IJPAM: Volume 74, No. 2 (2012)

COMPACTIFICATION APPLIED TO
A DISCRETE COMPETEING SYSTEM MODEL

Harry Gingold
Department of Mathematics
West Virginia University
Morgantown, WV 26506, USA


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

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 74
Issue: 2