### 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

**Global-in-time behavior of lotka-volterra system with diffusion : Skew-symmetric case.** / Suzuki, Takashi; Yamada, Yoshio.

Research output: Contribution to journal › Article

*Indiana University Mathematics Journal*, vol. 64, no. 1, pp. 181-216. https://doi.org/10.1512/iumj.2015.64.5460

}

TY - JOUR

T1 - Global-in-time behavior of lotka-volterra system with diffusion

T2 - Skew-symmetric case

AU - Suzuki, Takashi

AU - Yamada, Yoshio

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

KW - Blowup analysis

KW - Lotka-Volterra system

KW - Periodic-in-time solution

KW - Thermodynamic

KW - Ω-limit set

UR - http://www.scopus.com/inward/record.url?scp=84923762489&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84923762489&partnerID=8YFLogxK

U2 - 10.1512/iumj.2015.64.5460

DO - 10.1512/iumj.2015.64.5460

M3 - Article

AN - SCOPUS:84923762489

VL - 64

SP - 181

EP - 216

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 1

ER -