### Abstract

We study the global-in-time behavior of the Lotka- Volterra system with diffusion. In the first category, the interaction matrix is skew-symmetric and the linear terms are non-increasing. There, the solution exists globally in time with compact orbit, provided that n ≥ 2, where n denotes the space dimension. Under the presence of entropy, its Ö-limit set is composed of a spatially homogeneous orbit. Furthermore, any spatially homogeneous solution is periodic in time, provided with constant entropy. In the second category, the interaction matrix exhibits a dissipative profile. There, the solution exists globally in time with compact orbit if n ≥ 3. Its ω-limit set, furthermore, is contained in spatially homogeneous stationary states. In particular, no periodic-in-time solution is admitted.

Original language | English |
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Pages (from-to) | 181-216 |

Number of pages | 36 |

Journal | Indiana University Mathematics Journal |

Volume | 64 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 |

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### Keywords

- Blowup analysis
- Lotka-Volterra system
- Periodic-in-time solution
- Thermodynamic
- Ω-limit set

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Indiana University Mathematics Journal*,

*64*(1), 181-216. https://doi.org/10.1512/iumj.2015.64.5460