Analysis of a Stochastic Lotka-Volterra Competitive System with Distributed Delays

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1 Ieraoal Coferece o Appled Maheac Sulao ad Modellg (AMSM 6) Aaly of a Sochac Loa-Volerra Copeve Sye wh Drbued Delay Xagu Da ad Xaou L School of Maheacal Scece of Togre Uvery Togre 5543 PR Cha Correpodg auhor = = Abrac I h paper we coder a -pece ochac LoaVolerra copeve ye wh drbued delay Suffce codo for ably he ea ad exco of each populao are eablhed dx ( ) = x ( )[ r a x ( ) b x ( ) du ( )]d Keyword-copeve ye; exco; delay; ochac perurbao where r τ a b are pove coa σ are o- σ x ( )db ( ) () = I egave coa he fuco u ( ) a probably eaure ITRODUCTIO o [ ] have For he la decade dffereal equao wh delay have receved grea aeo owg o her exeve applcao a odel a varey of cefc area [] For exaple-he delay dffereal equao For plcy here we roduce he followg oao: Le e I J I are oepy e where I = { 3 } I = {l l l } ad afe J I = I = I J ad l < l < < l f = l up f ( ) f = l f f ( ) f ( ) = () f ()d R = {( x x x ) x > = } We ca eay o ee ha ye () ha o pove equlbru Hece a ereg ad pora o udy ha of wheher ye () ll ha oe ably aroud oe pove po I h paper we hall how MAI RESULTS II B ( ) B ( )) o odel he above oral drbuo eg Fr we roduce oe lea whch are ueful ad pora r r σ db ( ) where σ deoe he ey of he Lea ([4]) Suppoe ha z ( ) C (Ω R ) () If here ex wo pove coa T ad u uch ha oe ad B ( ) a adard Browa oo defed o a coplee probably pace (Ω F P) I h paper we hall coder a ochac copeve ye wh couou delay: 6 The auhor - Publhed by Ala Pre I pracce we uually eae he growh rae r by a error er plu a average value Accordg o he faou ceral l heore he error er ay be approxaed by a oral drbuo A he ae e he oe o r ay or ay o correlae o each oher Tag hee poble correlao o accou we hece ue depede Weer procee ( B ( ) = ( ) = = x (θ ) = φ (θ ) > θ [ τ ] = where φ (θ ) a couou fuco o [- τ ] However he real world populao ye are ofe ubec o evroeal oe May [] ha poed ou ha due o evroeal oe he growh rae populao ye hould be ochac[3-7] I [3] Meg Lu ad Ke Wag uded a delay Loa-Volerra copeve ye wh rado perurbao; a follow: The al codo of ye () dx( ) = x( )[r ax( τ )] d dx () = x ( ) [ r a x ( ) a x ( )] d σ x ( )db ( ) dx ( ) = x () [ r a x ( ) a x ()] d σ x ()db () τ du l z( ) λ u z()d α B ( ) for all T where α a coa he 49 =

2 λ z () af λ u l z( ) a f λ = < () If here ex hree pove coa λ T ad u uch ha l z ( ) λ u zd ( ) α B( ) for all T have = λ z () a u Lea For ay gve al value φ θ) = ( φ ( θ) φ ( θ) φ ( )) ( θ C([ ]R ) ye () have a uque global pove oluo ( ( ) ( ) ( )) rea x x x o ad he oluo wll R wh probably ad he oluo afe l x ( ) a = Proof: Our proof ovaed by he paper of [5] Sce he coeffce of ye () are locally Lpchz couou for ay gve al value φ ( θ) C ([ ] R ) here a uque axal local oluo x() o [ τ e ) where τ e he exploo e So we oly eed o how ha e = The proof lar o [6] by defg V( x)= ( x l x ) = Appled ô forula o V(x) have dv ( x) = ( x )[ r a x b x ( ) du ( )] τ = σ db () 5 σ = σ db () 5 σ Kd σ db () = r a x a x b x ( ) du ( ) where K a pove coa By he lar proof of [6] we ca oba ye () have a uque globally pove oluo ( x x x ) o τ we oed here For I coder he followg queo: y() θ = x() θ θ [- τ ] By vrue of he faou ochac coparo heore we ca oba ha y( ) x( ) he y ( ) have he followg explc repreeao repecvely exp{ r ( )} σ σ B y () = a r B d exp{ ( )} x () σ σ l ( ) The proof of x a = lar o he ehod of lea 3 [5] If r 5 σ > by () of lea ad (3) we have = r 5 l y( ) = a = a Therefore: σ = l ( ) ( ) l ( ) ( ) ( ) y p dpdu y p dp y p dp du = a he ogeher wh (3) have l x( p) dpdu( ) = a For coveece (4) = (5) Le R = r 5 σ A = ( a b ) B = ( a ) ll b ll = a b a b R a b a b a b a b R a b a b M = a b a b R a b a b = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) a b a( b( a( b( a b ) ) ) ) a b a b a b a b ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) A = ( ) a b a b a b a b ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) a b a b a b a b ( ) ( ) ( ) ( ) = = [ ] dy () y () r a y () d σ y () db () (3) 5

3 a b a b R a b a b ll ll ll ll l ll ll ll ll a b a b R a b a b ll ll ll ll l ll ll ll ll = a b a b R a b a b ll ll l l l l l l l l l l l ll = a b a b a b a b a b a b a b a b = ( ) a b a b a a b a b a b a b l l l l l l l l l l l l l l l l = ll ll ll ll ll ll ll ll l l l l ll ll l l l l ll ll b l l l l l l l l l l a l l l b l l l Lea 3 I pora o po ou ha f R > ad A > he M < = ca hold ulaeouly Proof If h aero o rue he M < = he we eay oba M = RA < becaue of a b are = pove coa he ( a b ) M = R( a b ) AK < = herefore we eay ee ha ( a b ) M = R( a b ) A = = = = = = R ( a b ) A = R A< o A < he coradco are Therefore f R > ad A > he M < ( = ) ca hold ulaeouly Theore () Suppoe R > ad A > = (a) If M > A > ad A ( ) he x able e average a l x ( ) = M / A a = (b) Suppoe M l > ad M < l I J f B > > > ad < ( = ) he x goe o exco alo urely ad xl able e average a l x ( ) = / B a l I = l () Suppoe R < he x goe o exco alo urely l x ( ) = a = () Suppoe R l > ad R < l I J f B > > > ad < ( = ) he xl able e average a ad x goe o exco alo urely l xl( ) = / B a l I = Proof: Applyg Iô' forula o () have l x/ x() = R b x ( p) dp x ( p) dp du ( ) So τ B () b x() a x() σ l x ( ) / x() l x ( ) / x () A A R A = = = A b x ( pdp ) x( ) ( ) pdpdu = ( a b) A A ( a b) x( ) = B () B() A σ A σ = = M A( a b) x( ) = A b x ( pdp ) x( ) pdp du () τ = A b x( pdp ) x( pdpdu ) ( ) A b x( pdp ) x( pdpdu ) ( ) B () B () A σ A σ = B () B() () σ σ = A b x ( pdp ) x( ) ( ) pdpdu = = M A x A A A b x( pdp ) x( pdpdu ) ( ) (a) By Lea ad (5) for arbrary > here a T > uch ha T have l x( ) / x() A < /3 = (6) (7) 5

4 A b x pdp x pdpdu < = ( ) ( ) ( ) /3 A b x( pdp ) x( pdpdu ) ( ) < /3 Subug he above equale o (7) we ca oba l x ( ) / x () A M A x A () σ = B () Sce M > ad A > he by vrue of lea ad he arbrare of we ge (8) x () M / A (9) Accordg o (6) ad (9) we eay ee ha l x( ) / () ( ) x a b M R ( a b) x( ) A σb () ( a ) B () b M = ( a b ) x ( ) σ A () By vrue of Lea have x() M / A a ogeher wh (9) we ca oba l x ( ) = M / A a = (b) Sce M < he we ca chooe uffcely all uch ha M < by (8) ad Lea we eay ge l x ( ) = J The for arbrary > here at > uch ha T have < < < < < b x ( p) dp x ( p) dp du ( ) < bl ( ) ( ) x dul J l x ( ) J ll l l ll = The we ubug he above equale o (6) we ca oba ha T l xl ( ) R ( a b ) x () ( a b ) x () l ll ll l ll ll l = σ lb () Appled he lar way of (7) o () we ca ge () l xl ( ) / x () l ( ) / () l x l x l = ( ) R [( a b ) ( a b ) ] x( ) l ll ll ll ll = = B() B() σl σl = B () B xl () σ = B () σl = I he ae way by () we have l x ( ) / x () l x ( ) / x () l l l l = B () ( Rl ) σl = ll ll l l l l l = B () σl = B () B xl () σ = B () σl = (( a b ) ( a b ) ) x ( ) () (3) By vrue of lea ad he arbrare of we ca oba fro () ad (3) ha l x ( ) = / B a l I = l () Suppoe R < = by (6) we have l x ( ) / () () x B R σ = I follow fro l B ( ) = a = ad R < lup l x( ) R a < = Therefore l x ( ) = a = () Suppoe R l > ad R < l I J = by he proof of () we ca how ha f R < he l x ( ) = a J ex he proof lar o ha of ()(b) hece we oed here III COCLUSIOS Th paper devoed o udy he dyac of a ochac copeve ye wh drbued delay Suffce codo for ably he ea ad exco of each populao are eablhed Thee reul are ueful ad 5

5 pora for udy pece coex Soe ereg equao deerve furher o udy REFERECES [] Y Kuag Delay dffereal equao wh applcao populao dyac Boo: Acadec Pre 993 [] R M May Sably ad Coplexy Model Ecoye Prceo Uv Pre [3] M Lu K Wag A oe o a delay Loa-Volerra copeve ye wh rado perurbao Appl Mah Le 6 (3) [4] M Lu K Wag Survval aaly of a ochac cooperao ye a pollued evroe J Bol Sye 9 () 83-4 [5] X R Mao G Maro E Rehaw Evroeal Browa oe uppre exploo populao dyac Sochac Procee ad her Applcao 97() 95- [6] M Lu Wag Q Wu Survval Aaly of Sochac Copeve Model a Pollued Evroe ad Sochac Copeve Excluo Prcple Bull Mah Bo l 73 () 969- [7] M Lu C Z Ba A rear o ochac Logc odel wh dffuo J Appled Maheac ad Copuao 8 (4)