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Volume 33, Issue 5
Integrable Discretisation of the Lotka-Volterra System

Yang He, Yajuan Sun & Zaijiu Shang

DOI: 10.4208/jcm.1504-m4543

J. Comp. Math., 33 (2015), pp. 468-494.

Published online: 2015-10

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  • Abstract

In this paper, we apply Hirota's discretisation to a three-dimensional integrable Lotka-Volterra system. By analyzing the three-dimensional modified equation of the resulting numerical method, we show that it is volume-preserving, and has two independent first integrals. Moreover, it can be formally reduced to a system in one dimension via a volume-preserving transformation. If the given initial value is located in the positive octant, we prove that the numerical solution is confined to a one-dimensional connected and compact space which is diffeomorphic to a circle.

  • Keywords

Integrable Lotka-Volterra system, Hirota's integrable discretisation, Backward error analysis, Modified differential equation.

  • AMS Subject Headings

65L12, 65P99, 37M05.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

heyang14@ustc.edu.cn (Yang He)

sunyj@lsec.cc.ac.cn (Yajuan Sun)

zaijiu@math.ac.cn (Zaijiu Shang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-33-468, author = {He , YangSun , Yajuan and Shang , Zaijiu}, title = {Integrable Discretisation of the Lotka-Volterra System}, journal = {Journal of Computational Mathematics}, year = {2015}, volume = {33}, number = {5}, pages = {468--494}, abstract = {

In this paper, we apply Hirota's discretisation to a three-dimensional integrable Lotka-Volterra system. By analyzing the three-dimensional modified equation of the resulting numerical method, we show that it is volume-preserving, and has two independent first integrals. Moreover, it can be formally reduced to a system in one dimension via a volume-preserving transformation. If the given initial value is located in the positive octant, we prove that the numerical solution is confined to a one-dimensional connected and compact space which is diffeomorphic to a circle.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1504-m4543}, url = {http://global-sci.org/intro/article_detail/jcm/9855.html} }
TY - JOUR T1 - Integrable Discretisation of the Lotka-Volterra System AU - He , Yang AU - Sun , Yajuan AU - Shang , Zaijiu JO - Journal of Computational Mathematics VL - 5 SP - 468 EP - 494 PY - 2015 DA - 2015/10 SN - 33 DO - http://doi.org/10.4208/jcm.1504-m4543 UR - https://global-sci.org/intro/article_detail/jcm/9855.html KW - Integrable Lotka-Volterra system, Hirota's integrable discretisation, Backward error analysis, Modified differential equation. AB -

In this paper, we apply Hirota's discretisation to a three-dimensional integrable Lotka-Volterra system. By analyzing the three-dimensional modified equation of the resulting numerical method, we show that it is volume-preserving, and has two independent first integrals. Moreover, it can be formally reduced to a system in one dimension via a volume-preserving transformation. If the given initial value is located in the positive octant, we prove that the numerical solution is confined to a one-dimensional connected and compact space which is diffeomorphic to a circle.

He , YangSun , Yajuan and Shang , Zaijiu. (2015). Integrable Discretisation of the Lotka-Volterra System. Journal of Computational Mathematics. 33 (5). 468-494. doi:10.4208/jcm.1504-m4543
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