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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

A global attractivity result for maps with invariant boxes

Pages: 97 - 110, Volume 6, Issue 1, January 2006      doi:10.3934/dcdsb.2006.6.97

 
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M. R. S. Kulenović - Department of Mathematics, University of Rhode Island, Kingston, Rhode Island 02881-0816, United States (email)
Orlando Merino - Department of Mathematics, University of Rhode Island, Kingston, Rhode Island 02881-0816, United States (email)

Abstract: We present a global attractivity result for maps generated by systems of autonomous difference equations. It is assumed that the map of the system leaves invariant a box, is monotone in a coordinate-wise sense (but not necessarily monotone with respect to a standard cone), and satisfies certain algebraic condition. It is shown that there exists a unique equilibrium, and that it is a global attractor. As an application, it is shown that a discretized version of the Lotka-Volterra system of differential equations of order $k$ has a global attractor in the positive orthant for certain range of parameters.

Keywords:  dynamical systems, global attractivity, Lotka-Volterra, monotonicity, stability.
Mathematics Subject Classification:  Primary: 37B25; Secondary: 39A11, 39A20.

Received: April 2005;      Revised: September 2005;      Available Online: October 2005.