## JMI2012A-2 On some properties of a discrete hungry Lotka-Volterra system of multiplicative type (pp.5-15)

Author(s)： Yosuke Hama, Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata and Yoshimasa Nakamura

J. Math-for-Ind. 4A (2012) 5-15.

Abstract
Two kinds of discrete hungry Lotka-Volterra systems (dhLV) are known as discretizations of the additive type hungry Lotka-Volterra system and the multiplicative one. By associating the dhLV of additive type (dhLV$_\mathrm{I}$) and the discrete hungry Toda equation (dhToda) with $LR$ transformations, some of the authors give a Bäcklund transformation between these two systems. In this paper, from the dhLV of multiplicative type (dhLV$_{\mathrm{II}}$), we first derive the qd-type dhLV$_{\mathrm{II}}$. Through finding the positivity of the qd-type dhLV$_{\mathrm{II}}$ and the $LR$ transformation associated with the dhLV$_{\mathrm{II}}$, we present Bäcklund transformations among the dhLV$_{\mathrm{I}}$, the dhLV$_{\mathrm{II}}$ and the dhToda. Moreover, by using one of the Bäcklund transformations, we show asymptotic convergence of the qd-type dhLV$_{\mathrm{II}}$.

Keyword(s).　 Bäcklund transformation, $LR$ transformation, asymptotic convergence, discrete hungry Toda equation, discrete hungry Lotka-Volterra system