Dynamics of a Rational System of Difference Equations in the Plane
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Research Article Dynamics of a Rational System of Difference Equations in the Plane ´ 2 Ignacio Bajo,1 Daniel Franco,2 and Juan Peran 1
Departamento de Matem´atica Aplicada II, E.T.S.E. Telecomunicaci´on, Universidade de Vigo, Campus Marcosende, 36310 Vigo, Spain 2 Departamento de Matem´atica Aplicada, E.T.S.I. Industriales, UNED, C/ Juan del Rosal 12, 28040 Madrid, Spain Correspondence should be addressed to Juan Per´an, [email protected] Received 10 December 2010; Accepted 21 February 2011 Academic Editor: Istvan Gyori Copyright q 2011 Ignacio Bajo 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. We consider a rational system of first-order difference equations in the plane with four parameters such that all fractions have a common denominator. We study, for the different values of the parameters, the global and local properties of the system. In particular, we discuss the boundedness and the asymptotic behavior of the solutions, the existence of periodic solutions, and the stability of equilibria.
1. Introduction In recent years, rational difference equations have attracted the attention of many researchers for varied reasons. On the one hand, they provide examples of nonlinear equations which are, in some cases, treatable but whose dynamics present some new features with respect to the linear case. On the other hand, rational equations frequently appear in some biological models, and, hence, their study is of interest also due to their applications. A good example of both facts is Ricatti difference equations; the richness of the dynamics of Ricatti equations is very well-known see, e.g., 1, 2, and a particular case of these equations provides the classical Beverton-Holt model on the dynamics of exploited fish populations 3. Obviously, higher-order rational difference equations and systems of rational equations have also been widely studied but still have many aspects to be investigated. The reader can find in the following books 4–6, and the works cited therein, many results, applications, and open problems on higher-order equations and rational systems. A preliminar study of planar rational systems in the large can be found in the paper 7 by Camouzis et al. In such work, they give some results and provide some open questions
2
Advances in Difference Equations
for systems of equations of the type xn1 yn1
α1 β1 xn γ1 yn A1 B1 xn C1 yn
⎫ ⎪ ⎪ ⎪ ⎬
,
α2 β2 xn γ2 yn ⎪ ⎪ ⎪ ⎭ A2 B2 xn C2 yn
n 0, 1, . . . ,
1.1
where the parameters are taken to be nonnegative. As shown in the cited paper, some of those systems can be reduced to some Ricatti equations or to some previously studied second-order rational equations. Further, since, for some choices of the parameters, one obtains a system which is equivalent to the case with some other parameters, Camouzis et al. arrived at a list of 325 nonequivalent systems to
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