Abstract
We investigate global dynamics of the following systems of difference equations , , , where the parameters , , , , , are positive numbers, and initial conditions and are arbitrary nonnegative numbers such that . We show that this system has up to three equilibrium points with various dynamics which depends on the part of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or nonhyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or of nonhyperbolic equilibrium points. We give an example of globally attractive nonhyperbolic equilibrium point and semistable non-hyperbolic equilibrium point.
1. Introduction
In this paper we consider the following rational system of difference equations where the parameters , , , , , are positive numbers, and initial conditions and are nonnegative numbers such that . System (1.1) was mentioned in [1] as one of three systems of open problem 3 which asked for the description of global dynamics of some rational systems of difference equations. In notation used to labels systems of linear fractional difference equations used in [1] system (1.1) is known as (3.19) and (4.1). In this paper, we provide the precise description of global dynamics of the system (1.1). We show that the system (1.1) may have between zero and three equilibrium points, which may have different local character. If the system (1.1) has one equilibrium point, then this point is either locally asymptotically stable or saddle point or nonhyperbolic equilibrium point. If the system (1.1) has two equilibrium points, then they are either locally asymptotically stable, and nonhyperbolic, or locally asymptotically stable and saddle point. If the system (1.1) has three equilibrium points then two of the equilibrium points are locally asymptotically stable and the third point, which is between these two points in South-East ordering defined below, is a saddle point. The major problem for global dynamics of the system (1.1) is determining the basins of attraction of different equilibrium points. The difficulty in analyzing the behavior of all solutions of the system (1.1) lies in the fact that there are many regions of parameters where this system possesses different equilibrium points with different local character and that in several cases the equilibrium point is nonhyperbolic. However, all these cases can be handled by using recent results in [2]. The dual of this system is the system where and replace their role, and it was labeled as system (4.1) and (3.19) in [1]. Dynamics of this system immediately follows from the results proven here, by simply replacing the roles of and .
System (1.1) is a competitive system, and our results are based on recent results about competitive systems in the plane, see [2, 3]. System (1.1) has a potential to be used as a mathematical model for competition. In fact, the first equation of (1.1) is of Leslie-Gower type, and the second equation can be considered to be of Leslie-Gower type with stocking (or immigration) represented with the term , see [4β7]. Here , are the inherent birth rates while and are related to the density-dependent effects on newborn recruitment. Finally, affects stocking for species with state variable .
In Section 2, we present some general results about competitive systems in the plane. In Section 3 contains some basic facts such as the nonexistence of period-two solution of system (1.1). In Section 4 analyzes local stability which is fairly complicated for this system. Finally, in Section 5 gives global dynamics for all values of parameters. This section finishes with an introduction of a new terminology for different type scenarios for competitive systems that can be used to give a simple classification of all possible global behavior for system (1.1). The interesting feature of this paper is that there are five regions of the parameters in which one of the equilibrium points is nonhyperbolic, and yet we are able to describe the global dynamics in all five cases. To achieve this goal, we use new method of proving stability of nonhyperbolic equilibrium points introduced in [2].
2. Preliminaries
Consider a first-order system of difference equations of the form where are continuous functions on an interval , is nondecreasing in and non-increasing in , and is non-increasing in and nondecreasing in . Such system is called competitive. One may associate a competitive map to a competitive system (2.1) by setting and considering on .
We now present some basic notions about competitive maps in plane. Define a partial order βͺ― on so that the positive cone is the fourth quadrant, that is, if and only if and . For the order interval is the set of all such that . A set is said to be linearly ordered if βͺ― is a total order on . If a set is linearly ordered by βͺ―, then the infimum and supremum of exist in . If both and belong to , then the linearly ordered set is bounded, and conversely. We note that the ordering βͺ― may be extended to the extended plane in a natural way. For example, if or . If , we denote with , , the four quadrants in relative to , that is, , , and so on.
A map on a set is a continuous function . The map is smooth on if the interior of is nonempty and if is continuously differentiable on the interior of . A set is invariant for the map if . A point is a fixed point of if , and a minimal period-two point if and . A period-two point is either a fixed point or a minimal period-two point. The orbit of is the sequence . A minimal period two orbit is an orbit for which and . The basin of attraction of a fixed point is the set of all such that . A fixed point is a global attractor on a set if is a subset of the basin of attraction of . A fixed point is a saddle point if is differentiable at , and the eigenvalues of the Jacobian matrix of at are such that one of them lies in the interior of the unit circle in , while the other eigenvalue lies in the exterior of the unit circle. If is a map on , define the sets and . For and , define the distance from to as .
A map is competitive if whenever , and is strongly competitive if implies that . If is differentiable, a sufficient condition for to be strongly competitive is that the Jacobian matrix of at any has the sign configuration For additional definitions and results (e.g., repeller, hyperbolic fixed points, stability, asymptotic stability, stable and unstable manifolds) see [8, 9] for competitive maps, and [10, 11] for difference equations.
If is any subset of , we shall use the notation to denote the closure of in , and to denote the interior of .
The next results are stated for order-preserving maps on and are known but given here for completeness. See [12] for a more general version valid in ordered Banach spaces.
Theorem 2.1. For a nonempty set and βͺ― a partial order on , let be an order-preserving map, and let be such that and . If and , then is invariant and(i)there exists a fixed point of in , (ii)if is strongly order preserving, then there exists a fixed point in which is stable relative to , (iii)if there is only one fixed point in , then it is a global attractor in and therefore asymptotically stable relative to .
Corollary 2.2. If the nonnegative cone of is a generalized quadrant in , and if has no fixed points in other than and , then the interior of is either a subset of the basin of attraction of or a subset of the basin of attraction of .
Define a rectangular region in to be the cartesian product of two intervals in .
Remark 2.3. It follows from the Perron-Frobenius theorem and a change of variables [9] that, at each 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 quadrant, respectively. Also, one can show that if the map is strongly competitive then no eigenvector is aligned with a coordinate axis.
Theorem 2.4. 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 axes. 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 .
We shall see in Theorem 2.7 and in the examples in [2] that the situation where the endpoints of are boundary points of is of interest. The following result gives a sufficient condition for this case.
Theorem 2.5. For the curve of Theorem 2.4 to have endpoints in , it is sufficient that at least one of the following conditions is satisfied. (i)The map has no fixed points nor periodic points of minimal period two in . (ii)The map has no fixed points in , , and has no solutions . (iii)The map has no points of minimal period two in , , and has no solutions .
In many cases, one can expect the curve to be smooth.
Theorem 2.6. Under the hypotheses of Theorem 2.4, suppose that there exists a neighborhood of in such that is of class on for some , and that the Jacobian matrix of at each is invertible. Then, the curve in the conclusion of Theorem 2.4 is of class .
In applications, it is common to have rectangular domains for competitive maps. If a competitive map has several fixed points, often the domain of the map may be split into rectangular invariant subsets such that Theorem 2.4 could be applied to the restriction of the map to one or more subsets. For maps that are strongly competitive near the fixed point, hypothesis (b) of Theorem 2.4 reduces just to . This follows from a change of variables [9] 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 quadrant, respectively. Also, one can show that in such case no associated eigenvector is aligned with a coordinate axes.
Smith performed a systematic study of competitive and cooperative maps in [9, 13, 14] and in particular introduced invariant manifolds techniques in his analysis [13β15] with some results valid for maps on -dimensional space. Smith restricted attention mostly to competitive maps that satisfy additional constraints. In particular, is required to be a diffeomorphism of a neighborhood of that satisfies certain conditions (this is the case if is orientation preserving or orientation reversing), and that the coordinate semiaxes are invariant under . For such class of maps (as well as for cooperative maps satisfying similar hypotheses), Smith obtained results on invariant manifolds passing through hyperbolic fixed points and a fairly complete description of the phase-portrait when , especially for those cases having a unique fixed point on each of the open positive semiaxes. In our results, presented here, we removed all these constraints and added the precise analysis of invariant manifolds of nonhyperbolic equilibrium points. The invariance of coordinate semiaxes seems to be serious restriction in the case of competitive models with constant stocking or harvesting, see [16] for stocking.
The next result is useful for determining basins of attraction of fixed points of competitive maps. Compare to Theoremβ4.4 in [13], where hyperbolicity of the fixed point is assumed, in addition to other hypotheses.
Theorem 2.7. Assume the hypotheses of Theorem 2.4, and let be the curve whose existence is guaranteed by Theorem 2.4. If the endpoints of belong to , then separates into two connected components, namely, such that the following statements are true: (i) is invariant, and as for every . (ii) is invariant, and as for every . If, in addition to the hypotheses of part (A), 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 .
Basins of attraction of period-two solutions or period-two orbits of certain systems or maps can be effectively treated with Theorems 2.4 and 2.7. See [2, 6, 11] for the hyperbolic case; for the nonhyperbolic case, see examples in [2, 17].
If is a map on a set and if is a fixed point of , the stable set of is the set , and unstable set of is the set When is noninvertible, the set may not be connected and made up of infinitely many curves, or may not be a manifold. The following result gives a description of the stable and unstable sets of a saddle point of a competitive map. If the map is a diffeomorphism on , the sets and are the stable and unstable manifolds of .
Theorem 2.8. In addition to the hypotheses of part (B) of Theorem 2.7, suppose that and that the eigenspace associated with is not a coordinate axes. If the curve of Theorem 2.4 has endpoints in , then is the stable set of , and the unstable set of 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 following result gives information on local dynamics near a fixed point of a map when there exists a characteristic vector whose coordinates have negative product and such that the associated eigenvalue is hyperbolic. This is a well-known result, valid in much more general setting which we include it here for completeness. A point is a subsolution if , and is a supersolution if . An order interval is the cartesian product of the two compact intervals and .
Theorem 2.9. Let be a competitive map on a rectangular set with an isolated fixed point such that . Suppose that has a extension to a neighborhood of . Let be an eigenvector of the Jacobian matrix of at , with associated eigenvalue . If , then there exists an order interval which is also a relative neighborhood of such that, for every relative neighborhood of , the following statements are true. (i)If , then contains a subsolution, and contains a supersolution. In this case, for every , there exists such that for . (ii)If , then contains a supersolution and contains a subsolution. In this case, for every .
In the nonhyperbolic case, we have the following result.
Theorem 2.10. Assume that the hypotheses of Theorem 2.9 hold, that is real analytic at , and that . Let , , be defined by the Taylor series Suppose that there exists an index such that and for . If either (a), or , is affine in t, or , is affine in , then there exists an order interval which is also a relative neighborhood of such that, for every relative neighborhood of , the following statements are true. (i)If is odd and , then contains a supersolution, and contains a subsolution. In this case, for every , there exists such that for . (ii)If is odd and , then contains a subsolution and contains a supersolution. In this case, for every . (iii)If is even and , then contains a subsolution and contains a subsolution. In this case, for every , and for every , there exists such that for .(iv)If is even and , then contains a supersolution and contains a supersolution. In this case, for every , and, for every there exists, such that for .
3. 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.
3.1. Equilibrium Points
The equilibrium points of the system (1.1) satisfy Solutions of System (3.1) are(i), when , that is, (ii) If , then using System (3.1), we obtain
Solutions of System (3.3) are where , which gives a pair of the equilibrium points and .
Geometrically, the equilibrium points are the intersections of two equilibrium curves: and . Depending on the values of parameters, may have between 0 and 3 intersection points with two lines which constitutes .
The algebraic criteria for the existence of the equilibrium points are summarized in Table 1.
Where
Remark 3.1. Observe the following: If the system (1.1) has two or three equilibrium points , , and then, . Indeed, consider the critical curve . Observe that , , and . It is obvious that the following holds . Since, the critical curve decreases, we have , that is, .
Lemma 3.2. Assume that , . Then the following statements are true for solutions of the system (1.1). (i)If , then , for all , and , .(ii)If , then , for all , and , .(iii) If , then , and , for all , , .
Assume that and . Then, the following statements are true for all :(iv). (v) andβ(a), , where is arbitrarily small positive number.β(b), , , where is arbitrarily small positive number.
Proof. Since (i)β(iv) are immediate consequences of the system (1.1), we will prove only (v).
Take and . Then, we have for all , and
Solution of (3.6), when is
which immediately implies (i) and (ii). Statement (iii) follows from (3.6).
Equation
implies that
Using the last inequality, we have
which by difference inequality theorem [18] implies the following
Furthermore, second equation in (1.1) implies that
which, by the difference inequalities argument, see [18], implies that , where satisfies (3.6). In view of (3.7) we obtain our conclusion.
3.2. Period-Two Solution
In this section, we prove that System (1.1) has no minimal period-two solution which will be essential for application of Theorems 2.5β2.7. The map associated to System (1.1) is given by
Lemma 3.3. System (1.1) has no minimal period-two solution.
Proof. We have
Period-two solution satisfies
We show that this system has no other positive solutions except the equilibrium points.
Equation (3.15) is equivalent to
If then we obtain the fixed point . So assume that . Then, using (3.17), we have
Equation (3.19) implies that
Substituting (3.20) into (3.18), we have
or
Equation (3.21) implies that, and (3.23) implies that
Replacing (3.23) into (3.19), we get
Solutions of (3.24) are the equilibrium points.
Consider (3.25). Discriminant of this equation is given by
Now, implies that
where
Using (3.23), we obtain
where
We prove the following claims.
Claim 3.4. For all values of parameters.Proof. If , then . Now, we assume that . Then,
Equation (3.32) implies that
Since if and only if
This implies that (3.33) is true. That is .Claim 3.5. Assume that . Then that .Proof. Assume that . This is equivalent to
Now
which is equivalent to
or
Also,
Since,
This inequality and (3.36) imply (3.40).
Since
Last inequality, (3.36) and (3.39) imply that . So we prove, if , then .
Assume that .
We have
Since,
we have that if and only if
which is true, because
Replacing with in the formula for , we obtain that .
Hence, there does not exist period-two solution.
4. Linearized Stability Analysis
The Jacobian matrix of the map , given by (3.13), has the form The determinant of (4.1) is given by The value of the Jacobian matrix of at the equilibrium point , is The determinant of (4.3) is given by and the trace of (4.3) is The characteristic equation has the form
Theorem 4.1. Assume that . Then there exists the equilibrium point and (i) is locally asymptotically stable if ,(ii) is a saddle point if . The corresponding eigenvalues are (iii) is nonhyperbolic if . The corresponding eigenvalues are and the corresponding eigenvectors are and , respectively.
Proof. The Jacobian matrix (4.1) at the equilibrium point ,
Note that the Jacobian matrix (4.9) implies that the map is not strongly competitive at the equilibrium point .
The determinant of (4.9) is given by
Note that, under the hypothesis of Theorem, the determinant is greater than zero.
The trace of (4.9) is
An equilibrium point is locally asymptotically stable if the following conditions are satisfied
Now, these two conditions become
Condition
implies that
If , then this condition is satisfied.
Condition is equivalent to
It is easy to see that the condition is satisfied if .
Next, we prove (ii).
An equilibrium point is a saddle if and only if the following conditions are satisfied
The first condition is equivalent to
which is satisfied if . The second condition is equivalent to
Finally, we prove (iii).
An equilibrium point is nonhyperbolic if the following conditions are satisfied
The first condition is equivalent to
which is satisfied if .
The second condition becomes
establishing part (iii).
We now perform a similar analysis for the other cases in Table 1.
Theorem 4.2. Assume that Then , , and exist and(i)the equilibrium point is locally asymptotically stable,(ii)the equilibrium point is a saddle point. Furthermore, if and , then the smaller eigenvalue belongs to the interval , and the larger eigenvalue belongs to . In all other cases, the smaller eigenvalues is in .That is is given by and the corresponding eigenvector is Eigenvalue , where , is given by (iii) The equilibrium point is locally asymptotically stable.
Proof. By Theorem 4.1, (i) holds.
Evaluating the Jacobian matrix (4.3) at the equilibrium point , we obtain
Note that the Jacobian matrix (4.27) implies that the map is strongly competitive.
The determinant of (4.27) is given by
and the trace of (4.27) is given by
The equilibrium point is a saddle if and only if (4.17) is satisfied. The first condition is equivalent to
which is equivalent to
In light of (3.3) , and by using (3.4), . Now, we have
This implies that which is true.
Condition
is equivalent to
which is true.
To prove the second part of the statement (ii), we use the characteristic equation (4.6) of System (1.1) at the equilibrium point. Now, we have
Since the map is strongly competitive, the Jacobian matrix (4.27) has two real and distinct eigenvalues, the larger one in absolute value being positive.
The first equation implies that either both eigenvalues are positive, or the smaller one is negative. First, we show that, under hypothesis (ii) of theorem, the product of these two eigenvalues is less than zero. In order to prove that, it is enough to prove that .
We have
Now if and only if
which holds if
Also, we have
In all other cases .
This proves that the smaller eigenvalue is negative. Since the equilibrium point is a saddle point, it has to belong to . The larger one belongs to . The proof of second statement is similar.
Now, we prove that is locally asymptotically stable.
Notice that
implies that which is true.
Theorem 4.3. Assume that Then exist and:(i)The equilibrium point is locally asymptotically stable. (ii)The equilibrium point is nonhyperbolic. The eigenvalues of the Jacobian matrix evaluated at are and the corresponding eigenvectors, respectively, are Furthermore, if , then . If , then .
Proof. By Theorem 4.1, is a locally asymptotically stable.
Now, we prove that is nonhyperbolic.
The Jacobian matrix (4.3) at the equilibrium point is
The eigenvalues of (4.44) satisfy
and are given as
Hence, is nonhyperbolic.
Notice that . Now, we show that can be in or .
We have
which is negative if , and positive if .
Theorem 4.4. Assume that Then, and exist and (i)the equilibrium point is a saddle point, (ii)the equilibrium point is locally asymptotically stable.
Proof. By Theorem 4.1, (i) holds. Observe that the assumption of Theorem 4.4 implies that the coordinate of the equilibrium point is less than zero.
The proof that the equilibrium point is locally asymptotically stable is similar to the corresponding proof of Theorem 4.2.
Theorem 4.5. The following statements are true. (a) Assume Then and exist and (i)the equilibrium point is a saddle. If , then the larger eigenvalue is in , the smaller eigenvalue is in . If , then the smaller eigenvalue is in . That is, is given by and the corresponding eigenvector is Eigenvalue , where , is given by (ii) The equilibrium point is locally asymptotically stable.(b)Assume thatββ,, .Then there exists a unique positive equilibrium which is nonhyperbolic. The eigenvalues are and , and the corresponding eigenvectors, respectively, are If , then . If then .
Proof. The proof of statements (a) is similar to the proof of the statements (ii) and (iii) of the Theorem 4.2.
Now, we prove statement (b).
The characteristic equation of the system (1.1) at the equilibrium point has the form
in which solutions are eigenvalues of
Hence, is nonhyperbolic. Notice that .
Now, we determine that the sign of
is negative if , and positive if .
Now, we consider the special case of the system (1.1) when .
In this case, the system (1.1) becomes
If the following condition holds then the system (4.57) has two positive equilibrium points
We prove the following.
Theorem 4.6. Assume that
Then the following statements hold. (i)If , then the system (4.57) has the unique equilibrium point which is nonhyperbolic. The following holds: β(a) If , then , and , and the corresponding eigenvectors, respectively, are
β(b) If , then and , and the corresponding eigenvectors, respectively, are
(c) If , then and , and the corresponding eigenvectors, respectively, are (ii) If , then the system (4.57) has two positive equilibrium points: is locally asymptotically stable and is a saddle point. The following holds. (d) If or and , then and . (e) If and , then and , where
where
and .
The corresponding eigenvector for both cases (c) and (d) is , where
where
Proof. Assume that . Then we have that
The characteristic equation associated to the system (4.57) at the equilibrium point is given by
Solutions of (4.69) are
It is easy to see that .
Now, assume that . Since , from , we have . The numerator of is . Assume the opposite, that is, . Then, we have
which is a contradiction. So, we confirmed (a).
Assume that . Then the numerator of . If , then we have
which is a contradiction. So, (b) holds.
Assume that . Then there are two positive equilibrium points
Now, we prove that is a locally asymptotically stable equilibrium point.
We check the conditions for locally asymptotically stable equilibrium point. We have
This implies that
which is equivalent to and
which is true.
Now, we check condition . We have that
This implies that which is true, since
Hence, is a locally asymptotically stable equilibrium point.
Now, we prove that is a saddle. We check the condition (4.17).
Condition is equivalent to . This is true, since
Condition
is equivalent to
Hence, is a saddle.
Now, we prove the statements (c) and (d).
The characteristic equation associated to the system (4.57) at the equilibrium point has the following form
Now, we have that
Consider . We have that
which implies statements (c) and (d).
5. Global Behavior
In this section, we present the results on global behavior of the system (1.1).
Theorem 5.1. Table 2 describes the global behavior of the systemββ(1.1)
Proof. Throughout the proof of theorem will denote . In view of Lemma 3.2, the map which corresponds to the system (1.1) has an attractive and invariant box , where , , which contains a unique fixed point . By Theorem 2.1, every solution of the system (1.1) converges to . Clearly, the basin of attraction of the equilibrium point is given by . By Lemma 3.2 implies that , for all , and , , which shows that -axes is a subset of the basin of attraction . Furthermore, every solution of (1.1) enters and stays in the box , and so we can restrict our attention to solutions that starts in . Clearly the set is an invariant set with a single equilibrium point , and so every solution that starts there is attracted to . In view of Corollary 2.2, the interior of rectangle is attracted to either or , and because is the local attractor, it is attracted to . If , then there exist the points and such that . Consequently, for all and so as , which completes the proof. The first part of this Theorem is proven in Theorem 4.4.Now, we prove a global result The eigenvalues of are given by and and so The eigenvector of at that corresponds to the eigenvalue is .The rest of the proof is similar to the proof of part () and uses some continuity arguments. The first part of this Theorem is proven in Theorem 4.2. Lemma 3.3 states that the system (1.1) has no minimal period-two solution. Take that . is strongly monotone in and differentiable in (interior of ). is a saddle point and . Then, all hypothesis (a)β(d) of Theorem 2.6 are satisfied. In light of Theorems 2.6 and 2.7, there exist the global stable manifold and the global unstable manifold which are the graphs of a continuous strictly monotonic functions. The global stable manifold separates the first quadrant into two invariant regions (below the stable) manifold and (above the stable manifold) which are connected. Each orbit starting above remains above and is asymptotic to . Each orbit starting below remains below and is asymptotic to . This implies that and are global attractors. Theorem 4.2 implies that they are globally asymptotically stable. Notice that in this case the eigenvector which corresponds to nonhyperbolic eigenvalue at is , see Theorem 4.3. Thus, the hypotheses of Theorems 2.4 and 2.7 are satisfied at the equilibrium point , and the conclusions of Theorems 2.4, 2.5, and 2.7 follow. Let , and be the sets given in the conclusion of Theorems 2.4 and 2.7. Let . Since for , , , the map satisfies . Thus, , which implies that is bounded. In view of Theorem 2.7, every solution which starts in eventually enters , and so is in rectangle , which by Theorem 2.1, implies that all such solutions converge to the equilibrium point .If is in , by Theorem 2.7, the orbit of eventually enters . Assume (without loss of generality) that .In view of Corollary 2.2 and the fact that is a local attractor is a subset of the basin of attraction of . Let be any point in . Then there exists such that . Now , which implies that is an increasing sequence, and so is a decreasing sequence and thus is convergent to . In view of , we conclude that and so eventually enters , and so it converges to . Let be any point in . Then there exist points and , such that . This gives for all . Clearly, , which implies that is a decreasing sequence bounded below by and so is convergent to . Proof that is convergent to is carried in a same way as in the proof of (). In view of we conclude that converges to . Put , . Take , where . It is known that is an invariant set, see [11].Then we have This implies By using monotonicity . By using induction . This implies that sequence is non-increasing and is nondecreasing. By Lemma 3.2, is bounded, hence it must converges. By using equation for we see that the limit is zero. Since, is unbounded and nondecreasing then , .By Theorems 2.4 and 2.7, all orbits below this manifold are attracted to the equilibrium point . Since the hypotheses of Theorems 2.4, and 2.7 are satisfied at the equilibrium point , the conclusions of Theorems 2.4, 2.5, and 2.7 follow. Let , and be the sets given in the conclusion of Theorems 2.4 and 2.7. Let . Since for , , , the map satisfies . Thus , which implies that is bounded. In view of Theorem 2.7 every solution which starts in eventually enters and so is in rectangle , which by Theorem 2.1, implies that all such solutions converge to the equilibrium point .If is in , by Theorem 2.7 the orbit of eventually enters . Assume (without loss of generality) that .A calculation gives for all and Since in expansion (2.5) we have with and is affine in , by Theorem 2.10 in any relative neighborhood of there exists a subsolution , that is, . Choose one such so that . Since is competitive, for . The monotonically decreasing sequence in is unbounded below, since if it were not it would converge to the unique fixed point in , namely , which is not possible. Let , . Then for , hence is bounded. It follows that is monotone and unbounded. From (1.1) it follows that . Since , it follows that . The eigenvalues of the map are and . The corresponding eigenvectors are The existence of the continuous curve follows as in the proof of () as well as convergence of all points which start in .If is in , by Theorem 2.7 the orbit of eventually enters . Assume (without loss of generality) that .βA straightforward calculation gives for all and Since in expansion (2.5) we have with and is affine in , by Theorem 2.10 in any relative neighborhood of there exists a subsolution , that is, . The rest of the proof is same as the proof of the case ().The proof, which is similar to the proof of (), follows as an immediate application of Lemmas 3.2 andβ3.3, and Theorems 2.4, 2.7, and 4.6. For every point , there exists such that . Clearly, , which implies that is a decreasing sequence, and so is an increasing sequence. If would be convergent, then would converge to an equilibrium of system (1.1) which is impossible. Thus, , which implies that . In view of , we obtain that and so follows.
Remark 5.2. We can see from Theorem 5.1 that system (1.1) exhibits variety of behaviors in different ranges of parameters. These behaviors can be classified in a few categories that verbally describe situation. Here, we use some terminology introduced in [19].
A coexistence attractor is one in which both species are present. An exclusion attractor is one in which one species is absent and the other species is present. By multiple mixed-type attractors, we mean a scenario that includes at least one coexistence attractor and at least one exclusion attractor. Park in [20, 21] observed the coexistence case in an experimental treatment that also included cases of competitive exclusion, that is, he observed a case termed to be multiple mixed-type attractors. Competition theory is primarily an equilibrium theory that is exemplified, by its limited number of asymptotic outcomes: a globally attracting coexistence equilibrium; a globally attracting exclusion equilibrium; at least two attracting exclusion equilibria; at least two attracting coexistence equilibria; a continuum of nonhyperbolic equilibria. (In this context, globally attracting means within the positive cone of state space.) Four of these five asymptotic alternatives are illustrated by the LeslieGower model (the discrete analog of the famous LotkaVolterra differential equation model), see [2, 5, 17]
and fifth alternative, precisely two attracting coexistence equilibria are illustrated by the LeslieGower model with stockings, see [16]
where the parameters , , , , , and are positive numbers, and the initial conditions , are arbitrary nonnegative numbers. Here, and are the inherent birth rates, and the density-dependent effects on newborn recruitment, and are constant stockings. With this in mind, we introduce the following terminology. Let , and be three equilibrium points of general competitive system (2.1) in south-east ordering and assume that and are attractors, and is a saddle point or nonhyperbolic equilibrium having one characteristic values in . If and belong to -axes and -axes and is a saddle point (resp., nonhyperbolic equilibrium having one characteristic values in ) then we say that system (2.1) exhibits saddle competitive exclusion (resp., nonhyperbolic competitive exclusion). If and belong to the interior of first quadrant and is a saddle point (respectively, nonhyperbolic equilibrium having one characteristic values in ) then we say that system (2.1) exhibits saddle competitive coexistence (resp., nonhyperbolic competitive coexistence). If one of the equilibrium points and is on the axes and the other is in the interior of first quadrant and is a saddle point (resp., nonhyperbolic equilibrium having one characteristic values in ), then we say that system (2.1) exhibits saddle competitive exclusion to coexistence (resp., nonhyperbolic competitive exclusion-to-coexistence). If there exists a single attractor in the interior of first quadrant which attracts all points where it is defined except eventually the points on the axes, we say that system (2.1) exhibits global competitive coexistence or global competitive exclusion depending on whether the attractor is on the axes or in the interior of first quadrant. If any of the attractors one the axes is the point or , such a situation will be named singular.
Using this terminology, we can describe the global behavior of system (1.1) in a concise way as follows. System (1.1) exhibits global competitive exclusion if the parameters belong to the regions (), (), or () and global competitive coexistence if the parameters belong to the region ().
System (1.1) exhibits saddle competitive exclusion to coexistence if the parameters belong to the region (). System (1.1) exhibits nonhyperbolic competitive exclusion to coexistence if the parameters belong to the region (). System (1.1) exhibits singular saddle competitive exclusion to coexistence if the parameters belong to the regions () or (). System (1.1) exhibits singular nonhyperbolic competitive exclusion to coexistence if the parameters belong to the region () or (). Finally, System (1.1) exhibits singular competitive exclusion when the parameters belong to the region (). With this terminology the possible scenarios for this system are limited, and transition from one scenario to another could be possible explained by using the bifurcation theory. In particular, transition from global competitive exclusion to global competitive coexistence was explained in [3], and some related results can be found in [19], where an attempt has been made to explain the transitions from one scenario to another by using evolutionary game theory.