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

doi: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

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

Research output: Chapter in Book/Report/Conference proceeding › Conference 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 language	English
Title of host publication	Eigenvalue Problems
Subtitle of host publication	Algorithms, Software and Applications in Petascale Computing - EPASA 2015
Editors	Yoshinobu Kuramashi, Takeo Hoshi, Tetsuya Sakurai, Toshiyuki Imamura, Shao-Liang Zhang, Yusaku Yamamoto
Publisher	Springer Verlag
Pages	157-169
Number of pages	13
ISBN (Print)	9783319624242
DOIs	https://doi.org/10.1007/978-3-319-62426-6_11
Publication status	Published - 2017
Event	1st InternationalWorkshop on Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing, EPASA 2015 - Tsukuba, Japan Duration: 2015 Sept 14 → 2015 Sept 16

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	117
ISSN (Print)	1439-7358

Other

Other	1st InternationalWorkshop on Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing, EPASA 2015
Country/Territory	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

Access to Document

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

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 Y. Kuramashi, T. Hoshi, T. Sakurai, T. Imamura, S-L. Zhang, & Y. Yamamoto (Eds.), 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 et al.

Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing - EPASA 2015. ed. / Yoshinobu Kuramashi; Takeo Hoshi; Tetsuya Sakurai; Toshiyuki Imamura; Shao-Liang Zhang; Yusaku Yamamoto. Springer Verlag, 2017. p. 157-169 (Lecture Notes in Computational Science and Engineering; Vol. 117).

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

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 Y Kuramashi, T Hoshi, T Sakurai, T Imamura, S-L Zhang & Y Yamamoto (eds), 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

Tobita A, Fukuda A, Ishiwata E, Iwasaki M, Nakamura Y. Monotonic convergence to eigenvalues of totally nonnegative matrices in an integrable variant of the discrete lotka-volterra system. In Kuramashi Y, Hoshi T, Sakurai T, Imamura T, Zhang S-L, Yamamoto Y, editors, Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing - EPASA 2015. Springer Verlag. 2017. p. 157-169. (Lecture Notes in Computational Science and Engineering). doi: 10.1007/978-3-319-62426-6_11

Tobita, Akihiko ; Fukuda, Akiko ; Ishiwata, Emiko et al. / Monotonic convergence to eigenvalues of totally nonnegative matrices in an integrable variant of the discrete lotka-volterra system. Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing - EPASA 2015. editor / Yoshinobu Kuramashi ; Takeo Hoshi ; Tetsuya Sakurai ; Toshiyuki Imamura ; Shao-Liang Zhang ; Yusaku Yamamoto. Springer Verlag, 2017. pp. 157-169 (Lecture Notes in Computational Science and Engineering).

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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.",

author = "Akihiko Tobita and Akiko Fukuda and Emiko Ishiwata and Masashi Iwasaki and Yoshimasa Nakamura",

note = "Funding Information: Acknowledgements This work is supported by the Grants-in-Aid for Scientific Research (C) Nos. 23540163 and 26400208 and Encouragement of Young Scientists (B) No. 16K21368 from the Japan Society for the Promotion of Science. Publisher Copyright: {\textcopyright} Springer International Publishing AG 2017.; 1st InternationalWorkshop on Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing, EPASA 2015 ; Conference date: 14-09-2015 Through 16-09-2015",

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N1 - Funding Information: Acknowledgements This work is supported by the Grants-in-Aid for Scientific Research (C) Nos. 23540163 and 26400208 and Encouragement of Young Scientists (B) No. 16K21368 from the Japan Society for the Promotion of Science. Publisher Copyright: © Springer International Publishing AG 2017.

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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.

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

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