Asymptotic behavior of a competitive system of linear fractional difference equations

• Main Authors: Nurkanović M, Kulenović MRS
• Format: Article
• Language: English
• Published: SpringerOpen 2006-01-01
• Series: Advances in Difference Equations
• Online Access: http://www.advancesindifferenceequations.com/content/2006/019756

<p/> <p>We investigate the global asymptotic behavior of solutions of the system of difference equations <it>x</it>_<it>n</it>+1 = (<it>a</it>+<it>x</it>_<it>n</it>)/(<it>b</it>+<it>y</it>_<it>n</it>), <it>y</it>_<it>n</it>+1 = (<it>d</it>+<it>y</it>_<it>n</it>)/(<it>e</it>+<it>x</it>_<it>n</it>), <it>n</it> = 0,1,..., where the parameters <it>a</it>, <it>b</it>, <it>d</it>, and <it>e</it> are positive numbers and the initial conditions <it>x</it>₀ and <it>y</it>₀ are arbitrary nonnegative numbers. In certain range of parameters, we prove the existence of the global stable manifold of the unique positive equilibrium of this system which is the graph of an increasing curve. We show that the stable manifold of this system separates the positive quadrant of initial conditions into basins of attraction of two types of asymptotic behavior. In the case where <it>a</it> = <it>d</it> and <it>b</it> = <it>e</it>, we find an explicit equation for the stable manifold to be <it>y</it> = <it>x</it>.</p>