Document Type

Article

Date of Original Version

2011

Abstract

We investigate global dynamics of the following systems of difference equations 𝑥𝑛+1=𝛽1𝑥𝑛/(𝐵1𝑥𝑛+𝑦𝑛), 𝑦𝑛+1=(𝛼2+𝛾2𝑦𝑛)/(𝐴2+𝑥𝑛), 𝑛=0,1,2,…, where the parameters 𝛽1, 𝐵1, 𝛽2, 𝛼2, 𝛾2, 𝐴2 are positive numbers, and initial conditions 𝑥0 and 𝑦0 are arbitrary nonnegative numbers such that 𝑥0+𝑦0>0. We show that this system has up to three equilibrium points with various dynamics which depends on the part of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or nonhyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or of nonhyperbolic equilibrium points. We give an example of globally attractive nonhyperbolic equilibrium point and semistable non-hyperbolic equilibrium point.

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Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.

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