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Báo cáo hóa học: "Dynamics of a two-dimensional system of rational difference equations of Leslie–Gower type"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Dynamics of a two-dimensional system of rational difference equations of Leslie–Gower type | Kalabusic et al. Advances in Difference Equations 2011 2011 29 http www.advancesindifferenceequations.eom content 2011 1 29 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Dynamics of a two-dimensional system of rational difference equations of Leslie-Gower type S Kalabusic1 MRS Kulenovic2 and E Pilav1 Correspondence mkulenovic@mail.uri.edu department of Mathematics University of Rhode Island Kingston RI 02881-0816 USA Full list of author information is available at the end of the article Springer Abstract We investigate global dynamics of the following systems of difference equations a 1 PiXn Xn 1 A II _ Ayn yn n 0 1 2 . yn 1 T z Á2 B2Xn yn where the parameters a1 b1 A1 g2 A2 B2 are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. We show that this system has rich dynamics which depends on the region of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or non-hyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or non-hyperbolic equilibrium points. We give examples of a globally attractive non-hyperbolic equilibrium point and a semi-stable non-hyperbolic equilibrium point. We also give an example of two local attractors with precisely determined basins of attraction. Finally in some regions of parameters we give an explicit formula for the global stable manifold. Mathematics Subject Classification 2000 Primary 39A10 39A11 Secondary 37E99 37D10 Keywords Basin of attraction Competitive map Global stable manifold Monotonicity Period-two solution 1 Introduction In this paper we study the global dynamics of the following rational system of difference equations ai P1Xn Xn 1 _ Any n 0 1 2 . 1 yn 1 Á2 B2Xn yn where the parameters a1 b1 A1 g2 A2 B2 are positive numbers and initial conditions x0 and y0 are arbitrary nonnegative numbers. System 1 was mentioned in 1 as one of three systems of