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

Akihiko Tobita, Akiko Fukuda, Emiko Ishiwata, Masashi Iwasaki, Yoshimasa Nakamura

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish
Title of host publicationEigenvalue Problems
Subtitle of host publicationAlgorithms, Software and Applications in Petascale Computing - EPASA 2015
PublisherSpringer Verlag
Pages157-169
Number of pages13
ISBN (Print)9783319624242
DOIs
Publication statusPublished - 2017 Jan 1
Event1st InternationalWorkshop on Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing, EPASA 2015 - Tsukuba, Japan
Duration: 2015 Sep 142015 Sep 16

Publication series

NameLecture Notes in Computational Science and Engineering
Volume117
ISSN (Print)1439-7358

Other

Other1st InternationalWorkshop on Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing, EPASA 2015
CountryJapan
CityTsukuba
Period15/9/1415/9/16

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ASJC Scopus subject areas

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

Cite this

Tobita, A., Fukuda, A., Ishiwata, E., Iwasaki, M., & Nakamura, Y. (2017). Monotonic convergence to eigenvalues of totally nonnegative matrices in an integrable variant of the discrete lotka-volterra system. In 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