Multiple limit cycles in the standard model of three species competition for three essential resources

Steven Baer, Bingtuan Li, Hal Smith

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


We consider the dynamics of the standard model of 3 species competing for 3 essential (non-substitutable) resources in a chemostat using Liebig's law of the minimum functional response. A subset of these systems which possess cyclic symmetry such that its three single-population equilibria are part of a heteroclinic cycle bounding the two-dimensional carrying simplex is examined. We show that a subcritical Hopf bifurcation from the coexistence equilibrium together with a repelling heteroclinic cycle leads to the existence of at least two limit cycles enclosing the coexistence equilibrium on the carrying simplex- the "inside" one is an unstable separatrix and the "outside" one is at least semi-stable relative to the carrying simplex. Numerical simulations suggest that there are exactly two limit cycles and that almost every positive solution approaches either the stable limit cycle or the stable coexistence equilibrium, depending on initial conditions. Bifurcation diagrams confirm this picture and show additional features. In an alternative scenario, we show that the subcritical Hopf together with an attracting heteroclinic cycle leads to an unstable periodic orbit separatrix.

Original languageEnglish (US)
Pages (from-to)745-760
Number of pages16
JournalJournal Of Mathematical Biology
Issue number6
StatePublished - Jun 1 2006


  • Carrying simplex
  • Coexistence
  • Heteroclinic cycle
  • Hopf bifurcation
  • Limit cycle
  • Persistence
  • Resource competition

ASJC Scopus subject areas

  • Modeling and Simulation
  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics


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