Global Dynamics of a Competitive System of Rational Difference Equations in the Plane
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Research Article Global Dynamics of a Competitive System of Rational Difference Equations in the Plane ˇ c, ´ 1 M. R. S. Kulenovic, ´ 2 and E. Pilav1 S. Kalabusi 1 2
Department of Mathematics, University of Sarajevo, 71 000 Sarajevo, Bosnia and Herzegovina Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USA
Correspondence should be addressed to M. R. S. Kulenovi´c, [email protected] Received 26 August 2009; Accepted 8 December 2009 Recommended by Panayiotis Siafarikas We investigate global dynamics of the following systems of difference equations xn1 α1 β1 xn /yn , yn1 α2 γ2 yn /A2 xn , n 0, 1, 2, . . ., where the parameters α1 , β1 , α2 , γ2 , and A2 are positive numbers and initial conditions x0 and y0 are arbitrary nonnegative numbers such that y0 > 0. 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. Copyright q 2009 S. Kalabuˇsi´c 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.
1. Introduction and Preliminaries In this paper, we study the global dynamics of the following rational system of difference equations: xn1 yn1
α1 β1 xn , yn
α2 γ2 yn , A2 xn
n 0, 1, 2, . . . ,
1.1
where the parameters α1 , β1 , α2 , γ2 , and A2 are positive numbers and initial conditions x0 ≥ 0 and y0 > 0 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 21, 29. In this paper, we provide the precise description of global dynamics of system 1.1. We show that system
2
Advances in Difference Equations
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
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