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

Transcription

1 Shno et al., Cogent Mathematcs 5, : APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE An asymptotc analyss for an ntegrable varant of the Lota Volterra prey predator model va a determnant expanson technque Receved: 6 August 4 Accepted: Aprl 5 Publshed: 3 June 5 *Correspondng author: Masash Iwasa, Faculty of Lfe and Envronmental Scences, Kyoto Prefectural Unversty, Kyoto 66-85, Japan E-mal: masa@pu.ac.p Revewng edtor: Ko Lay Teo, Curtn Unversty, Australa Addtonal nformaton s avalable at the end of the artcle Masato Shno, Masash Iwasa *, Ao Fuuda 3, Emo Ishwata 4, Yusau Yamamoto 5 and Yoshmasa Naamura Abstract: The Hanel determnant appears n representatons of solutons to several ntegrable systems. An asymptotc expanson of the Hanel determnant thus plays a ey role n the nvestgaton of asymptotc analyss of such ntegrable systems. Ths paper presents an asymptotc expanson formula of a certan Casorat determnant as an extenson of the Hanel case. Ths Casorat determnant s then shown to be assocated wth the soluton to the dscrete hungry Lota Volterra dhlv system, whch s an ntegrable varant of the famous prey predator model n mathematcal bology. Fnally, the asymptotc behavor of the dhlv system s clarfed usng the expanson formula for the Casorat determnant. Subects: Appled Mathematcs; Dynamcal Systems; Mathematcal Bology; Mathematcs & Statstcs; Scence Keywords: Casorat determnant; dscrete hungry Lota Volterra system; asymptotc expanson AMS subect classfcaton: 39A; 34E5; 5A5. Introducton Integrable systems are often classfed as nonlnear dynamcal systems whose solutons can be explctly expressed. Such an ntegrable system s the Toda equaton whch descrbes the current voltage functon n an electrc crcut. A tme dscretzaton, called the dscrete Toda equaton Hrota, 98, s smply equal to the recurson formula of the qd algorthm for computng egenvalues of a symmetrc trdagonal matrx Henrc, 988; Rutshauser, 99 and sngular values of a bdagonal matrx Parlett, 995. Another commonly nvestgated ntegrable system s the ntegrable Lota Volterra LV system, whch s a prey predator model n mathematcal bology Yamaza, 987. The dscrete LV dlv system was shown n Iwasa and Naamura to be applcable to computng for bdagonal ABOUT THE AUTHOR Masato Shno s a doctoral student n the Department of Appled Mathematcs and Physcs, Graduate School of Informatcs, Kyoto Unversty. He studes asymptotc analyss of nonlnear dynamcal systems nown as ntegrable systems. PUBLIC INTEREST STATEMENT In ths paper, we present a powerful new technque for an asymptotc expanson of the Casorat determnant. The Casorat determnant s assocated wth several dfference equatons appearng n mathematcal physcs, and plays a role smlar to the Wronsan n the theory of dfferental equatons. Our technque wll be useful for asymptotcally analyzng not only the dscrete hungry Lota Volterra system, but also other dynamcal systems assocated wth the Casorat determnant. 5 The Authors. Ths open access artcle s dstrbuted under a Creatve Commons Attrbuton CC-BY 4. lcense. Page of 4

2 Shno et al., Cogent Mathematcs 5, : sngular values. The hungry Lota Volterra hlv system s a varant that captures a more complcated prey predator relatonshp n comparson wth the orgnal LV system Bogoyavlensy, 988; Itoh, 987. Tme dscretzaton of ths system leads to the dscrete hungry Lota Volterra dhlv system. It was shown n Fuuda, Ishwata, Iwasa, and Naamura 9, Fuuda, Ishwata, Yamamoto, Iwasa, and Naamura 3, Yamamoto, Fuuda, Iwasa, Ishwata, and Naamura that the dhlv system can generate LR matrx transformatons for computng egenvalues of a banded totally nonnegatve TN matrx whose mnors are all nonnegatve. The determnant solutons to both the dscrete Toda equaton and the dlv system can be expressed usng the Hanel determnant, H n : =, Hn : = a n a n+ a n+ a n+ a n+ a n+ a n+ a n+ a n+, =,,. where and n correspond to the dscrete spatal and dscrete tme varables, respectvely Tsumoto,. Here, the formal power seres f z = n= an z n assocated wth H n s assumed to be analytc at z = and meromorphc n the ds D ={z z <ζ}. The fnte or nfnte types of poles u, u, of f z are ordered such that < u < u < <ζ. Then, there exsts a nonzero constant c ndependent of n such that, for some ρ satsfyng u >ρ>u +, H n n = c u u u n ρ + O u as n Henrc, 988. The asymptotc expanson. as n enables the asymptotc analyss of the dscrete Toda equaton and the dlv system as n Henrc 988, Rutshauser 99 and n Iwasa and Naamura, respectvely. A generalzaton of the Hanel determnant H n s gven n the below determnant of a nonsymmetrc square matrx of order,. : =, Cn : =,, a n a n+ a n+ a n a n + + a n+ a n+ + + a n+ a n+ + +, =,,, =,,.3 whch s called the Casorat determnant or Casoratan. The Casorat determnant s useful n the theory of dfference equatons, partcularly n mathematcal physcs, and plays a role smlar to the Wronsan n the theory of dfferental equatons Ven & Dale, 999. No one wonder here that the formal power seres f z = n= an z n s assocated wth the Casorat determnant for each. The formal power seres, f z dffers from f z n that not only the superscrpt, but also the subscrpt, appears n the coeffcents. To the best of our nowledge, from the vewpont of the formal power seres f z, the asymptotc analyss for the Casorat determnant has not yet been dscussed n the lterature. The frst purpose, of ths paper s to present an asymptotc expanson of the Casorat determnant as n. The, asymptotc behavor of the dhlv system was dscussed n Fuuda et al. 9, 3 n the case where the dscretzaton parameter δ n s restrcted to be postve. However, t was suggested n Yamamoto et al. that the choce δ n < n the dhlv system yelds a convergence acceleraton of the LR transformatons. The dscrete tme evoluton n the dhlv system wth δ n < corresponds to a reverse of the contnuous-tme evoluton n the hlv system. It s nterestng to note that such artfcal dynamcs are useful for computng egenvalues of a TN matrx. The second purpose of ths paper s to provde an asymptotc analyss for the dhlv system wthout beng lmted by the sgn of δ n. Page of 4

3 Shno et al., Cogent Mathematcs 5, : The remander of ths paper s organzed as follows. In Secton, we frst observe that the entres n, can be expressed usng poles of f z. We then gve an asymptotc expanson of the Casorat determnant n terms of the poles of f z as n by expandng the theorem analytcty for the Hanel determnant gven n Henrc 988. In Secton 3, we fnd the determnant soluton to the dhlv system through relatng the dhlv system to a three-term recurson formula. Wth the help of the resultng theorem for the Casorat determnant, we explan n Secton 4 that the determnant soluton to, the dhlv system can be rewrtten usng the Casorat determnant, and we clarfy the asymptotc, behavor of the soluton to the dhlv system. Fnally, we gve concludng remars n Secton 5.. An asymptotc expanson of the Casorat determnant In ths secton, we frst gve an expresson of the entres of the Casorat determnant n terms of, poles of the formal power seres f z assocated wth. Referrng to the theorem on analytcty for, the Hanel determnant gven n Henrc 988, we present an asymptotc expanson of the Casorat determnant, as n usng the poles of f z. We also descrbe the case where some restrcton s mposed on the poles of f z. Let f z = n= an z n, whch s the formal power seres assocated wth for =,,, be analytc at z = and meromorphc n the ds D ={z z <ζ}. Moreover, let r,, r,,,, denote the poles of f z such that r <, r <, <ζ. By extractng the prncpal parts n f z, we derve f z = α, r, z + α, r, z + + α, p r, p z + n= b n z n. where p s an arbtrary postve nteger, α,, α,,, α, p are some nonzero constants, and b n, whch contans the terms wth respect to r,, p+ r, p+,, satsfes b n μ ρ n. for some nonzero postve constants μ and ρ wth r, p+ <ρ < r, p. The proof of. s gven n Henrc 988 utlzng the Cauchy coeffcent estmate. We now gve a lemma for an expresson of a n appearng n f z = n= an z n. Lemma. Let us assume that the poles r,, r,,, r of f, p z are not multple. Then, an can be expressed usng r,, r,,, r, p as p a n = c, l r n, l + bn.3 where c,, c,,, c, p are some nonzero constants. Proof The crucal element s the replacement α, = c, r, α = c,,, r,, α = c,, p, p r, p n., namely, f z = c, r, r z +, c, r, r, z + + c, p r, p r z +, p n= b n z n.4 Snce each c, l r, l r, l z n.4 can be regarded as the summaton of a geometrc seres, we obtan f z = c, r n, zn + c, r n, zn + + c, p r n, p zn + b n z n n= n= n= n= [ p ] = c, l r n, l + b n z n n= Page 3 of 4

4 Shno et al., Cogent Mathematcs 5, : whch mples.3. Smlarly to the asymptotc expanson as n of the Hanel determnant H n 988, we have the followng theorem for the Casorat determnant., gven n Henrc Theorem. Let us assume that the poles r,, r,,, r of f, p z are not multple. Then there exsts some constant c, σκ, κ,, κ ndependently of n such that, as n,, = σ [ n c, σκ, κ,, κ r, κ r +, κ r +, κ + ρ+l n] O r +l, κl.5 where σ denotes the mappng from {κ, κ,, κ } to {,,, p} and ρ +l s some constant such that r +l, p+ <ρ +l < r +l, p. Proof By applyng Lemma. and the addton formula of determnants to the Casorat determnant, we derve,, = σ D n, σκ, κ,, κ + σ D n, σκ, κ,, κ.6 where n the frst summaton : =, σκ, κ,, κ D n c, κ r n, κ c +, κ r n +, κ c +, κ r n +, κ c, κ r n+, κ c +, κ r n+ +, κ c +, κ r n+ +, κ c, κ r n+, κ c +, κ r n+ +, κ c +, κ r n+ +, κ and n D n the second summaton denotes a determnant of the same form as Dn, σκ, κ,, κ, σκ, κ,, κ except that the lth column s replaced wth b l : =b n +l, bn+ +l,, bn+ +l for at least one of l. Evaluatng the frst summaton n.6, we obtan σ D n, σκ, κ,, κ = σ where c, σκ, κ,, κ r, κ r +, κ r +, κ n c, κ c +, κ c +, κ c : = c, κ r, κ c +, κ r +, κ c +, κ r +, κ, σκ, κ,, κ c, κ r c, κ +, κ r c +, κ +, κ r +, κ.7.8 To estmate the second summaton n.6, for example, we consder the case where the st column s replaced wth b. It mmedately follows from. that b n c +,κ r n +,κ c +,κ r n +,κ b n+ c +,κ r n+ +,κ c +,κ r n+ +,κ b n+ c +,κ r n+ +,κ c +,κ r n+ +,κ n = O ρ r +,κ r +,κ It s also easy to chec Or, κ r +, κ r +l, κl ρ +l r +l, κl+ r +, κ n f the lth column s replaced wth b l. Smlarly, by examnng the case where some columns are replaced wth some of b, b,, b, we can see that D =, σκ, κ,, κ c r, σκ, κ,, κ, κ r +, κ r +, κ n O σ σ ρ+l r +l, κl n.9 Page 4 of 4

5 Shno et al., Cogent Mathematcs 5, : Thus, from.7.9, we obtan.5 Now, let us consder the restrcton r, = r, r, = r,, r, = r n f z. Then, by replacng r, l wth r l n.3, we easly obtan p a n = c, l r n l + bn. As a specalzaton of Theorem., we derve the followng theorem for an asymptotc expanson of the Casorat determnant wth restrcted an as n., Theorem.3 Let us assume that the poles r, r,, r of f z are not multple. Then there exsts some constant c, ndependently of n such that, for r + <ρ < r, as n,, = c, r r r n + O ρ+l r n. Proof Replacng r, = r, r, = r,, r, p = r p n.8 gves c = c, σκ, κ,, κ, κ c +, κ c +, κ r κ r κ r κ r κ r κ r κ. Thus, by tang nto account that c only n the case where κ, κ,, κ σκ, κ,, κ are dstnct to each other, we can smplfy.7 as π D n, πκ, κ,, κ =r r r n π c, κ c +, κ c +, κ r κ r κ r κ r κ r κ r κ.3 where π denotes the becton from {κ, κ,, κ } to {,,, }. It s noted here that the becton π s equal to the mappng σ wth p =. Moreover, there exsts a constant ρ, whch s not equal to one n Theorem., such that r + <ρ < r. Ths s because ρ and ρ + do not always satsfy ρ = ρ + even f r, = r, r, = r,, r, = r n Theorem.. Thus,.9 becomes n O r r r ρ +l.4 Therefore, from.3 and.4, we obtan.. Theorem.3 covers an asymptotc expanson of the Hanel determnant H n. Theorems. and.3 should be useful for the asymptotc analyss of dynamcal systems wth solutons expressed n terms of the Casorat determnant,. 3. The dhlv system and ts determnant soluton In ths secton, smlarly to wor n Tsumoto and Kondo, Sprdonov and Zhedanov 997, we derve the dhlv system from a three-term recurson formula, and then clarfy the determnant expresson of an auxlary varable n the soluton to the dhlv system through nvestgatng the three-term recurson formula. Page 5 of 4

6 Shno et al., Cogent Mathematcs 5, : Let us consder a three-term recurson formula wth respect to the polynomals T n at the dscrete tme n, x, Tn x, { T n + T n x =xtn x vn Tn x, = M, M +,, M x: =, Tn x: = x,, Tn x: = xm M 3. where M s a postve nteger and, M vn, do not depend on x. Accordngly, Tn M+ are all monc. Moreover, let us prepare a tme evoluton from n to n +, T n+ x = x M+ δ n M+ x, Tn x,, where V n : = Tn +M+ δn T n δn. Then, by replacng n wth n + n 3. and usng 3., we obtan T n +M+ x Vn + Tn x =x T n + +M+ x Vn Tn x v n+ T n + x Vn M Tn x M 3.3 Thus, t s observed that T n +M+ x Vn Tn x By usng 3. agan for deletng except for terms wth respect to T n + V n + v n+ +M+ Vn T n x = V n + + vn v n+ V n M x and Tn x n 3.3, we derve T n M x M 3. V n Vn + v n+ = +M+ + Vn + = v n+ V n M Let us ntroduce a new varable u n such that M = u n M V n = = δ n + u n M M δ n + u n = Then, t follows from that M v n+ = u n δ n + u n M + = Moreover, from 3.6 and 3.8, we see that v n+ +M+ = M = δ n + u n M = δ n + u n It s obvous from 3.7 that the rght-hand sde of 3.9 s equal to V n + Vn. Ths mples that vn+ n 3.8 also satsfes 3.4. Consequently, by combnng 3.6 and 3.8, notng that M = δn + u n = M + = δn + u n and replacng M wth, we have the dscrete system M+ u n+ M δ n+ + u n+ = = u n M δ n + u n + = 3. Equaton 3. can be regarded as a dscretzaton of the hlv system whch dffers from the smple LV system n that more than one food exsts for each speces. Thus, 3. s the dhlv system and M Page 6 of 4

7 Shno et al., Cogent Mathematcs 5, : corresponds to the number of the speces of foods for each speces. Clearly, from the defnton, 3. wth M = s smply equal to the dlv system. The dhlv system 3. s essentally equal to the dhlv system n Fuuda et al. 9, u n+ Ths s because 3. s derved by replacng u n n =,, n 3.. Let M + δ n+ u n+ = T n wth [ δ n M ]u n n, T, be polynomals satsfyng a three-term recurson formula, = u n M + δ n u n + = 3. and δ n M+ wth δ n for { Tn + T n T n M x =xm Tn M+ x wn x, = M, M +,, x: =, Tn x: = x,, Tn x: = xm M 3. where w n, M wn n n,, do not depend on x. It s obvous from 3. that T x, T M+ M M+ x, are also all monc. Moreover, let us ntroduce a lnear functonal form n, n [T n xm T n x]: = T n xm T n xωn xdx R l l = { h n = l l 3.3 where ω n x s a weght functon. The lnear functonal n wth M = s equvalent to that n Chhara 978. Further, n wth arbtrary M s a specalzaton of a lnear functon appearng n Maeda, M, and Tsumoto 3. Snce t follows from 3., 3., and 3.3 that [T n xm x M Tn x] = hn and [x M T n M xm T n x] = vn M hn, we easly derve M = hn h n M Let μ n : = n [x ] for =,,. From 3., t turns out that x s expressed as T n x =sn + +, sn, x + x where s n,,, sn are some constants at each and each n., Snce t s clear from 3. that T n x l can be gven as the summaton of xl and the lnear combnaton of T n x, Tn x,, T n l x, we see from 3.3 that n [T n l xm T n x] = n lm [x Tn x]. Thus, t follows that T n 3.4 n [T n xm T n x] = sn, μn n [T n xm T n x] = sn n [T n xm T n x] = sn, μn, μn + + sn M + + sn M + + sn, μn + μn, μn + M+ μn M+, μn + M+ μn M+ By combnng the above wth 3.3, we derve a system of lnear equatons μ n μ n μ n μ n μ n M M+ μ n μ n M M+ μ n M+ μ n M+ s n, s n, = h n 3.5 Page 7 of 4

8 Shno et al., Cogent Mathematcs 5, : Snce s n,,, sn are unquely determned, the coeffcent matrx n 3.5 s nonsngular. Ths, suggests that 3.5 can be transformed nto s n, s n, = + μ n μ n μ n μ n M M+ μ n μ n M M+ μ n μ n M+ μ n M+ h n 3.6 where the hat denotes cofactors of the coeffcent matrx n 3.5 and + : = μ n μ n μ n μ n M M+ μ n μ n M M+ It s of sgnfcance to note that μ n we fnd μ n μ n M+ μ n M+ = M+ τn 3.7. Thus, by examnng the last row for both sdes of 3.6, h n = τn Equatons 3.4 and 3.8 therefore lead to = τn + τn M τn M+ 3.9 Snce we can easly obtan the soluton to the dhlv system 3., by combnng 3.6 wth 3.9, the determnant expresson of s mportant for the asymptotc analyss of the dhlv system 3. n the next secton. Let us defne the tme evoluton of the lnear functonal from n to n+ by n+ [Px] = n [ x MM+ δ n M+ Px ] 3. where Px s an arbtrary polynomal. Then, t s easy to chec that T n+ x M and x are orthogonal to each other wth respect to n+. Equaton 3. yelds a tme evoluton wth respect to l μ s, T n μ n+ = μ n +MM+ δn M+ μ n 3. Notng 3. and 3., we fnd that n [T n xm T n x] wth = l can be expressed as the lnear l combnaton of μ n, μn M+,, μ n. Thus, by combnng t wth 3.3, we derve M+ μ n M+, =,,, 3. Smlarly, n the case where n [T n μ n =, =,,, M, =,,, +M+ xm T n x] wth l, we have l 3.3 Tang nto account that the sequence {μ n } M+ n =,, wth 3. s a specalzaton of the sequence } n =,, appearng n the prevous secton, we may replace μ n n the followng {a n dscusson. Thus, we can rewrte as M+ wth an Page 8 of 4

9 Shno et al., Cogent Mathematcs 5, : : =, τn : = M+, M, M, M M, M M+, M M+,M M, M M+, M M+, M +M+ : =, M, M, M M, M M+, M M+, M M, M M,, M+, M+, M M+, M M+, M+, M+, M+,, =,,, M where : = s, t dagan, s an,, s+ an s an t + -by-t + dagonal matrx wth the relatonshp s+t concernng the evoluton from n to n +, a n+ = a n +M δn M+ a n Asymptotc analyss of the dhlv system Ths secton begns by explanng that the auxlary varable n the dhlv system can be rewrtten n terms of the Casorat determnant. By usng Theorem., we clarfy the asymptotc behavor of the dhlv varables as n. The st, nd,, th row and column blocs n are M-by-M matrces, but the th row +M+ and column blocs are -by- matrces. The followng lemma gves the representaton of n terms of the appearng n Secton., Lemma 4. The auxlary varable s expressed as +M+ = M+M+ =, + Cn +,, Cn +, M, + Cn, M, Cn, +, =,,, M, =,,, m, =,,, m Proof Let us ntroduce a new determnant of a square matrx of order, G n : =, Gn : =,, a n a n +M a n a n + + a n a n +M+ +M+ a n +M a n +M + a n +M+, =,, We begn by showng that can be transformed nto a bloc dagonal determnant wth respect M+ to G n,, Gn,,, Gn. By nterchangng the nd, 3rd, th rows and columns wth the [ +M + ]th, M, [ + M + ]th,, [ + M + ]th rows and columns n, we observe that the same form M+ of G n appears n the st dagonal bloc of τn. The entres n the st, nd,, th rows and columns, M+ n are smultaneously all, except for those n the dagonal bloc secton. Permutatons smlar M+ to the above provde the forms of G n,, Gn,,, Gn as the nd, 3rd,, M + th blocs n. Thus, M, M+ can be expressed n terms of Gn, M+, Gn,,, Gn as M, M+ = M l= G n l, 4.4 Page 9 of 4

10 Shno et al., Cogent Mathematcs 5, : Smlarly, can be transformed nto the determnant of a bloc dagonal matrx whose M + +M+ blocs are G n,, + Gn,,, + Gn and Gn,, +, Gn,, +, Gn M, M +M+ = l= G n l, + l= G n l,. Thus, t follows that 4.5 The cases where = + M + and = M + M + n 3.9 become +M+ = +M++ τn + M++ +M+ τn + M++ M+M+ = +M+ τn M+ M+M+ τn M By combnng them wth 4.4 and 4.5, we obtan G n =, + Gn +,, =,,, M +M+ M+M+ = G n G n, Gn +, M, + Gn, G n M, Gn, + The entres n the th row of G n are gven by the lnear combnaton, a n = +M +l an+ +M +l +δn M+ a n for l =,,,. By multplyng the th row by +M +l δ n M+ and then addng t to the th, we get row a n+, +M an+,, +M + an+ as the new th +M+ + row. Smlarly, for the th, th,, nd rows, t follows that G n =, a n a n+ a n a n + + a n+ a n+ + + a n+ +M a n+ +M + a n+ +M+ + It s worth notng here that the subscrpt M can be regarded as be transformed nto the superscrpt. Thus, G n n 4.3 s equal to the Casorat determnant Cn n.3. Then, by accountng for t n 4.6 and,, 4.7, we have 4. and 4.. Lemma 4. wth Theorem. leads to the followng theorem for asymptotc behavor of M+M+ as n. Theorem 4. The auxlary varable converges to some constant ĉ M+M+ as n. Proof Let σ be the mappng from {κ, κ,, κ } to {κ, κ,, κ} where κ, κ,, κ are postve ntegers such that r, κ r +, κ r +, κ = maxr σ, κ r +, κ r +, κ. Then, t follows from Theorem. that lm n r, κ r +, κ = lm n, r +, κ { c, σ κ, κ,, κ + σ σ [c, σκ, κ,, κ n + ρ+l n O r +l, κ r, κ r +, κ r +, κ r, κ r +, κ r +, κ l n + ρ+l n]} O r +l, κl 4.7 = c, σ κ, κ,, κ 4.8 Page of 4

11 Shno et al., Cogent Mathematcs 5, : It s of sgnfcance to note the relatonshp between f z and f +M z s derved from 3.4, f +M z = [ +δ n M+ z ] f z a z 4.9 Equaton 4.9 mples that the poles of f z and f +M z are equal to each other, namely, r, = r +M,, r, = r +M,,. Thus, by combnng them wth Theorem., we derve lm n r, κ r +, κ +M, r +, κ n = lm n n C+M, r +M, κ r +M+, κ = c +M, σ κ, κ,, κ r +M+, κ n 4. Snce 4.8 and 4. mply that, Cn c c M,, σ κ, κ,, κ M, σ κ, κ,, κ as n, we can conclude that ĉ M+M+ =c c c c, σ κ, κ,, κ M, σ κ, κ,, κ + M, σ κ, κ,, κ, σ κ, κ,, κ + as n. By consderng the postvty of M+, vn +M+,,, we derve the followng theorem for the M +M+ asymptotc behavor of M+, vn +M+,, as n. M +M+ Theorem 4.3 Let us assume that >, vn >,, > for n =,,. Then vn, M+ +M+ M +M+ M+,, converge to as n. +M+ M + M+ Proof From the Jacob determnant dentty Hrota, 3, t follows that, + Cn+ = +, Cn, Cn+ C n+ +,, +, 4. Equaton 4. allows us to smplfy M = vn +M+ as M = M = +M+ = = C n+ M, M, C n+ +, +, C n+,, C n+,, 4. From 4.8, we derve C n+, lm n, = r, κ r +, κ r +, κ 4.3 Thus, by combnng 4.3 and r M, κ = r, κ, r M+, κ = r, κ,, r M+, κ = r, κ wth 4., we have M lm n = +M+ = 4.4 Therefore, by tang nto account that >, vn >,, > n 4.4, we fnd that M+ +M+ M +M+, vn,, as n M+ +M+ M +M+ By recallng the relatonshp of the dhlv varable u n followng theorem concernng an asymptotc convergence of u n to the auxlary varable n 3.6, we have the as n. Theorem 4.4 As n, the dhlv varable u n converges to some nonzero constant c M+, and un, +M+ M u n,, u n go to, provded that +M+ M M+M+ δn satsfy u n M M = δn + u n > for n =,, and the M lmt of δ n as n exsts. Page of 4

12 Shno et al., Cogent Mathematcs 5, : Fgure. A graph of the dscrete tme n x-axs and the value of u n y-axs n the dhlv 7 system 3. wth M = 3 and m = 3. Cross : δ n =, Crcle : δ n = δ n = δ n = Proof The proof s gven by nducton for. Wthout loss of generalty, let us assume that lm n δ n = δ where δ denotes some constant. From 3.6, t holds that u n = M +M δ n + u n l 4.5 By tang the lmt as n of both sdes of 4.5 wth = and usng M ĉ as n, we obtan lm n un = c 4.6 where c = ĉ δ M. By consderng Theorem 4. wth 4.6 n the case where =,,, M n 4.5, we successvely chec that u n, u n,, u n as n. M Let us assume that u n c M+ and un, un,, un as n. Equaton +M+ +M+ M+M+ 4.5 wth = + M + becomes u n +M+ = M M++M+ δ n + u n +M+ l 4.7 It s clear that the denomnator on the rght-hand sde of 4.7 converges to δ M as n under ths assumpton. By combnng t wth ĉ M++M+ + as n, we observe that un c = ĉ +M+ + + δm as n. Moreover, t follows that lm n un ++M++ = lm n M ++M+ δ n + u n ++M++ l =, =,,, M snce M δn + u n ++M++ l δm δ + c + and ++M+ as n 4.8 The convergence theorem concernng the dhlv system 3. n Fuuda et al. 9 s restrcted to the case where the dhlv varable u n s postve and the dscretzaton parameter δ n s fxed postve for every n. Theorem 4.4 clams that the [M + ]th speces survves and the [ + M + ]th, [ + M + ]th,..., [M + M + ]th speces vansh as n even n the case where δ n s a changeable negatve for each n. Although the case of negatve u n s not longer recognzed as a vald bologcal model, we note that the convergence s not dfferent from the postve case f the values of δ n are sutable for n =,,. Page of 4

13 Shno et al., Cogent Mathematcs 5, : To observe the asymptotc convergence numercally, we consder two cases where δ n = and δ n =.69 n the dhlv system 3.. The ntal values are set as u =δ n M M δn + u l for =,,,8 n the dhlv system 3. wth M = 3 and m = 3. Fgure shows the behavor of u n 7 for n =,,, 5 n the case where δn = and δ n =.69. Ths fgure demonstrates that u n 7 tends to as n grows larger even f δn <. We also see that the case where δ n =.69 has a superor convergence speed n comparson wth the case where δ n =. Smlarly, the asymptotc behavor of u n, un,, un can be seen to follow Theorem Concludng remars In ths paper, we assocated a formal power seres f z =, n= an z n wth the Casorat determnant, and gave asymptotc expansons of the Casorat determnants as n n Theorems. and.3. By mang use of Theorem., we then clarfed the asymptotc behavor of the dhlv varables as n Theorem 4.4. Theorems. and.3 may contrbute to asymptotc analyss for other dscrete ntegrable systems. One possble applcaton s the dscrete hungry Toda dhtoda equaton derved from the numbered box and ball system through nverse ultra-dscretzaton Tohro, Naga, & Satsuma, 999. The dhtoda equaton has a relatonshp of varables to the dhlv system whose soluton s gven n the Casorat determnant Fuuda, Yamamoto, Iwasa, Ishwata, & Naamura,. The Casorat determnant drectly appears n, for example, the soluton to the dscrete Darboux Pöschl Teller equaton whch s a dscretzaton of a dynamcal system concernng a specal class of potentals for the -dmensonal Schrödnger equaton Gallard & Matveev, 9. It was proved n Fuuda et al. 3 that the dhlv system 3. wth a fxed postve δ n s assocated wth the LR transformaton for a TN matrx. The paper Yamamoto et al., also suggested that the dhlv system 3. wth changeable negatve δ n generates the shfted LR transformaton for a TN matrx. Egenvalues of an m-by-m TN matrx correspond to the constants ĉ = δ M c, ĉ = δ M c,, ĉ m = δ M c m n Theorem 4.4. Theorems wll be useful for nvestgatng the convergence of the sequence of the shfted LR transformatons based on the dhlv system 3. n the changeable negatve case of δ n. Acnowledgements The authors would le to than Prof. S. Tsumoto for helpful dscussons on the determnant expresson. The authors also than the revewers for ther careful readng and nsghtful suggestons. Fundng Ths wor s supported by the Grant-n-Ad for Scentfc Research C [grant number 648] from the Japan Socety for the Promoton of Scence. Author detals Masato Shno E-mal: mshno@amp..yoto-u.ac.p Masash Iwasa E-mal: masa@pu.ac.p Ao Fuuda 3 E-mal: afuuda@shbaura-t.ac.p Emo Ishwata 4 E-mal: shwata@rs.tus.ac.p Yusau Yamamoto 5 E-mal: yusau.yamamoto@uec.ac.p ORCID ID: Yoshmasa Naamura E-mal: ynaa@.yoto-u.ac.p Graduate School of Informatcs, Kyoto Unversty, Kyoto, 66-85, Japan. Faculty of Lfe and Envronmental Scences, Kyoto Prefectural Unversty, Kyoto, 66-85, Japan. 3 College of Systems Engneerng and Scence, Shbaura Insttute of Technology, Satama, , Japan. 4 Department of Mathematcal Informaton Scence, Toyo Unversty of Scence, Shnuu, Toyo, 6-86, Japan. 5 Department of Communcaton Engneerng and Informatcs, The Unversty of Electro-Communcatons, Chofu, Toyo, , Japan. Ctaton nformaton Cte ths artcle as: An asymptotc analyss for an ntegrable varant of the Lota Volterra prey predator model va a determnant expanson technque, Masato Shno, Masash Iwasa, Ao Fuuda, Emo Ishwata, Yusau Yamamoto & Yoshmasa Naamura, Cogent Mathematcs 5, : References Bogoyavlensy, O. I Integrable dscretzatons of the KdV equaton. Physcs Letters A, 34, Chhara, T. S An ntroducton to orthogonal polynomals. New Yor, NY: Golden and Breach Scence Publsher. Fuuda, A., Ishwata, E., Iwasa, M., & Naamura, Y. 9. The dscrete hungry Lota Volterra system and a new algorthm for computng matrx egenvalues. Inverse Problems, 5, 57. Fuuda, A., Ishwata, E., Yamamoto, Y., Iwasa, M., & Naamura, Y. 3. Integrable dscrete hungry Page 3 of 4

14 Shno et al., Cogent Mathematcs 5, : system and ther related matrx egenvalues. Annal d Matematca Pura ed Applcata, 9, Fuuda, A., Yamamoto, Y., Iwasa, M., Ishwata, E., & Naamura, Y.. A Bäclund transformaton between two ntegrable dscrete hungry systems. Physcs Letters A, 375, Gallard, P., & Matveev, V. B. 9. Wronsan and Casorat determnant representatons for Darboux-Pӧschl- Teller potentals and ther dfference extensons. Journal of Physcs A: Mathematcal and Theoretcal, 4, 449. Henrc, P Appled and computatonal complex analyss Vol.. New Yor, NY: Wley. Hrota, R. 98. Dscrete analogue of a generalzed Toda equaton. Journal of the Physcal Socety of Japan, 5, Hrota, R. 3. Determnant and Pfaffans. Sūraseenyūsho Kōyūrou, 3, 4. Itoh, Y Integrals of a Lota Volterra system of odd number of varables. Progress of Theoretcal Physcs, 78, Iwasa, M., & Naamura, Y.. On the convergence of a soluton of the dscrete Lota Volterra system. Inverse Problems, 8, Maeda, K., M, H., & Tsumoto, S. 3. From orthogonal polynomals to ntegrable systems [n Japanese]. Transactons of the Japan Socety for Industral and Appled Mathematcs, 3, Parlett, B. N The new qd algorthm. Acta Numerca, 4, Rutshauser, H. 99. Lectures on numercal mathematcs. Boston: Brhäuser. Sprdonov, V., & Zhedanov, A Dscrete-tme Volterra chan and classcal orthogonal polynomals. Journal of Physcs A: Mathematcal and General, 3, Tohro, T., Naga, A., & Satsuma, J Proof of soltoncal nature of box and ball systems by means of nverse ultradscretzaton. Inverse Problems, 5, Tsumoto, S., & Kondo, K.. Molecule solutons to dscrete equatons and orthogonal polynomals [n Japanese]. Sūraseenyūsho Kōyūrou, 7, 8. Tsumoto, S., Naamura, Y., & Iwasa, M.. The dscrete Lota Volterra system computes sngular values. Inverse Problems, 7, Ven, R., & Dale, P Determnants and ther applcatons n mathematcal physcs Appled mathematcal scences, Vol. 34. New Yor, NY: Sprnger. Yamamoto, Y., Fuuda, A., Iwasa, M., Ishwata, E., & Naamura, Y.. On a varable transformaton between two ntegrable systems: The dscrete hungry Toda equaton and the dscrete hungry Lota Volterra system. AIP Conference Proceedngs, 8, Yamaza, S On the system of non-lnear dfferental equatons ẏ = y y + y -. Journal of Physcs A: Mathematcal and General,, The Authors. Ths open access artcle s dstrbuted under a Creatve Commons Attrbuton CC-BY 4. lcense. You are free to: Share copy and redstrbute the materal n any medum or format Adapt remx, transform, and buld upon the materal for any purpose, even commercally. The lcensor cannot revoe these freedoms as long as you follow the lcense terms. Under the followng terms: Attrbuton You must gve approprate credt, provde a ln to the lcense, and ndcate f changes were made. You may do so n any reasonable manner, but not n any way that suggests the lcensor endorses you or your use. No addtonal restrctons You may not apply legal terms or technologcal measures that legally restrct others from dong anythng the lcense permts. Cogent Mathematcs ISSN: s publshed by Cogent OA, part of Taylor & Francs Group. Publshng wth Cogent OA ensures: Immedate, unversal access to your artcle on publcaton Hgh vsblty and dscoverablty va the Cogent OA webste as well as Taylor & Francs Onlne Download and ctaton statstcs for your artcle Rapd onlne publcaton Input from, and dalog wth, expert edtors and edtoral boards Retenton of full copyrght of your artcle Guaranteed legacy preservaton of your artcle Dscounts and wavers for authors n developng regons Submt your manuscrpt to a Cogent OA ournal at Page 4 of 4