An asymptotic analysis for an integrable variant of the Lotka–Volterra prey–predator model via a determinant expansion technique

Masato Shinjo,Yusaku Yamamoto,Masashi Iwasaki,Emiko Ishiwata,Yoshimasa Nakamura,Akiko Fukuda

doi:10.1080/23311835.2015.1046538

Abstract

The Hankel determinant appears in representations of solutions to several integrable systems. An asymptotic expansion of the Hankel determinant thus plays a key role in the investigation of asymptotic analysis of such integrable systems. This paper presents an asymptotic expansion formula of a certain Casorati determinant as an extension of the Hankel case. This Casorati determinant is then shown to be associated with the solution to the discrete hungry Lotka–Volterra (dhLV) system, which is an integrable variant of the famous prey–predator model in mathematical biology. Finally, the asymptotic behavior of the dhLV system is clarified using the expansion formula for the Casorati determinant.

Highlights

Integrable systems are often classified as nonlinear dynamical systems whose solutions can be explicitly expressed
The second purpose of this paper is to provide an asymptotic analysis for the discrete hungry Lotka–Volterra (dhLV) system without being limited by the sign of (n)
Rf ethfeerrCinagsotroatthi deettheeromreinmanotnCai(,nnj)ainlytteicrimtysfoorf the Hankel determinant given in Henrici (1988), we present an asymptotic expansion of the Casorati determinant Ci(,nj) as n → ∞ using the poles of fi(z)

Summary

Introduction

Integrable systems are often classified as nonlinear dynamical systems whose solutions can be explicitly expressed. A time discretization, called the discrete Toda equation (Hirota, 1981), is equal to the recursion formula of the qd algorithm for computing eigenvalues of a symmetric tridiagonal matrix (Henrici, 1988; Rutishauser, 1990) and singular values of a bidiagonal matrix (Parlett, 1995). Another commonly investigated integrable system is the integrable Lotka–Volterra (LV) system, which is a prey–predator model in mathematical biology (Yamazaki, 1987). The discrete LV (dLV) system was shown in Iwasaki and Nakamura (2002) to be applicable to computing for bidiagonal

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Proof Let

Equation allows us to simplify