Spectral properties of a near-periodic row-stochastic Leslie matrix

Transcription

1 Linear Algebra an its Applications ) wwwelseviercom/locate/laa Spectral properties of a near-perioic row-stochastic Leslie matrix Mei-Qin Chen a Xiezhang Li b a Department of Mathematics an Computer Science The Citael Charleston SC USA b Department of Mathematical Sciences Georgia Southern University Statesboro GA USA Receive October 2004; accepte July 2005 Available online 30 August 2005 Submitte by S Kirklan Abstract Leslie matrix moels are iscrete moels for the evelopment of age-structure populations It is known that eigenvalues of a Leslie matrix are important in escribing the asymptotic behavior of the corresponing population moel It is also known that the ratio of the spectral raius an the secon largest subominant) eigenvalue in moulus of a non-perioic Leslie matrix etermines the rate of convergence of the corresponing population istributions to a stable age istribution In this paper we further stuy the spectral properties of a row-stochastic Leslie matrix A with a near-perioic fecunity pattern of type ks) base on Kirklan s results in 993 Intervals containing arguments of eigenvalues of A on the upper-half plane are given Sufficient conitions are erive for the argument of the subominant eigenvalue of A to be in the interval 2π 2π for the cases where k = A computational [ scheme ] is suggeste to approximate the subominant eigenvalue when its argument is in 2π 2π 2005 Elsevier Inc All rights reserve AMS classification: 5A8 Keywors: Leslie matrix; Spectral property; Subominant eigenvalue; Near-perioic fecunity pattern; Row-stochastic matrix This research was partially supporte by a grant from the Citael Founation Corresponing author aress: chenm@citaeleu M-Q Chen) xli@georgiasoutherneu X Li) /$ - see front matter 2005 Elsevier Inc All rights reserve oi:006/jlaa

2 Introuction M-Q Chen X Li / Linear Algebra an its Applications ) A Leslie matrix arises in a iscrete age-epenent moel for population growth [37] It is a matrix of the form m m 2 m 3 m n m n p p L = 0 0 p p n 0 where p j > 0j = 2n are age-specific survival probabilities an m j 0 j = 2n are age-specific fecunity rates with at least one m j > 0 for the population being moele Let x 0 be the initial population vector Then x τ = L τ x 0 an x τ x τ are the population vector an age istribution vector at time τ respectively where enotes the l -norm of a vector The asymptotic behavior of x τ epens on the ratio of the spectral raius ρ an the secon largest eigenvalue in moulus of L By a similarity transformation with a iagonal matrix S { S = iag p ρ p } p 2 ρ 2 p p 2 p n ρ n L can be normalize to a so-calle row-stochastic Leslie matrix [245] a a 2 a 3 a n a n A = ρ S LS = where a = m ρ 0 a j = m j p p 2 p j 0 for j = 2 3nan n ρ j j= a j = Let λ j = r j e iθ j for j = T be all eigenvalues of A whose arguments lie in [0π] where r r 2 r T The following facts are known: i) λ = an e)is an eigenpair of A where e =[ ] T ii) When A is not perioic λ 2 the subominant eigenvalue of A etermines the rate of convergence of population istributions to a stable population istribution vector x Since λ = we are intereste in eigenvalues λ j for j 2 especially λ 2

3 68 M-Q Chen X Li / Linear Algebra an its Applications ) In this paper we will stuy the spectral properties of a class of row-stochastic Leslie matrices with a near-perioic fecunity pattern which is first introuce for some population moels by Kirklan in [45] Consier a class of row-stochastic Leslie matrices A whose top rows [a a 2 a n ] where n = k has the following property: for some k an 0 s a q > 0 only if q = j i for some j k an 0 i s ) That is the top row can have positive entries only in positions s s + 2 s 2 s + 2k s k s + k When s = 0 the top row of A is of the form 0 0 a 0 0 a a k ) 0 0 a k where = gc{i a i > 0} 2 an A is perioic with perio In this case A has k sets of eigenvalues an each of these sets consists of eigenvalues which are evenly istribute on a circle centere at the origin One of the sets inclues an consists of the eigenvalues e 2πji/ j = 0 When k = the top row of A is of the form [0 ] 0 a a + a a where at least a an a are positive It is assume that the value of s is much smaller than We restrict ourselves further to values of k an s such that s< k 2) 2k + A row-stochastic Leslie matrix A satisfying ) an 2) is sai to have a near-perioic fecunity pattern of type ks)or we say A is a near-perioic row-stochastic Leslie matrix of type k s) for simplicity) For example the matrix A with the top row 0 0 }{{} 8of0 s an with the top row is a near-perioic row-stochastic Leslie matrix of type 2) 0 0 }{{} 8of0 s }{{} 8of0 s 8 4 is a near-perioic row-stochastic Leslie matrix of type 2 0 ) Throughout this paper A is a near-perioic rowstochastic Leslie matrix of type ks) Note that some perioic row-stochastic Leslie matrices with a small perturbation will not be consiere in this paper eg the matrix A with the top row 000 } {{ 000 } 099 is consiere as the perioic 9of000 s matrix B with the top row 0 } 0 {{} with a small perturbation However it will be 9of0 s

4 M-Q Chen X Li / Linear Algebra an its Applications ) interesting to investigate the spectral properties of such a class of matrices with the matrix perturbation theory For a near-perioic row-stochastic Leslie matrix of type ks) one also wants to know in what perio it is near to In [5] Kirklan presente an example of a nearperioic row-stochastic Leslie matrix A of type 2) to illustrate its near-perioic pattern of convergence reflecte in the age istributions Here we further explain this phenomenon an give an estimate of the near-perio Let { λ j u j ) } n be a set of n j= eigenpairs of A an assume that the set {u u 2 u n } is linearly inepenent We write an initial population vector as x 0 = n j= c j u j with assuming c /= 0 Then the population vector x τ at the time τ is given by n n x τ = c u + λ τ j c j u j = c u + rj τ eiτθ j c j u j j=2 j=2 For simplicity let λ 2 an λ 3 = λ 2 be a pair of simple conjugate subominant eigenvalues of A If the argument θ 2 of λ 2 is close to 2π t for some positive integer t then for a sufficiently large integer p x p+)t x pt = n j=2 = r pt 2 + r pt 2 λ pt j λt j )c j u j [ ) e iθ 2pt r2 t eitθ 2 n j=4 rj r 2 ) ] c 2 u 2 + e iθ 2pt r2 t e itθ 2 c 3 u 3 ) pt ) rj t eiθ j t c j u j r pt 2 rt 2 )c 2u 2 + c 3 u 3 ) Thus x p+2)t x p+)t r2 t x p+)t x pt ) or ) x p+2)t x p+)t r t 2 x p+)t x pt ) 0 for any vector norm It means that for a sufficiently large positive integer Pthe subsequence x P x P +t x P +2t x P +pt of {x τ } has nearly the same asymptotically convergent behavior Hence {x τ } behaves asymptotically very much like a sequence with a perio t The following example illustrates this phenomenon Example Let A A 2 an A 3 be near-perioic row-stochastic Leslie matrices of type 2) where the first rows of A A 2 an A 3 are 0 } 0 {{} of0 s 0 0 }{{} 8of0 s an 0 0 }{{} 8of0 s respectively In all three cases θ 2

5 70 M-Q Chen X Li / Linear Algebra an its Applications ) π 2π 9 So θ 2 can be consiere close to 2π 2π 0 or 2π 9 Results are liste in the following table where M = 600 an R = x M x M t ) r2 t x M t x M 2t ) A λ 2 t θ 2 2π t R x M x M A e 06277i A e 05799i A e 0686i The ata suggest that A has a near-perio 0 A 2 has a near-perio an A 3 has a near-perio 9 We conclue that the rate of convergence of a near-perioic row-stochastic Leslie matrix is etermine by the moulus of λ 2 while the perio of the near-perioic convergence is etermine by θ 2 the argument of λ 2 So it is also important to fin θ 2 or an estimate of θ 2 for stuying the convergence behavior of a near-perioic row-stochastic Leslie matrix For any k an s satisfying 2) efine { if 2ks s) l := l ks) = 2ks 2ks if 2ks s) It means that l is the largest integer less than 2ks Note that with the efinition we have 2πl l an s < π ks The following results give l isjoint intervals containing arguments of l eigenvalues of A on the upper-half plane an a lower boun on the moulus of the eigenvalue in each interval Theorem [5 Theorem 2] For ] each m { 2l}Ahas exactly one eigenvalue re iθ such that θ an ˆr m θ) r where ˆr m θ) solves the equation [ ˆr k m [ sin s)θ) ˆr m sinkθ) + sin[k ) + s)θ]=0 ˆr m > 0 3) Clearly ] for each m { 2l} the solution ˆr m θ) of the equation 3) for θ epens only on s k an m but is inepenent on the actual values of the non-zero top-row elements a j of A If r m = min ˆr m θ) θ

6 M-Q Chen X Li / Linear Algebra an its Applications ) for some m {l} as pointe out in [5] we can conclue that λ 2 is close to an therefore x τ x very slowly for that population moel However in the cases where r m is not close to for all m = l what o we know about λ 2 from r m?isθ 2 in one of these l isjoint intervals? Where are the arguments of other eigenvalues on the upper-half plane? In the cases where the actual values of the nonzero top-row elements a j are known can we obtain a better estimate of ˆr m θ) an a better estimate of λ 2? With these questions in min we further stuy the spectral properties of A In this paper we first escribe intervals containing arguments of kl + eigenvalues of A on the upper-half plane in Section 2 We then consier the special case k = In Section 3 properties of ˆr m θ) are investigate an base on the finings a computational scheme using Newton s metho is evelope to efficiently fin the eigenvalue r + e iθ + for θ + without computing all eigenvalues of A Sufficient conitions in terms of the actual values [ of the non-zero ] top-row elements a j of A an θ = 2π for θ 2 to be in the interval 2π 2π are erive in Section 4 Conclusions an a iscussion of future work on cases where k> are given in Section 5 2 Intervals for arguments of kl + eigenvalues As given ] in Theorem A has exactly one eigenvalue re iθ such that θ for each m { 2l} From l< 2πl 2ks it follows that < ks π [ an thus these isjoint l intervals are subsets of 0 π ks ) As an extension of this result the intervals containing arguments of a total of kl + eigenvalues of A are given in the following theorem Theorem 2 A has exactly ) k eigenvalues whose arguments lie on each of the isjoint intervals for m { 2l} [ ) Proof For a fixe m { 2l} let ɛ>0be sufficiently small an let Γ ɛ m) = Γ m ɛ) Γ 2 m ɛ) Γ 3 m ɛ) where { ) Γ m ɛ)= re i ) } ɛ 0 r + ɛ ; { Γ 2 m ɛ)= + ɛ)e iθ ) ɛ θ } ɛ ; an { Γ 3 m ɛ)= re i ɛ ) } 0 r + ɛ

7 72 M-Q Chen X Li / Linear Algebra an its Applications ) π ji/k) * e j=0 Γ ε 3) Γ ε ) raius= Fig Γ ε ) Γ ε 3) an e 2πji/k) Γ ɛ ) Γ ɛ 3) an e 2πji/k) j = 0 ) are shown in Fig for k = 2 = 5 an s = Let B be a row-stochastic Leslie matrix of type k0) that is all elements of the top row are zero except the kth element which is For each 0 α let Cα) = α)b + αa an Fλα) = etλi Cα)) Zeros of Fλα)are eigenvalues of Cα) In particular zeros of Fλ0) are eigenvalues of B an zeros of Fλ) are eigenvalues of A By Remark 24 in [5] for a sufficiently small ɛ>0 no near-perioic row-stochastic Leslie matrix can have an eigenvalue with argument equal to ɛ Then the function Fλα) is continuous in α is analytic in λ an has no zeros on Γ ɛ m) Thus the number of zeros of F insie Γ ɛ m) is Nα) = 2πi Γ ɛ m) Fλα) F λ α) λ ) λ Since the integran is uniformly continuous on Γ ɛ m) [0 ] Nα)is a continuous an integer-value function on [0 ] an therefore Nα) must be a constant It is known that k eigenvalues of B are evenly istribute on the unit circle an the length of interval [ ) ] is 2π It follows that k of these eigenvalues are insie Γ ɛ m) So N0) = k an then N) = k Therefore A has exactly k eigenvalues insie Γ ɛ m)

8 M-Q Chen X Li / Linear Algebra an its Applications ) Remark From Theorem an Theorem 2 we have the intervals for arguments of kl + eigenvalues of A: i) A has exactly eigenvalue whose argument lies on for each m { 2l}; ii) A has exactly k eigenvalues whose arguments lie on ) each m { 2l}; an iii) A has exactly kl + eigenvalues whose arguments lie on 0 2πl ) for 3 Properties of ˆr m θ) As state in Theorem for each m] { 2l} A has exactly one eigenvalue λ + = r + e iθ + where θ + an ˆr m θ + ) is a lower boun of r + Since θ + is [ unknown ˆr m θ + ) cannot be compute explicitly in general In this section we further stuy the properties of the solution function ˆr m θ) in the case when k = From now on ˆr m in 3) an ˆr m θ) are enote for simplicity by r an rθ) respectively Eq 3) becomes r sin s)θ) r sinθ) + sinsθ) = 0 r > 0 4) For each[ m { 2l} ] as shown in [5] the solution function rθ) is continuous in θ on an satisfies equalities ) ) r = r = s ) More properties of rθ) on are given in the following lemma Lemma For each m { 2l} there is a unique θ r θ ) = 0 Moreoverr θ )>0 Proof For each m { 2l} let α = α = 2mπ β = 2mπ + sβ s)α = 2mπ sα an s)β = 2mπ It follows from θ 2πl < π s that for θ [α β] 0 <sθ<π 2m )π < s)θ 2mπ an 2mπ θ < 2mπ + π ) such that an β = Observe that

9 74 M-Q Chen X Li / Linear Algebra an its Applications ) Hence sinsθ) > 0 sin s)θ) 0 an sinθ) 0 for θ [α β] Define Gr θ) = r sin s)θ) r sinθ) + sinsθ) for α θ β We have G θ r θ)=r s)cos s)θ) r cosθ) + s cossθ) G r r θ)=r sin s)θ) s)r sinθ) an Since sin s)θ) 0 sinθ) 0 an both sin s)θ) = 0 an sinθ) = 0 o not hol simultaneously G r r θ) < 0 5) It follows from implicit ifferentiation that rθ) is ifferentiable on α β) an r θ) = G θ r θ) G r r θ) > 0 < 0) if an only if G θ r θ) > 0 < 0) 6) When θ = α we have that cosα) = cos2mπ) = an cos s)α) = cos2mπ sα) = cossα) It follows from rα) = that G θ α)= s)cossα) + s cossα) = + cossα) < 0 When θ = β we have that cos s)β) = cos2mπ) = an cosβ) = cos2mπ + sβ) = cossβ) It follows from rβ) = that G θ β)= s) cossβ) + s cossβ) = s) cosαs)) > 0 Hence lim θ α r θ) < 0 an lim + θ β r θ) > 0 Since r θ) is continuous on α β) there is at least one θ α β) such that r θ ) = 0 Now we show r θ )>0 From 6) r Gθθ + G θr r θ) ) G r G θ Grθ + G rr r θ) ) θ) = G 2 r

10 M-Q Chen X Li / Linear Algebra an its Applications ) Since r θ ) = 0 an G r rθ ) θ ) /= 0 G θ rθ ) θ ) = 0 Hence r θ ) = G θθ rθ ) θ ) G r rθ ) θ ) Observe that G θθ rθ)θ) = rθ) s) 2 sin s)θ) + rθ) 2 sinθ) s 2 sinsθ) From 4) sinsθ) = rθ) sin s)θ) + rθ) sinθ) Substituting this expression into G θθ wehave G θθ rθ)θ) = rθ) s) 2 s 2 ) sin s)θ) + rθ) 2 s 2 ) sinθ) > 0 because sin s)θ) < 0 an sinθ) > 0 for θ α β) Combining this inequality with 5) we have r θ )>0 Because rθ) is ifferentiable the fact that r θ )>0 whenever r θ ) = 0 implies that θ is unique in α β) This completes the proof [ Lemma ] shows that rθ ) is the unique absolute minimum [ value of ] rθ) on Fig 2 gives an example of the graph of rθ) on for some m {l} an the relations of the eigenvalue λ + = r + e iθ + rθ + ) an rθ ) where θ + Because λ + = r + e iθ + is a solution of the characteristic equation s λ a p λ p = 0 p=0 r + θ + ) is the unique solution of the system of equations: ur θ) = r cosθ) s a p r p cospθ) = 0 vr θ) = r sinθ) p=0 s p= a p r p sin pθ) = 0 { for r θ) R m = r θ) } θ an ˆr mθ) r an can be solve numerically It is known that Newton s metho converges locally an quaratically Since each R m is usually a small region Newton s metho can be use to efficiently

11 76 M-Q Chen X Li / Linear Algebra an its Applications ) λ=r + e iθ+ 098 r=rθ) 097 r r=rθ + ) 094 r=rθ * ) 093 θ=θ θ Fig 2 The graph of r = rθ) solve r + θ + ) with an initial estimate in R m We list the steps to approximate r + θ + ) in R m as follows Algorithm An algorithm for computing r + θ + ): θ=θ * i) Input ε>0; set K = r 0 = an θ 0 = 2 ii) Let r = r K an θ = θ K Compute r K an θ K by [ rk r ur r θ) u = θ r θ) ur θ) ; θ K θ] v r r θ) v θ r θ) vr θ) ) + ; iii) If max { ur K θ K ) vr K θ K ) } >ε then K = K + an go to ii); Otherwise the algorithm is terminate an r + θ + ) r K θ K ) Remark 2 When the values of the non-zero top row elements a j of A are known ] Algorithm computes efficiently the eigenvalue λ = r + e iθ + for θ + [ without computing all eigenvalues of A Computation of all of the eigenvalues is numerically intensive especially when the size of A is large

12 M-Q Chen X Li / Linear Algebra an its Applications ) Example 2 [5 Example 32] Let ks)= 2) an a 9 = 4 a 0 = 2 a = 4 Let m = Then θ 2π 2π 9 Forε = 0000 Algorithm takes only 2 iterations to satisfy the stopping criterion an r + θ + ) ) The return time T of r + ie the number of time units it takes to reuce a small perturbation from equilibrium by the factor e isgivenbyt = log ) = 9974 It means that the convergence of population istribution to a stable population istribution vector is very slow 4 Subominant eigenvalue The eigenvalue λ 2 = r 2 e iθ 2 also calle the subominant eigenvalue of A etermines the rate of convergence of population istributions to a stable [ population ] istribution vector The properties of the solution function rθ) on for each m { 2l} [ an the] numerical metho to approximate the eigenvalue of A with argument in suggest a way to approximate λ 2 if its argument is in one of these l isjoint intervals A natural question arises: Is θ 2 in one of these [ l isjoint ] intervals? Many applications an numerical results have shown that θ 2 2π 2π However we can also manufacture some examples for which θ 2 2π 2π In the following we give one example for each case Example 3 [5 Example 3;6] The species Lasioerma serricorne the cigarette beetle): k s) = 45 2) The characteristic polynomial of the corresponing Leslie matrix is given by We have et L λi)=λ λ 2 774λ 397λ λ λ 8 θ 2 = λ λ λ 5 008λ 4 002λ λ λ π 2π = [ ] s Example 4 In this example = an s = 2 3 The last s + nonzero elements of the first rows of corresponing matrices A are liste in the following table:

13 78 M-Q Chen X Li / Linear Algebra an its Applications ) s a a The [ locations ] of λ 2 for s = 2 3 respectively shown in Fig 3 inicate that θ 2 2π 2π when s = 2 3 It is natural to ask when θ 2 is in 2π 2π for a given near-perioic row-stochastic Leslie matrix A of type s) Several sufficient conitions are iscusse in the following two theorems an one corollary Theorem 3 Let A be a near-perioic [ row-stochastic ] Leslie matrix of type s) The eigenvalue whose argument lies on 2π 2π is the largest eigenvalue in moulus of A for all arguments over 0 π ] s Especially for s = the eigenvalue with the argument on 2π 2π is the subominant eigenvalue of A Proof Let λ = re iθ be an eigenvalue of A Then r e iθ = a r s e isθ + +a 2 r 2 e i2θ + a re iθ + a Location of = s=3 =2 / 3) =2 / 2) =2 / ) 2 s= s=2 =2 / Fig 3 θ 2 [ 2π 2π ] for s = 2 an 3

14 M-Q Chen X Li / Linear Algebra an its Applications ) Squaring both sies of the real an imaginary parts of the equation: ) 2 r 2 cos 2 θ= a r s cos sθ + +a 2 r 2 cos 2θ + a r cos θ + a = r 2 sin 2 θ= = s a p 2 r2p cos 2 pθ + 2 p=0 0 p<q s a p a q r p+q cos pθ cos qθ a r s sin sθ + +a 2 r 2 sin 2θ + a r sin θ s a p 2 r2p sin 2 pθ + 2 p=0 an aing both sies we have r 2 = s a p 2 r2p + 2 p=0 0 p<q s 0 p<q s ) 2 a p a q r p+q sin pθ sin qθ a p a q r p+q cos q p) θ The expression of the right han sie of the above equation will be use many times later For convenience it is enote by frθ)as frθ)= s a p 2 r2p + 2 p=0 0 p<q s For any θ 0 π ] s f0θ)= a 2 0 an s fθ)< a p a q r p+q cos q p) θ 7) a p p=0 2 = It is clear that r 2 is monotonically increasing for r [0 ] So for any θ [ 0 π ] s the equation r 2 = frθ)has a positive solution for r [0 ] Note that fr0) = a r s ) 2 + +a r + a It follows from 0 p<q s that 0 q p)θ π an frθ) is a monotonically ecreasing function of θ on [ 0 π ] s ie whenever θ o >θ frθ o )<frθ ) for 0 r The graphs of y = r 2 y = fr0) y = frθ o ) an y = frθ ) are given [ in Fig ] 4 Let λ i2 = r i2 e iθ i 2 an λ o = r o e iθ o be eigenvalues of A where θ i2 2π 2π

15 80 M-Q Chen X Li / Linear Algebra an its Applications ) r i2 fr i2 θ i2 )) 07 y=r 2 06 frθ) y=frθ * ) r o fr o θ o )) y=fr0) y=frθ 0 o ) r Fig 4 Graphs of y = r 2 y = fr0) y = frθ o ) an y = frθ ) r o r i2 ] an θ o 2π π s It is known from Theorem that r i2 is the unique positive solution of the equation r 2 = frθ i2 ) Note that the points r i2 fr i2 θ i2 )) an r o fr o θ o )) are intersections of the graphs of y = r 2 an y = frθ) when θ = θ i2 an θ = θ o respectively Because the graph of y = frθ o ) is always below the one of y = frθ i2 ) the point r o fr o θ o )) locates at the left lower corner of the point r i2 fr i2 θ i2 )) in the r y)-plane ie ) r o <r i2 By Remark there is no eigenvalue of A whose argument lies on 0 2π Hence the eigenvalue λ i2 = r i2 e iθ i 2 where θ i2 2π 2π is the largest eigenvalue in moulus of A on 0 π ] s This completes the proof Notice that [ 0 π ] s can be a small interval What can we say about f rθ)θ) for θ> π s? What are conitions for θ 2 2π 2π when s>? Before answering these questions we first erive a recurrence formula of cosine functions It follows from the sum-to-prouct formula cos mx + cosm 2)x = 2 cosm )x cos x

16 M-Q Chen X Li / Linear Algebra an its Applications ) for any integer m 2 that cos mx cos my = 2 cosm )x cos x cosm 2)x 2 cosm )y cos y+cosm 2)y = 2 cosm )x cos x 2 cosm )y cos x + 2 cosm )y cos x 2 cosm )y cos y cosm 2)x + cosm 2)y = 2 cos x [cosm )x cosm )y] cosm 2)x cosm 2)y) + 2 cosm )ycos x cos y) Notice that cos mx cos my has a factor cos x cos y Defineg m x y) by cos mx cos my = cos x cos y)g m x y) Then we have the following recurrence formula for g m x y) for cos x/= cos y: { g0 x y) = 0 g x y) = g m x y) = 2 cos xg m x y) g m 2 x y) + 2 cosm )y m 2 8) For example g 2 x y) = 2cos x + cos y); g 3 x y) = 4cos 2 x + cos x cos y + cos 2 y) 3; g 4 x y) = 8cos x + cos y)cos 2 x + cos 2 y ) The sufficient conitions for θ 2 2π 2π are state in the following theorem Theorem 4 Let θ = 2π an r be the unique absolute minimum value of rθ) on 2π 2π If for each m {2 3s}θ [ π s π ] an r r< a m ) r m + a m 2) r m 2 g 2 θ θ)+ +a g m θ θ) 0 9) where g m θ θ) satisfies the recurrence relationship in 8) then θ 2 2π 2π Proof We first enote frθ)in 7) byhθ) for simplicity an rewrite the terms in the orer of cos θcos sθ as follows: hθ) :=frθ)= a 2 + a r) 2 ++a 2 r 2 ) 2 + +a r s ) a a r + a a 2 r 3 + +a s ) a r 2s ) cos θ + 2 a a 2 r 2 + a a 3 r 4 + +a s 2) a r 2s 2) cos 2θ + 2 a a 3 r 3 + a a 4 r 5 + +a s 3) a r 2s 3) cos 3θ + +2a a cos sθ

17 82 M-Q Chen X Li / Linear Algebra an its Applications ) It is easy to see that hθ) is monotonically ecreasing for θ [ 0 π ] s Itisknown ) from Theorem 2 that there is no eigenvalue whose argument is in 0 2π an there is only one eigenvalue whose argument is in 2π 2π 0 π ) s As shown in the proof of Theorem 3 the value of the r-coorinate of the intersection of y = r 2 an y = frθ)ecreases as θ increases on [ 0 π ] s Hence if hθ) h θ) [ π ] for θ s π an r r< 0) then θ 2 2π 2π The inequality 0) is equivalent to the inequality: hθ) h θ) = a a r + a a 2 r 3 + +a s ) a r 2s ) cos θ cos θ) + a a 2 r 2 + a a 3 r 4 + +a s 2) a r 2s 2) cos 2θ cos 2 θ) + a a 3 r 3 + a a 4 r 5 + +a s 3) a r 2s 3) 0 cos 3θ cos 3 θ) + +a a r s cos sθ cos s θ) or is equivalent to the following inequality using the notation g m θ θ) in 8): hθ) h θ) cos θ cos θ = a a r + a a 2 r 3 + +a s ) a r 2s ) + a a 2 r 2 + a a 3 r 4 + +a s 2) a r 2s 2) g 2 θ θ) + a a 3 r 3 + a a 4 r 5 + +a s 3) a r 2s 3) g 3 θ θ) + +a a r s g s θ θ) 0 ) because cos θ cos θ <0 from s< 3 in 2) It suffices to show that 9) implies ) We have from 9)

18 M-Q Chen X Li / Linear Algebra an its Applications ) m = 2 a r + a g 2 θ θ) 0; m = 3 a 2 r 2 + a rg 2 θ θ)+ a g 3 θ θ ) 0; m = 4 a 3 r 3 + a 2 r 2 g 2 θ θ)+ a rg 3 θ θ)+ a g 4 θ θ ) 0; m = s a s ) r s + a s 2) r s 2 g 2 θ θ)+ +a g s θ θ) 0 Multiplying a 2 r 2 a 3 r 3 a 4 r 4 a r s on both sies of each inequality in 2) respectively we have 2) a a 2 r 3 + a a 2 r 2 g 2 θ θ) 0; a 2 a 3 r 5 + a a 3 r 4 g 2 θ θ)+ a a 3 r 3 g 3 θ θ ) 0; a s ) a r 2s + a s 2) a r 2s 2 g 2 θ θ) + +a a r s g s θ θ) 0 3) The inequality ) follows by aing both sies of all inequalities in 3) an the inequality a a r 0 This completes the proof Note that the left sie of each inequality in 9) contains r [r ) To check the sufficient conitions in 9) we may nee an estimate of r Observe that the equation in 4) can be written as Since r = sin sθ sin θ r s sin s)θ 2π s 2 s π θ 2π + 2 r min 2π θ 2π = min 2π θ 2π sin sθ sin θ sin s)θ cos sθ 2 cos 2)θ 2 s s π an s ) /) ) /) cos π s 2 θ ) sπ /) s s s π<π 2 Hence r = cos sπ ) /) is a lower bounof r an will be use later in erivations of inequalities For the case 2) an alternative sufficient conition can be erive as follows

19 84 M-Q Chen X Li / Linear Algebra an its Applications ) Corollary Let A be a near-perioic row-stochastic Leslie matrix of type 2) If a cos then θ 2 2π 2 2π ) 2π a a 2 2 Proof It is well-known that n + n 2 n n 2 2 for any two nonnegative numbers n an n 2 So a + a 2 r 2 2r a a 2 Let θ [ π 2 π ] an θ = 2 2π Notice that cos θ + cos θ + cos θ The inequality in ) hols since a a r + a a 2 r 3 + 2a a 2 r 2 cos θ + cos θ) 2a a a 2 r 2 + 2a a 2 r 2 cos θ + cos θ) > 2r 2 a a 2 a cos θ ) ) a a 2 0 It follows from the proof of Theorem 4 that θ 2 2π 2 2π The sufficient conition given in Corollary is inepenent of r an is easily checke Two examples are given below to illustrate the results in Theorem 4 an Corollary Example 5 [5 Example 32] Let ks)= 2) an a 9 = 4 a 0 = 2 a = 4 In this example the sufficient conition in Corollary hols: a = ) 2 > 2π a cos a 2 = So we conclue that θ 2 2π 2π 9 Example 6 Let ks)= 3) an a 8 = 25 a 9 = 9 40 a 0 = 9 40 a = 00 For this example θ = π 4 θ [ π 3 π ] r = cos 3π 8 g 2 θ θ) = 2cos θ + cos θ) ) /8 =

20 M-Q Chen X Li / Linear Algebra an its Applications ) ) g 3 θ θ)=4 cos 2 θ + cos θ cos θ + cos 2 θ 3 = 4 cos 2 θ cos θ 40) ) = 2 2 The sufficient conitions 9) in Theorem 4 hol: a r + a g 2 θ θ) 9 ) ) ) 00 a 2 r 2 + a rg 2 θ θ)+ a g 3 θ θ) ) ) 2 + ) 2 = > 0 2π π 4 So we conclue that θ ) 2 00 = > 0 5 Conclusion The spectral properties of a near-perioic row-stochastic Leslie matrix A of type ks) [ are stuie ] in this paper We present the result that each of the intervals ) for m = 2lcontains the arguments of exactly k eigenvalues of A It is shown by Kirklan [5] that each of the intervals for m = 2l contains the argument of exactly one eigenvalue λ + = r + e iθ + of A In the special case k = we show that as a lower boun of r + the solution ] function rθ) attains uniquely its absolute minimum value r on for each [ m { 2l} Base on the finings we evelop a simple computation proceure using Newton s metho to approximate efficiently the pair r + θ + ) We then erive sufficient conitions for which the argument of the subominant eigenvalue λ 2 of A is on [ 2π 2π ] Using these sufficient conitions an the esigne computation proceure we can fin λ 2 efficiently without computing all eigenvalues of A A further stuy of the spectral properties of A an the location of the argument of λ 2 for k> is ongoing There might be other irections to attack the problem of how to etermine the rate of convergence of a row-stochastic Leslie moel The matrix perturbation theory for example may be use for a perioic row-stochastic Leslie matrix with a small perturbation as mentione in the Introuction Acknowlegements The authors sincerely thank referees for their many valuable comments an suggestions that have greatly improve the paper

21 86 M-Q Chen X Li / Linear Algebra an its Applications ) References [] P Cull The perioic limit for the Leslie moel Math Biosci 2 974) [2] P Cull A Vogt Mathematical analysis of the asymptotic behavior of the Leslie population matrix moel Bull Math Biol ) [3] PE Hansen Leslie matrix moels Math Population Stu 2 ) 989) [4] S Kirklan An eigenvalue region for Leslie matrices SIAM J Matrix Anal Appl 3 2) 992) [5] S Kirklan On the spectrum of a Leslie matrix with a near-perioic fecunity pattern Linear Algebra Appl ) [6] LP Lefkovitch The stuy of population growth in organisms groupe by stages Biometrika 2 965) 8 [7] JB Pick Natural an spectral convergence measures of Leslie matrices: aitive norms an the imprimitive cases Math Comput Moel 26 6) 997) 25 37