1.7, and 1.8 and Lemmas 2.1 and 2.2.

Theorem 4.2.

Then system (1.1) has two equilibrium points: Open image in new window which is a saddle point and Open image in new window which is a sink. Furthermore, there exists the global stable manifold Open image in new window that separates the positive quadrant so that all orbits below this manifold are asymptotic to Open image in new window and all orbits above this manifold are attracted to equilibrium Open image in new window All orbits that start on Open image in new window are attracted to Open image in new window The global unstable manifold Open image in new window is the graph of a continuous, unbounded, strictly decreasing function with end point Open image in new window

Proof.

The existence of the global stable manifold Open image in new window with the stated properties follows from Theorems 1.5, 1.7, and 1.8 and Lemmas 2.1 and 2.2.

Theorem 4.3.

Then system (1.1) has a unique equilibrium Open image in new window which is non-hyperbolic. The sequences Open image in new window , and Open image in new window are eventually monotonic. Every solution that starts in Open image in new window is asymptotic to Open image in new window and every solution that starts in Open image in new window is asymptotic to the equilibrium Open image in new window Furthermore, there exists the global stable manifold Open image in new window that separates the positive quadrant into three invariant regions, so that all orbits below this manifold are asymptotic to Open image in new window and all orbits that start above this manifold are attracted to the equilibrium Open image in new window All orbits that start on Open image in new window are attracted to Open image in new window

Proof.

The existence of the global stable manifold Open image in new window with the stated properties follows from Theorems 1.5, 1.7, and 1.8 and Lemmas 2.1 and 2.2.

First we prove that for all points Open image in new window the following holds:

Observe that Open image in new window is actually an arbitrary point on the curve Open image in new window , which represents one of two equilibrium curves for system (1.1).

Now we have
The last inequality is equivalent to
This is equivalent to

which always holds since the discriminant of the quadratic polynomial on the left-hand side is zero.

Note that Open image in new window and Open image in new window for Open image in new window

Monotonicity of the map Open image in new window implies

Set Open image in new window Then the sequence Open image in new window is increasing and bounded by Open image in new window coordinate of the equilibrium, and the sequence Open image in new window is decreasing and bounded by Open image in new window coordinate of the equilibrium. This implies that Open image in new window converges to the equilibrium as Open image in new window

Now, take any point Open image in new window Then there exists point Open image in new window such that Open image in new window By using monotonicity of the map Open image in new window we obtain
Letting Open image in new window in (4.10), we have
Now, we consider Open image in new window By choosing Open image in new window such that Open image in new window , we note that
By using monotonicity of the map Open image in new window we have

Set Open image in new window Then the sequence Open image in new window is increasing, and the sequence Open image in new window is decreasing and bounded by Open image in new window coordinate of equilibrium and has to converge. If Open image in new window converges, then Open image in new window has to converge to the equilibrium, which is impossible. This implies that Open image in new window Since Open image in new window then Open image in new window

Now, take any point Open image in new window in Open image in new window . Then there is point Open image in new window such that Open image in new window Using monotonicity of the map Open image in new window we have

Since, Open image in new window is asymptotic to Open image in new window then Open image in new window

Theorem 4.4.

Then system (1.1) has a unique equilibrium Open image in new window which is a saddle point. Furthermore, there exists the global stable manifold Open image in new window that separates the positive quadrant so that all orbits below this manifold are asymptotic to Open image in new window and all orbits above this manifold are asymptotic to Open image in new window All orbits that start on Open image in new window are attracted to Open image in new window The global stable manifold Open image in new window is the graph of a continuous, unbounded, strictly increasing function.

Proof.

The existence of the global stable manifold Open image in new window with the stated properties follows from Theorems 1.5, 1.7, and 1.8 and Lemmas 2.1 and 2.2.

Theorem 4.5.

Then system (1.1) does not possess an equilibrium point. Its global behavior is described as follows:

Proof.

If the conditions of this theorem are satisfied, then (2.6) implies that there is no real (if the first condition of this theorem is satisfied) or positive equilibrium points (if the second condition of this theorem is satisfied).

Consider the second equation of system (1.1). That is,
Note the following
Now, consider equation
Its solution is given by
Since Open image in new window then letting Open image in new window we obtain that Open image in new window Now, (4.21) implies

This means that sequence Open image in new window is bounded for Open image in new window

In order to prove the global behavior in this case, we decompose System (1.1) into the system of even-indexed and odd-indexed terms as

for Open image in new window .

Lemma 2.1 implies that subsequences Open image in new window and Open image in new window are eventually monotone.

Since sequence Open image in new window is bounded, then the subsequences Open image in new window and Open image in new window must converge. If the sequences Open image in new window and Open image in new window 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 Open image in new window and Open image in new window tends to Open image in new window . Assume that Open image in new window as Open image in new window . In view of third equation of (4.25), Open image in new window and in view of first equation of (4.25), Open image in new window which by fourth equation of (4.25) implies that Open image in new window as Open image in new window .

Now, we prove the case when Open image in new window and Open image in new window

In this case System (1.1) becomes
The map Open image in new window associated to System (4.26) is given by
Equilibrium curves Open image in new window and Open image in new window can be given explicitly as the following functions of Open image in new window

It is obvious that these two curves do not intersect, which means that System (4.26) does not possess an equilibrium point.

Similarly, as in the proof of Theorem 4.3, for all points Open image in new window the following holds:
Now, we have
The last inequality is equivalent to

which always holds.

Monotonicity of Open image in new window implies

Set Open image in new window Then the sequence Open image in new window is increasing and the sequence Open image in new window is decreasing. Since Open image in new window is decreasing and Open image in new window then it has to converge. If Open image in new window converges, then Open image in new window has to converge to the equilibrium, which is impossible. This implies that Open image in new window The second equation of System (4.26) implies that Open image in new window

Now, take any point Open image in new window Then there exists point Open image in new window such that
Monotonicity of Open image in new window implies
Then, we have
we conclude, using the inequalities (4.37), that

Similarly, we can prove the case Open image in new window

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Copyright information

© S. Kalabušić et al 2009

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  • S. Kalabušić
    • 1
  • M. R. S. Kulenović
    • 2
  • E. Pilav
    • 1
  1. 1.Department of MathematicsUniversity of SarajevoBosnia and Herzegovina
  2. 2.Department of MathematicsUniversity of Rhode IslandKingstonUSA

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