### Abstract

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 language | English (US) |
---|---|

Pages (from-to) | 745-760 |

Number of pages | 16 |

Journal | Journal of Mathematical Biology |

Volume | 52 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2006 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Agricultural and Biological Sciences (miscellaneous)
- Mathematics (miscellaneous)

### Cite this

**Multiple limit cycles in the standard model of three species competition for three essential resources.** / Baer, Steven; Li, Bingtuan; Smith, Hal.

Research output: Contribution to journal › Article

*Journal of Mathematical Biology*, vol. 52, no. 6, pp. 745-760. https://doi.org/10.1007/s00285-005-0367-x

}

TY - JOUR

T1 - Multiple limit cycles in the standard model of three species competition for three essential resources

AU - Baer, Steven

AU - Li, Bingtuan

AU - Smith, Hal

PY - 2006/6

Y1 - 2006/6

N2 - 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.

AB - 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.

KW - Carrying simplex

KW - Coexistence

KW - Heteroclinic cycle

KW - Hopf bifurcation

KW - Limit cycle

KW - Persistence

KW - Resource competition

UR - http://www.scopus.com/inward/record.url?scp=33646676284&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33646676284&partnerID=8YFLogxK

U2 - 10.1007/s00285-005-0367-x

DO - 10.1007/s00285-005-0367-x

M3 - Article

C2 - 16463185

AN - SCOPUS:33646676284

VL - 52

SP - 745

EP - 760

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 6

ER -