Cycle chains and the LPA model

Research output: Contribution to journalArticle

18 Citations (Scopus)

Abstract

The LPA model is a three dimensional system of nonlinear difference equations that has found many applications in population dynamics and ecology. In this paper, we consider a special case of the model (a case that approximates that used in experimental bifurcation and chaos studies) and prove several theorems concerning the existence and stability of periodic cycles, invariant loops, and chaos. Key is the notion of synchronous orbits (i.e. orbits lying on the coordinate planes). The main result concerns the existence of an invariant loop of synchronous orbits that bifurcates, in a nongeneric way, from the trivial equilibrium. The geometry and dynamics of this invariant loop are characterized. Specifically, it is shown that the loop is a cycle chain consisting of synchronous heteroclinic orbits connecting the three temporal phases of a synchronous 3-cycle. We also show that a period doubling route to chaos occurs within the class of synchronous orbits.

Original languageEnglish (US)
Pages (from-to)655-670
Number of pages16
JournalJournal of Difference Equations and Applications
Volume9
Issue number7
DOIs
StatePublished - Jul 1 2003

Fingerprint

Orbits
Orbit
Cycle
Chaos theory
Invariant
Chaos
Cartesian plane
Bifurcation and Chaos
Heteroclinic Orbit
Nonlinear Difference Equations
Period Doubling
Ecology
Population Dynamics
Population dynamics
Model
Difference equations
Trivial
Three-dimensional
Theorem
Geometry

Keywords

  • Bifurcation
  • Difference equations
  • Equilibria
  • Invariant loop
  • Periodic cycles
  • Stability

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Applied Mathematics
  • Analysis

Cite this

Cycle chains and the LPA model. / Cushing, Jim M.

In: Journal of Difference Equations and Applications, Vol. 9, No. 7, 01.07.2003, p. 655-670.

Research output: Contribution to journalArticle

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