### Abstract

The Lotka-Volterra (LV) system describes a simple predator-prey model in mathematical biology. The hungry Lotka-Volterra (hLV) system assumed that each predator preys on two or more species is an extension; those involving summations and products of nonlinear terms are referred to as summation-type and product-type hLV systems, respectively. Time-discretizations of these systems are considered in the study of integrable systems, and have been shown to be applicable to computing eigenvalues of totally nonnegative (TN) matrices. Monotonic convergence to eigenvalues of TN matrices, however, has not yet been observed in the time-discretization of the product-type hLV system. In this paper, we show the existence of a center manifold associated with the time-discretization of the product-type hLV system, and then clarify how the solutions approach an equilibrium corresponding to the eigenvalues of TN matrices in the final phase of convergence.

Original language | English |
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Title of host publication | Eigenvalue Problems |

Subtitle of host publication | Algorithms, Software and Applications in Petascale Computing - EPASA 2015 |

Publisher | Springer Verlag |

Pages | 157-169 |

Number of pages | 13 |

ISBN (Print) | 9783319624242 |

DOIs | |

Publication status | Published - 2017 Jan 1 |

Event | 1st InternationalWorkshop on Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing, EPASA 2015 - Tsukuba, Japan Duration: 2015 Sep 14 → 2015 Sep 16 |

### Publication series

Name | Lecture Notes in Computational Science and Engineering |
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Volume | 117 |

ISSN (Print) | 1439-7358 |

### Other

Other | 1st InternationalWorkshop on Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing, EPASA 2015 |
---|---|

Country | Japan |

City | Tsukuba |

Period | 15/9/14 → 15/9/16 |

### ASJC Scopus subject areas

- Modelling and Simulation
- Engineering(all)
- Discrete Mathematics and Combinatorics
- Control and Optimization
- Computational Mathematics

### Cite this

*Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing - EPASA 2015*(pp. 157-169). (Lecture Notes in Computational Science and Engineering; Vol. 117). Springer Verlag. https://doi.org/10.1007/978-3-319-62426-6_11

**Monotonic convergence to eigenvalues of totally nonnegative matrices in an integrable variant of the discrete lotka-volterra system.** / Tobita, Akihiko; Fukuda, Akiko; Ishiwata, Emiko; Iwasaki, Masashi; Nakamura, Yoshimasa.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing - EPASA 2015.*Lecture Notes in Computational Science and Engineering, vol. 117, Springer Verlag, pp. 157-169, 1st InternationalWorkshop on Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing, EPASA 2015, Tsukuba, Japan, 15/9/14. https://doi.org/10.1007/978-3-319-62426-6_11

}

TY - GEN

T1 - Monotonic convergence to eigenvalues of totally nonnegative matrices in an integrable variant of the discrete lotka-volterra system

AU - Tobita, Akihiko

AU - Fukuda, Akiko

AU - Ishiwata, Emiko

AU - Iwasaki, Masashi

AU - Nakamura, Yoshimasa

PY - 2017/1/1

Y1 - 2017/1/1

N2 - The Lotka-Volterra (LV) system describes a simple predator-prey model in mathematical biology. The hungry Lotka-Volterra (hLV) system assumed that each predator preys on two or more species is an extension; those involving summations and products of nonlinear terms are referred to as summation-type and product-type hLV systems, respectively. Time-discretizations of these systems are considered in the study of integrable systems, and have been shown to be applicable to computing eigenvalues of totally nonnegative (TN) matrices. Monotonic convergence to eigenvalues of TN matrices, however, has not yet been observed in the time-discretization of the product-type hLV system. In this paper, we show the existence of a center manifold associated with the time-discretization of the product-type hLV system, and then clarify how the solutions approach an equilibrium corresponding to the eigenvalues of TN matrices in the final phase of convergence.

AB - The Lotka-Volterra (LV) system describes a simple predator-prey model in mathematical biology. The hungry Lotka-Volterra (hLV) system assumed that each predator preys on two or more species is an extension; those involving summations and products of nonlinear terms are referred to as summation-type and product-type hLV systems, respectively. Time-discretizations of these systems are considered in the study of integrable systems, and have been shown to be applicable to computing eigenvalues of totally nonnegative (TN) matrices. Monotonic convergence to eigenvalues of TN matrices, however, has not yet been observed in the time-discretization of the product-type hLV system. In this paper, we show the existence of a center manifold associated with the time-discretization of the product-type hLV system, and then clarify how the solutions approach an equilibrium corresponding to the eigenvalues of TN matrices in the final phase of convergence.

UR - http://www.scopus.com/inward/record.url?scp=85041499847&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85041499847&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-62426-6_11

DO - 10.1007/978-3-319-62426-6_11

M3 - Conference contribution

AN - SCOPUS:85041499847

SN - 9783319624242

T3 - Lecture Notes in Computational Science and Engineering

SP - 157

EP - 169

BT - Eigenvalue Problems

PB - Springer Verlag

ER -