Research Article The Stationary Distribution and Extinction of Generalized Multispecies Stochastic Lotka-Volterra Predator-Prey System

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1 Hidawi Publishig Corporaio Maheaical Probles i Egieerig Volue 5, Aricle ID 47936, pages hp://dx.doi.org/.55/5/47936 Research Aricle The Saioary Disribuio ad Exicio of Geeralized Mulispecies Sochasic Loka-Volerra Predaor-Prey Syse Facheg Yi ad Xiaoya Yu CollegeofScieces,HohaiUiversiy,Najig98,Chia Correspodece should be addressed o Facheg Yi; yfc586@sia.co Received 9 May 5; Revised Augus 5; Acceped 3 Augus 5 Acadeic Edior: Leoid Shaikhe Copyrigh 5 F. Yi ad X. Yu. This is a ope access aricle disribued uder he Creaive Coos Aribuio Licese, which peris uresriced use, disribuio, ad reproducio i ay ediu, provided he origial work is properly cied. This paper is cocered wih he exisece of saioary disribuio ad exicio for ulispecies sochasic Loka-Volerra predaor-prey syse. The coribuios of his paper are as follows. (a) By usig Lyapuov ehods, he sufficie codiios o exisece of saioary disribuio ad exicio are esablished. (b) By usig he space decoposiio echique ad he coiuiy of probabiliy, weaker codiios o exicio of he syse are obaied. Fially, a uerical experie is coduced o validae he heoreical fidigs.. Iroducio The dyaic relaioship bewee he predaors ad he preyshaslogbeeadwillcoiueobeoeofhe doia hees i boh ecology ad aheaical ecology due o is uiversal exisece ad iporace []. The classic predaor-prey odel is he Loka-Volerra odel, govered by he followig differeial equaio: x=x(a by), y = y ( c + fx), where x() ad y() deoehepreyadpredaorpopulaio size, respecively, a ie. For he prey copoe, he paraeers a ad b are he fixed growh ad oraliy raes, respecively. For he predaor copoe, he paraeers c ad f are he fixed growh ad oraliy raes, respecively. Sice he, varias of he wo-species Loka-Volerra syse have bee frequely ivesigaed o describe populaio dyaics wih predaor-prey relaios; see, for exaple, [ 4]. Recely, he ulispecies predaor-prey syses have received a grea deal of research aeio sice hey ook he differeces aog idividual growh ad oraliy io accou (see [5 8]). I order o udersad he aure of he copeiive ieracios ad relaioships bewee predaor () ad prey, Yag ad Xu [8] cosidered he followig periodic -prey ad -predaor Loka-Volerra differeial syse wih periodic coefficies: dx i =x i (b i () dx i =x i ( b i () + a ij () x j )d, a ij () x j j=+,...,, a ij () x j )d, i=+,...,, where x i (),,...,, deoes he desiy of prey species a ie ad x i (), i = +,...,, deoes he desiy of predaor species a ie. Uder he assupio ha b i () > (i =,...,), b i () (i = +,...,), a ii () > (i =,...,), a ij (i = j) are coiuous periodic fucios wih a coo periodic T>,aseofsufficie codiios o he exisece ad global araciveess of he periodic soluio o syse () are obaied. Recely, Che ad Shi [5] furher cosidered he alos periodic case of ore coplicaed syses ha syse () uder he alos periodic case. By cosrucig a suiable Lyapuov fucio, hey obaied a se of sufficie codiios which guaraees he exisece of a uique globally aracive posiive alos periodic soluio o he correspodig syse. ()

2 Maheaical Probles i Egieerig O he oher had, fro he biological poi of view, populaio syses i he real world are ieviably affeced by eviroeal oise. I he pas decades, he dyaics of sochasic populaios ad relaed opic have received a grea deal of research aeio (see [9 ]), sice hey have bee successfully used i a variey of applicaio fields, icludig biology (see [ 8]), epideiology (see [9, 3]), ad eural eworks (see [3 33]). More recely, he asypoic properies of sochasic predaor-prey syses have received a lo of aeio; he readers ca refer o [,, 34] ad he refereces herei. For exaple, he dyaics of he desiy depede sochasic predaor-prey syse wih differe fucioal respose have bee sudied by Ji ad Jiag i [, ]. Vasilova [34] has ivesigaed a sochasic Gilpi-Ayala predaor-prey odel wih iedepede delay, ad cerai asypoic resuls regardig he log-ie behavior of rajecories of he soluio ad sufficie crieria for exicio of species for a special case of he cosidered syse are give. I his paper, cosiderig he effec of eviroeal oise, we iroduce sochasic perurbaio io he growh rae of he prey ad he predaor i syse () ad assue ha paraeers b i ad a ij are cosa. The we obai he followig -prey ad -predaor sochasic Loka- Volerra syse wih cosa coefficies: dx i =x i (b i dx i =x i ( b i + a ij x j )d+σ i x i db i (), +σ i x i db i (), a ij x j j=+ a ij x j )d i=+,...,,,...,, where B() = (B (), B (),...,B ()) is a -diesioal Browia oio ad σ i will be called he oise iesiy. Throughou his paper, we always assue ha he followig hypohesis holds: b i >, b i, a ii >,,...,, i=+,...,,,...,, a ij (i=j). I he sudy of sochasic populaio syses, exicio ad exisece of saioary disribuio are wo ipora ad ieresig properies, respecively, eaig ha he populaio syse will die ou or he disribuio of he soluio coverges weakly o he probabiliy easure i he fuure, which have received a lo of aeio (see [, 35 37]). The oe quesio arises aurally: uder wha codiio ca syse (3) have a saioary disribuio ad becoe exic, respecively? This issue cosiues he firs oivaio of his paper. (3) (4) I addiio, he exisig lieraures (see [, 35]) show clearly ha if he oise iesiy of every prey species is ore ha wice he correspodig irisic growh rae, he populaio will becoe exic expoeially. The oe ieresig quesio is as follows: Wha will happe if he oise iesiy equals wice he irisic growh rae? Thus, he secod purpose of his paper is o solve his ieresig proble. The orgaizaio of he paper is as follows. Secio describes soe preliiaries. The ai resuls are saed i Secios 3 ad 4. I Secio 3, sufficie codiios are obaied uder which here is a saioary disribuio o syse (3). By uilizig soe ovel sochasic aalysis echiques, sufficie crieria for esurig he exicio of syse (3) are obaied i Secio 4. Secio 5 provides soe uerical exaples o check he effeciveess of he derived resuls. Coclusio is ade i Secio 6.. Noaio Throughou his paper, uless oherwise specified, le (Ω, F,{F }, P) be a coplee probabiliy space wih a filraio {F } saisfyig he usual codiios (i.e., i is icreasig ad righ coiuous while F coais all P-ull ses). Le B() = (B (), B (),...,B ()) be a -diesioal Browia oio defied o he probabiliy space. If A R is syeric, is larges ad salles eigevalues are deoed by λ ax (A) ad λ i (A).Lex =(x,x,...,x ) be he posiive equilibriu of he correspodig deeriisic predaor-prey syse o syse (3), ha is, he soluio o he followig equaio: b i + a ij x j b i j=+ a ij x j =,,,...,, a ij x j =, i=+,...,. (5) I he sae way as Zhu ad Yi [38] ad Liu e al. [39] did, we ca also show he followig resul o he exisece of global posiive soluio. Lea. Suppose ha codiio (4) holds; he oe has he followig asserios: (i) For ay give iiial value x R +,hereisauique soluio x(, x ) o syse (3) ad he soluio will reai i R + wih probabiliy ; aely, P {x (, x ) R +, } =, (6) for ay x R +. (ii) For ay give iiial value x R + ad ay β>, alos every saple pah of x β (, x ) is locally bu uiforly Holder coiuous. Lea (see [4]). Le f() be a oegaive fucio defied o [, + ) such ha f() is iegrable o [, + ) ad is uiforly coiuous o [, + );heli f() =.

3 Maheaical Probles i Egieerig 3 3. Saioary Disribuio I his secio, we aily show ha syse (3) has a saioary disribuio. Le us give a lea ha will be used i he followig proof. Le X() be a hoogeeous Markov process i E R described by he followig sochasic equaio: dx () =b(x) d + The diffusio arix is d k= A (x) =(a ij (x)), a ij (x) = σ k (X) db k (). (7) d k= σ i k (x) σj k (x). (8) Lea 3 (see [4]). Oeassueshahereisaboudedope subse G E wih a regular (i.e., sooh) boudary such ha is closure G E,ad (i) i he doai G ad soe eighborhood herefore, he salles eigevalue of he diffusio arix A(x) is boudedawayfrozero; (ii) if x E \G, he ea ie τ a which a pah issuig fro x reaches he se G is fiie, ad sup x K E x τ<+ for every copac subse K E ad hroughou his paper oe ses if =. Oe he has he followig asserios: () The Markov process X() has a saioary disribuio μ( ) wih desiy i E. () Le f(x) be a fucio iegrable wih respec o he easure μ( ).The P { li f (x (s)) ds = f(y)μ(dy)}=. (9) E Reark 4. The proof is give by [4] i deail. Precisely, he exisece of a saioary disribuio wih desiy is obaied i Theore 4.3 o pp. 7. The ergodic propery is referred o Theore 4. o pp.. To validae (i), we ca direcly show ha λ i {A(x)} >. To validae (ii), i suffices o prove ha here is soe eighborhood U ad a oegaive C - fucio V such ha, for ay x E \U, LV(x) is egaive (for deails refer o [4], pp. 63). Theore 5. Le codiio (4) hold ad le x(, x ) be he global soluio o syse (3) wih ay posiive iiial value x R +. Assue ha here exiss c=(c,c,...,c ) such ha c i a ii (c i a ij +c j a ji )>, c i x i σ i,,...,, { < i (c i { i a ii (c i a ij +c j a ji )) (x } j ). } { } The here is a saioary disribuio for syse (3). () Proof. Le x i () = x i (, x ) for sipliciy. Applyig Iô s forula o V(x) = c i(x i x i x i l(x i /x i )) yields LV = c i a ii (x i x i ) + j=+ i=+ c i a ij (x i x i )(x j x j ) c i a ij (x i x i )(x j x j ) c i a ij (x i x i )(x j x j ) i=+ j=+ + c i x i σ i. c i a ij (x i x i )(x j x j ) By he iequaliy ab (a +b ),wehave LV i a ii (x i x c i ) + c i a ij (x i x i ) i=+ j=+ i=+ j= j=+ j=+ i=+ i=+ c i a ij (x j x j ) c i a ij (x i x i ) c i a ij (x j x j ) c i a ij (x i x i ) c i a ij (x j x j ) c i a ij (x i x i ) i a ij (x j x c j ) + c i x i σ i = i a ii (x i x c i ) + c i a ij (x i x i ) ()

4 4 Maheaical Probles i Egieerig + c j a ij (x i x i ) + c i x i σ i Theore 7. Le codiio (4) hold ad le x(, x ) be he global soluio o syse (3) wih ay iiial value x R +. Assue ha here exiss a ieger k such ha = [ c i a ii (c i a ij +c j a ji ) ] (x i x i ) [ ] + c i x i σ i. () b i < σ i,,...,k, b i = σ i, i=k+,...,. Oe he has he followig asserios: (5) Noe ha (); he he ellipsoid [ c i a ii (c i a ij +c j a ji ) ] (x i x i ) [ ] = c i x i σ i (3) lies eirely i R +.NowwecaakeU o be a eighborhood of he ellipsoid wih U E =R +,suchha,forx R + \U, LV <, which eas codiio (ii) of Lea 3 is verified. Now we begi o verify codiio (i) i Lea 3. Le us defie H(x) = diag(σ x,σ x,...,σ x ),sohe diffusio arix is A(x) = H T (x)h(x). Iisclearha λ i {H T (x)h(x)}.ifλ i {H T (x)h(x)} = holds, here exiss ξ = (ξ,ξ,...,ξ ) T R such ha ξ = ad ξ T H T (x)h(x)ξ =, which iplies ha H(x)ξ =. Byhe defiiio of σ i, i =,,...,,adx R + \U,weseeξ=, bu i coradics wih ξ =. Soλ i {H T (x)h(x)} > for x R + \Uus hold. Tha eas codiio (i) of Lea 3 is verified. Therefore, we ca say ha sochasic syse (3) has a saioary disribuio. Reark 6. Theore 5 shows ha syse (3) has a uique saioary disribuio whe he perurbaio is sall i he sese ha c i x i σ i { < i (c i { i a ii (c i a ij +c j a ji )) (x } j ). } { } 4. Exicio (4) Exicio is oe of he os basic quesios ha ca be sudied i he populaio dyaics, which eas he populaio syse will die ou. Mos of he ie we eed o kow he exicio rae of he species for which we have o ake a suiable policy i advace ad o ake useful easures o proec he fro becoig exic. (i) For,...,k,hesoluiox i (, x ) o syse (3) has he propery ha li i (, x ) =b i σ i a.s. (6) Tha is, for each i =,...,k, he species i will becoe exic expoeially wih probabiliy oe ad he expoeial exicio rae is (σ i / b i ). (ii) For i=k+,...,,hesoluiox i (, x ) o syse (3) has he propery ha li x i (, x )=, li i (, x ) = a.s. (7) Tha is, for each i = k +,...,,hespeciesi sill becoes exic wih zero expoeial exicio rae. (iii) For i=+,...,,hesoluiox i (, x ) o syse (3) has he propery ha Li i (, x ) = b i σ i a.s. (8) Tha is, for each i = +,...,,hespeciesi will becoe exic expoeially wih probabiliy oe ad he expoeial exicio rae is (σ i / + b i ). Proof. Le x i () = x i (, x ) for sipliciy. To ake he proof clear, we are goig o divide i io four seps. The firs sepadhehirdsepareoshowheleasupperboud of expoeial exicio rae for he op k preys ad he predaors of syse (3), respecively. The secod sep is o show he exicio for he boo kpreys of syse (3) i he case of σ i =b i, i=k+,...,.thefourhsepiso accoplish he proof of asserios (i) (iii) based o he proof of Seps 3.

5 Maheaical Probles i Egieerig 5 Sep. Applyig Iô s forula o i () yields i () = i () + (b i σ i )ds a ij x j (s) ds + M i (),,...,, (9) Sep. Theaiaiofhissepisoshowli x i () = a.s. for i=k+,...,.wheσ i =b i,()ursobehe followig for: i () = i () a ij x j (s) ds + M i (), i=k+,...,. (5) i () = i () + ( a ij j=+ a ij )x j (s) ds + ( b i σ i )ds+m i (), i=+,...,, () where M i () = σ idb i (s), i =,...,, is he real-valued coiuous local arigale vaishig a =,wihhe quadraic variaio M i (), M i () = σ i. Dividig boh sides of (9) ad () by,wehaveha Accordig o he covergece of x i (s)ds, wecadecoposehesaplespaceiowoexclusiveevesspacesas follows: J i, ={ω: x i (s) ds < }, J i, ={ω: x i (s) ds = }, i=k+,...,. (6) Oheoherhadwecadividehesaplespaceiohree uually exclusive eves as follows: i () = i () + (b i σ i )ds a ij x j (s) ds + M i (), () E i, ={ω: li sup x i () li if x i () =γ i >}, E i, ={ω: li sup x i () > li if x i () =}, (7) i () = i () + ( + a ij j=+,...,, a ij )x j (s) ds ( b i σ i )ds+ M i (), i=+,...,. () E i,3 ={ω: li x i () =}. Fro he above, he proof of li x i () = a.s. is equivale o show J i, E i,3 a.s. ad J i, E i,3 a.s. Now we give he process i wo pars. Par of Sep. Now we show J i, E i,3 a.s. I follows fro Lea ha alos every saple pah of x i () is locally bu uiforly Holder coiuous ad herefore alos every saple pah of x i (, x ) us be uiforly coiuous. Cosiderig he defiiio of J i, ad Lea, we obai Usighesroglawoflargeubersforarigales[43], we obai ha M li i () = a.s.,,...,. (3) For,,...,k,leig i () yields ha li sup i () li < a.s.,,...,k. (b i σ i )ds=b i σ i (4) li x i () = a.s., i=k+,...,, (8) which eas J i, E i,3 a.s. ParofSep.The ai of his par is o prove ha J i, E i,3 a.s. I is sufficie o show P(J i, E i, )=ad P(J i, E i, )=. If his P(J i, E i, )=is o rue, for ay ω i (J i, E i, ) ad ε i (,γ i /) here exiss T(ε i,ω i ) such ha >T(ε i,ω i ) x i () >γ i ε i > γ i, i=k+,..., a.s. (9)

6 6 Maheaical Probles i Egieerig Siple copuaios show ha a ij x j (s) ds = T a ij x j (s) ds + a ij x j (s) ds T T Tγ a ij x j (s) ds + a i ij a.s., i=k+,...,. Leig o boh sides of (3) yields li if This iplies ha a ij x j (s) ds > a ijγ i > a.s., i=k+,...,. (3) (3) BasedohehypohesisP(J i, E i, )>, we ca clai ha here exiss θ>such ha P(H θ )>. I is herefore o see ha, for ay ω H θ, a ij x j (s) ds = + a ij x j (s) ds H θ a ij x j (s) ds [,]\H θ a ij x j (s) ds H θ a ij θ (Bθ ) Leig o boh sides of (37) yields a.s., i=k+,...,. (37) li sup i () a ij γ i a.s., i=k+,...,, (3) which coradics wih he defiiio of J i, ad E i,.so P(J i, E i, )=us hold. Now we proceed o show P(J i, E i, )>is false. Now we eed ore oaios such as B ε := { s :x(s) ε}, h ε := (Bε ), h ε := li sup h ε, H ε := {ω J i, E i, :h ε >}, (33) where (B ε ) eas he legh of Bε. Fro he defiiio of H ε,wecaeasilygehah =J i, E i,.thefollowigis righ for ay ε <ε : which yields B ε B ε, (B ε ) (Bε ), h ε h ε, (34) a ij x j (s) ds which eas li sup i () a ij θh θ a ij θh θ a.s., i=k+,...,, (38) a.s., i=k+,...,. (39) This also coradics wih he defiiio of J i, ad E i,.i iediaely yields ha he asserio P(J i, E i, )=us hold. Now we ca clai ha P(J i, E i, )=ad P(J i, E i, )=,whicheasj i, E i,3. Cobiig wih he fac J i, E i,3 ad J i, E i,3,wehave li x i () = a.s., i=k+,...,. (4) Sep 3. I follows fro (4) ad (4) ha This iplies li x i () = a.s.,,...,. (4) li a ij x j (s) ds = a.s. (4) Now leig o boh sides of () yields h ε h ε, H ε H ε, (35) li sup i () b i σ i < a.s., i=+,...,. (43) ε <ε. Fro he coiuiy of probabiliy, we ca obviously ge P(H ε ) P(H )=P(J i, E i, ), as ε. (36) Sep 4. I is iediae fro (4) ad (43) ha li x i () = a.s.,,...,. (44)

7 Maheaical Probles i Egieerig 7 This iplies li a ij x j (s) ds = a.s.,,...,. (45) By akig lii o boh sides of () ad (), we have li i () =b i σ i a.s.,,...,k, li i () = b i σ i a.s., i=+,...,. So asserios (i) (iii) of Theore 7 us hold. (46) Reark 8. Noe ha, for =, syse (3) becoes he followig classic sochasic Loka-Volerra copeiive syses, which have received uch aeio (see [, 36, 38]): dx i =x i (b i a ij x j )d+σ i x i db i (), Ad codiio (4) becoes he followig for: b i >,,...,, a ii >,,...,, a ij (i=j).,...,. (47) (48) Thus, by Theore 7, we have he sufficie codiios o exicio for syse (45) as Corollary 9. Corollary 9. Le codiio (48) hold ad le x(, x ) be he global soluio o syse (47) wih ay iiial value x. Assue ha here exiss a ieger k such ha b i < σ i,,...,k, b i = σ i, i=k+,...,. (49) Oe he has he followig asserios: (i) For,...,k,hesoluiox i (, x ) o syse (47) has he propery ha li i () =b i σ i a.s. (5) Tha is, for each i =,...,k, he species i will becoe exic expoeially wih probabiliy oe ad he expoeial exicio rae is (σ i / b i ). (ii) For i=k+,...,,hesoluiox i (, x ) o syse (47) has he propery ha li x i () =, (5) li i () = a.s Figure : Copuer siulaio of x (), x (), adx 3 () geeraed by he Heu schee for ie sep Δ = 3 for syse (5) o [, ],respecively. Tha is, for each i=k+,...,,hespeciesi sill becoes exic wih zero expoeial exicio rae. By usig soe ovel sochasic aalysis echiques, we poi ou ha species i is sill exic whe σ = b i.i copariso wih Theore 4. i [], he codiios iposed o he exicio are weaker. 5. Exaple ad Siulaios I his secio, we prese a uerical exaple o illusrae he usefuless ad flexibiliy of he heore developed i he previous secio. Exaple. Cosider a 3-diesioal sochasic Loka- Volerra predaor-prey syse as follows: dx =x (.9.8x.x.4x 3 )d +σ x db (), dx =x (.8.3x.9x.5x 3 )d +σ x db (), dx =x 3 (. +.3x +.x.6x 3 )d +σ 3 x 3 db 3 (). (5) Syse (5) is exacly syse (3) wih =3, =, a =.8 >, a =. >, a 3 =.4 >, a =.3 >, a =.9 >, a 3 =.5 >, a 3 =.3 >, a 3 =. >, a 33 =.6 >, b =.9>, b =.8>,ad b 3 =. >. We copue ha he equilibriu (x,x,x 3 ) = (.97,.488,.88). The exisece ad uiqueess of he soluio follow fro Lea. O he codiio of he suiable paraeers, we ca ge he siulaio figures wih iiial value (x (), x (), x 3 ()) = (.6,.4,.4) by MATLAB. I Figures 3, he blue lie represes he populaio of x (),

8 8 Maheaical Probles i Egieerig Figure : Copuer siulaio of a sigle pah of x i (), i =,, 3, geeraed by he Heu schee for ie sep Δ= 3 for syse (5) o [, ],respecively Figure 3: Copuer siulaio of ( i ())/, i =,, 3, geeraed by he Heu schee for ie sep Δ= 3 for syse (5) o [, ],respecively. he gree lie represes he populaio of x (),adhered lie represes he populaio of x 3 (). (i) (σ,σ,σ 3 ) = (.,.,.5) :choosigc =, c =,c 3 =.5,wefurhercopueha c a (c a +c a +c a 3 +c 3 a 3 ) =.5 >, c a (c a +c a +c a 3 +c 3 a 3 ) =.35 >, c 3 a 33 (c 3a 3 +c a 3 +c 3 a 3 +c a 3 ) =. >, c a (ii) (σ,σ,σ 3 ) = (.38,.6,.5) :oehaσ =.944 > b =.8, σ =.6 = b =.6;byvirueof Theore 7, syse (5) is exicive. Fro Figure, we ca see ha he predaor populaio will die ou hough i suffers hesallwhieoisewhehepreypopulaiobecoes exic. Buwecaoseehevalueofheexicioraeof he hree populaios clearly. So we give Figure 3 o show (logx i ())/ for i =,, 3. Accordig o Theore 7, we ca copue ha, for i =, he growh rae is.5 (i is said ha he exicio rae is.5). By he sae ehod, we ca kow haheexiciorae is.fori =ad he exicio rae is.5 for i=3. [(c a +c a )(x ) +(c a 3 +c 3 a 3 )(x 3 ) ] = , c a [(c a +c a )(x ) +(c a 3 +c 3 a 3 )(x 3 ) ] =.6845, c 3 a 33 (53) 6. Coclusio This paper is devoed o he exisece of saioary disribuio ad exicio for ulispecies sochasic Loka- Volerra predaor-prey syse. Firsly, by applyig Lyapuov ehods, sufficie codiios for esurig he exisece of saioary disribuio of he syse are obaied. Secodly, soe ovel echiques have bee developed o derive weaker sufficie codiios uder which he syse is exicive. Fially, uerical experie is provided o illusrae he effeciveess of our resuls. [(c 3a 3 +c a 3 )(x ) +(c 3 a 3 +c a 3 )(x ) ] 3 = , c i x i σ i =.979 < i { ,.6845, }. i 3 By virue of Theore 5, syse (5) has a uique saioary disribuio. Coflic of Ieress The auhors declare ha here is o coflic of ieress regardig he publicaio of his paper. Ackowledges The projec repored here is suppored by he Naioal Sciece Foudaio of Chia (Gra os ad 746) ad he Fudaeal Research Fuds for he Ceral Uiversiies of Chia (Gra o. 3B64).

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