IJPAM: Volume 66, No. 3 (2011)

COMPACTIFICATION APPLIED TO
A DISCRETE COMPETEING SYSTEM MODEL

Harry Gingold
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 $y_{n+1}=f(y_{n})$, with large population values $\Vert y_{n}\Vert$, is carried out with the aid of the unconventional compactification, $y=\frac{x}{1-x^{\dagger}x}$. Its utilization makes it possible to define fixed points at infinity, ``$\infty p$'', and match them to finite fixed points $p$ on a certain boundary sphere in a compacted space. It is shown that all fixed points ``$\infty p$'', 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 ``$\infty p$'', 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 $y_{n}=\infty p_{n}$, $p_{n}^{\dagger}p_{n}=1$, $n=0,1,2,...$, are defined. Moreover, it is shown that in every such ideal sequence, every $p_{n},n=N+1,N+2,N+3,...$, for some $N>0$, has at least one negative component. A linearization of a nonlinear system is carried out in the compacted space, about a fixed point $p$ on the unit sphere, and its dependence on $f(y_{n})$ is given. The large magnitudes $\Vert y_{n}\Vert$ 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