Dynamics of a two-dimensional system of rational difference equations of Leslie--Gower type

S Kalabušić,Mrs Kulenović,E Pilav

doi:10.1186/1687-1847-2011-29

S Kalabušić, Mrs Kulenović + Show 1 more

Open Access

https://doi.org/10.1186/1687-1847-2011-29

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Abstract

Abstract We investigate global dynamics of the following systems of difference equations x n + 1 = α 1 + β 1 x n A 1 + y n y n + 1 = γ 2 y n A 2 + B 2 x n + y n , n = 0 , 1 , 2 , … where the parameters α 1, β 1, A 1, γ 2, A 2, B 2 are positive numbers, and the initial conditions x 0 and y 0 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

Highlights

We study the global dynamics of the following rational system of difference equations
We provide a precise description of the global dynamics of the System (1)
We show that System (1) may have between zero and three equilibrium points, which may have different local character

Summary

Introduction

We study the global dynamics of the following rational system of difference equations (1). The following theorems were proved by Kulenović and Merino [2] for competitive systems in the plane, when one of the eigenvalues of the linearized system at an equilibrium (hyperbolic or non-hyperbolic) is by absolute value smaller than 1 while the other has an arbitrary value These results are useful for determining basins of attraction of fixed points of competitive maps. The following result gives information on local dynamics near a fixed point of a map when there exists a characteristic vector whose coordinates have negative product and such that the associated eigenvalue is hyperbolic. This is a well-known result, valid in much more general setting that we include it here for completeness.

Period-two solutions

Global behavior