### Abstract

The classical Lotka-Volterra equations for two competing species have constant coefficients. In this paper these equations are studied under the assumption that the coefficients are periodic functions of a common period. As a generalization of the existence theory for equilibria in the constant coefficient case, it is shown that there exists a branch of positive periodic solutions which connects (i.e. bifurcates from) the two nontrivial periodic solutions lying on the coordinate axes. This branch exists for a finite interval or "spectrum" of bifurcation parameter values (the bifurcation parameter being the average of the net inherent growth rate of one species). The stability of these periodic solutions is studied and is related to the theory of competitive exclusion. A specific example of independent ecological interest is examined by means of which it is shown under what circumstances two species, which could not coexist in a constant environment, can coexist in a limit cycle fashion when subjected to suitable periodic harvesting or removal rates.

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

Pages (from-to) | 385-400 |

Number of pages | 16 |

Journal | Journal of Mathematical Biology |

Volume | 10 |

Issue number | 4 |

DOIs | |

State | Published - 1980 |

### Fingerprint

### Keywords

- Bifurcation
- Competition
- Competitive exclusion
- Periodic environment

### ASJC Scopus subject areas

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

### Cite this

**Two species competition in a periodic environment.** / Cushing, Jim M.

Research output: Contribution to journal › Article

*Journal of Mathematical Biology*, vol. 10, no. 4, pp. 385-400. https://doi.org/10.1007/BF00276097

}

TY - JOUR

T1 - Two species competition in a periodic environment

AU - Cushing, Jim M

PY - 1980

Y1 - 1980

N2 - The classical Lotka-Volterra equations for two competing species have constant coefficients. In this paper these equations are studied under the assumption that the coefficients are periodic functions of a common period. As a generalization of the existence theory for equilibria in the constant coefficient case, it is shown that there exists a branch of positive periodic solutions which connects (i.e. bifurcates from) the two nontrivial periodic solutions lying on the coordinate axes. This branch exists for a finite interval or "spectrum" of bifurcation parameter values (the bifurcation parameter being the average of the net inherent growth rate of one species). The stability of these periodic solutions is studied and is related to the theory of competitive exclusion. A specific example of independent ecological interest is examined by means of which it is shown under what circumstances two species, which could not coexist in a constant environment, can coexist in a limit cycle fashion when subjected to suitable periodic harvesting or removal rates.

AB - The classical Lotka-Volterra equations for two competing species have constant coefficients. In this paper these equations are studied under the assumption that the coefficients are periodic functions of a common period. As a generalization of the existence theory for equilibria in the constant coefficient case, it is shown that there exists a branch of positive periodic solutions which connects (i.e. bifurcates from) the two nontrivial periodic solutions lying on the coordinate axes. This branch exists for a finite interval or "spectrum" of bifurcation parameter values (the bifurcation parameter being the average of the net inherent growth rate of one species). The stability of these periodic solutions is studied and is related to the theory of competitive exclusion. A specific example of independent ecological interest is examined by means of which it is shown under what circumstances two species, which could not coexist in a constant environment, can coexist in a limit cycle fashion when subjected to suitable periodic harvesting or removal rates.

KW - Bifurcation

KW - Competition

KW - Competitive exclusion

KW - Periodic environment

UR - http://www.scopus.com/inward/record.url?scp=0019202916&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0019202916&partnerID=8YFLogxK

U2 - 10.1007/BF00276097

DO - 10.1007/BF00276097

M3 - Article

AN - SCOPUS:0019202916

VL - 10

SP - 385

EP - 400

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 4

ER -