Time-delay version of the integrable discrete Lotka-Volterra system in terms of the LR transformations

To introduce time delay into integrable discrete systems, we present a time-delay version of the discrete Lotka-Volterra (dLV) system, which is a time-discretization of the famous predator-prey Lotka-Volterra system. Focusing on the LR transformations, which has been designed for solving symmetric eigenvalue problems, plays a key role in deriving the essential properties of the original dLV system. We also clarify asymptotic convergence in the resulting time-delay system and present an application for computing matrix eigenvalues and singular values.


Introduction
The Lotka-Volterra (LV) system is a well-known mathematical model of predator-prey interactions. A skillful discretization of the LV system describing the interaction of 2m−1 species leads to the integrable discrete LV (dLV) system with discretization parameter δ ( n) : where the subscript k and the superscript (n) respectively denote species index and discrete-time and the variable ( ) u k n corresponds to the population of the kth species at discrete-time n. In this paper, for simplicity, we focus on the case where δ ( n) =1 in the dLV system (1). Here, 'integrable' means that the solution can be explicitly expressed. Using the Hankel determinants:  the discrete relativistic Toda equation [4,5] and the discrete hungry Toda equation [6], which are extensions of the discrete Toda equation. Similarly, we can derive the determinantal solution to the discrete hungry Lotka-Volterra system [6], which is an extension of the dLV system (1). The discrete Korteweg-de Vries equation, which often appears in the study of integrable systems, is also shown to have determinantal solution [7]. The viewpoint of determinant structure is futhermore applied in finding the solutions to the discrete Painlevé equations [8,9]. With respect to the continuous LV system, various types of time delay have been considered, and the preservability and convergence of the solutions have been investigated [10][11][12]. However, to the best of our knowledge, the discrete analogues have been extensively clarified and time delays in discrete integrable systems have not yet been discussed. One of the authors proved [13][14][15] that, under the initial settings The key point in this asymptotic analysis is to relate the dLV system (1) to a sequence of the LR transformations for computing eigenvalues of tridiagonal matrices. In this paper, we introduce a time delay into the dLV system (1) in terms of the LR transformations. We also show asymptotic convergence of the delay dLV (ddLV) variables to matrix eigenvalues and singular values.
The remainder of this paper is organized as follows. In section 2, we briefly explain the LR transformations related to the dLV system (1) and their applications to computing matrix eigenvalues and singular values. In section 3, we derive a time-delay version of the dLV system (1) by considering that of the LR transformations related to the dLV system (1). We then clarify the matrix structure to which the delay LR transformation can be applied. In section 4, we show asymptotic convergence in the ddLV system, and present an example for numerical verification. Finally, we give concluding remarks in section 5.

LR transformations derived from the dLV system
In this section, we briefly review [13][14][15] an application of the dLV system (1) to computing eigenvalues and singular values of matrices.
Let us introduce new variables ( ) q k n and ( ) e k n given using the dLV variables ( ) u k n as: Since the dLV system (1) immediately leads to: we can easily derive a recursion formula with respect to discrete-time evolutions from ( ) q k n to ( ) + q k n 1 and from ( ) e k n to ( ) + e k n 1 :  (5) is simply the discrete Toda equation [16], which coincides with the recursion formula of the quotient-difference (qd) algorithm [17] for computing tridiagonal eigenvalues. Equation (4) is called the Miura transformation and can be regarded as the transformation from the dLV system (1) to the discrete Toda equation (5). Now, let us prepare lower and upper bidiagonal matrices with the discrete Toda variables ( ) q k n and ( ) e k n : Noting that the inverse ( ) , where  is the identity matrix, we obtain: which implies that the discrete Toda equation (5) generates a similarity transformation from ( )  n to ( ) q k n and ( ) e k n are given from the entries of ( )  n . The dLV variables ( ) u k n are also uniquely determined from the entries of ( )  n because the latter can be expressed in terms of the dLV variables ( ) u k n as:  as: 1 converge to eigenvalues of ( )  s 0 as  ¥ n . From the Cholesky decomposition: we further find that

Discrete-time delay in the dLV system
In this section, we introduce a time delay into the dLV system (1) by considering the related LR transformations. Let us introduce ( )  + + + t+ m m m 1 2 1 -by-(m 1 +m 2 +L+m τ+1 ) block diagonal matrices:  We note that (9) with τ=0 is equivalent to the discrete Toda equation (5). Equation (9) with nonzero τ differs from the discrete Toda equation (5) in that it requires the q and e values at discrete-time n−τ rather than discrete-time n. Thus, we can regarded (9) as a time-delay extension of the discrete Toda equation (5). Since it is obvious from (7) that ( )  Combining this with (9), we obtain: It is important to note here that (10) holds if: , , where 1 . Then, using the diagonal matrices D ( n) , we can symmetrize block diagonal matrices Y ( n) as: 1 . Then, we can derive: 1 . This implies that the ddLV system (12) also generates similarity The case where τ=0, of course, equates to that the dLV system (1), which gives similarity transformations of symmetric tridiagonal matrices. Since the blocks ( ) Y s j n , can be decomposed as: the singular values of ( )

Asymptotic convergence
In this section, we investigate asymptotic convergence as  ¥ n in the ddLV system (12), and then relate it to eigenvalues and singular values of matrices.
First, we split the ddLV system (12) as:  which is equivalent to the dLV system (1). The other equations of (13) are similarly equivalent. Thus, from the asymptotic convergence in the dLV system (1), it is obvious that, for j=0,1, K, τ, the sequences { }

Concluding remarks
In this paper, we proposed a time-delay version of the integrable discrete Lotka-Volterra (dLV) system by introducing a time delay into the LR transformations derived from the dLV system. The resulting delay dLV (ddLV) system is also an integrable system. This is because the solution can be explicitly expressed using the Hankel determinants, similarly to the original dLV system. We clarified the asymptotic convergence as discretetime goes to infinity in the ddLV system, and showed an application for computing eigenvalues of block diagonal matrices whose diagonal blocks are tridiagonal matrices. In future work, by focusing on the LR transformations of other structured matrices, we plan to design time-delay versions of more complicated integrable discrete systems, such as the discrete hungry Toda equation, discrete hungry Lotka-Volterra system, and the discrete relativistic Toda equation. These integrable studies are also expected to support the development of discrete time-delay frameworks in mathematical and theoretical biology.