Abstract
We investigate global dynamics of the following systems of difference equations , , , where the parameters , , , , , and are positive numbers and the initial conditions and are arbitrary nonnegative numbers. This system is a version of the Leslie-Gower competition model for two species. We show that this system has rich dynamics which depends on the part of parametric space.
1. Introduction
In this paper we study the global dynamics of the following rational system of difference equations:where the parameters , , , , , and are positive numbers and initial conditions and are arbitrary nonnegative numbers.
System (1) is a competitive system, and our results are based on recent results about competitive systems in the plane; see [1]. System (1) can be used as a mathematical model for competition in population dynamics. System (1) is related to Leslie-Gower competition modelwhere the parameters , , , , , and are positive numbers and initial conditions and are arbitrary nonnegative numbers, considered in [2]. System (2) globally exhibits three dynamic scenarios in five parametric regions which are competitive exclusion, competitive coexistence, and existence of an infinite number of equilibrium solutions; see [1–3]. System (2) does not exhibit the Allee effect, which is desirable from modeling point of view. The simplest variation of system (2) which exhibits the Allee effect is probably systemwhere the parameters , , , , , and are positive numbers and initial conditions and are arbitrary nonnegative numbers, considered in [4]. System (3) has between 1 and 9 equilibrium points and exhibits nine dynamics scenarios part of each is the Allee effect. In the case of the dynamic scenario with nine equilibrium points system (3) exhibits both competitive exclusion and competitive coexistence as well as the Allee effect. Another system with quadratic terms iswhere the parameters , , , , and are positive numbers and initial conditions and are arbitrary nonnegative numbers such that , considered in [5]. System (4) exhibits seven scenarios part of each is singular Allee’s effect, which means that the origin as the singular point of this system still has some basin of attraction. First systematic study for a system with quadratic terms was performed in [6] for system which exhibits nine dynamic scenarios and whose dynamics is very similar to the corresponding system without quadratic terms considered in [7].
In general, it seems that an introduction of quadratic terms in equations of the Leslie-Gower model (2) generates the Allee effect. We will test this hypothesis in this paper by introducing the quadratic terms only in the second equation. System (1) can be considered as the competitive version of the decoupled systemwhere the parameters , , , and are positive numbers and initial conditions and are arbitrary nonnegative numbers, whose dynamics can be directly obtained from two separate equations. Unlike system (2) which has five regions of parameters with distinct local behavior system (1) has eighteen regions of parameters with distinct local behavior, which is caused by the geometry of the problem, that is, by the geometry of equilibrium curves. More precisely, the equilibrium curves of system (2) are lines while the equilibrium curves of system (1) are a line and a parabola. In the case when , all equilibrium points are hyperbolic and all solutions are attracted to the three equilibrium points on the -axis and we can describe this situation as competitive exclusion case. When , the equilibrium point is nonhyperbolic and dynamics is analogous to the case when . In both cases the Allee effect is present. When , there exist 11 regions of parameters with different global dynamics. In nine of these regions the global dynamics is in competitive exclusion case, which means that all solutions converge to one of the equilibrium points on the axes and in only two situations we have competitive coexistence case, which means that the interior equilibrium points have substantial basin of attraction. In all 11 cases, the zero equilibrium has some basin of attraction which is a part of -axis so we can say that in these cases system (1) exhibits weak Allee’s effect. Figure 3 gives the bifurcation diagram showing the transition from different global dynamics situations when , since the cases are simple and do not need graphical interpretation.
The paper is organized as follows. Section 2 contains some necessary results on competitive systems in the plane. Section 3 provides some basic information about the number of equilibrium points. Section 4 contains local stability analysis of all equilibrium solutions. Section 5 contains some global results on injectivity of the map associated with system (1). Section 6 gives global dynamics of system (1) in all regions of the parameters.
2. Preliminaries
A first-order system of difference equationswhere , , , are continuous functions is competitive if is nondecreasing in and nonincreasing in , and is nonincreasing in and nondecreasing in . If both and are nondecreasing in and , system (7) is cooperative. Competitive and cooperative maps are defined similarly. Strongly competitive systems of difference equations or strongly competitive maps are those for which the functions and are coordinate-wise strictly monotone.
Competitive and cooperative systems have been investigated by many authors; see [1–3, 7–16]. Special attention to discrete competitive and cooperative systems in the plane was given in [1–3, 16, 17]. One of the reasons for paying special attention to two-dimensional discrete competitive and cooperative systems is their applicability and the fact that many examples of mathematical models in biology and economy which involve competition or cooperation are models which involve two species. Another reason is that the theory of two-dimensional discrete competitive and cooperative systems is very well developed, unlike such theory for three-dimensional and higher systems. Part of the reason for this situation is de Mottoni-Schiaffino theorem given below, which provides relatively simple scenarios for possible behavior of many two-dimensional discrete competitive and cooperative systems. However, this does not mean that one can not encounter chaos in such systems as has been shown by Smith; see [16].
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 from to 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:
For standard definitions of attracting fixed point, saddle point, stable manifold, and related notions see [10].
We now state three results for competitive maps in the plane. The following definition is from [16].
Definition 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-Schiaffino [17] for the Poincaré map of a periodic competitive Lotka-Volterra system of differential equations. Smith generalized the proof to competitive and cooperative maps [13, 14].
Theorem 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 [16], 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 3. 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 .
The following result is a direct consequence of the Trichotomy Theorem of Dancer and Hess (see [18]) and is helpful for determining the basins of attraction of the equilibrium points.
Corollary 4. 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 .
Next result is well known global attractivity result which holds in partially ordered Banach spaces as well; see [18].
Theorem 5. 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 [1] for competitive systems in the plane, when one of the eigenvalues of the linearized system at an equilibrium (hyperbolic or nonhyperbolic) is 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.
Theorem 6. 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 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 .
The situation where the endpoints of are boundary points of is of interest. The following result gives a sufficient condition for this case.
Theorem 7. For the curve of Theorem 6 to have endpoints in , it is sufficient that at least one of the following conditions is satisfied(i)The map has no fixed points or 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 .
The next result is useful for determining basins of attraction of fixed points of competitive maps.
Theorem 8. (A) Assume the hypotheses of Theorem 6, and let be the curve whose existence is guaranteed by Theorem 6. 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 .(B) 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 .
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 9. In addition to the hypotheses of part (B) of Theorem 8, suppose that and that the eigenspace associated with is not a coordinate axis. If the curve of Theorem 6 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 .
3. Number of Equilibria
In this section we give some basic facts which are used later. Let be the map associated with system (1) given byLet The equilibrium points of system (1) satisfy equationsFor we have from which we obtain three equilibrium points where
Assume that Then, from the first equation of system (12) we have By substituting this into the second equation we obtain orfrom which we obtain the other three equilibrium points where
Lemma 10. The following hold:(i)The equilibrium points and exist if and only if and if and only if (ii)The equilibrium point exists if and only if and if and only if (iii)Assume that The equilibrium point exists if and only if and or (iv)Assume that The equilibrium point exists if and only if and
Proof. The proof of the statements (i) and (ii) is trivial and we skip it. Now we prove the statement (iii). In view of Descartes’ rule of signs we obtain that (17) has no positive solutions if Now, we suppose that One can see that for all values of parameters. We consider two cases: (1)Assume that which is equivalent to Since we have that if and only if which is equivalent to From (27) and (28) it follows if and only if (2)Assume that which is equivalent to Then if and only if which is equivalent to Since then from (33) and we have Since we have that (31) and (36) are equivalent to orNow, the proof of the statement (iii) follows from (28), (38), and (39). The proof of the statement (iv) is similar and we skip it.
We now introduce the following notation for regions in parameter space (see Figure 1):
Figure 1 gives a graphical representation of above sets. The following result gives a complete classification for the number of equilibrium solutions of system (1).
Proposition 11. Let , , , , , and be positive real numbers. Then, the number of positive equilibrium solutions of system (1) with parameters , , , , , and can be from 1 to 6. The different cases are given in Table 1.
Proof. The proof follows from Lemma 10.
4. Linearized Stability Analysis
The Jacobian matrix of the map has the formThe determinant of (41) at the equilibrium point is given byand the trace of (41) at the equilibrium point is given by The characteristic equation has the form
Lemma 12. The following statements hold: (a) is locally asymptotically stable if .(b) is a saddle point if .(c) is a nonhyperbolic equilibrium point if .
Proof. We have that, for the equilibrium point , and . The characteristic equation of (50) at has the form , from which the proof follows.
Lemma 13. The following statements hold: (a) is locally asymptotically stable if .(b) is a nonhyperbolic equilibrium point if .
Proof. We have that, for the equilibrium point , and . The characteristic equation of (50) at has the form , from which the proof follows.
The equilibrium points and are intersection points of the curves Let for
Lemma 14. Let be the map defined by (11). Then , , Let Then, and are zeros of and for
Proof. The first derivative of is given by Since , , we get , Similarly, one can see that Since , we get Further, from which the proof follows.
Lemma 15. Let be the map associated with system (1) andbe the Jacobian matrix of at fixed point Then the Jacobian matrix (50) has real and distinct eigenvalues and such that Furthermore, the following hold:
Proof. Implicit differentiation of the equations defining and at gives Characteristic equations associated with the Jacobian matrix of at are given by Since the map is competitive, then the eigenvalues of the Jacobian matrix of the map , at the equilibrium , are real and distinct and furthermore By (53), we have In view of Lemma 14 and from we get The map is competitive, which implies In view of Lemma 14 we get from which it follows (51). Similarly, from and we obtain (52).
The following lemma describes the local stability of the equilibrium points and
Lemma 16. Assume that and . Then the following hold: (i)If and exists then it is locally asymptotically stable.(ii)If and exists then it is a saddle point.(iii)If then Furthermore, if exists then it is nonhyperbolic equilibrium point. The eigenvalues of are given by and
Proof. Assuming that , then and are zeros of multiplicity one of and . From this we have for and for
By Lemmas 6 and 7 from [19] the equilibrium curves and intersect transversally at and , that is, By this and Lemma 14 and by continuity of function there exists a neighborhood of such that for and for . This implies that and . By Lemma 15 we have that is locally asymptotically stable and is a saddle point whenever equilibrium points and exist.
Assume that Then is zero of of multiplicity two. In view of Lemmas 6 and 7 from [19] we have that The rest of the proof follows from the proof of Lemma 15.
Lemma 17. Assume that . The following statements are true: (a) is a saddle point if , and or and (b) is a repeller if , , and (c) is a nonhyperbolic equilibrium point if or If then the eigenvalues of are given by with corresponding eigenvectors If (57) holds then the eigenvalues of are given by with corresponding eigenvectors
Proof. One can see that (a)Since and , the equilibrium is a saddle point if and only if If it is obvious that Assume that Then if and only if from which the proof of the statement follows.(b)Since and , the equilibrium is repeller if and only if and The proof of the statement follows from the facts (c)Since and , the equilibrium is nonhyperbolic if and only if or and From the proof of the statements (a) and (b) if we obtain Now, assume that and This implies The rest of the proof follows from the fact that if then and if (57) holds then
Lemma 18. Assume that . The following statements are true: (a) is locally asymptotically stable if , and or and (b) is a saddle point if , and , (c) is a nonhyperbolic equilibrium point if or If then the eigenvalues of are given by with corresponding eigenvectors If (57) holds then the eigenvalues of are given by with corresponding eigenvectors
Proof. Since the proof of this lemma is similar to the proof of Lemma 17, it is omitted.
We summarize results about local stability in the following theorem.
Theorem 19. Let , , , , , and be positive real numbers. Then, local stability of the equilibrium points for different parameter regions is given by Table 2.
Proof. The proof follows from Theorem 8 and Lemmas 17 and 18.
Figure 2 illustrates visually local stability of all equilibrium points of system (1).
5. Injectivity and Convergence to Equilibrium Points
In this section we prove some global properties of the map such as injectivity and property and give global behavior on the coordinate axes.
Lemma 20. The map is injective.
Proof. Assume that Then, we have Equation (74) is equivalent toEquation (75) impliesBy substituting this into (76) we obtainfrom which it follows that . From (77) we have , which complete the proof.
The global behavior of on the coordinate axes is described with the following result.
Lemma 21. The following statements hold: (i)If and then and for and for (ii)If and then and for (iii)If and then and for (iv)If and then and for (v).
Proof. (i)From (11) it is easy to see that if then for Since we obtain that Take Then Since we obtain as Similarly, if then which implies as If then which implies as (ii)If then has only equilibrium on -axis and for all Since is monotone map we get which implies as from which the proof follows.(iii)The proof of the statements (iii) and (iv) is similar to the proof of statements (i) and (ii) and follows from the fact that and will be omitted.(v)The proof follows from the facts that and
Lemma 22. Let . Then , , and and the following hold: (i)If then for and for (ii)If then for and for
Proof. The proof follows from the fact where is given by (17) and and
Theorem 23. Every solution of system (1) converges to an equilibrium point.
Proof. The map associated with the system is injective. Relation (42) implies that determinant of Jacobian (41) is positive for all By using Lemma 20 we have that condition of Theorem 3 is satisfied for the map ( is competitive). Theorem 2 implies that is eventually componentwise monotone for all . The statement (v) of Lemma 21 implies that every solution enters in compact set , from which the proof follows.
Remark 24. In view of Theorem 23 the main objective in determining the global dynamics of system (1) is to characterize the basins of attractions of all equilibrium points. As we will see in Theorem 25 the boundaries of these basins of attractions will be the global stable manifolds of the saddle or nonhyperbolic equilibrium points, whose existence is guaranteed by Theorems 7, 8, and 9.
6. Global Behavior
In this section we give results which precisely describe global dynamics of system (1) including precise characterization of basins of attraction of different equilibrium points. The main result of this paper is the following.
Theorem 25. The global behavior of system (1) is given by Table 3. See Figure 3 for visual illustration of dynamic scenarios.
Proof. We will prove statements (i)–(x) listed in the second column of Table 3 in the given order. The proof of other statements is similar. Let (i)Suppose . By Proposition 11, in there exist six equilibria , , , , , and . By Theorem 19 equilibria and are locally asymptotically stable; , , and are the saddle points and is repeller. In view of (41) the map is competitive on and strongly competitive on . It follows from the Perron-Frobenius Theorem and a change of variables [16] 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. Hence, all conditions of Theorems 7, 8, and 9 are satisfied, which yields the existence of the global stable manifold , with endpoint at point , which is graph of an increasing function. Let and . By Lemma 21 and uniqueness of the global stable manifold we have and . Take . By Theorem 8 and Lemma 21 we have that there exists such that , . In view of Corollary 4 and Since is competitive, this implies Take By Theorem 8 and Lemma 21 we have that there exists such that , . Since is competitive, in view of Corollary 4 . This completes the proof of statement (i).(ii)Suppose . By Proposition 11, in there exist five equilibria , , , , and . By Theorem 19 and are the saddle points, is repeller, and is nonhyperbolic. Similarly as in the proof of the statement (i), all conditions of Theorems 7, 8, and 9 are satisfied, which yields the existence of the invariant curve with one endpoint at and which is passing through , and it is graph of an increasing function. Let and By Lemma 21 and uniqueness of the global stable manifold we have and Take By Theorem 8 we have that there exists such that , In view of Corollary 4 Take . By Theorem 8 and Lemma 21 we have that there exists such that , . Since is competitive, in view of Corollary 4 . This completes the proof of statement (ii)(iii)The proof is similar to the proof of case (i) and we skip it.(iv)Suppose By Proposition 11, in there exist four equilibria , , , and . By Theorem 19 and are saddle points; is repeller; is locally asymptotically stable. By Lemma 21 and uniqueness of the global stable manifold we have and Since, by Theorem 23, every solution of system (1) converges to an equilibrium point, we have that (v)Suppose By Proposition 11, in there exist four equilibrium points , , , and . By Theorem 19 and are the saddle points; and are locally asymptotically stable. By Lemma 21 and uniqueness of the global stable manifold we have , and and . Similarly as in the proof of case (i), all conditions of Theorems 7, 8, and 9 are satisfied, which yields the existence of the global stable manifold , which is a graph of an increasing function. The rest of the proof follows from the facts that and are invariant sets, , , uniqueness of the global stable manifold and Theorem 23.(vi)The proof is similar to the proof of case (i) and we skip it.(vii)The proof is the same as the proof of case (viii) and we skip it.(viii)Suppose . By Proposition 11, in there exist four equilibrium points , , , and . By Theorem 19 we have that is locally asymptotically stable; is a saddle point; is nonhyperbolic; and is repeller. By Lemma 21 and uniqueness of the global stable manifold we have . In view of Lemma 18, for we have that the eigenspace associated with is a coordinate axis, so we can not use Theorem 6. By Lemma 21 we obtain Similarly as in the proof of Theorem 8 (see [1] for more details) one can prove that for every there exists such that for . By Lemmas 21 and 22 for , there exists and such that Since we have that Since we obtain This implies , which completes the proof.(ix)Suppose . By Proposition 11, there exist four equilibrium points , , , and . By Theorem 19 we have that and are locally asymptotically stable; is a saddle point; is nonhyperbolic. By Lemma 21 and uniqueness of the global stable manifold we have . In view of Lemma 18, for , we have that and , so we can not use Theorem 6. By Lemmas 21 and 22 for , there exists and such that . Since we have that . Since we obtain . This implies Let denote the boundary of considered as a subset of and let denote the boundary of considered as a subset of It is easy to see by using Lemmas 21 and 22 that Since , following the proof of Claims 1 and 2 [20], one can see that and for for . Further, are graphs of the continuous strictly increasing functions. If is point above the curve and below the curve then there exists and such that . Since and and as we have as (x)Suppose By Proposition 11, in there exist three equilibrium points , , and . By Theorem 19 is nonhyperbolic, is a saddle point, and is locally asymptotically stable. By Lemma 21 and uniqueness of the global stable manifold we have In view of Lemma 18 we have that and if and if , so we can use Theorem 6 if . In this case there exists strictly increasing curve with endpoint at The rest of the proof follows from Theorems 7, 8, and 9 and Lemma 21. Now, we assume that . By Lemmas 21 and 22 for , there exists and such that Since we have that . Since we obtain . This implies Let denote the boundary of considered as a subset of It is easy to see by using Lemmas 21 and 22 that . Since , following the proof of Claims 1 and 2 [20], one can see that Further, is graph of strictly increasing function. By Theorem 23 if then If is point above the curve then there exists and such that Since , , and as we have as
Based on a series of numerical simulations we propose the following conjectures.
Conjecture 26. Suppose that all assumptions of the statement (ix) of Theorem 25 are satisfied; then the following holds:
Conjecture 27. Assume that There exist three equilibrium points , , and , where is locally asymptotically stable; is a saddle point; and is nonhyperbolic. The basins of attraction of and are given by and and the basin of attraction of is
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.