American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Error estimates for a nonlinear local projection stabilization of transient convection--diffusion--reaction equations
  • DCDS-S Home
  • This Issue
  • Next Article
    On spiral solutions to generalized crystalline motion with a rotating tip motion
October  2015, 8(5): 889-899. doi: 10.3934/dcdss.2015.8.889

Conserved quantities of the integrable discrete hungry systems

Sonomi Kakizaki 1, , Akiko Fukuda 2, , Yusaku Yamamoto 3, , Masashi Iwasaki 4, , Emiko Ishiwata 5, and  Yoshimasa Nakamura 6,

1. 

Department of Mathematical Science for Information Sciences, Graduate School of Science, Tokyo University of Science, Tokyo 162-8601, Japan
2. 

Department of Mathematical Sciences, Shibaura Institute of Technology, Saitama 337-8570, Japan
3. 

Department of Communication Engineering and Informatics, The University of Electro-Communications, Tokyo 182-8585, Japan/JST CREST, Tokyo, Japan
4. 

Faculty of Life and Environmental Sciences, Kyoto Prefectural University, Kyoto 606-8522, Japan
5. 

Department of Mathematical Information Science, Tokyo University of Science, Tokyo 162-8601, Japan
6. 

Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan

Received  December 2013 Revised  December 2013 Published  July 2015

In this paper, conserved quantities of the discrete hungry Lotka-Volterra (dhLV) system are derived. Our approach is based on the Lax representation of the dhLV system, which expresses the time evolution of the dhLV system as a similarity transformation on a certain square matrix. Thus, coefficients of the characteristic polynomial of this matrix constitute conserved quantities of the dhLV system. These coefficients are calculated explicitly through a recurrence relation among the characteristic polynomials of its leading principal submatrices. The conserved quantities of the discrete hungry Toda (dhToda) equation is also derived with the help of the Bäcklund transformation between the dhLV system and the dhToda equation.
Keywords: leading principal submatrix., discrete hungry Lotka-Volterra system, Conserved quantities, discrete hungry Toda equation, characteristic polynomial.
Mathematics Subject Classification: Primary: 37K10; Secondary: 39A9.
Citation: Sonomi Kakizaki, Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata, Yoshimasa Nakamura. Conserved quantities of the integrable discrete hungry systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 889-899. doi: 10.3934/dcdss.2015.8.889
References:
[1]

LAPACK:, http://www.netlib.org/lapack/, ., ().   Google Scholar

[2]

A. Fukuda, E. Ishiwata, M. Iwasaki and Y. Nakamura, The discrete hungry Lotka-Volterra system and a new algorithm for computing matrix eigenvalues,, Inverse Probl., 25 (2009).  doi: 10.1088/0266-5611/25/1/015007.  Google Scholar

[3]

A. Fukuda, E. Ishiwata, Y. Yamamoto, M. Iwasaki and Y. Nakamura, Integrable discrete hungry systems and their related matrix eigenvalues,, Annal. Mat. Pura Appl., 192 (2013), 423.  doi: 10.1007/s10231-011-0231-0.  Google Scholar

[4]

A. Fukuda, Y. Yamamoto, M. Iwasaki, E. Ishiwata and Y. Nakamura, A Bäcklund transformation between two integrable discrete hungry systems,, Phys. Lett. A, 375 (2011), 303.  doi: 10.1016/j.physleta.2010.11.029.  Google Scholar

[5]

R. Hirota, S. Tsujimoto and T. Imai, Difference scheme of soliton equations,, Sūrikaisekikenkyūsho Kōkyūroku, 822 (1993), 144.   Google Scholar

[6]

M. Iwasaki and Y. Nakamura, On the convergence of a solution of the discrete Lotka-Volterra system,, Inverse Probl., 18 (2002), 1569.  doi: 10.1088/0266-5611/18/6/309.  Google Scholar

[7]

M. Iwasaki and Y. Nakamura, Accurate computation of singular values in terms of shifted integrable schemes,, Jpn. J. Indust. Appl. Math., 23 (2006), 239.  doi: 10.1007/BF03167593.  Google Scholar

[8]

T. Tokihiro, A. Nagai and J. Satsuma, Proof of solitonial nature of box and ball systems by means of inverse ultra-discretization,, Inverse Problems, 15 (1999), 1639.  doi: 10.1088/0266-5611/15/6/314.  Google Scholar

show all references

References:
[1]

LAPACK:, http://www.netlib.org/lapack/, ., ().   Google Scholar

[2]

A. Fukuda, E. Ishiwata, M. Iwasaki and Y. Nakamura, The discrete hungry Lotka-Volterra system and a new algorithm for computing matrix eigenvalues,, Inverse Probl., 25 (2009).  doi: 10.1088/0266-5611/25/1/015007.  Google Scholar

[3]

A. Fukuda, E. Ishiwata, Y. Yamamoto, M. Iwasaki and Y. Nakamura, Integrable discrete hungry systems and their related matrix eigenvalues,, Annal. Mat. Pura Appl., 192 (2013), 423.  doi: 10.1007/s10231-011-0231-0.  Google Scholar

[4]

A. Fukuda, Y. Yamamoto, M. Iwasaki, E. Ishiwata and Y. Nakamura, A Bäcklund transformation between two integrable discrete hungry systems,, Phys. Lett. A, 375 (2011), 303.  doi: 10.1016/j.physleta.2010.11.029.  Google Scholar

[5]

R. Hirota, S. Tsujimoto and T. Imai, Difference scheme of soliton equations,, Sūrikaisekikenkyūsho Kōkyūroku, 822 (1993), 144.   Google Scholar

[6]

M. Iwasaki and Y. Nakamura, On the convergence of a solution of the discrete Lotka-Volterra system,, Inverse Probl., 18 (2002), 1569.  doi: 10.1088/0266-5611/18/6/309.  Google Scholar

[7]

M. Iwasaki and Y. Nakamura, Accurate computation of singular values in terms of shifted integrable schemes,, Jpn. J. Indust. Appl. Math., 23 (2006), 239.  doi: 10.1007/BF03167593.  Google Scholar

[8]

T. Tokihiro, A. Nagai and J. Satsuma, Proof of solitonial nature of box and ball systems by means of inverse ultra-discretization,, Inverse Problems, 15 (1999), 1639.  doi: 10.1088/0266-5611/15/6/314.  Google Scholar

[1]

Xiaoli Liu, Dongmei Xiao. Bifurcations in a discrete time Lotka-Volterra predator-prey system. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 559-572. doi: 10.3934/dcdsb.2006.6.559
[2]

Lih-Ing W. Roeger, Razvan Gelca. Dynamically consistent discrete-time Lotka-Volterra competition models. Conference Publications, 2009, 2009 (Special) : 650-658. doi: 10.3934/proc.2009.2009.650
[3]

Rui Xu, M.A.J. Chaplain, F.A. Davidson. Periodic solutions of a discrete nonautonomous Lotka-Volterra predator-prey model with time delays. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 823-831. doi: 10.3934/dcdsb.2004.4.823
[4]

Lih-Ing W. Roeger. Dynamically consistent discrete Lotka-Volterra competition models derived from nonstandard finite-difference schemes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 415-429. doi: 10.3934/dcdsb.2008.9.415
[5]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[6]

Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953
[7]

Linping Peng, Zhaosheng Feng, Changjian Liu. Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4807-4826. doi: 10.3934/dcds.2014.34.4807
[8]

Fuke Wu, Yangzi Hu. Stochastic Lotka-Volterra system with unbounded distributed delay. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 275-288. doi: 10.3934/dcdsb.2010.14.275
[9]

Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083
[10]

Qi Wang, Chunyi Gai, Jingda Yan. Qualitative analysis of a Lotka-Volterra competition system with advection. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1239-1284. doi: 10.3934/dcds.2015.35.1239
[11]

Anthony W. Leung, Xiaojie Hou, Wei Feng. Traveling wave solutions for Lotka-Volterra system re-visited. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 171-196. doi: 10.3934/dcdsb.2011.15.171
[12]

Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195
[13]

Juan Luis García Guirao, Marek Lampart. Transitivity of a Lotka-Volterra map. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 75-82. doi: 10.3934/dcdsb.2008.9.75
[14]

Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072
[15]

Xiao He, Sining Zheng. Protection zone in a modified Lotka-Volterra model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2027-2038. doi: 10.3934/dcdsb.2015.20.2027
[16]

Yoshiaki Muroya. A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model). Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 999-1008. doi: 10.3934/dcdss.2015.8.999
[17]

Yuzo Hosono. Traveling waves for the Lotka-Volterra predator-prey system without diffusion of the predator. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 161-171. doi: 10.3934/dcdsb.2015.20.161
[18]

Hélène Leman, Sylvie Méléard, Sepideh Mirrahimi. Influence of a spatial structure on the long time behavior of a competitive Lotka-Volterra type system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 469-493. doi: 10.3934/dcdsb.2015.20.469
[19]

Yubin Liu, Peixuan Weng. Asymptotic spreading of a three dimensional Lotka-Volterra cooperative-competitive system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 505-518. doi: 10.3934/dcdsb.2015.20.505
[20]

Li-Jun Du, Wan-Tong Li, Jia-Bing Wang. Invasion entire solutions in a time periodic Lotka-Volterra competition system with diffusion. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1187-1213. doi: 10.3934/mbe.2017061

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences