# IJPAM: Volume 66, No. 3 (2011)

**COMPACTIFICATION APPLIED TO**

A DISCRETE COMPETEING SYSTEM MODEL

A DISCRETE COMPETEING SYSTEM MODEL

Department of Mathematics

Eberly College of Arts and Sciences

West Virginia University

320 Armstrong Hall, P.O. Box 6310, Morgantown, WV 26506-6310, USA

e-mail: gingold@math.wvu.edu

**Abstract.**A study of sequence solutions of a discrete competing species model
of difference equations
, with large population
values , is carried out with the aid of the unconventional
compactification,
. Its utilization
makes it possible to define fixed points at infinity, ``'', and match them to finite fixed points on a certain boundary
sphere in a compacted space. It is shown that all fixed points ``'', of the discrete competing species model, do not lie in the first
quadrant of the ``extended'' plane and have at least one strictly
negative component. It is also shown that the basin of divergence
of almost all critical points ``'', of the discrete model
contain a one dimensional manifold. On the unit sphere of the compacted
system, a family of solutions that correspond to ideal solutions
,
, ,
are defined. Moreover, it is shown that in every such
ideal sequence, every
, for
some , has at least one negative component. A linearization
of a nonlinear system is carried out in the compacted space, about
a fixed point on the unit sphere, and its dependence on
is given. The large magnitudes of the populations of
species, could be impacted dramatically by the linear terms of the
model.

**Received: **November 27, 2010

**AMS Subject Classification: **92-08, 39A10

**Key Words and Phrases: **competing species, population, model, logistic equation, continuous model, discrete model, difference equations, difference systems, nonlinear, polynomial, compactification, fixed point, fixed point at infinity, asymptotic, stability, global, globally asymptotically stable, Jacobian

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2011

**Volume:** 66

**Issue:** 3