Analysis of a non-autonomous mutualism model driven by Lévy jumps

Transcription

1 Analysis of a non-auonomous muualism model driven by Lévy jumps arxiv: v [mah.ap] 4 Jul 5 Mei Li a,b, Hongjun Gao a,, Binjun Wang a a Insiue of mahemaics, Nanjing Normal Universiy, Nanjing 3, PR China b School of Applied Mahemaics, Nanjing Universiy of Finance and Economics, Nanjing 3, PR China limei@njue.edu.cn Absrac. This aricle is concerned wih a muualism ecological model wih Lévy noise. The local exisence and uniqueness of a posiive soluion are obained wih posiive iniial value, and he asympoic behavior o he problem is sudied. Moreover, we show ha he soluion is sochasically bounded and sochasic permanence. The sufficien condiions for he sysem o be exinc are given and he condiions for he sysem o be persisence in mean are also esablished. MSC: primary: 34K5, 6H; secondary: 9B5 Keywords: Iô s formula; Muualism model; Persisen in mean; Exincion; Sochasic permanence The workis suppored by Suppored in par by a NSFC Gran No. 758,NSF of Jiangsu Educaion Commiee No. KJA, PAPD of Jiangsu Higher Educaion Insiuions and he Jiangsu Collaboraive Innovaion Cener for Climae Change and Projec of Graduae Educaion Innovaion of Jiangsu Province No. KLX 79. The corresponding auhor. gaohj@njnu.edu.cn

2 Inroducion Muualism is an imporan biological ineracion in naure. I occurs when one species provides some benefi in exchange for some benefi, for example, pollinaors and flowering plans, he pollinaors obain floral necar (and in some cases pollen) as a food resource while he plan obains non-rophic reproducive benefis hrough pollen dispersal and seed producion. Anoher insance is ans and aphids, in which he ans obain honeydew food resources excreed by aphids while he aphids obain increased survival by he non-rophic service of an defense agains naural enemies of he aphids. Los of auhors have discussed hese models [,, 5, 7,, 4, 3,, 4, 36]. One of he simples models is he classical Loka-Volerra wo-species muualism model as follows: { ẋ() = x() ( a b x()+c y() ), ẏ() = y() ( a b y()+c x() ). (.) Among various ypes muualisic model, we should specially menion he following model which was proposed by May [3] in 976: { ẋ() = x() ( r b x() K +y() ε x() ), ẏ() = y() ( r b y() K +x() ε y() ), (.) where x(),y() denoe populaion densiies of each species a ime, r i,k i,b i,ε i (i=, ) are posiive consans, r,r denoe he inrinsic growh rae of species x(),y() respecively, K is he capabiliy of species x() being shor of y(), similarly K is he capabiliy of species y() being shor of x(). For (.), here are hree rivial equilibrium poins E = (,), E = ( r ε + b K,), E 3 = (, r ε + b K ), and a unique posiive inerior equilibrium poin E = (x,y ) saisfying he following equaions { r b x() K +y() ε x() =, r b y() K +x() ε y() =, (.3)

3 where E is globally asympoically sable. In addiion, populaion dynamics is ineviably affeced by environmenal noises, May[33] poined ou he fac ha due o environmenal flucuaion, he birh raes, carrying capaciy, and oher parameers involved in he model sysem exhibi random flucuaion o a greaer or lesser exen. Consequenly he equilibrium populaion disribuion flucuaes randomly around some average values. Therefore los of auhors inroduced sochasic perurbaion ino deerminisic models o reveal he effec of environmenal variabiliy on he populaion dynamics in mahemaical ecology [8,, 8, 7, 6,, 5, 6, 7, 34, 35]. Li and Gao e al ook ino accoun he effec of randomly flucuaing environmen in [3], where hey considered whie noise o each equaion of he problem (.). Suppose ha parameer r i is sochasically perurbed, wih r i r i +α i Ẇ i (), i =,, where W (),W () are muually independen Brownian moion, α i, i =, represen he inensiies of he whie noise. Then he corresponding deerminisic model sysem (.) may be described by he Iô problems: { dx() = x() ( r b x() K +y() ε x() ) d+α x()dw (), dy() = y() ( r b y() K +x() ε y() ) d+α y()dw (). (.4) On he oher hand, populaion sysems may suffer abrup environmenal perurbaions, such as epidemics, earhquakes, hurricanes, ec. As a consequence, hese sysems are very complex and heir sample pahs may no be coninuous, which yields he sysem (.4) fail o cope wih hem. I is recognized ha inroducing Lévy noise ino he underlying populaion sysem may be quie suiable o describe such disconinuous sysems. There exiss some ineresing lieraures concerned wih SDEs wih jumps. We here only menion Bao e al [4, 3], Liu and Wang [8], Liu and Liang []. Moivaed by hose sudies, in his paper we consider he following non-auonomous sysem wih jumps: 3

4 dx() = x( ) [( r () b ()x() K ()+y() ε ()x() ) d+α ()dw () + γ (,u)ñ(d,du)], dy() = y( ) [( r () b ()y() K ()+x() ε ()y() ) d+α ()dw () + γ (,u)ñ(d,du)], (.5) where x( ) and y( ) are he lef limi of x() and y() respecively, r i (),b i (), K i (),α i (),i =, are all posiive, coninuous and bounded funcions on [,+ ). N is a Poisson random measure wih compensaor Ñ and characerisic measure µ on a measurable subse of (,+ ) wih µ() < +, Ñ(d,du) = N(d,du) µ(du)d, γ i : Ω R is bounded and coninuous wih respec o µ, and is B() F -measurable, i=,. In he nex secion, he global exisence and uniqueness of he posiive soluion o problem (.4) are proved by using comparison heorem for sochasic equaions. Secions 3 is devoed o sochasic boundedness. Secion 4 deals wih sochasic permanence. Secion 5 discusses he persisence in mean and exincion, sufficien condiions of persisence in mean and exincion are obained. Throughouhispaper, wele(ω,f,{f },P)beacompleeprobabiliyspace wih a filraion {F } saisfying he usual condiions. For convenience, we assume ha X() = (x(),y()) and X() = x ()+y (). + γ i (,u) >,u,i =,, here exiss a consan k > such ha ln(+γ i (,u)) [ln(+γ i (,u))] µ(du) < k, β i () =.5α i ()+ Q i () = [γ i (,u) ln(+γ i (,u))]µ(du), i =,, ln(+γ i (s,u))ñ(ds,du), i =,, ˆf = inf f(), f = sup f(). We end his secion by recalling hree definiions which we will use in he forhcoming secions. 4

5 Definiion. [9] If for any < ε <, here is a consan δ(ε) > such ha he soluion X() of (.5) saisfies P{ X() < δ} ε, for any iniial value (x,y ) > (,), hen we say he soluion X() be sochasically ulimae boundedness. Definiion. [9] If for arbirary ε (, ), here are wo posiive consans ζ := ζ (ε) and ζ := ζ (ε) such ha lim inf P{x() ζ } ε, liminf P{y() ζ } ε. lim inf P{x() ζ } ε, liminf P{y() ζ } ε. Then soluion of problem (.5) is said o be sochasically permanen. Definiion.3 [6] If x(), y() saisfy he following condiion lim x(s)ds >, lim The problem of (.5) is said o be persisence in mean. y(s)ds > a.s. Exisence and uniqueness of he posiive soluion Firs, we show ha here exiss a unique local posiive soluion of (.5). Lemma. For he given posiive iniial value (x,y ), here is τ > such ha problem (.5) admis a unique posiive local soluion X() a.s. for [,τ). Proof: We firs se a change of variables : u() = lnx(),v() = lny(), hen problem (.5) deduces o du() = ( r () β () b ()e u() K ε ()+e v() ()e u()) d+α ()dw () + ln(+γ (u))ñ(d,du), dv() = ( r () β () b ()e v() K ε ()+e u() ()e v()) d+α ()dw () + ln(+γ (u))ñ(d,du) 5 (.)

6 on wih iniial value u() = lnx,v() = lny. Obviously, he coefficiens of (.) saisfy he local Lipschiz condiion, hen making use of he heorem [9, 3] abou exisence and uniqueness for sochasic differenial equaion here is a unique local soluion (u(),v()) on [,τ), where τ is he explosion ime. Hence, by Iô s formula, (x(), y()) is a unique posiive local soluion o problem (.5) wih posiive iniial value. Nex we need o prove soluion is global, ha is τ =. Theorem. For any posiive iniial value (x,y ), here exiss a unique global posiive soluion (x(), y()) o problem (.5), which saisfies λ() x() Λ(), θ() y() Θ(),, a.s. where λ(), Λ(), θ() and Θ() are defined as (.4), (.3), (.7) and (.6). Proof: The reference of [7] was he main source of inspiraion for is proof. Because of (x(),y()) is posiive, from he firs equaion of (.5), we can define he following problem { dλ() = Λ( )[ ( r () ε ()Λ() ) d+α ()dw ()+ γ (,u)ñ(d,du)], hen Λ() = x, Λ() = e (r (s) β (s))ds+ α (s)dw (s)+q () x + s e (r (u) β (u))du+ s α (u)dw (u)+q (s) ε (s)ds (.) is he unique soluion of (.), and i follows from he comparison heorem for sochasic equaions ha On he oher hand, λ() = x() Λ(), [,τ), a.s. (.3) e (r (s) β (s))ds+ α (s)dw (s)+q () x + s e (r (u) β (u))du+ s α (u)dw (u)+q (s) (ε (s)+ b (s) )ds K (s) 6

7 is he soluion o he problem dλ() = λ( )[ ( r () ( b () +ε K () ())λ() ) d+α ()dw () + γ(,u)ñ(d,du)], λ() = x, hen (.4) x() λ(), [,τ), a.s. (.5) Similarly, we can ge y() Θ(), [,τ), a.s, (.6) where and, Θ() = e (r (s) β (s))ds+ α (s)dw (s)+q () y + s e (r (u) β (u))du+ s α (u)dw (u)+q (s) ε (s)ds where θ() = y() θ(), [,τ), a.s. (.7) e (r (s) β (s))ds+ α (s)dw (s)+q () y + s e (r (u) β (u))du+ s α (u)dw (u)+q (s) (ε (s)+ b (s) )ds. K (s) Combining (.3), (.5), (.6) wih (.7), we obain λ() x() Λ(), θ() y() Θ(),, a.s. By Lemma 4. in [4], we know ha Λ(),λ(),Θ(),θ() will no be exploded in any finie ime, i follows from he comparison heorem for sochasic equaions [5] ha (x(), y()) exiss globally. 3 Sochasically ulimae boundedness In a populaion dynamical sysem, he nonexplosion propery is ofen no good enough bu he propery of ulimae boundedness is more desired. Now, le us presen a heorem abou he sochasically ulimae boundedness of (.5) for any posiive iniial value. 7

8 Theorem 3. Assume ha here exiss a consan L(q) > such ha γ i (s,u) q µ(du) L(q), q >, i =,. Then for any posiive iniial value (x,y ), he soluion X() of problem (.5) is sochasically ulimae boundedness. Proof: As he reference of [4] we define a Lyapunov funcion U(x) = x q. By he Iô formula: E(e U(x)) = U(x )+E es [U(x)+qx q dx+ q(q )xq (dx) ]ds = U(x )+E es {U(x)+q[r (s) b (s)x ε K (s)+y (s)x ( q)α (s) = + [(+γ (s,u)) q qγ (s,u)]µ(du)]u(x)}ds U(x )+E es {[ ε (s)x++qr (s)+ q(q )α (s) + [(+γ (s,u)) q qγ (s,u)]µ(du)]u(x)}ds. If q >, we can deduce ha here exiss consan L (q) > by assumpion such ha U(x) { [(+qr (s)+ q(q ) α (s) ) qε ()x] + [(+γ (s,u)) q qγ (s,u)]µ(du) } L (q). If < q <, using (+γ (s,u)) q qγ (s,u), hen we have Therefore, U(x) { [(+qr (s)+ q(q ) α (s) ) qε ()x] + [(+γ (s,u)) q qγ (s,u)]µ(du) } U(x)[+qr (s) qε ()x]. Thus, Similarly, we have E(e U(x)) E(U(x ))+L (q)(e ). Ex q L (q). (3.) Ey q L (q). (3.) 8

9 We now combine (3.), (3.) wih he formula [ x() +y() ]q [ q x() q +y() q] o yield E X q q [L (q)+l (q)] < +. By he Chebyshev s inequaliy [3] and he above inequaliy we can complee he proof. 4 Sochasic permanence In he sudy of populaion models, sochasic permanence is one of he mos ineresing and imporan opics. We will discuss his propery by using he mehod as in [8] in his secion. Theorem 4. If min{ˆr β,ˆr β } >, hen soluion of problem (.5) is sochasically permanen. Proof: For a posiive consan < η <, we se a funcion Z(x) = x,v(x) = eλ Z η (x). Sraighforward compuaion dv(x) by Iô, s formula shows ha dv() = ηe λ Z η () { Z ()[r () α () γ (,u)µ(du) (η )α () ( ) η(+γ (,u)) η η µ(du) λ η ]+Z ()(ε ()+ b () )} K d ()+y() ηα e λ Z η ()dw ()+e λ Z η () [ ( (+γ ] Ñ(d,du) (,u)) η ηe λ LZ η ()d ηα e λ Z η ()dw ()+e λ Z η () [ ( (+γ ] Ñ(d,du), (,u)) η Due o {(η )α ln(+γ (,u))µ(du) = lim () + η + (+γ (,u)) η η(+γ (,u)) η µ(du) }, hen when ˆr β >, we can choose a sufficienly small η o saisfy r () α () γ (,u)µ(du) { (η )α () (+γ (,u)) η + µ(du) } >. η(+γ (,u)) η Le us choose λ > sufficienly small o saisfy λ η < r () α () γ (,u)µ(du) { (η )α () + 9 (+γ (,u)) η η(+γ (,u)) η µ(du) }.

10 Then, here is a posiive consan L saisfying r () α () ( γ (,u)+ (+γ (,u)) η ) µ(du) η(+γ (,u)) η (η )α () λ η > L. dv() ηe λ LZ η ()d ηα e λ Z η ()dw () + e λ Z η () [ ( (+γ ] Ñ(d,du), (,u)) η where L := L + ε + b ˆK. Inegraing and hen aking expecaions yields Therefore, E[V()] = e λ E(Z η (x)) ( x ) η + ηl λ (eλ ). + Similarly, when ˆr β >, we have + E[ x η () ] ηl λ. E[ y η () ] ηl λ. For arbirary ε (,), choosing ζ (ε) = ( λε ηl ) η and using Chebyshev inequaliy, we yield he following inequaliies, Hence, hen, + P{x() < ζ } = P{ x η () > ζ η P{y() < ζ } = P{ y η () > ζ η ] x η () ζ η } E[ ] y η () ζ η } E[ P{x() < ζ } ε, limsupp{y() < ζ } ε. + lim inf + P{x() ζ } ε, liminf + P{y() ζ } ε. Combining Chebyshev s inequaliy wih(3.), (3.), we can prove ha for arbirary ε (,), here is a posiive consan ζ such ha,. lim inf + P{x() ζ } ε, liminf + P{x() ζ } ε. This complees he proof.

11 5 Persisence in mean and exincion In he descripion of populaion dynamics, i is criical o discuss he propery of persisence in mean and exincion. Firs, we give a Lemma using he argumen as in [5, 6] wih suiable modificaions. Lemma 5. Suppose ha x() C(Ω [,+ ),R + ). (A) If here exis hree posiive consans T,η and η such ha lnx() η η x(s)ds + for all T, hen + x(s)ds σ i (s)dw i (s)+q() i = or η η a.s. (B) If here exis hree posiive consans T,η and η such ha lnx() η η x(s)ds + for all T, hen lim inf + σ i (s)dw i (s)+q() i = or x(s)ds η a.s. η Proof: (A)DenoeM i () = α i(s)dw i (s),q i () = ln(+γ i(s,u))ñ(ds,du), hen M i (), Q i (),i =, are real valued local maringales vanishing a =. One can see ha he quadraic variaions of M () and Q () are Q i (),Q i () = M i (),M i () = α i(s)ds ᾰ i, (ln(+γ i (s,u))) µ(du)ds k, i =, where M, M is Meyer s angle bracke process, and ρ M () = d M,M (s) (+s) < max{k,ᾰ } ds (+s) <. By he srong law of large numbers for local maringales [3], we have lim α i(s)dw i (s) Q i () =, lim =,a.s. i =,.

12 Then for arbirary ε >, here exiss a T > such ha for > T ε < α i (s)dw i (s)+q i () < ε. Se g() = x(s)ds for all > T, hen we have ln dg d (η +ε) η g, T = max{t,t }. Tha is o say : for T,e η gdg d e(η+ε), inegraing his inequaliy from T o, we can ge g() ln( e ηt + η η+ε (e(η+ε) e (η+ε)t ) ). η Therefore + x(s)ds limsup + Using he arbirariness of ε we have he asserion. ln( 3η e (η+ε) η+ε ) η The proof of (B) is similar o (A). The proof is compleed. Using Lemma 5., we have following heorem. = η +ε η. Theorem 5. Suppose ha ˆr i > β i,(i =,), X() is he posiive soluion o (.5) wih posiive iniial value (x,y ), hen he problem (.5) is persisen in mean. Proof: The mehod is similar o []. We firs deduce ln x(), limsup ln y(). a.s. Making use of Iô s formula o e lnx, we deduce e lnx lnx = e s [lnx(s)+r (s) β (s) ε (s)x(s) b (s)x(s) K (s)+y(s) ]ds+l ()+L (), where L () = es α (s)dw (s), L () = es ln( + γ (s,u))ñ(ds,du) are maringales wih he quadraic forms L (),L () = e s α (s)ds,

13 L (),L () = e s (ln(+γ (s,u))) µ(du)ds k e s ds. By he exponenial maringale inequaliy [3], for any posiive consans k, γ, δ, we can ge ha P { [ sup Li ().5e γk L i (),N i () ] > δe γk lnk } k δ, γk i follows from he Borel-Canelli lemma ha for almos all ω Ω, here is k (ω) such ha for each k k (ω), Hence L i ().5e γk L i (),L i () +δe γk lnk, γk. e lnx lnx es [lnx(s)+r (s) β (s) ε ( s)x(s) b (s)x(s) ]ds K (s)+y(s) + e γk es α (s)ds+ ke γk es ds+δe γk lnk = es [lnx(s)+r (s) (γ (s,u) ln(+γ (s,u)))µ(du) ε(s)x(s) b (s)x(s) K (s)+y(s) α (s)[ es γk ] k[ es γk ]]ds + δe γk lnk es [lnx(s)+r (s)+ ( γ (s,u) + ln(+γ (s,u)) )µ(du) α (s)[ e s γk ] k[ es γk ]]ds+δe γk lnk. Obviously, for any s γk and x >, here is a consan A which is independen of k such ha lnx(s)+r (s)+ ( γ (s,u) + ln(+γ (s,u)) )µ(du).5α (s)[ es γk ].5k[ e s γk ] A. Then for γk,k > k (ω), we derive e lnx lnx A[e ]+δe γk lnk. Tha is lnx() e lnx +A[ e ]+e δe γk lnk. Leing, we have ln x() 3.

14 Similarly, we ge ln y(). On he oher hand, applying Iô s formula o (.4) we have: Tha is dlnλ() = (r () β () (ε ()+ b () K () )λ())d+α ()dw () + ln(+γ (u))ñ(d,du). lnλ() = lnx()+ ( r (s) β (s) (ε (s)+ b (s) K (s) )λ(s)) ds For T, we have + α (s)w (s)ds+q (). lnλ() ( r ˆβ +ε) (ˆε + ˆb K ) λ(s)ds + lnλ() (ˆr β ε) ( ε + b ) λ(s)ds + ˆK α (s)w (s)ds+q (), α (s)w (s)ds+q (). Le ε be sufficienly small such ha ˆr β ε >, hen applying Lemma 5. o above wo inequaliies, we ge ˆK (ˆr β ε) b + ε ˆK lim inf λ(s)ds λ(s)ds K ( r ˆβ +ε). ˆb + ˆε K Making use of he arbirariness of ε we ge λ(s)ds ˆK (ˆr β ε). b + ε ˆK Then Therefore lnx() lnλ(), a.s. lim sup lnλ(),a.s. 4

15 To sum up, we have Similarly, we yield ha ln x() lim =. ln y() lim =. Inegraing he firs equaion of (.) from o, we yield b (s)x(s) ds = dlnx(s)++ (r K (s)+y(s) (s) β (s))ds + α (s)dw (s)+q () ε (s)x(s)ds. Because of b (s)x(s)ds K (s)b (s)x(s) K ds, we obain (s)+y(s) which is b (s)x(s)ds ˆK [ (lnx() lnx )+ (r (s) β (s))ds + α (s)dw (s)+q () ε (s)x(s)ds ], (b (s)+ ˆK ε (s))x(s)ds ˆK [ (lnx() lnx ) +(ˆr β )+ Since ha lim α (s)dw (s) Similarly, we yield lim lim α (s)dw (s)+q () ]. =,lim Q () =, and lim lnx() x(s)ds ˆr β >, a.s. b + ε ˆK y(s)ds ˆr β >, a.s. b + ε ˆK =, we ge This complees he proof. Theorem 5.3 Le X() be a posiive soluion of (.5) wih posiive iniial value X(), hen (A) If r < ˆβ, r < ˆβ, hen x(),y() be exincion. (B) If ˆr > β, r < ˆβ, hen y() is exincion, x() is persisen in mean. (C) If r < ˆβ,ˆr > β, hen x() is exincion, y() is persisen in mean. 5

16 Proof: We firs prove Case (A) of he heorem. Making use of Iô, s formula o lnx,x [,+ ) yields Because of lnx() lnx() lim (r (s) β (s))ds+ α (s)dw (s) and r ˆβ <, we can deduce Similarly =, lim Q () lim x() =, a.s. lim y() =, a.s. α (s)dw (s)+q (). =, a.s. Case (B). Since ha r < ˆβ, we have lim y() =, a.s. Then lnx() lnx() ( r ˆβ ) ˆε x(s)ds lnx() lnx() (ˆr β ) ε x(s)ds b Making use of Lemma 5., we obain x(s)ds ˆK (ˆr β ) ε ˆK + b lim inf Hence, we ge lim inf ˆb x(s) K ds+ lim sup x(s) ˆK ds+ x(s)ds x(s)ds ˆK (ˆr β ) >. ε ˆK + b α (s)dw (s)+q (), α (s)dw (s)+q (). K ( r ˆβ ) K ˆε +ˆb,a.s. Case (C). Similar o he argumens in Case (A) and (B), i is easy o find ha: x() is exincion, y() is persisen in mean, if r < ˆβ,ˆr > β. References [] E. S. Allman and J. A. Rhodes, Mahemaical Models in Biology: An Inroducion, Cambridge Universiy Press, 4. [] D. H. Boucher, S. James and K. H. Keeler, The ecology of muualisms, Annual Review of Ecology and Sysemaics 3 (98),

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