IJPAM: Volume 66, No. 3 (2011)
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: [email protected]
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