Research Article Asymptotic Behaviour and Extinction of Delay Lotka-Volterra Model with Jump-Diffusion

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1 Hdaw Publshg Corporao Joural of Appled Mahemacs Volume 4, Arcle ID 4954, 6 pages hp://dx.do.org/.55/4/4954 Research Arcle Asympoc Behavour ad Exco of Delay Loka-Volerra Model wh Jump-Dffuso Da L, Jg a Cu, ad Guohua Sog Foudao Deparme, Hoha Uversy Wea College, Maasha, Ahu 43, Cha School of Scece, Bejg Uversy of Cvl Egeerg ad Archecure, Bejg 44, Cha Correspodece should be addressed o Jg a Cu; cujga@bucea.edu.c Receved 8 Sepember 3; Acceped December 3; Publshed Jauary 4 Academc Edor: Xyu Sog Copyrgh 4 Da L e al. Ths s a ope access arcle dsrbued uder he Creave Commos Arbuo Lcese, whch perms uresrced use, dsrbuo, ad reproduco ay medum, provded he orgal work s properly ced. Ths paper sudes he effec of jump-dffuso radom evromeal perurbaos o he asympoc behavour ad exco of Loka-Volerra populao dyamcs wh delays. The corbuos of hs paper le he followg: (a) o cosder delay sochasc dffereal equao wh jumps, we roduce a proper al daa space, whch he al daa may be dscouous fuco wh dowward jumps; (b) we show ha he delay sochasc dffereal equao wh jumps assocaed wh our model has a uque global posve soluo ad gve suffce codos ha esure sochascally ulmae boudedess, mome average boudedess me, ad asympoc polyomal growh of our model; (c) he suffce codos for he exco of he sysem are obaed, whch geeralzed he former resuls ad showed ha he suffcely large radom jump magudes ad esy (average rae of jump eves arrval) may lead o exco of he populao.. Iroduco Populaos of bologcal speces some regos are ofe subjec o sudde evromeal shocks, for example, earhquakes, floods, suam, hurrcaes, ad so forh. As s well kow, occurrece of hese dsasers has he properes of radom upredcably ad grea desruco. To llusrae our mahemacal resuls erms of her ecologcal mplcaos,wecosderheproecoofwldlferarespeces, he souhwes rego Schua Cha, bes-preserved pada haba o earh, whch belogs o Logme Sha acve faul zoe. Boh he 8 M8. Wechua ad he 3 M7. a a earhquakes occurred hs rego. Earhquakes ad secodary dsasers caused by he earhquake such as mudsldes, ladsldes, barrer lake, ad oher geologcal dsasers may desroy aural haba of wldlfe ad eve may lead o exco of edagered wldlfe. Thus, s very eresg o reveal how hese sudde evromeal shocks have a effec o he populaos hrough sochasc aalyss of he uderlyg dyamc sysems. The classcal deermsc Loka-Volerra model wh delays s geerally descrbed by he egrodffereal equao dx () d =x () b a j x j () c j b j x j ( τ j ) x j (θ) dμ j (θ), =,,...,, whch s used o descrbe he populao dyamcs of speces wh eracos, where x () represes he populao sze of he h speces; b, a j, b j,adc j (,,,...,) are cosa parameers; b s he here e brh rae of he h speces; a j, b j,adc j represe he eraco raes; τ j ad μ j ( ) s a probably measure o (, ha may be ay fuco defed o (, of bouded ()

2 Joural of Appled Mahemacs varaos. There s a exesve leraure cocered wh he dyamcs of model () or sysems smlar o(), regardg aracvy, perssece, global sables of equlbrum, ad oher dyamcs, ad we here oly meo Gopalsamy, Kuag ad Smh, Berekeoglu ad Győr 3, He 4, Teg ad Che 5, Fara 6, ad Che 7 amogmayohers. I parcular, he books by Gopalsamy 8 ad Kuag9 are good refereces hs area. The model meoed above s a deermsc model, whch assumes ha parameers he model are all deermsc rrespecve of evromeal flucuaos. However, populao sysems he real word are ofe evably affeced by evromeal oses, whch are mpora facors a ecosysem (see, e.g., ). I s herefore useful o reveal how he ose affecs he delay populao sysems. I mos prevously suded sochasc populao models, was assumed ha populaos chage sze couously, as, for example, dffuso processes whch cosdered small evromeal ose, amely, he whe ose. The ose arses from a early couous seres of small or moderae perurbaos ha smlarly affec he brh ad deah raes of all dvduals (wh each age or sage class) a populao 3. Recall ha b s he here e brh rae of he h speces. I pracce we usually esmae by a average value plus a error whch follows a ormal dsrbuo. If we sll use b o deoe he here e brh rae, he b b β w (), () where w () s a whe ose ad β s used o measure he esy of he whe ose mposed o he here e brh rae of he h speces. By he same way, every eraco parameer a j s sochascally perurbed, wh a j a j σ j w (), (3) where w () s aoher whe ose ad σ j measures he esy of he ose mposed o he eraco raes a j. The ()akeshesochascform dx () =x () b a j x j () c j β x () dw () b j x j ( τ j ) x j (θ) dμ j (θ) d σ j x () x j () dw (), =,,...,, where w () ad w () are muually depede Browa moos wh w () =, =,, defed o a complee probably space (Ω, F, P) wh a flrao {F } sasfyg he usual codos (.e., s rgh couous ad creasg whle F coas all P-ull ses). I rece years, several auhors roduced whe oses o deermsc sysems wh delays o reveal he effec of evromeal varably (4) o he populao dyamcs (see, e.g., 4 8). Parcularly, has also bee revealed by Bahar ad Mao 4adWuad 7 ha he evromeal ose ca suppress a poeal populao exploso of he Loka-Volerra model wh delays. These dcae clearly ha evromeal ose may chage he properes of populao sysems sgfcaly. However, all real populaos experece sudde caasrophc dsasers of varous magudes, ragg sze from very small o large. These adverse pheomea ca have may causes: earhquakes, volcaoes, floods, suam, hurrcaes, severe weaher, fre, epdemcs, ad so forh. Sochasc dyamc sysem (4) s pure dffuso-ype sochasc process ad s populao desy chage couously, whch cao expla such large, occasoal caasrophc dsurbaces. To expla hese pheomea, roducg a jump process o he uderlyg populao dyamcs provdes a feasble ad more realsc model. Haso ad Tuckwell 9 have aalyzed he mpacs of such radom dsasers o perssece me of populao by employg a sochasc dffereal equao whch cosss of a smple couous deermsc model ad a pure Posso jump compoe. Bao e al., 3 suggesed ha hese pheomea ca be descrbed by a Lévy jump process ad hey cosdered sochasc Loka-Volerra populao sysems wh jumps for he frs me. Campllo e al. 4 cosdered he sochasc model wh jump perurbaos he chemosa crcumsace. I 5, Lu ad Wag suded he dyamcs of a Lesle-Gower Hollg-ype II predaor-prey sysem wh Lévy jumps. We also refer he readers o, ad he refereces here for more reasos why he dsasers should be cosdered populao dyamcs modelg. To he bes of our kowledge, o resuls relaed o delay Loka-Volerra model wh jumps have bee repored. I hs paper, we hus sudy possble superposos of jump ad dffuso processes, amely, wha s called jump-dffuso processes. Jump-dffuso models have also some uve appeal ha hey le populao chage couously mos ofheme,buheyalsoakeoaccouhefacha from me o me larger jumps may occur ha cao be adequaely modeled by pure dffuso-ype processes of sysem (4). Movaed by hese, le us ow ake a furher sep; hs paper we focus o sochasc populao dyamcs (4) whch suffer occasoal caasrophc dsurbaces; ha s, dx () =x () b a j x j () c j β x () dw () b j x j ( τ j ) x j (θ) dμ j (θ) d γ (, u) x ( )N(d, du), σ j x () x j () dw () =,,...,, (5)

3 Joural of Appled Mahemacs 3 where x ( ), =,,...,,areheleflmofx () ad b, a j, b j, c j, τ j, μ j, β,adσ j aredefedasmodel (4). Le w (), =,, be muually depede Browa moo defed o he probably space (Ω, F, P) wh he flrao {F } sasfyg he usual codo; le N(d, du) be he Posso radom measure ad depede of w (), whle compesaed Posso radom measure s deoed by N(d, du) := N(d, du) (du)d,where s a characersc measure o a measurable bouded below subse of R \ wh () <. γ : R R beg a measurable fuco, ad <γ (, u) <, γ (, u), u s couous perodc fuco of me, ) wh a perod ω >. ThesymbolN(d, du) cous he umber of jumps of a compoud Posso process wh amplude x ( )γ (, u) of he h speces geeraed he mark erval (u, u du) durg he me erval (, d). Whe he effec of a dsaser s proporoal o populao sze, he dsaser s sad o be desy depede, because he relave dsaser sze s depede of he populao sze. The desy depede dsaser s usually due o aboc causes whch affec he populao as a whole, whereas bologcal causes due o speces eracos, such as epzoocs, may lead o olear or oher desy depede dsasers 6. I hs paper, we oly cosder aboc caasrophes such as earhquakes, suam, hurrcaes, floods, ad fre. Uder hs assumpo, each caasrophe reduces saaeously he populao by a proporo γ (, u); has,apopulao of sze x ( ) jus pror o a caasrophe s reduced o sze ( γ (, u))x ( ) jus afer he caasrophe. Noe ha ay dsaser greaer ha he populao sze has he same effec as a dsaser wpg ou he whole populao. Cosequely, our assumpo ha he loss magude of caasrophes he las erm of sysem (5) srepreseedbyx ( )γ (, u) ad < γ (, u) < s reasoable aboc caasrophc crcumsaces. I referece o he exsg resuls, our corbuos are as follows. () We use jump-dffuso process o model Loka- Volerra sysems wh delays whch suffer sudde caasrophc dsurbaces ad roduce a proper al daa space, whch he al daa may be dscouous fuco wh dowward jumps. () Usg he Khasmsk-Mao heorem ad approprae Lyapuov fucos, we show ha he delay sochasc dffereal equao wh jumps assocaed wh he modelhasauqueglobalposvesoluo. () We gve suffce codos ha esure sochascally ulmae boudedess, mome average boudedess me, ad asympoc polyomal growh of our model. (v) We show ha he suffcely large radom jump magudes ad esy (average rae of jump eves arrval) may lead o exco of he populao. Ths paper s orgazed as follows. I he ex seco, we provde some oaos ad show ha here exss a uque posve global soluo wh ay al posve daa. I Secos 3 ad 4, wedscussheasympocmome esmao ad asympoc pahwse esmao, respecvely. We oba he suffce codos for he exco of he populao Seco 5, ad we ry o erpre our mahemacal resuls erms of her ecologcal mplcaos compared wh he former resuls he fal seco.. Global Posve Soluo As he dyamcs of sysem (5) are assocaed wh he bologcal speces, should be oegave. Moreover, order o guaraee ha (5) has a uque global (.e., o exploso a fe me) soluo for ay gve al daa, he coeffces ofheequaoaregeerallyrequredosasfyhelear growh ad local Lpschz codos (cf. 7). However, he drf coeffce of (5) doesosasfyheleargrowh codo, hough s locally Lpschz couous, so he soluo of (5) may explode a fe me. I s herefore ecessary o provde some codos uder whch he soluo of (5) o oly s posve bu also does o explode o fy ay fe me. Khasmsk 8, Theorem4. ad Mao 9 gave he Lyapuov fuco argume, whch s a powerful es for oexploso of soluos whou he lear growh codo ad s referred o as he Khasmsk- Mao heorem. I hs seco, we wll apply he Khasmsk- Mao approach o show ha whe ose of he eracos amog bologcal speces ca suppress he exploso o our ew model. Throughou hs paper, uless oherwse specfed, we use he followg oaos. x deoes a vecor of Eucldea space R ;has,x=(x,x,...,x ).Leb=(b,b,...,b ), β= (β,β,...,β ), σ=σ j,ad deoes he Eucldea orm of a vecor x R.IfA s a vecor or marx, s raspose s deoed by A.IfA s a marx, s race orm s deoed by A = race(a A).LeR ={x R :x > for all }.Forayx R, deoe x = x. If x() s a R -valued sochasc process o R, wele x = {x( θ) : θ (, } for. To cosder delay sochasc dffereal equao wh jumps, we here also eed o roduce a al daa space. Le A deoe he famly of fucos ha map (, o R sasfyg he followg. () For every φ() A, φ () s rgh-couous o (, wh fe lef-had lms ad has oly fely may dowward jumps over ay fe me erval, =,,...,. () For every φ() A, sup φ (s) <, =,,...,. (6) s I addo, we mpose he followg assumpo o he probably measure μ j. Assumpo. There exss a suffcely small cosa λ> such ha μ j := e λθ dμ j (θ) <,,,...,. (7)

4 4 Joural of Appled Mahemacs Clearly, he above assumpo may be sasfed whe μ j (θ) = e kλθ (k > ) for θ, so here exs a large umber of hese probably measures. Clearly, he smaller λ makes Assumpo easer o sasfy sce dμ j =<. To carry ou he aalyss, we also eed he followg smple assumpos o he eraco ose esy ad he jump-dffuso coeffce. Assumpo. (H) σ >f whls σ j f =j. (H) There exss a fuco δ:r (, ) such ha for each, ) γ (, u) δ(, u), u, =,,...,, sup { l ( δ(, u)) (du)} <. For coveece of referece, we recall wo fudameal equales saed as a lemma. Lemma 3. The followg equales hold: (8) x r r(x ), x, r, (9) ( (p/)) x p = x p ( p/) x p, () p >, x R. The he followg heorem o he global posve soluo follows. Theorem 4. Uder Assumpo, forayaldaaξ A, here s a uque posve soluo x(, ξ) of sysem (5),adhe soluo wll rema R wh probably ; amely, x(, ξ) for all a.s. R Proof. Sce he drf coeffce does o sasfy he lear growh codo, he geeral heorems of exsece ad uqueess cao be mplemeed for sysem (5). However, s locally Lpschz couous; herefore, for ay gve al daa ξ(θ) A, here s a uque maxmal local soluo x(, ξ) for,τ e ),whereτ e s he exploso me. For he sake of smplcy, we wre x(, ξ) = x() hereafer. To show ha he soluo s global, we oly eed o prove ha τ e = a.s. Noe ha ξ() <.Lek be a suffcely large posve umber such ha /k <ξ () < k for all =,,...,.For each eger k k, defe he soppg me τ k = f {, τ e ):x () ( k,k) for some =,,...,} () wh he radoal coveo f =,where deoes he empy se. Clearly, τ k s creasg as k ad τ k τ τ e a.s. If we ca show ha τ =,heτ e = a.s., whch mples he desred resul. Ths s also equvale o provg ha, for ay >, P(τ k ) as k. To prove hs saeme, le us defe a C -fuco U:R R by U (x) = = x.5l x. () I s easy o see ha U(x) for all x R. Applyg Iô s formula o (5) yelds du (x) =.5 Compue (x.5 = =.5.5 (x.5 ) b = =.5 = = a j x j c j (.5x.5 ) β ( β (x.5 )dw () = (x.5 b j x j ( τ j ) x j (θ) dμ j (θ) d )σ j x j dw () σ j x j ) d ( γ (, u)).5 x.5 N (d, du).5 l ( γ (, u))n(d, du). )(b a j x j ) b (x.5 ) = b (x.5 ) = a j x j = a j x j =.5 a j (x x j ) = = = b (x.5 ) ( a j.5 a j )x =.5 a j x, a j x.5 x j (3)

5 Joural of Appled Mahemacs 5 (x.5 = ) b j x j ( τ j ).5 b j =.5 = (x.5 = (x.5 b j ).5x j ( τ j) x.5x j ( τ j), = ).5 c j =.5.5 = c j (x.5 =.5 c j.5 = b j x.5 = x j (θ) dμ j (θ) ) x j (θ) dμ j (θ) x c j c j.5 b j x.5 x j (θ) dμ j (θ). Moreover, by assumpo (H), = (.5x.5 ) β ( = =.5.5 = =.5β x.5 β (.5x.5 x.5 ( = σ j x j ) ( σ j x j ) σ j x j ) σ x.5 = = ( = σ j x j ) ( σ j.5σ x.5 σ x β. σ j x j ) x j ) β = β = (4) (5) The above equales mply du (x) { { = {.5σ x a j.5 a j a j.5 σ b j.5 c j.5 b.5 b j.5 c j.5 b.5 b j.5 c j β.5x j ( τ j)d =.5.5 = =.5 = } } d } x j (θ) dμ j (θ) d β (x.5 )dw () = (x.5 )σ j x j dw ().5 l ( γ (, u))n(d, du), sce ( γ (, u)).5.defe U (x, ) =U(x).5 Noe ha =.5 = = x j (s) ds τ j x x.5 x j (s) ds dμ j (θ). θ.5 l ( γ (, u))n(d, du) = =.5 l ( γ (, u)) N (d, du) =.5 l ( γ (, u)) (du) d. x (6) (7) (8)

6 6 Joural of Appled Mahemacs By assumpo (H), we have The du (x, ) { { = {.5 = = = <..5 l ( γ (, u)) (du).5 l ( γ (, u)) (du).5 l ( δ(, u)) (du).5σ x a j.5 σ.5 a j.5 a j b j.5 c j.5 b.5 b j.5 c j.5 b.5 b j =.5 =.5 c j β β (x.5 )dw () = Kd.5 (x.5 } } d } )σ j x j dw ().5 l ( γ (, u))n(d, du).5 = = = β (x.5 )dw () (x.5 )σ j x j dw ().5 l ( γ (, u)) N (d, du), x x x.5 (9) () where K s a posve cosa. We herefore have EU(x( τ k )) EU (x ( τ k ), τ k ) Defe for each u> τ k U (ξ (),) E Kds =: K. () ρ (u) := f {U (x) :x (,u) for some =,,...,}. u () Due o he propery of he fuco h(x) := x.5 l x, x>,weseeha ad hece lm h (x) =, lm x x h (x) =, (3) lm u ρ (u) =. (4) By he defo of τ k, x (τ k ) kor x (τ k )/kfor some =,,...,,so whch mples ha P (τ k )ρ(k) P (τ k ) U (x ( τ k )) EU(x( τ k )) K, K (5) lm sup P (τ k ) lm =, (6) k k ρ (k) as requred. Ths complees he proof of Theorem Asympoc Mome Properes Theorem 4 shows ha he soluo of sysem (5) remas heposvecoer wh probably. Thscepropery perms us o furher exame how he soluos vary R more deal. Compared wh he oexploso propery of he soluo, he mome properes are more eresg from he bologcal po of vew. I hs seco, we wll show ha he soluo of he sysem (5) ssochascallyulmae boudedess, ad he average me of he mome of he soluo o (5) s also bouded. To dscuss sochascally ulmae boudedess, we frs exame he ph mome boudedess, whch s also eresg. Defo 5. Equao (5) ssadobesochascallyulmaely bouded, f for ay ε (, ), here exss a posve cosa χ(= χ(ε)), such ha, for ay al daa ξ A, he soluo of (5)has he propery ha lm sup P{ x () >χ}<ε, (7) where x() s he soluo of (5) wh ay al daa ξ A.

7 Joural of Appled Mahemacs 7 Theorem 6. Le Assumpo hold. Uder Assumpo, for ay p (, ), here exss a posve cosa K p depede of he al daa ξ A such ha he global posve soluo x(, ξ) of sysem (5) has he propery ha lm sup E x () p K p. (8) Proof. By Theorem 4, he soluo of sysem (5) remas R a.s. for all.forayp (, ), defe a C -fuco V p : R R by V p (x) = = x p. (9) For ay ε (, λ), applyg Iô s formula o e ε V p (x) ad akg expecao yeld EV p (x ()) =e ε EV p (ξ ()) e ε E e εs LV p (x s )εv p (x (s))ds, (3) p (a j p b j c j )a j = p e ετ j b j μ j c j xp b j (xp = c j p ( = Moreover,seasyoshowha = x p = ( τ j ) e ετ j x p ) x p (θ) dμ j (θ) μ j x p ). ( γ (, u)) p x p (du), β ( = = σ j x j ) β xp σ j σ k x p x jx k =k= (3) (33) where LV p s defed as LV p (x ) =p x p = b p(p ) = By oug s equaly, x p = a j x j x p = c j a j x j c j x p = b j x j ( τ j ) x j (θ) dμ j (θ) β ( σ j x j ) ( γ (, u)) p x p (du). a j x j b j x j ( τ j ) x j (θ) dμ j (θ) c j b j x j ( τ j ) x j (θ) dμ j (θ) (3) = As a resul, we oba LV p (x )εv p (x ()) where = β xp R (x ) p p p c j p ( = σ xp. = b j (xp = R (x) = p( p) σ xp ( τ j ) e ετ j x p ) x p (θ) dμ j (θ) μ j x p ), p p (a j p b j c j )a j eετ j b j μ j c j xp pb p(p ) β εx p. (34) (35) Nog ha σ > ad p (, ), by he boudedess propery of polyomal fucos, here exss a posve cosa K(= K(p, ε)) such ha = R (x ) K,whch mples ha EV p (x ()) e ε EV p (ξ ()) ε K( e ε )

8 8 Joural of Appled Mahemacs e ε e ε p b j p E = p c j p E = I s easy o show ha e εs (x p τ j = τ j e εs (x p (s τ j ) e εs ( (s τ j ) e ετ j x p (s))ds e ε(sτ j) x p ε e ετ j supξ p (u). u (s) ds e ετ j x p (s))ds x p (sθ) dμ j (θ) μ j x p (s) )ds. e ε(sτj) x p (s) ds (36) (37) Usg a smlar argume as Theorem 3. 6, by Fub s Theorem, we may esmae ha e εs = x p (sθ) dμ j (θ) ds e εs s ds e εs s ds sup u ξ p μ j whch mples λ ε sup u e εs μ j s dμ j (θ) θ (u) dμ j (θ) x p λ ε sup u ξ p x p (sθ) dμ j (θ) x p (sθ) dμ j (θ) x p (sθ) e λ(sθ) dμ j (θ) e εs x p (sθ) ds e (λ ε)s ds e λθ dμ j (θ) e ε(s θ) x p (s) ds (u) μ j e εs x p (s) ds, (sθ) dμ j (θ) μ j x p (s)ds ξ p (u). (38) (39) Hece EV p (x ()) e ε EV p (ξ ()) ε K( e ε ) e ε whch mples ξ p = u p sup p μ j (u) ε b j eετ j c j λ ε, (4) lm sup EV p (x ()) ε K. (4) Sce p (, ),byequaly(), we oba ha lm supe x () p lm supev p (x ()) ε K, (4) ad he assero (8)followsbysegK p =ε K. Theorem 7. Le Assumpo hold. Uder Assumpo, he soluo of (5) s sochascally ulmaely bouded. The proof of Theorem 7 s a smple applcao of he Chebyshev equaly ad Theorem 6. The followg resul shows ha he average me of he mome of he soluo o sysem (5)wllbebouded. Theorem 8. Le Assumpo hold. Uder Assumpo, for ay p (, ), here exss a posve cosa K p depede of he al daa ξ A such ha he global posve soluo x(, ξ) of (5) has he propery ha lm sup E x p = (s) ds K p. (43) Proof. Applyg Iô s formula o V p (x) defed by (9) yelds EV p (x ()) = EV p (ξ ()) E LV p (x s )ds, (44) where LV p s defed as LV p (x ) =p x p = b p(p ) = a j x j c j x p = b j x j ( τ j ) x j (θ) dμ j (θ) β ( σ j x j ) ( γ (, u)) p x p (du). (45)

9 Joural of Appled Mahemacs 9 By a smlar compuao o (34), we have LV p (x ) p( p) 4 = σ xp = Ψ (x ) p p p c j p ( = b j (xp = where Ψ (x) = p( p) σ 4 xp p p ( τ j ) x p ) x p (θ) dμ j (θ) x p ), p (a j b j c j )a j b j c j xp pb p(p ) β xp. (46) (47) Nog ha p (, ), hepolyomal = Ψ (x ) has a upper posve boud, say R p ;equaly(46)yelds EV p (x ()) p( p) 4 = EV p (ξ ()) R p ds p b j p E = p c j p E = σ E (x p ( x p (s) ds (s τ j ) x p (s))ds x p (sθ) dμ j (θ) x p (s))ds, sce ( γ (, u)) p,for<γ (, u) <.Compue (x p (s τ j ) x p (s))ds τ j = x p τ j τ j supξ p (u), u = (s) ds x p (sθ) dμ j (θ) ds s ds s x p (s) ds x p (sθ) dμ j (θ) x p (sθ) dμ j (θ) (48) (49) s ds sup u ξ p μ j λ sup whch mples Hece x p (sθ) e λ(sθ) dμ j (θ) dμ j (θ) θ (u) dμ j (θ) ξ p u ( x p (sθ) ds e λs ds e λθ dμ j (θ) (u) x p x p (s) ds x p (s) ds, (sθ) dμ j (θ) x p (s))ds μ j λ sup ξ p (u). u EV p (x ()) p( p) 4 = σ E EV p (ξ ()) R p ds p b j p τ j sup = hs mples lm sup E = where ξ p u x p (s) ds (u) p c j λ(p) μ j sup ξ p (u) ; = x p u (s) ds (5) (5) (5) 4R p σp ( p), (53) σ =m {σ,}. (54) The requred assero (43)followsmmedaely. 4. Asympoc Pahwse Esmao I Seco 3 we have dscussed how he soluos vary he R probably or mome. I hs seco, we coue oexamehepahwseproperesofhesoluososysem (5). The followg resul shows ha hs sochasc populao sysem wll grow a mos polyomally. Theorem 9. Le Assumpo hold. Uder Assumpo, for ay al daa ξ A,hesoluoof (5) sasfes l x () lm sup a.s. (55) l Tha s, x() grows wh a mos polyomal speed.

10 Joural of Appled Mahemacs Proof. By Theorem 4, he soluo of (5) remasr a.s. for all.defeac -fuco V : R R by V (x ()) = Applyg Iô s formula yelds dv (x ()) = = x b = = = a j x j c j x. (56) b j x j ( τ j ) x j (θ) dμ j (θ) d β x dw () σ j x x j dw () = x ( )γ (, u) N (d, du). The, applyg Iô s formula o l V (x) yelds d l V (x) where = x V (x) (b = a j x j c j Z () Z () d Z () dw () Z () dw () b j x j ( τ j ) x j (θ) dμ j (θ)) l ( = x ( )γ (, u) V (x ( )N(d, du), )) Z () = β V (x) x (), = Z () = σ V (x) j x () x j (). = (57) (58) (59) Hece, for ay ε (, λ), applyg Iô s produc formula o e ε l V (x) gves l V (x ()) =e ε l V (ξ ()) e ε e ε e εs ε l V (x (s)) Z (s) Z (s) V (x) e ε e εs = x (b a j x j c j b j x j (s τ j ) x j (sθ) dμ j (θ)) ds e εs Z (s) dw (s) e ε e εs Z (s) dw (s) l ( = x (s )γ (s, u) V (x (s )N(ds, du). )) (6) By vrue of he expoeal margale equaly, for ay T, α, β>,wehave P { sup T e αβ. e εs Z (s) dw (s) e εs Z (s) dw (s) α e εs (Z (s) Z (s))ds β} (6) Choose T=kε, α=pe kε,adβ=p θe kε l k,whek N, <p<,adθ> he above equao. Sce k= k θ <, we ca deduce from he Borel-Caell lemma 3,..4 ha here exss a Ω Ωwh P(Ω )=such ha, for ay ω Ω, a eger k=k(ω, p) ca be foud such ha e εs Z (s) dw (s) e εs Z (s) dw (s) p θe kε l k pe kε p θe kε l k p e εs (Z (s) Z (s))ds e εs (Z (s) Z (s))ds (6) wheever k k, kε.thus,forω Ω ad (k )ε kεwh k k,wehave l V (x ()) l l V (ξ ()) e ε l p θe kε l k e (k )ε l ((k ) ε) l ε l V (x (s)) p Z (s) x V (x) (b = a j x j c j e ε(s ) b j x j (s τ j ) x j (sθ) dμ j (θ)) ds

11 Joural of Appled Mahemacs l e ε(s ) l ( = x (s )γ (s, u) V (x (s ) )) N(ds, du). (63) Noe ha l u u for ay u>.hece,byequaly() we have Clearly, I s easy o show ha l V (x) V (x) /4 x /. (64) a V (x) j x x j = b V (x) x max = {b }. (65) a j V (x) x x j max,j {a j }V (x) = max,j {a j } x, b V (x) j x x j ( τ j ) = b j V (x) x x j ( τ j ) = max,j {b j } x j ( τ j ), V (x) = c j x V (x) = x j (θ) dμ j (θ) c j x x j (θ) dμ j (θ) max,j {c j } x j (θ) dμ j (θ). By equaly (), s also easy o see ha Z () σ V (x) x () Noe ha l e ε(s ) = (66) (67) (68) m {σ } x (). (69) l ( = x (s )γ (s, u) V (x (s ) )) N(ds, du) ; (7) hs, ogeher wh (64) (69), mples l V (x ()) l where l V (ξ ()) e ε l l p θe kε l k e (k )ε l ((k ) ε) e ε(s ) Ψ (x (s)) ds l max,j {b j } e ε(s ) x j (s τ j ) e ετ j x j (s)ds l max,j {c j } e ε(s ) x j (sθ) dμ j (θ) μ j x j (s)ds, Ψ (x) = p m {σ } x max,j {a j } x max,j {b j } e ετ j x j max,j {c j } μ j x j /4 x / max {b }. (7) (7) Recallg ha p (, ), by he boudedess of polyomal fucos, here exss a posve cosa K such ha Ψ(x) K. I addo, og ha ε (, λ), usgasmlar argume as (37)ad(39), we may esmae ha Hece, e εs x j (s τ j ) e ετ j x j (s)dsε e ετ j supξ j (u), u e εs x j (sθ) dμ j (θ) μ j x j (s)ds l V (x ()) l μ j λ ε sup ξ j (u). u l V (ξ ()) e ε l l ε K( e ε ) p θe kε l k e (k )ε l ((k ) ε) e ε l max,j {b j } ε e ετ j supξ j (u) μ j u (73) e ε l max,j {c j } λ ε sup ξ j (u), u (74)

12 Joural of Appled Mahemacs whch mples ha by equaly () l x () lm sup θeε l p. (75) Leg p, θ,adε gves he desred resul. 5. Exco Udersadg he mpacs of whe oses ad radom caasrophes o he exco of he populao s mpora bohpureadappledecology,adheformulaoof effecve coservao plas for hreaeed ad edagered speces (Lade 3; Glp ad Hask 3). I hs seco, he suffce codos for he exco of he sysem (5) are esablshed. We show ha f he esy of whe oses, he magude of caasrophes, or he frequecy of dsasers s suffcely large, he soluo o sysem (5) wll become exc wh probably, alhough he soluo o he orgal deermsc model () may be persse. For laer applcaos, le us ce a srog law of large umber for local margales (see, e.g., 33) as he followg lemma. Lemma. Le M(),,bealocalmargalevashg a me ad defe ρ M () := d M (s),, (76) (s) where M () := M, M () s Meyer s agle bracke process. The M () lm = a.s. provded ha lm ρ M () < Remark. Suppose ha Ψ loc ad for Ψ Ψ loc, a.s. := {Ψ (, z) Ψ(, z) s predcable, Ψ (s, z) (du) ds < } (77) (78) M () := Ψ (s, z) N (ds, du). (79) The, by, for example, Kua 34,Proposo.4, M () = Ψ (s, z) (du) ds. (8) Theorem. Le Assumpos ad hold. Assume moreover ha he followg codos are sasfed: () here exss a fuco γ d : R (, ) such ha for each, ) γ d (, u) γ (, u) δ(, u), u, =,,...,, () here exss a cosa <p<such ha κ Γ (8) > (max,j {a j }max,j {b j }max,j {c j }) ( p)m {σ }. (8) The, for ay al daa ξ A,hesoluoof (5) sasfes l x () lm sup η a.s., (83) where η s a posve cosa, ad κ=m {β β j b b j,,j}, (84) Γ= max { l ( γ d (, u)) (du)} ; (85),ω amely, he populao wll decay expoeally wh probably. Proof. Recall from (58)ha where d l V (x) = x V (x) (b = Z () Z () a j x j c j b j x j ( τ j ) x j (θ) dμ j (θ)) l ( = x γ (, u) ) (du)d V (x) Z () dw () Z () dw () l ( = x ( )γ (, u) V (x ( ) N (d, du), )) Z () = β V (x) x (), = Z () = σ V (x) j x () x j (). = (86) (87)

13 Joural of Appled Mahemacs 3 Iegrag boh sdes of hs equaly yelds l V (x) = l V (ξ ()) x V (x) (b = a j x j c j b j x j (s τ j ) x j (sθ) dμ j (θ)) l ( = x γ (s, u) ) (du) V (x) Z (s) Z (s)ds M () M () M 3 (), (88) where M (), =,, 3, are local margales defed, respecvely, as follows: M () = Z (s) dw (s) = M () = Z (s) dw (s), M 3 () = β V (x) x (s) dw (s), = l ( = x (s )γ (s, u) V (x (s ) N (ds, du) )) = l ( H (x (s ),s,u)) N (ds, du) wh M () =, =,, 3,whereH s defed by (89) H (x,, u) = = x γ (, u). (9) V (x) The quadrac varao of he couous local margale M () s M,M () = Z (s) ds. (9) For ay θ > ad each eger k, < p <,he expoeal margale equaly yelds P { sup M () p k M θ l k,m () p }. (9) kθ Sce k= k θ <, by he Borel-Caell lemma 3,..4, here exss a Ω Ωwh P(Ω )=such ha, for ay ω Ω, here exss a eger k(ω), whek > k(ω), ad k k, M () p Z θ l () (s) ds. (93) p Clearly, x V (x) b V (x)( β x ) = = = = V (x) (x b x x ββ x) V (x) (x (b b )x x ββ x) =: V Kx, (x)x (94) where =(,,...,)ad K=ββ b b.noehahe jh eleme of he marx K s β β j b b j κ by (84). By he equaly (), for x R,seasyofdha Hece x Kx κ x κ V (x). (95) x V (x) b = V (x)( β x ) κ =. (96) By (8) ad Jese s equaly, we may oba ha l ( = x γ (s, u) ) (du) V (x) l ( γ d (s, u)) (du). (97) Ths, ogeher wh (93), (96), ad (66) (69), gves from (88) ha l V (x) l V (ξ ()) Φ (x (s)) ds where θ l () p M () M 3 () max,j {b j } x j (s τ j ) x j (s)ds max,j {c j } Φ (x) = p m {σ } x x j (sθ) dμ j (θ) x j (s)ds, (98) (max,j {a j } max,j {b j } max,j {c j }) x κ max { l ( γ d (, u)) (du)}.,ω (99)

14 4 Joural of Appled Mahemacs Le A= p m {σ }, B=(max,j {a j } max,j {b j } max,j {c j }), C= κ max { l ( γ d (, u)) (du)}.,ω The, Φ(x) s deoed by () Φ (x) =A x B x C. () Recall ha <p<,soa<.sceb ad C>,by (8), we may oba ha he dscrma Δ=B 4AC<. Hece, here exss a posve cosa η>such ha Therefore, we oba from (98) ha l V (x) l V (ξ ()) η Φ (x) η, x R. () θ l () p max,j {b j } x j (s τ j ) x j (s)ds max,j {c j } M () M 3 (). x j (sθ) dμ j (θ) x j (s)ds By (8) ad he defo of H,forx R,weobaha Ths mples ha (3) H (x,, u) δ(, u). (4) l ( H(x,, u)) l ( δ(, u)). (5) The, by (H), here exss a posve cosa K such ha M 3,M 3 () = l ( H(x (s),s,u)) (du) ds O he oher had, M,M () = Z (s) ds l ( δ(, u)) (du) ds K. ( = β x (s)) V (x) ds max {β }. (6) (7) Hece, by Lemma,wehave M lm () = a.s. =,3. (8) I addo, we may esmae ha from (49) ad(5) x j (s τ j ) x j (s)dsτ j sup ξ j (u), u x j (sθ) dμ j (θ) x j (s)ds μ j λ sup ξ j (u), u (9) whch, ogeher wh (8), mply l V lm sup (x ()) η a.s. () Cosequely, by equaly ()wecompleeheproofofhs Theorem. 6. Dscusso Tradoally, he populao dyamcs are modeled by he deermsc models, whch assume ha parameers he model are all deermsc rrespecve of evromeal flucuaos. However, populao sysems he real word are ofe evably affeced by evromeal oses, whch are mpora facors a ecosysem. I parcular, he populao may suffer sudde aboc caasrophes such as earhquakes, suam, hurrcaes, floods, ad fre. I s herefore useful o reveal how he evromeal oses affec he populao sysems. I hs paper, we cosder he effec of jump-dffuso radom evromeal perurbaos o he asympoc properes ad exco of delay Loka-Volerra populao dyamcs. From codo (8) Theorem, we ca observehafesesβ, σ of whe oses, he radom dowward jump magude γ (, u), oresy (average rae of jump eves arrval) s suffcely large, he speces wll be exc. Compared wh he former resuls 7, hs resul gves a eresg ad mpora codo uder whch he freque aure dsasers ca force he populao o become exc. Recallg proeco of wldlfe rare speces, he souhwes rego Schua Cha, s bes-preserved pada haba o earh, whch belogs o Logme Sha acve faul zoe. Boh he 8 M8. Wechua ad he 3 M7. a a earhquakes occurred hs rego. Earhquakes ad secodary dsasers caused by he earhquake such as mudsldes, ladsldes, barrer lake, ad oher geologcal dsasers may desroy a lo of woods ad bamboo, whch s aural habaofedageredwldlfespeces.becausehepopulao does o have eough me o adjus o hese sudde ad severe evromeal perurbaos, he deah rae of hese edagered wldlfe speces may crease suddely ad grealy. I hs crcumsace, γ (, u) represes radom loss magude of he h speces caused by earhquake ad sadsforheesyofoccurreceofearhquake(average

15 Joural of Appled Mahemacs 5 rae of earhquake arrval). By Theorem, f he earhquake desruco ad he frequecy of occurrece are large eough such ha codo (8) s sasfed, he he wldlfe speces face exco ad we should do somehg o avod s exco. Coflc of Ieress The auhors declare ha here s o coflc of eress regardg he publcao of hs paper. Ackowledgmes Ths work s suppored by he Naoal Naural Scece Foudao of Cha (3748, 7) ad he Fudg Projec for Academc Huma Resources Developme Isuos of Hgher Learg uder he Jursdco of Bejg Mucpaly (PHR73). Refereces K. Gopalsamy, Global asympoc sably Volerra s populao sysems, Joural of Mahemacal Bology, vol.9,o., pp.57 68,984.. Kuag ad H. L. Smh, Global sably for fe delay Loka-Volerra ype sysems, JouralofDfferealEquaos, vol. 3, o., pp. 46, H. Berekeoglu ad I. Győr, Global asympoc sably a oauoomous Loka-Volerra ype sysem wh fe delay, Joural of Mahemacal Aalyss ad Applcaos,vol.,o.,pp.79 9, X.-Z. He, The Lyapuov fucoals for delay Loka-Volerraype models, SIAMJouraloAppledMahemacs,vol.58,o. 4, pp. 36, Z. Teg ad L. Che, Global asympoc sably of perodc Loka-Volerra sysems wh delays, Nolear Aalyss: Theory, Mehods & Applcaos,vol.45,o.8,pp.8 95,. 6 T. Fara, Sharp codos for global sably of Loka-Volerra sysems wh dsrbued delays, Joural of Dffereal Equaos,vol.46,o.,pp ,9. 7 F. Che, Global asympoc sably -speces oauoomous Loka-Volerra compeve sysems wh fe delays ad feedback corol, Appled Mahemacs ad Compuao,vol.7,o.,pp ,5. 8 K. Gopalsamy, Sably ad Oscllaos Delay Dffereal Equaos of Populao Dyamcs, vol.74ofmahemacs ad s Applcaos, Kluwer Academc, Dordrech, The Neherlads, Kuag, Delay Dffereal Equaos wh Applcaos Populao Dyamcs, vol.9ofmahemacs Scece ad Egeerg, Academc Press, Boso, Mass, USA, 993. T. C. Gard, Perssece sochasc food web models, Bulle of Mahemacal Bology,vol.46,o.3,pp ,984. T. C. Gard, Sably for mulspeces populao models radom evromes, Nolear Aalyss: Theory, Mehods & Applcaos, vol., o., pp. 4 49, 986. R. Lade, Rsks of populao exco from demographc ad evromeal sochascy ad radom caasrophes, The Amerca Naurals,vol.4,pp.9 97, R. M. May, Sably ad Complexy Model Ecosysems, Prceo Uversy Press, A. Bahar ad X. Mao, Sochasc delay Loka-Volerra model, Joural of Mahemacal Aalyss ad Applcaos,vol.9,o.,pp ,4. 5 X. Mao, C. ua, ad J. Zou, Sochasc dffereal delay equaos of populao dyamcs, Joural of Mahemacal Aalyss ad Applcaos,vol.34,o.,pp.96 3,5. 6 F. Wu ad. Xu, Sochasc Loka-Volerra populao dyamcs wh fe delay, SIAM Joural o Appled Mahemacs, vol.7,o.3,pp ,9. 7 F. Wu ad G., Evromeal ose mpac o regulary ad exco of populao sysems wh fe delay, Joural of Mahemacal Aalyss ad Applcaos,vol.396,o., pp ,. 8 M. Lu ad K. Wag, Global asympoc sably of a sochasc Loka-Volerra model wh fe delays, Commucaos Nolear Scece ad Numercal Smulao, vol.7,o.8,pp ,. 9 F. B. Haso ad H. C. Tuckwell, Perssece mes of populao wh large radom flucuaos, Theorecal Populao Bology,vol.4,o.,pp.46 6,978. F. B. Haso ad H. C. Tuckwell, Logsc growh wh radom desy depede dsasers, Theorecal Populao Bology, vol.9,o.,pp. 8,98. F. B. Haso ad H. C. Tuckwell, Populao growh wh radomly dsrbued jumps, Joural of Mahemacal Bology, vol.36,o.,pp.69 87,997. J. Bao, X. Mao, G., ad C. ua, Compeve Loka- Volerra populao dyamcs wh jumps, Nolear Aalyss: Theory, Mehods & Applcaos, vol.74,o.7,pp ,. 3 J. Bao ad C. ua, Sochasc populao dyamcs drve by Lévy ose, Joural of Mahemacal Aalyss ad Applcaos, vol. 39, o., pp ,. 4 F. Campllo, M. Joades, ad I. Larramedy-Valverde, Sochasc modelg of he chemosa, Ecologcal Modellg, vol., pp ,. 5 M. Lu ad K. Wag, Dyamcs of a Lesle-Gower Hollg-ype II predaor-prey sysem wh Lévy jumps, Nolear Aalyss: Theory, Mehods & Applcaos,vol.85,pp.4 3,3. 6 F. B. Haso, Boecoomc model of he Lake Mchga alewfe fshery, Caada Joural of Fsheres ad Aquac Sceces,vol. 44, pp , X. Mao ad C. ua, Sochasc Dffereal Equaos wh Markova Swchg, Imperal College Press, Lodo, UK, 6. 8 R. Z. Khasmsk, Sochasc Sably of Dffereal Equaos, vol. 7 of Moographs ad Texbooks o Mechacs of Solds ad Fluds: Mechacs ad Aalyss, Sjhoff & Noordhoff, Alphe aa de Rj, The Neherlads, X. Mao, A oe o he LaSalle-ype heorems for sochasc dffereal delay equaos, Joural of Mahemacal Aalyss ad Applcaos,vol.68,o.,pp.5 4,. 3 R. B. Ash, Probably ad Measure Theory, Harcour/Academc Press, d edo,. 3 R. Lade, Geecs ad demography bologcal coservao, Scece, vol. 4, pp , M. Glp ad I. Hask, Eds., Meapopulao Dyamcs: Emprcal ad Theorecal Ivesgaos, Academc Press, New ork, N, USA, 99.

16 6 Joural of Appled Mahemacs 33 R. Sh. Lpser, A srog law of large umbers for local margales, Sochascs,vol.3,o.3,pp.7 8, H. Kua, Iô s sochasc calculus: s surprsg power for applcaos, Sochasc Processes ad her Applcaos, vol., o. 5, pp. 6 65,.

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