### Abstract

We consider a semilinear reaction diffusion system with time periodic coefficients. The system models the competitive interaction of three species, which inhabit a bounded domain. We make assumptions that may be loosely stated (cyclically for i ? {1,2,3}) as: Species i outcompetes species i + 1 in the absence of species i + 2. Under those assumptions we prove the existence of a time periodic solution, which is strictly positive in each of its three components. Moreover, we obtain a new result on the nonexistence of positive time-periodic solutions and the extinction of one species for the related two-species subsystem. Our discussion includes a situation, known as cyclic competition in the autonomous ODE-case, in a more general frame-work that includes temporal and spatial heterogeneity. © Springer-Verlag 1996.

Original language | English |
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Pages (from-to) | 789-809 |

Number of pages | 21 |

Journal | Journal of Mathematical Biology |

Volume | 34 |

Issue number | 7 |

Publication status | Published - 1996 |

### Keywords

- Coexistence
- Cyclic competition -persistence
- Periodic parabolic problems

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## Cite this

*Journal of Mathematical Biology*,

*34*(7), 789-809.