PERMANENCE (or persistence) is an important property

Transcription

1 Engineering Leers, 5:, EL_5 6 Permanence and Exincion of Delayed Sage-Srucured Predaor-Prey Sysem on Time Scales Lili Wang, Pingli Xie Absrac This paper is concerned wih a delayed Leslie- Gower predaor-prey sysem wih sage-srucure on ime scales By using he heory of calculus on ime scales and some mahemaical mehods as well as some dynamic inequaliies on ime scales, sufficien condiions which guaranee he predaor and he prey o be permanen are obained Moreover, under some suiable condiions, we show ha he prey species will be driven o exincion Finally, an example and is corresponding numerical simulaions are presened o explain our heoreical resuls Index Terms Permanence; Exincion; sage-srucure; Dynamic inequaliy; Time scale I INTRODUCTION PERMANENCE (or persisence) is an imporan propery of dynamical sysems and of he sysems arising in e- cology, epidemics ec, since permanence addresses he is of growh for some or all componens of a sysem, while persisence also deals wih he long-erm survival of some or all componens of he sysem As is well known, mos prey species have a life hisory ha includes muliple sages juvenile and aduls or immaure and maure In paricular, we have in mind mammalian populaions and some amphibious animals In he pas few years, permanence (or persisence) and exincion of differen ypes of coninuous or discree predaor-prey sysems wih sage-srucure have been sudied wildly boh in heories and applicaions; see, for example, [-4] and he references herein Noice ha, in he naure world, here are many species whose developing processes are boh coninuous and discree Hence, using he only differenial equaion or difference equaion can accuraely describe he law of heir developmens Therefore, here is a need o esablish corresponden dynamic models on new ime scales A ime scale is a nonempy closed subse of R The heory of calculus on ime scales (see [5]) was iniiaed by Sefan Hilger in his PhD hesis in 988 (see [6]) The ime scales approach no only unifies differenial and difference equaions, bu also solves some oher problems such as a mix of sop-sar and coninuous behaviors powerfully, see [7-] However, o he bes of he auhors knowledge, here are few papers considered permanence of predaor-prey sysem wih sage-srucure on ime scales Manuscrip received November 7, 6; revised February 5, 7 This work is suppored by he Naional Naural Science Foundaion of China (No 363); he Key Projec of Scienific Research in Colleges and Universiies in Henan Province (No 6A8) L Wang is wih he School of Mahemaics and Saisics, Anyang Normal Universiy, Anyang, Henan, 455 China ay wanglili@6com P Xie is wih he School of Science, Henan Universiy of Technology, Zhengzhou, Henan, 45 China Moivaed by he above saemens, in his paper, we firs esablish a iaion heorem on ime scales by using he heory of calculus on ime scales and some mahemaical mehods, hen, based on he heorem and some inequaliies obained in [7], as an applicaion, we shall sudy he permanence and he exincion of he following delayed Leslie- Gower predaor-prey sysem wih sage-srucure on ime scales: x () = αx () x () αe rτ x (δ (τ, )), x () = αe rτ x (δ (τ, )) r x () x () a y ()x (), x () + y () = ( β a () y () ) y () x () + k + D (y () y ()), y () = (β r 4 y ())y () + D (y () y ()), where T, T is a ime scale x () and x () represen he densiies of immaure and maure individual prey in pach a ime, y i () denoe he densiy of predaor species in pach i, i =, a ime The prey only lives in pach For immaure prey, α is birh rae, is deah rae, and he erm αe rτ x (δ (τ, )) represens he number of immaure prey ha was born a ime δ (τ, ), which sill survive a ime and are ransferred from he immaure sage o he maure sage a ime For maure prey, r is deah rae, is he inraspecific compeiion rae of maure prey, a is he maximum value of he per-capia reducion rae of x due o y, and (resp, k ) measures he exen o which environmen provides proecion o prey x (resp, o he predaor y ) For he predaor, β i is he birh rae of predaor in pach i, i =,, D i is he dispersion rae of predaor beween wo paches, r 4 is deah rae of predaor in pach, a has a similar meaning o a For he ecological jusificaion of (), one can refer o [,] δ (τ, ) is a delay funcion wih T and τ [, + ) T, where δ is a backward shif operaor on he se T, and T is a non-empy subse of he ime scale T More deails abou backward shif operaor, one may see [3] The iniial condiions of () are of he form x i () = ϕ i (), y i () = ψ i (), () ϕ i () >, ψ i () >, i =,, where (ϕ (), ϕ (), ψ (), ψ ()) C([δ (τ, ), ] T, R 4 +), he Banach space of coninuous funcion mapping he inerval [δ (τ, ), ] T ino R 4 +, where R 4 + = (x, x, y, y ) : x i >, y i >, i =, } (Advance online publicaion: 4 May 7)

2 Engineering Leers, 5:, EL_5 6 The organizaion of his paper is as follows In secion, we inroduce some noaions and definiions and sae some preinary resuls needed in laer secions; a iaion heorem on ime scales is esablished, which plays an imporan role in his paper Secion 3 is devoed o discussing he permanence and he exincion of sysem () An Example ogeher wih is numerical simulaions is provided in Secion 4; and a conclusion is made in secion 5 II PRELIMINARIES Le T be a nonempy closed subse (ime scale) of R The forward and backward jump operaors σ, ρ : T T and he graininess µ : T R + are defined, respecively, by σ() = infs T : s > }, ρ() = sups T : s < }, µ() = σ() A poin T is called lef-dense if > inf T and ρ() =, lef-scaered if ρ() <, righ-dense if < sup T and σ() =, and righ-scaered if σ() > If T has a lef-scaered maximum m, hen T k = T\m}; oherwise T k = T If T has a righ-scaered minimum m, hen T k = T\m}; oherwise T k = T For he basic heory of calculus on ime scales, see [5] A funcion p : T R is called regressive provided + µ()p() for all T k The se of all regressive and rd-coninuous funcions p : T R will be denoed by R = R(T) = R(T, R) We define he se R + = R + (T, R) = p R : + µ()p() >, T} If r is a regressive funcion, hen he generalized exponenial funcion e r is defined by } e r (, s) = exp ξ µ(τ) (r(τ)) τ for all s, T, wih he cylinder ransformaion Log(+hz) ξ h (z) = h, if h, z, if h = s Le p, q : T R be wo regressive funcions, define p q = p + q + µpq, p = p + µp, p q = p ( q) Lemma (see [5]) Assume ha p, q : T R be wo regressive funcions, hen (i) e (, s) and e p (, ) ; (ii) e p (σ(), s) = ( + µ()p())e p (, s); (iii) e p (, s) = e p (s,) = e p(s, ); (iv) e p (, s)e p (s, r) = e p (, r); (v) (e p (, s)) = ( p)()e p (, s) In order o discuss he permanence of sysem (), we need he following lemmas Lemma (see [7]) Assume ha a >, b > and a R + Then implies x () ( )b ax(), x() >, [, ) T x() ( ) b a [ + (ax( ) b )e a (, )], [, ) T Lemma 3 Consider he following equaion: x () = ax(δ (τ, )) bx() cx (), where a, b, c, and τ are posiive consans, x() > for [δ (τ, ), ] T Then (i) if a > b, hen (ii) if a < b, hen a b x() = x() = c ; Proof: Case (i) I is easy o show ha x() is posiive and bounded for all [, + ) T Clearly, x = a b c is he unique posiive equilibrium of he equaion Suppose ha x() is evenually monoonic, hen x() exiss Denoe L = hen x(), we show ha L = x, oherwise, if L > x, x () = al bl cl = cl(x L) <, which implies x() =, a conradicion, herefore L = x Now, suppose ha x() is no evenually monoonic, since x() is bounded, le η = sup x() x, hen η is bounded, we can show ha η =, oherwise, if η >, hen here exiss a sequence x( i )( i > i, i = + ) such ha x( i) = x + η or x( i) = x η, (x η) Wihou loss of generalizaion, we only consider he firs case, hen here exiss an ε ( < ε c(x +η)η a+b+c(x +η) ) such ha a(x + η + ε) c(x + η ε) b(x + η ε) < (3) For his ε, here exiss a T = T (ε) > τ such ha for i (δ (τ, T ), + ) T, we have x( i ) < x + η + ε (4) We also know ha here exiss a i (T, + ) T such ha x ( i ) =, x( i ) x > η ε This implies ha Thus ax(δ (τ, i )) = cx ( i ) + bx( i ) ax(δ (τ, i )) > c(x + η ε) + b(x + η ε) By (3), we have Hence ax(δ (τ, i )) > a(x + η + ε) x(δ (τ, i )) > x + η + ε, a conradicion o (4), hen η =, ha is x() = x Case (ii) If x() is evenually monoonic, hen x() exiss Denoe L = x(), hen L, we show ha L =, oherwise, if L >, hen which implies x () = al bl cl <, x() = x() =, a conradicion, herefore (Advance online publicaion: 4 May 7)

3 Engineering Leers, 5:, EL_5 6 Now, suppose ha x() is no evenually monoonic, we can show ha u = sup x() =, oherwise, if u >, hen here exiss a sequence x( i )( i > i, i = + ) such ha x( i) = u Since au bu cu <, hen here exiss an ε > such ha a(u + ε) c(u ε) b(u ε) < (5) For his ε, here exiss a T = T (ε) > τ such ha for i (δ (τ, T ), + ) T, we have x( i ) < u + ε (6) We also know ha here exiss a i (T, + ) T such ha x ( i ) =, x( i ) > u ε This implies ha Thus By (5), we have ax(δ (τ, i )) = cx ( i ) + bx( i ) ax(δ (τ, i )) > c(u ε) + b(u ε) x(δ (τ, i )) > u + ε, a conradicion o (6), hen u =, ha is This complees he proof x() = III PERMANENCE AND EXTINCTION As an applicaion, based on he resuls obained in secion and [7], we shall discuss he permanence and he exincion of sysem () Definiion Sysem () is said o be permanen if here exis a compac region D InR 4 +, such ha for any posiive soluion (x (), y (), x (), y ()) of sysem () wih iniial condiions () evenually eners and remains in region D Proposiion Assume ha (x (), x (), y (), y ()) be a posiive soluion of sysem () wih iniial condiions () If R +, hen here exiss a T 4 > such ha x i () M, y i () M, i =,, [T 4, + ) T where M is a consan and M > maxm, M, M }, M = αm ( e τ ) + ε, M = αe τ + ε, M = α + (A + D + β ) (M + k ) 4A 4Aa + (A + D + β ) + ε, 4Aa A = min, r } Proof: Assume ha (x (), x (), y (), y ()) be any posiive soluion of sysem () wih iniial condiions () I follows from he second equaion of sysem () ha for [τ, + ) T, x () αe τ x (δ (τ, )) x () Consider he following auxiliary equaion: u () = αe rτ u(δ (τ, )) u (), by Lemma 3, we can ge αe rτ u() = According o he comparison principle, i follows ha sup x () αe τ Therefore, for arbirary small ε >, here exiss a T > τ such ha x () αe τ + ε := M, [T, + ) T (7) Seing T = T + τ, from he firs equaion of sysem () and (7), for [T, + ) T, x () α( e rτ )M x (), by Lemma, for arbirary small ε >, here exiss a T 3 > T such ha x () αm ( e τ ) + ε := M, [T 3, + ) T Define v() = x () + x () + y () + y (), [T 3, + ) T, calculaing he derivaive of v() along he soluions of (), we have v () Av() + α + (A + D + β ) (M + k ) 4A 4Aa + (A + D + β ), (8) 4Aa where A = min, r } By Lemma, here exiss a T 4 > T 3, i follows from (8) ha, for [T 4, + ) T, v() α 4A + (A + D + β ) (M + k ) 4Aa + (A + D + β ) 4Aa + ε := M (9) Le M > maxm, M, M }, hen x i () M, y i () M, i =,, [T 4, + ) T This complees he proof Proposiion Assume ha (x (), x (), y (), y ()) be a posiive soluion of sysem () wih iniial condiions () If αe rτ > r + a M, where M is defined in Proposiion, hen here exiss a T 8 > such ha x i () m, y i () m, i =,, [T 8, + ) T where m is a consan and < m < minm, m, m 3 }, m = αm ( e τ ) ε, m = αe rτ r a M/ ε, m 3 = m 3 ε, m 3 = min β (m + k ) a, β r 4 } (Advance online publicaion: 4 May 7)

4 Engineering Leers, 5:, EL_5 6 Proof: Assume ha (x (), y (), x (), y ()) be any posiive soluion of sysem () wih iniial condiions () I follows from he second equaion of sysem () ha for [T 4, + ) T, ( x () αe rτ x (δ (τ, )) r + a ) M x () x () Consider he following auxiliary equaion: u () = αe τ u(δ (τ, )) u (), by Lemma 3, we can ge ( r + a M u() = αe rτ r a M/ ) u() According o he comparison principle, i follows ha inf x () αe rτ r a M/ Therefore, for arbirary small ε >, here exiss a T 5 > T 4 such ha x () αe rτ r a M/ ε := m, () for [T 5, + ) T By he hird and forh equaion of sysem (), we have y () ( β a y ()) m y +k () + D (y () y ()), y () = (β r 4 y ())y () + D (y () y ()), for [T 5, + ) T Consider he following auxiliary equaion: u () = ( β a u ()) m u +k () + D (u () u ()), u () = (β r 4 u ())u () + D (u () u ()), for [T 5, + ) T Define v() = minu (), u ()} Using a similar argumen in he proof of [4, Lemma ], we can obain β (m + k ) inf v() min, β } := m a r 3 4 Therefore, for arbirary small ε >, here exiss a T 6 > T 5 such ha y () m 3 ε := m 3, y () m 3 ε := m 3 Seing T 7 = T 6 + τ, from he firs equaion of sysem () and (), for [T 7, + ) T, x () α( e τ )m x (), by Lemma, for arbirary small ε >, here exiss a T 8 > T 7 such ha x () αm ( e τ ) ε := m, [T 8, + ) T Le < m < minm, m, m 3 } hen x i () m, y i () m, i =,, [T 8, + ) T This complees he proof Togeher wih Proposiions and, we can obain he following heorem Theorem If condiions in Proposiions and hold, hen sysem () is permanen Theorem If αe τ < r, hen he maure and immaure prey populaion in sysem () will go o exincion Proof: Assume ha (x (), x (), y (), y ()) be any posiive soluion of sysem () wih iniial condiions () I follows from he second equaion of sysem () ha x () αe τ x (δ (τ, )) r x () x () Consider he following auxiliary equaion: u () = αe rτ u(δ (τ, )) r u() u (), by Lemma 3, we can ge u() = A sandard comparison argumen shows ha x () = Therefore, for arbirary small ε >, here exiss a T 9 > τ such ha ε < x () < α( e rτ ), [T 9, + ) T () Seing T = T 9 + τ, from he firs equaion of sysem () and (), for [T, + ) T, x () ε x (), by Lemma, here exiss a T > T such ha ha is x () ε + ε = ε, [T, + ) T, x () = This complees he proof IV NUMERICAL EXAMPLE AND SIMULATIONS Consider he following sysem on ime scales x () = 5x () 5x () 5e 5τ x (δ (τ, )), x () = 5e 5τ x (δ (τ, )) r x () 3x () 8y ()x () x ()+8, y () = ( 5y () x ()+5) y () +5(y () y ()), y () = (5 y ())y () +5(y () y ()), () ha is α = 5, = 5, = 3, r 4 =, a = 8, a = 5, = 8, k = 5, β =, β = 5, D = 5, D = 5, τ is a posiive consan Consider sysem () on T = R and T = Z wih = Obviously, R + Le τ =, hen δ (τ, ) = Case Le r = in (), by a direc calculaion, we can ge αe rτ = 337, r + a M = 36 Therefore, all condiions in Proposiions and hold, by Theorem, sysem () is permanen, see Figures and Case Le r = 4 in (), by a direc calculaion, we can ge αe rτ = 337 < 4 = r The condiion in Theorem holds, hen he maure and immaure prey populaion in sysem () will go o exincion, see Figures 3 and 4 (Advance online publicaion: 4 May 7)

5 Engineering Leers, 5:, EL_5 6 y y Fig T = R Dynamics behavior of sysem () wih iniial condiions x () = 3, x () = 58, y () = 76, y () = 3 y y Fig T = Z Dynamics behavior of sysem () wih iniial condiions x () = 8, x () = 5, y () =, y () = Fig 3 T = R Dynamics behavior of sysem () wih iniial condiions x () = 3, x () = 58, y () = 76, y () = namely, permanence and exincion I is imporan o noice ha he mehods used in his paper can be exended o oher ypes of biological models [5-7] Fuure work will include biological or epidemic dynamic sysems modeling and analysis on ime scales REFERENCES [] X Wang, C Huang, Permanence of a sage-srucured predaorcprey sysem wih impulsive socking prey and harvesing, Appl Mah Compu, vol 35, pp 3-4, 4 [] Z Ma, S Wang, W Wang, Z Li, Permanence of a sage-srucured predaor-prey sysem wih a class of funcional responses, CR Biol, vol 334, no, pp , [3] C Huang, M Zhao, L Zhao, Permanence of periodic predaor-prey sysem wih wo predaors and sage srucure for prey, Nonlinear Anal RWA, vol, no, pp 53-54, 9 [4] J Wang, Q Lu, Z Feng A nonauonomous predaorcprey sysem wih sage srucure and double ime delays, J Compu Appl Mah, vol 3, no, pp 83-99, 9 [5] M Bohner, A Peerson, Dynamic equaions on ime scales, An Inroducion wih Applicaions, Boson: Birkhauser, [6] S Hilger, Analysis on measure chains-a unified approach o coninuous and discree calculus, Resul Mah, vol 8, pp 8-56, 99 [7] M Hu, L Wang, Dynamic inequaliies on ime scales wih applicaions in permanence of predaor-prey sysem, Discree Dyn Na Soc, vol, Aricle ID 85, [8] M Hu, L Wang, Exisence and nonexisence of posiive periodic soluions in shifs dela(+/-) for a Nicholson s blowflies model on ime scales, IAENG Inernaional Journal of Applied Mahemaics, vol 46, no 4, pp , 6 [9] Z Zhang, D Hao, D Zhou, Global asympoic sabiliy by complexvalued inequaliies for complex-valued neural neworks wih delays on period ime scales, Neurocompuing, vol 9, pp 494-5, 7 [] H Zhang, F Zhang, Permanence of an N-species cooperaion sysem wih ime delays and feedback conrols on ime scales, J Appl Mah Compu, vol 46, no, pp 7-3, 4 [] A Nindjin, M Aziz-Alaoui and M Cadivel, Analysis of a predaorprey model wih modified Leslie-Gower and Holling-ype II schemes wih ime delay, Nonlinear Anal RWA, vol 7, no 5, pp 4-8, 6 [] X Song, Y Li, Dynamic behaviors of he periodic predaor-prey model wih modified Leslie- Gower Holling-ype II schemes and impulsive effec, Nonlinear Anal RWA, vol 9, no, pp 64-79, 8 [3] M Adivar, Y Raffoul, Shif operaors and sabiliy in delayed dynamic equaions, Rend Sem Ma Univ Poliec Torino, vol 68, no 4, pp , [4] R Xu, L Chen, Persisence and sabiliy for a wo-species raiodependen predaor-prey sysem wih ime delay in a wo-pach environmen, Compu Mah Appl, vol 4, no 4-5, pp , [5] Y Yang, T Zhang, Dynamics of a harvesing schoener s compeiion model wih ime-varying delays and impulsive effecs, IAENG Inernaional Journal of Applied Mahemaics, vol 45, no 4, pp63-7, 5 [6] Y Guo, Exisence and exponenial sabiliy of pseudo almos periodic soluions for Mackey-Glass equaion wih ime-varying delay, IAENG Inernaional Journal of Applied Mahemaics, vol 46, no, pp7-75, 6 [7] W Du, S Qin, J Zhang, J Yu, Dynamical Behavior and Bifurcaion Analysis of SEIR Epidemic Model and is Discreizaion, IAENG Inernaional Journal of Applied Mahemaics, vol 47, no, pp-8, Fig 4 T = Z Dynamics behavior of sysem () wih iniial condiions x () = 6, x () =, y () =, y () = 5 V CONCLUSION Two problems for a delayed Leslie-Gower predaor-prey sysem wih sage-srucure on ime scales have been sudied, (Advance online publicaion: 4 May 7)