### Abstract

The classical Volterra-Lotka system of differential equations modeling the interaction of two species is considered under the assumption that the interaction and/or density inhibition terms may possess time lags (of a very general nature). Under the assumption that there exists a positive equilibrium, sufficient conditions are given which guarantee the existence of nonconstant, positive periodic solution. These conditions are algebraic in nature and are easily and straightforwardly applied. A few specific examples (namely, some predator-prey and competing species models) are discussed and are used to illustrate how lags are able to alter significantly the general dynamics of the classical equations. The mathematical approach is to construct periodic solutions which bifurcate from the equilibrium (using the net birth rates as free parameters) by making use of Liapunov-Schmidt type expansions, an approach which requires the proof of a Fredholm-type alternative for Stieltjes integrodifferential systems.

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

Pages (from-to) | 143-156 |

Number of pages | 14 |

Journal | Mathematical Biosciences |

Volume | 31 |

Issue number | 1-2 |

DOIs | |

State | Published - 1976 |

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### ASJC Scopus subject areas

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

### Cite this

**Periodic solutions of two species interaction models with lags.** / Cushing, Jim M.

Research output: Contribution to journal › Article

*Mathematical Biosciences*, vol. 31, no. 1-2, pp. 143-156. https://doi.org/10.1016/0025-5564(76)90046-8

}

TY - JOUR

T1 - Periodic solutions of two species interaction models with lags

AU - Cushing, Jim M

PY - 1976

Y1 - 1976

N2 - The classical Volterra-Lotka system of differential equations modeling the interaction of two species is considered under the assumption that the interaction and/or density inhibition terms may possess time lags (of a very general nature). Under the assumption that there exists a positive equilibrium, sufficient conditions are given which guarantee the existence of nonconstant, positive periodic solution. These conditions are algebraic in nature and are easily and straightforwardly applied. A few specific examples (namely, some predator-prey and competing species models) are discussed and are used to illustrate how lags are able to alter significantly the general dynamics of the classical equations. The mathematical approach is to construct periodic solutions which bifurcate from the equilibrium (using the net birth rates as free parameters) by making use of Liapunov-Schmidt type expansions, an approach which requires the proof of a Fredholm-type alternative for Stieltjes integrodifferential systems.

AB - The classical Volterra-Lotka system of differential equations modeling the interaction of two species is considered under the assumption that the interaction and/or density inhibition terms may possess time lags (of a very general nature). Under the assumption that there exists a positive equilibrium, sufficient conditions are given which guarantee the existence of nonconstant, positive periodic solution. These conditions are algebraic in nature and are easily and straightforwardly applied. A few specific examples (namely, some predator-prey and competing species models) are discussed and are used to illustrate how lags are able to alter significantly the general dynamics of the classical equations. The mathematical approach is to construct periodic solutions which bifurcate from the equilibrium (using the net birth rates as free parameters) by making use of Liapunov-Schmidt type expansions, an approach which requires the proof of a Fredholm-type alternative for Stieltjes integrodifferential systems.

UR - http://www.scopus.com/inward/record.url?scp=0017151799&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0017151799&partnerID=8YFLogxK

U2 - 10.1016/0025-5564(76)90046-8

DO - 10.1016/0025-5564(76)90046-8

M3 - Article

AN - SCOPUS:0017151799

VL - 31

SP - 143

EP - 156

JO - Mathematical Biosciences

JF - Mathematical Biosciences

SN - 0025-5564

IS - 1-2

ER -