A limit set trichotomy for monotone nonlinear dynamical systems

Krause, Ulrich; Ranft, Peter

doi:10.1016/0362-546x(92)90182-e

Cited by 40 publications

(40 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The boundedness of orbits are established and then strong persistence and weak persistence are investigated. By exploiting the main theorem in [4,8] and the stability trichotomy results on monotone and continuous operator in [12] we furthermore obtain the sufficient conditions for permanence of systems (1.2) and (1.4).…”

Section: Discussionmentioning

confidence: 99%

“…In Section 4, some definitions on persistence are given and the strong persistence and weak persistence of populations are investigated. In Section 5, sufficient conditions for permanence of systems (1.2) and (1.4) are obtained by exploiting the main theorem in [4,8] and the stability trichotomy results on monotone and continuous operator in [12]. Section 6 gives some applications of the main theorems and numerical results which show the dynamic behavior is very complex in the permanent region.…”

Section: X(t + 1) = X(t)g(x(t) + βY 2 (T))mentioning

confidence: 99%

See 1 more Smart Citation

A discrete predator-prey system with age-structure for predator and natural barriers for prey

Tang

Chen

2001

ESAIM: M2AN

View full text Add to dashboard Cite

Abstract.We analyze a two species discrete predator-prey model in which the prey disperses between two patches of a heterogeneous environment with barriers and the mature predator disperses between the patches with no barrier. By using the discrete dynamical system generated by a monotone, concave maps for subcommunity of prey, we obtain the subcommunity of prey exists an equilibrium which attracts all positive solutions, and using the stability trichotomy results on the monotone and continuous operator, we obtain some sufficient conditions for the permanence of species. These results are applied to the models with rational growth functions and exponential growth functions. We also present numerical examples to illustrate the dynamic complexity of systems.Mathematics Subject Classification. 34C35, 39A10.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: X(t + 1) = X(t)g(x(t) + βY 2 (T))mentioning

confidence: 99%

A discrete predator-prey system with age-structure for predator and natural barriers for prey

Tang

Chen

2001

ESAIM: M2AN

View full text Add to dashboard Cite

show abstract

“…Let K be a polyhedral cone with N facets and let ψ i : X → R, with 1 ≤ i ≤ N , be the linear functionals that define the facets of K. We define for each x ∈ K a set I x by (19) I…”

Section: Theorem 41 ([20])mentioning

confidence: 99%

“…Order preserving subhomogeneous maps have been studied intensively in nonlinear Perron-Frobenius theory. They arise in various fields, such as optimal control and game theory [1,24,29], idempotent analysis [17,23], the analysis of monotone dynamical systems [15,16,18,19,32,33], and discrete event systems [4,12,13]. In this list we have quoted only a few recent works and we suggest the reader to consult [25,26] for further references.…”

Section: Introductionmentioning

confidence: 99%

Iteration of order preserving subhomogeneous maps on a cone

Akian

Gaubert

Lemmens

et al. 2006

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

Abstract. We investigate the iterative behaviour of continuous order preserving subhomogeneous maps f : K → K, where K is a polyhedral cone in a finite dimensional vector space. We show that each bounded orbit of f converges to a periodic orbit and, moreover, the period of each periodic point of f is bounded bywhere N is the number of facets of the polyhedral cone. By constructing examples on the standard positive cone in R n , we show that the upper bound is asymptotically sharp.These results are an extension of work by Lemmens and Scheutzow concerning periodic orbits in the interior of the standard positive cone in R n .

show abstract

“…Maps that are order preserving and subhomogeneous arise in various areas of mathematics, including the theory of discrete event systems [3,5], optimal control and game theory [1], idempotent analysis [8], nonlinear Perron-Frobenius theory [4,12], and in the analysis of monotone dynamical systems [6,7,9,11,12,13]. A particularly interesting class of continuous, order preserving, homogeneous maps on the standard positive cone are the so-called min-max maps.…”

Section: Face Is the Dimension Of Its Linear Span And Is Denoted By Dmentioning

confidence: 99%

A note on periodic points of order preserving subhomogeneous maps

Lemmens

Sparrow

2005

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Let R n + be the standard closed positive cone in R n and let Γ(R n + ) be the set of integers p ≥ 1 for which there exists a continuous, order preserving, subhomogeneous map f : R n + → R n + , which has a periodic point with period p. It has been shown by Akian, Gaubert, Lemmens, and Nussbaum that Γ(R n + ) is contained in the set B(n) consisting of those p ≥ 1 for which there exist integers q 1 and q 2 such that p = q 1 q 2 , 1 ≤ q 1 ≤ n m , and 1 ≤ q 2 ≤ m m/2 for some 1 ≤ m ≤ n. This note shows that Γ(R n + ) = B(n) for all n ≥ 1.

show abstract

A limit set trichotomy for monotone nonlinear dynamical systems

Cited by 40 publications

References 10 publications

A discrete predator-prey system with age-structure for predator and natural barriers for prey

A discrete predator-prey system with age-structure for predator and natural barriers for prey

Iteration of order preserving subhomogeneous maps on a cone

A note on periodic points of order preserving subhomogeneous maps

Contact Info

Product

Resources

About