Asymptotic behavior of a competitive system of linear fractional difference equations

Abstract

We investigate the global asymptotic behavior of solutions of the system of difference equations x n+1 = (a + x n )/(b + y n ), y n+1 = (d + y n )/(e + x n ), n = 0,1,..., where the parameters a,b,d, and e are positive numbers and the initial conditions x 0 and y 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 a = d and b = e, we find an explicit equation for the stable manifold to be y = x.

Invariant Manifolds for Competitive Discrete Systems in the Plane

“…The examples are of nonhyperbolic type. The hyperbolic case is well known and has been treated in [7,8,22,23,24]. Example 1 A system with a continuum of non-hyperbolic equilibria along a vertical line.…”

“…Strongly competitive systems of difference equations or maps are those for which the functions f and g are coordinate-wise stricly monotone. Competitive and cooperative systems of difference equations of the form (3) have been studied by many authors [3,4,5,6,7,8,9,10,12,13,14,15,17,18,22,23,24,25,28,29,30,31] and others. A classical result of Poincaré, Hadamard, and Sternberg (see Lemma 5.1 in page 234 of [16] and the Notes section in page 271 therein) gives conditions for the existence of a local smooth curve through the fixed point of a smooth map on the plane when the characteristic values are real and distinct and such that one of them is smaller than 1.…”

Basins of attraction of an anti-competitive discrete rational system

“…Competitive system of the form (2) were studied by many authors and there is an extensive literature, see [1,3,6,8,11,12,17,18,21,22].…”

Dynamical behavior of a system of three-dimensional nonlinear difference equations

“…Also, for similar results in the area of difference equations and systems, see [4,10,[16][17][18][19][20][21].…”