### Abstract

We show the existence of a periodic solution in which four species coexist in competition for three essential resources in the standard model of resource competition. By assuming that species i is limited by resource i for each i near the positive equilibrium, and that the matrix of contents of resources in species is a combination of cyclic matrix and a symmetric matrix, we obtain an asymptotically stable periodic solution of three species on three resources via Hopf bifurcation. A simple bifurcation argument is then employed which allows us to add a fourth species. In principle, the argument can be continued to obtain a periodic solution adding one new species at a time so long as asymptotic stability can be assured at each step. Numerical simulations are provided to illustrate our analytical results. The results of this paper suggest that competition can generate coexistence of species in the form of periodic cycles, and that the number of coexisting species can exceed the number of resources in a constant and homogeneous environment.

Original language | English (US) |
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Pages (from-to) | 115-135 |

Number of pages | 21 |

Journal | Mathematical Biosciences |

Volume | 184 |

Issue number | 2 |

DOIs | |

State | Published - Aug 2003 |

### Fingerprint

### Keywords

- Bifurcation
- Coexistence
- Periodic orbit
- Resource competition

### ASJC Scopus subject areas

- Agricultural and Biological Sciences(all)
- Ecology, Evolution, Behavior and Systematics

### Cite this

**Periodic coexistence of four species competing for three essential resources.** / Li, Bingtuan; Smith, Hal.

Research output: Contribution to journal › Article

*Mathematical Biosciences*, vol. 184, no. 2, pp. 115-135. https://doi.org/10.1016/S0025-5564(03)00060-9

}

TY - JOUR

T1 - Periodic coexistence of four species competing for three essential resources

AU - Li, Bingtuan

AU - Smith, Hal

PY - 2003/8

Y1 - 2003/8

N2 - We show the existence of a periodic solution in which four species coexist in competition for three essential resources in the standard model of resource competition. By assuming that species i is limited by resource i for each i near the positive equilibrium, and that the matrix of contents of resources in species is a combination of cyclic matrix and a symmetric matrix, we obtain an asymptotically stable periodic solution of three species on three resources via Hopf bifurcation. A simple bifurcation argument is then employed which allows us to add a fourth species. In principle, the argument can be continued to obtain a periodic solution adding one new species at a time so long as asymptotic stability can be assured at each step. Numerical simulations are provided to illustrate our analytical results. The results of this paper suggest that competition can generate coexistence of species in the form of periodic cycles, and that the number of coexisting species can exceed the number of resources in a constant and homogeneous environment.

AB - We show the existence of a periodic solution in which four species coexist in competition for three essential resources in the standard model of resource competition. By assuming that species i is limited by resource i for each i near the positive equilibrium, and that the matrix of contents of resources in species is a combination of cyclic matrix and a symmetric matrix, we obtain an asymptotically stable periodic solution of three species on three resources via Hopf bifurcation. A simple bifurcation argument is then employed which allows us to add a fourth species. In principle, the argument can be continued to obtain a periodic solution adding one new species at a time so long as asymptotic stability can be assured at each step. Numerical simulations are provided to illustrate our analytical results. The results of this paper suggest that competition can generate coexistence of species in the form of periodic cycles, and that the number of coexisting species can exceed the number of resources in a constant and homogeneous environment.

KW - Bifurcation

KW - Coexistence

KW - Periodic orbit

KW - Resource competition

UR - http://www.scopus.com/inward/record.url?scp=0037696076&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037696076&partnerID=8YFLogxK

U2 - 10.1016/S0025-5564(03)00060-9

DO - 10.1016/S0025-5564(03)00060-9

M3 - Article

VL - 184

SP - 115

EP - 135

JO - Mathematical Biosciences

JF - Mathematical Biosciences

SN - 0025-5564

IS - 2

ER -