Positivity 5: 95–114, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
A Limit Set Trichotomy for Order-Preserving
and IGOR CHUESHOV
Institut für Dynamische Systeme, Fachbereich 3, Universität, Postfach 33 04 40, 28334 Bremen,
Department of Mechanics and Mathematics,
Kharkov University, 4 Svobody Sq. 310077 Kharkov, Ukraine. E-mail:firstname.lastname@example.org
(Received 16 November 1998; accepted 25 August 1999)
Abstract. We study the asymptotic behavior of order-preserving (or monotone) random systems
which have an additional concavity property called sublinearity (or subhomogeneity), frequently
encountered in applications. Sublinear random systems are contractive with respect to the part metric,
hence random equilibria are unique and asymptotically stable in each part of the cone. Our main
result is a random limit set trichotomy, stating that in a given part either (i) all orbits are unbounded,
or (ii) all orbits are bounded but their closure reaches out to the boundary of the part, or (iii) there
exists a unique, globally attracting equilibrium. Several examples, including afﬁne and cooperative
systems, are given.
Mathematics Subject Classiﬁcations (2000): primary 34F05, 37H05; secondary 47B80, 60H25,
Key words: afﬁne, cooperative, equilibrium, limit set trichotomy, long-term behavior, monotone
random dynamical system, order-preserving, random attractor, sublinear, sub- and super-equilibrium
The subject of this paper is the investigation of the long-term behavior of order-
preserving (or monotone) random dynamical systems which possess a certain kind
of concavity (which we call sublinearity) arising naturally in numerous mathemat-
ical models in ecology, epidemiology, economics and biochemistry.
Deterministic versions of these systems were studied by many authors. The
ground for their qualitative theory was laid by Krasnoselskii in his two books [11,
12]. Hirsch [8–10], Smith [16, 17], Taka
c , Krause et al. [13, 14] and many
others made further important contributions.
In  we introduced the general concept of an order-preserving random dynam-
ical system, gave numerous examples and studied the properties of their random
equilibria and attractors. This program is continued here with the investigation of
sublinear random systems.
Our proofs rely crucially on the concept of sub- and super-equilibria which for
the deterministic case is well-known , and for the random case was introduced