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

Abstract

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

1. Introduction and Preliminaries

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

(1.1)

where the parameters   and are positive numbers and initial conditions and 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 . 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 and be intervals of real numbers. Consider a first-order system of difference equations of the form

(1.2)

where

When the function is increasing in and decreasing in and the function is decreasing in and increasing in , the system (1.2) is called competitive. When the function is increasing in and increasing in and the function is increasing in and increasing in the system (1.2) is called cooperative. A map that corresponds to the system (1.2) is defined as . 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 and are coordinate-wise strictly monotone.

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

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

(1.3)

Competitive systems were studied by many authors; see [4–19], 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 be a nonempty subset of . A competitive map is said to satisfy condition () if for every , in , implies , and is said to satisfy condition () if for every , in , implies .

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 be a nonempty subset of . If is a competitive map for which () holds, then for all , is eventually componentwise monotone. If the orbit of has compact closure, then it converges to a fixed point of . If instead () holds, then for all , is eventually componentwise monotone. If the orbit of has compact closure in , 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 () and ().

Theorem 1.3 (Smith [18]).

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

Theorem 1.4.

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

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 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 is rectangular if it is the Cartesian product of two intervals in .

Theorem 1.5.

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

• (a) The map has a extension to a neighborhood of .

• (b) The Jacobian matrix of at has real eigenvalues , such that , where , and the eigenspace associated with is not a coordinate axis.

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

Corollary 1.6.

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

For maps that are strongly competitive near the fixed point, hypothesis b. of Theorem 1.5 reduces just to . 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 and that the eigenspace associated with is not a coordinate axis. If the curve of Theorem 1.5 has endpoints in , then is the global stable manifold of , and the global unstable manifold is a curve in that is tangential to at and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of in are fixed points of .

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

(1.4)

such that the following statements are true.

• (i) is invariant, and as for every .

• (ii) is invariant, and as for every .

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

• (iii) For every there exists such that for .

• (iv) For every there exists such that for .

2. Some Basic Facts

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

2.1. Equilibrium Points

The equilibrium points of system (1.1) satisfy

(2.1)

First equation of System (2.1) gives

(2.2)

Second equation of System (2.1) gives

(2.3)

Now, using (2.2), we obtain

(2.4)

This implies

(2.5)

which is equivalent to

(2.6)

Solutions of (2.6) are

(2.7)

Now, (2.2) gives

(2.8)

The equilibrium points are:

(2.9)

where are given by the above relations.

Note that

(2.10)

The discriminant of (2.6) is given by

(2.11)

The criteria for the existence of equilibrium points are summarized in Table 1 where

Table 1

(2.12)

2.2. Condition and Period-Two Solution

In this section we prove three lemmas.

Lemma 2.1.

System (1.1) satisfies either or Consequently, the second iterate of every solution is eventually monotone.

Proof.

The map associated to system (1.1) is given by

(2.13)

Assume

(2.14)

then we have

(2.15)

(2.16)

Equations (2.15) and (2.16) are equivalent, respectively, to

(2.17)

(2.18)

Now, using (2.17) and (2.18), we have the following:

(2.19)

Lemma 2.2.

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

Proof.

Set

(2.20)

Then

(2.21)

Period-two solution satisfies

(2.22)

(2.23)

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

Equations (2.22) and (2.23) are equivalent, respectively, to

(2.24)

(2.25)

Equation (2.24) implies

(2.26)

Equation (2.25) implies

(2.27)

Using (2.26), we have

(2.28)

Putting (2.28) into (2.27), we have

(2.29)

This is equivalent to

(2.30)

Putting (2.30) into (2.24), we obtain

(2.31)

or

(2.32)

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

Discriminant of (2.32) is given by

(2.33)

Real solutions of (2.32) exist if and only if The solutions are given by

(2.34)

Using (2.30), we have

(2.35)

Claim.

Assume Then

• (i) for all values of parameters,

• (ii) for all values of parameters,

Proof.

() Assume Then it is obvious that the claim is true. Now, assume Then if and only if

(2.36)

which is equivalent to

(2.37)

This is true since

(2.38)

() Assume Then it is obvious that . Now, assume

(2.39)

Then if and only if

(2.40)

This is equivalent to

(2.41)

Using (2.39), we have

(2.42)

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 associated to System (1.1) satisfies the following:

(2.43)

Proof.

By using (2.1), we have

(2.44)

First equation implies

(2.45)

Second equation implies

(2.46)

Note the following

(2.47)

Using (2.47), Equations (2.45) and (2.46), respectively, become

(2.48)

Note that System (2.48) is linear homogeneous system in and The determinant of System (2.48) is given by

(2.49)

Using (2.1), the determinant of System (2.48) becomes

(2.50)

This implies that System (2.48) has only trivial solution, that is

(2.51)

3. Linearized Stability Analysis

The Jacobian matrix of the map has the following form:

(3.1)

The value of the Jacobian matrix of at the equilibrium point is

(3.2)

The determinant of (3.2) is given by

(3.3)

The trace of (3.2) is

(3.4)

The characteristic equation has the form

(3.5)

Theorem 3.1.

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

• (a) If then and

• (b) If and then and

• (c) If and then and

• (d) If and then and

Proof.

The equilibrium is a saddle point if and only if the following conditions are satisfied:

(3.6)

The first condition is equivalent to

(3.7)

This implies the following:

(3.8)

Notice the following:

(3.9)

That is,

(3.10)

Similarly,

(3.11)

Now, we have

(3.12)

This is equivalent to

(3.13)

The last condition is equivalent to

(3.14)

which is true since and

The second condition is equivalent to

(3.15)

This is equivalent to

(3.16)

establishing the proof of Theorem 3.1.

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

From (3.5) at we have

(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

(3.18)

where

(a) If then the smaller root is negative, that is,

If then

(3.19)

From the last inequality statements and follow.

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

Theorem 3.2.

Assume

(3.20)

Then exist. is a saddle point; is a sink. For the eigenvalues of the following holds.

• (a) If then

• (b) If and then

• (c) If and then

Proof.

Note that if and then and which implies , which is a contradiction.

The equilibrium is a sink if the following condition is satisfied:

(3.21)

The condition is equivalent to

(3.22)

This implies

(3.23)

Now, we prove that is a sink.

We have to prove that

(3.24)

Notice the following:

(3.25)

Similarly,

(3.26)

Now, condition

(3.27)

becomes

(3.28)

that is,

(3.29)

which is true. (see Theorem 3.1.)

Condition

(3.30)

is equivalent to

(3.31)

This implies

(3.32)

We have to prove that

(3.33)

Using (2.2), we have

(3.34)

This is equivalent to

(3.35)

which is always true since and the left side is always negative, while the right side is always positive.

Notice that conditions

(3.36)

imply that is a saddle point.

From (3.5) at we have

(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

(3.38)

We have

(3.39)

Inequality

(3.40)

is equivalent to

(3.41)

which is obvious if . Then inequality (3.41) holds. This confirms The other cases follow from (3.41).

Theorem 3.3.

Assume

(3.42)

Then there exists a unique positive equilibrium point

(3.43)

which is non-hyperbolic. The following holds.

• (a) If then and

• (b) If then and

Proof.

Evaluating the Jacobian matrix (3.2) at equilibrium we have

(3.44)

The characteristic equation of is

(3.45)

which is simplified to

(3.46)

Solutions of (3.46) are

(3.47)

Note that can be written in the following form:

(3.48)

Note that

The corresponding eigenvectors, respectively, are

(3.49)

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

Consider numerator of (3.48)

(3.50)

From

(3.51)

we have

(3.52)

Substituting from (3.52) in (3.50), we obtain

(3.53)

Now, (3.48) becomes

(3.54)

establishing the proof of the theorem.

Now, we consider the special case of System (1.1) when

In this case system (1.1) becomes

(3.55)

Equilibrium points are solutions of the following system:

(3.56)

The second equation implies

(3.57)

Now, the first equation implies

(3.58)

The map associated to System (3.55) is given by

(3.59)

The Jacobian matrix of the map has the following form:

(3.60)

The value of the Jacobian matrix of at the equilibrium point is

(3.61)

The determinant of (3.61) is given by

(3.62)

The trace of (3.61) is

(3.63)

Theorem 3.4.

Assume

(3.64)

Then there exists a unique positive equilibrium point

(3.65)

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

• (a) If then and

• (b) If then and

Proof.

We prove that is a saddle point.

We check the conditions

(3.66)

Condition is equivalent to

(3.67)

This implies

(3.68)

Condition

(3.69)

is equivalent to

(3.70)

Hence is a saddle point.

Now,

(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

(3.72)

establishing the proof of theorem.

4. Global Behavior

Theorem 4.1.

Assume

(4.1)

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

Proof.

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

Theorem 4.2.

Assume

(4.2)

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

Proof.

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

Theorem 4.3.

Assume

(4.3)

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

Proof.

The existence of the global stable manifold 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 the following holds:

(4.4)

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

Indeed,

(4.5)

Now we have

(4.6)

The last inequality is equivalent to

(4.7)

This is equivalent to

(4.8)

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

Note that and for

Monotonicity of the map implies

(4.9)

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

Now, take any point Then there exists point such that By using monotonicity of the map we obtain

(4.10)

Letting in (4.10), we have

(4.11)

Now, we consider By choosing such that , we note that

(4.12)

By using monotonicity of the map we have

(4.13)

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

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

(4.14)

Since, is asymptotic to then

Theorem 4.4.

Assume

(4.15)

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

Proof.

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

Theorem 4.5.

Assume

(4.16)

(4.17)

or

(4.18)

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

(4.19)

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,

(4.20)

Note the following

(4.21)

Now, consider equation

(4.22)

Its solution is given by

(4.23)

Since then letting we obtain that Now, (4.21) implies

(4.24)

This means that sequence is bounded for

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

(4.25)

for .

Lemma 2.1 implies that subsequences and are eventually monotone.

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

Now, we prove the case when and

In this case System (1.1) becomes

(4.26)

The map associated to System (4.26) is given by

(4.27)

Equilibrium curves and can be given explicitly as the following functions of

(4.28)

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 the following holds:

(4.29)

Indeed,

(4.30)

Now, we have

(4.31)

The last inequality is equivalent to

(4.32)

which always holds.

Monotonicity of implies

(4.33)

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

Now, take any point Then there exists point such that

(4.34)

Monotonicity of implies

(4.35)

Set

(4.36)

Then, we have

(4.37)

Since

(4.38)

we conclude, using the inequalities (4.37), that

(4.39)

Similarly, we can prove the case