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Abstract and Applied Analysis
Volume 2017 (2017), Article ID 3104512, 19 pages
https://doi.org/10.1155/2017/3104512
Research Article

Bifurcation and Global Dynamics of a Leslie-Gower Type Competitive System of Rational Difference Equations with Quadratic Terms

1Division of Mathematics, Faculty of Mechanical Engineering, University of Sarajevo, Bosnia and Herzegovina
2Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USA
3Department of Mathematics, University of Sarajevo, 75 000 Sarajevo, Bosnia and Herzegovina

Correspondence should be addressed to M. R. S. Kulenović

Received 1 April 2017; Accepted 4 June 2017; Published 2 August 2017

Academic Editor: Patricia J. Y. Wong

Copyright © 2017 V. Hadžiabdić et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [13]. 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 [13, 716]. Special attention to discrete competitive and cooperative systems in the plane was given in [13, 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 thatwhich is equivalent to Since we have that if and only if which is equivalent toFrom (27) and (28) it follows if and only if (2)Assume that which is equivalent toThen if and only if which is equivalent toSince then from (33) and we haveSince we have that (31) and (36) are equivalent toorNow, 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: Parameter regions in the -plane. The curves , , and are defined as part of the parabola and the curves and are defined as part of the parabola

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.

Table 1: The criteria for the existence of the equilibrium points.

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 orIf 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