Multiple Attractors for a Competitive System of Rational Difference Equations in the Plane
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,...
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Series: | Abstract and Applied Analysis |
Online Access: | http://dx.doi.org/10.1155/2011/295308 |
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doaj-11e1a3f9e9bf489db6ae19d8fe3398c42020-11-24T21:41:54ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092011-01-01201110.1155/2011/295308295308Multiple Attractors for a Competitive System of Rational Difference Equations in the PlaneS. Kalabušić0M. R. S. Kulenović1E. Pilav2Department of Mathematics, University of Sarajevo, 71000 Sarajevo, Bosnia and HerzegovinaDepartment of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USADepartment of Mathematics, University of Sarajevo, 71000 Sarajevo, Bosnia and HerzegovinaWe 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.http://dx.doi.org/10.1155/2011/295308 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
S. Kalabušić M. R. S. Kulenović E. Pilav |
spellingShingle |
S. Kalabušić M. R. S. Kulenović E. Pilav Multiple Attractors for a Competitive System of Rational Difference Equations in the Plane Abstract and Applied Analysis |
author_facet |
S. Kalabušić M. R. S. Kulenović E. Pilav |
author_sort |
S. Kalabušić |
title |
Multiple Attractors for a Competitive System of Rational Difference Equations in the Plane |
title_short |
Multiple Attractors for a Competitive System of Rational Difference Equations in the Plane |
title_full |
Multiple Attractors for a Competitive System of Rational Difference Equations in the Plane |
title_fullStr |
Multiple Attractors for a Competitive System of Rational Difference Equations in the Plane |
title_full_unstemmed |
Multiple Attractors for a Competitive System of Rational Difference Equations in the Plane |
title_sort |
multiple attractors for a competitive system of rational difference equations in the plane |
publisher |
Hindawi Limited |
series |
Abstract and Applied Analysis |
issn |
1085-3375 1687-0409 |
publishDate |
2011-01-01 |
description |
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. |
url |
http://dx.doi.org/10.1155/2011/295308 |
work_keys_str_mv |
AT skalabusic multipleattractorsforacompetitivesystemofrationaldifferenceequationsintheplane AT mrskulenovic multipleattractorsforacompetitivesystemofrationaldifferenceequationsintheplane AT epilav multipleattractorsforacompetitivesystemofrationaldifferenceequationsintheplane |
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1725920061722460160 |