DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

ON THE GLOBAL ATTRACTOR IN A CLASS
OF DYNAMIC SYSTEMS
R. RAUTMANN
We consider a class of dynamic systems (∗) (d/dt)x = f (x) with a continuous function
f : R n + → R n defined in the positive cone R n + of the Euclidean space R n , n ≥ 2. In the stable
case, from an observation concerning flow-invariant n-dimensional rectangles Q and
contractivity of a flow in Q, we find that the unique stationary point E of (∗)isglobalattractor
in R n +. In the unstable case for more specialized systems, we get explicit conditions
for blowing up and dying out by constructing lower and upper bounds for the solutions.
By the well-known comparison methods, these results apply to solutions of weakly coupled
quasimonotone parabolic systems with Dirichlet or Neumann boundary conditions.
In addition, the systems (∗) in question could be used as models of cooperative societies
in order to indicate future perspectives for such communities.
Copyright © 2006 R. Rautmann. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
1.1. Problems and results. In this contribution, we will consider dynamic systems,
ẋ = f (x),
(ẋ = d )
dt x , (1.1)
intheopenpositiveconeR n + ={x = (x i) ∈ R n | 0