Application of inequalities technique to dynamics analysis of a stochastic eco-epidemiology model

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1 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 DOI.86/s z R E S E A R C H Open Access Applicaion of inequaliies echnique o dynamics analysis of a sochasic eco-epidemiology model Tao Feng,, Xinzhu Meng,,3*,LidanLiu and Shujing Gao * Correspondence: mxz76@sdus.edu.cn College of Mahemaics and Sysems Science, Shandong Universiy of Science and Technology, Qingdao, 6659, P.R. China Key Laboraory of Jiangxi Province for Numerical Simulaion and Emulaion Techniques, Gannan Normal Universiy, Ganzhou, 34, P.R. China Full lis of auhor informaion is available a he end of he aricle Absrac This paper formulaes an infeced predaor-prey model wih Beddingon-DeAngelis funcional response from a classical deerminisic framework o a sochasic differenial equaion SDE. Firs, we provide a global analysis including he global posiive soluion, sochasically ulimae boundedness, he persisence in mean, and exincion of he SDE sysem by using he echnique of a series of inequaliies. Second, by using Iô s formula and Lyapunov mehods, we invesigae he asympoic behaviors around he equilibrium poins of is deerminisic sysem. The soluion of he sochasic model has a unique saionary disribuion, i also has he characerisics of ergodiciy. Finally, we presen a series of numerical simulaions of hese cases wih respec o differen noise disurbance coefficiens o illusrae he performance of he heoreical resuls. The resuls show ha if he inensiy of he disurbance is sufficienly large, he persisence of he SDE model can be desroyed. Keywords: sochasic eco-epidemiology model; Hölder inequaliy and Chebyshev inequaliy; asympoic behavior; persisence in mean; saionary disribuion Inroducion Mahemaical inequaliies play a large role in mahemaics analysis and is applicaion. Recenly, he inequaliy echnique was applied o impulsive differenial sysems, and sochasic differenial sysems 3 5, hussomenewresulswereobained. Predaion can have far-reaching effecs on biological communiies. Thus many scieniss have sudied he ineracion beween predaor and prey 6. Ineracion beween predaor and prey is hard o avoid being influenced by some facors. One of he mos common facors is he disease. Therefore, here are many scholars who have sudied he infeced predaor-prey sysems 7. For insance, Hadeler and Freedman 6 considered a predaor-prey sysem wih parasiic infecion. They proved he epidemic hreshold heorem for here is coexisence of he predaor wih he uninfeced prey. Han and Ma 5 analyzed four modificaions of a predaor-prey model o include an SIS or SIR parasiic infecion. They obained he hresholds and global sabiliy resuls of he four sysems. Species may be subjec o uncerain environmenal disurbances, such as flucuaions of birh rae and deah rae, food, habia and waer, ec. These phenomena can be de- The Auhors 6. This aricle is disribued under he erms of he Creaive Commons Aribuion 4. Inernaional License hp://creaivecommons.org/licenses/by/4./, which permis unresriced use, disribuion, and reproducion in any medium, provided you give appropriae credi o he original auhors and he source, provide a link o he Creaive Commons license, and indicae if changes were made.

2 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page of 9 scribed by sochasic processes. Recenly, he sochasic predaor-prey sysems have received much aenion from scholars 8. Zhang and Jiang 8 sudied a sochasic hree species eco-epidemiological sysem. They analyzed he sochasic sabiliy and asympoic behaviors around he equilibrium poins of is deerminisic model. Liu and Wang 9 considered a wo-species non-auonomous predaor-prey model wih whie noise. They obained he sufficien crieria for exincion, non-persisence in he mean, and weak persisence in he mean. The funcional response of predaor is a very imporan facor of predaor-prey sysem, which reflecs he average consumpion rae of predaor o prey. Therefore, many scholars prefer o sudy he predaor-prey sysem wih funcional response 5. For insance, Wang and Wei explored a predaor-prey sysem wih srong Allee effec and an Ivlev-ype funcional response. Liu and Berea 3 sudied a predaor-prey model wih a Beddingon-DeAngelis funcional response. Some biologiss have argued ha in many insances, especially when predaors have o hun for food and, herefore, have o share or compee for food, he funcional response in a prey-predaor model should be predaor-dependen. This view has been suppored by some pracical observaions 6, 7. Skalski and Gilliam 6 colleced observaion daa from 9 predaor-prey communiies, hey found ha hree predaor-dependen funcional responses Crowley-Marin 8, Hassell-Varley 9 and Beddingon-DeAngelis 3, 3 were in agreemen wih he observaion daa, and in many insances, he Beddingon-DeAngelis ype looked beer han he oher wo. The Beddingon-DeAngelis ype funcional response of per capia feeding rae can be expressed as follows: Fx, y= axy px qy, a unis: ime represens he effecs of capure rae on he feeding rae, p unis: prey denoes he effecs of handling ime on he feeding rae, q unis: predaor represens he magniude of inerference among predaors. Compared wih he Holling II funcional response, he Beddingon-DeAngelis ype funcional response has an addiional erm qy in he denominaor. In oher words, his ype of funcional response is affeced by boh predaor and prey. Therefore, he effec of muual inerference on he dynamics of populaion is worh sudying. To he bes of our knowledge, he research on global asympoic behaviors of a sochasic infeced predaor-prey sysem wih Beddingon-DeAngelis has no gone very far ye. Therefore, according o a deerminisic predaor-prey model, his paper invesigaes he saionary disribuion and ergodic propery of a sochasic infeced predaor-prey wih Beddingon-DeAngelis and explores he influence of whie noise on he persisence in mean and exincion of he predaor-prey-disease sysem. Firs of all, a deerminisic predaor-prey sysem is described in 3by a Ẋ=Xb a X S pxqs, a Ṡ=S c a S X pxqs βi, İ=I d a 33 IβS,

3 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 3 of 9 X is he populaion densiy of prey a ime, S andi, respecively, sand for he densiies of suscepible predaor and infeced predaor a ime, b is he inrinsic growh rae of X, c is he naural moraliy rae of S, d is he diseased deah rae of I. a, a, a 33, respecively, sand for he densiy coefficiens of X, S andi. a is he capured rae of X, a a isheconversionraefromx os, β represens he infecion rae from S oi, p, q > are consan coefficiens. Second, he world is full of uncerainy and random phenomena, so species in he ecosysem may be subjec o differen forms of random inerference. In his paper, we assume ha he disurbance in he environmen affecs no only he rae of predaion bu also he infecion rae of he disease, so ha a a σ B, a a σ B, β β σ B, B andb are sandard Brownian moions, σ, σ,andσ are he inensiies of he Brownian moions. Taking ino accoun he effecs of random inerference gives a dx=xb a X S pxqs d σ SX pxqs db, a ds=s c a S X βi d pxqs σ SX pxqs db σsi db, di=i d a 33 IβS d σ SI db. The res of his paper is organized as follows. In he nex secion, we consider he exisence of a global posiive soluion and he sochasically ulimae boundedness of model. In Secion 3, we sudy he global asympoic behaviors of model around he equilibrium poins of is deerminisic sysem. In addiion, we explore he saionary disribuion and ergodic propery of model. In Secion 4, we obain he condiions for he persisence in mean and exincion of model. In he las secion, we summarize our main resuls and give some numerical simulaions. Throughou his paper, le, F, {F}, P be a complee probabiliy space wih a filraion {F } saisfying he usual condiions i.e. i is increasing and righ coninuous while F conains all P-null ses. The funcion B i i =, is a Brownian moion defined on he complee probabiliy space. For an inegrable funcion X on,, we define X = Xs ds, X = lim inf X, X = X. Global posiive soluion and sochasically ulimae boundedness. Global posiive soluion Due o he physical meaning, variables S, I, and Y inmodel shouldremain nonnegaive for. We nex prove ha his is acually he case and, furhermore, he posiive soluion is unique. Lemma. For any iniial value X, S, I R 3, model has a local unique posiive soluion X, S, I on, τ e, τ e is he explosion ime. Theorem. For any iniial value X, S, I R 3, model has a unique posiive soluion X, S, I R 3 on wih probabiliy.

4 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 4 of 9 Proof By Lemma., we only need o prove ha τ e = a.s. To his end, le k >bea sufficienly large consan such ha X, S and I all lie in k, k. For each k k k N, define he sopping ime { } τ k = inf, τ e :X /, k, S /, k or I /, k. k k k As is easy o see, τ k is a monoonically increasing funcion when k.leτ = lim k τ k,husτ τ e a.s. Now we need o prove τ = a.s., oherwise, here exis wo consans T >andɛ, such ha P{τ T} > ɛ. Thus, here is an ineger k k such ha P{τ T} > ɛ, k k. 3 Define a C 3 -funcion V : R 3 R, VX, S, I=X ln X S ln S I ln I. The non-negaiviy of he funcion VX, S, Icanbeseenbyu ln u, u >. Applying Iô s formula o he sochasic differenial sysem yields dv = LV d σ X S px qs db σ S X px qs db σ S IdB σ I SdB, LV =X b a X S c a S a S px qs a X px qs βi I d a 33 I βs σ S σ I = bx a X cs a S σ X a SX px qs b a X σ S px qs σ X px qs a S px qs a XS px qs c a a X S px qs βi σ S px qs px qs σ I di a 33 I d a 33 I βs σ S a X b a X a q σ a q S a a S c σ p p a 33 I β a 33 I d σ S I.

5 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 5 of 9 Since d σ σ X S I d = σ σ X σ X S I σ b a X I d a 33 I βs σ σ a X b X a S px qs { σ b 3 max, a p, 4σ a 4a C, S c a S a S a p 4a 33 } S a 33 I I a X px qs βi C is a posiive consan. Then we have σ σ XSI X S I e C e σ σ e σ S I C e σ { } σ max X S I, C σ and σ XSI C. 4 σ Therefore, we have LV a X b a X a q σ a q S a a S c σ p p K, a 33 I β a 33 I d σ C K is a posiive consan. So we have dv K d σ X S px qs db σ S X px qs db σ S IdB σ I SdB. 5 Inegraing 5fromoτ k T and aking expecaion on boh sides, we have EV Xτ k T, Sτ k T, Iτ k T V X, S, I K T. 6

6 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 6 of 9 Le k = {τ k T}, frominequaliy3 wecanseehap k ɛ. Noeha,forevery ω k,hereexissaleasoneofxτ k, ω, Sτ k, ω, Iτ k, ω ha equals eiher k or k.asa resul, we have V Xτ k T, Sτ k T, Iτ k T k ln k k ln. 7 k Applying equaion 6 and equaion 7, we ge V X, S, I K T E k ωv Xτ k T, Sτ k T, Iτ k T ɛk ln k k ln, k k is he indicaor funcion of k. When k,wehave > V X, S, I K T =. This is a conradicion. So τ =.. Sochasically ulimae boundedness Theorem. shows ha R 3 is he posiive invarian se of model. Now we prove he sochasically ulimae boundedness of model. Definiion. Le X, S, I be he soluion of model wih iniial value X, S, I R 3.If,foranyε,, here exiss a χ= χω > such ha he soluion of model saisfies P { X, S, I } > χ < ε, hen model has sochasically ulimae boundedness. Lemma. The following elemenary inequaliy will be used frequenly in he sequel. x r rx,x, r, n p/ x p n i= xp i n p/ x p, R n := {x Rn : x i >, i n}, n R, p >. Theorem. Le X,S,I be he soluion of model wih iniialvalue X,S, I R 3, hen X, S, I is sochasically ulimae boundedness. Proof Define VX, S, I=X S I, X, S, I R 3.

7 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 7 of 9 Applying Iô s formula o sochasic differenial sysem yields dv = LV d σ X S px qs db σ S X px qs db σ S IdB σ I SdB, LV = X a S σ b a X X S px qs 8 px qs S c a S a X px qs βi I d a 33 I βs 8 σ S I 8 σ I S σ S X 8 px qs a X 3 bx a S 3 a p S a 33I 3 βi S. Applying he Hölder inequaliy ab ap p bq q, p =p, q >,wehave q I S 3 I 3 3 S 3. Therefore, LV a X 3 b X a 3 β S 3 a p S a 33 3 β I 3 I X S I H VX, S, I, H > is a posiive consan. Thus dv H VX, S, I d σ S IdB σ I SdB. σ X S px qs db σ S X px qs db Applying Iô s formula o e VX, S, Iyields d e VX, S, I = e VX, S, I d dvx, S, I e H d e σ X S px qs db σ S X px qs db σ S IdB σ I SdB.

8 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 8 of 9 So we have and e EV X, S, I V X, S, I H e EV X, S, I H. Applying he second inequaliy of Lemma. and leing n =3,p =,wehave X, S, I VX, S, I. Thus, we obain E X, S, I H. Therefore, for any ε >,seχ = H ε, applying he Chebyshev inequaliy, we have P { X, S, I > χ } E X, S, I χ, ha is, P { X, S, I } > χ ε. 3 Asympoic behaviors Sysem has hree equilibrium poins 3: i when R = an equilibrium poin E K,,; ii when R = a b ca pb a b ca pb >andr = <,sysem has a b < c da β a pb qda β, sysem has anoher disease free equilibrium poin E X, S, ; iii when R = a b >,sysem has a posiive equilibrium poin E c da 3X, S, I. For is β a pb qda β sochasic sysem, however, hese equilibrium poins do no exis. In his secion, we sudy he asympoic behaviors of model around he hree equilibrium poins E K,,, E X, S,, and E 3 X, S, I of is deerminisic model, respecively. 3. Asympoic behaviors around he equilibrium poin E of sysem When R <,sysem has an equilibrium poin E K,,= b a,,, bu i is no he equilibrium poin of sysem. In his subsecion, we sudy he asympoic behaviors of sysem arounde K,,. Theorem 3. Le X,S,I be he soluion of model wih iniial value X,S, I R 3. If R <and K = b a c a, hen W = min{a, a a a, a a 33 a }. Xθ K Sθ I θ dθ σ K q W,

9 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 9 of 9 Proof Noe ha K,,isheequilibriumpoinofsysem, K = b a. Define VX, S, I= X K K ln X a S I. K a Applying Iô s formula o sochasic differenial sysem yields dv =LV d σ X KS px qs db a σ SX a px qs db, 8 LV =X K b a X a S c a S a =X K a S px qs a X px qs βi σ KS px qs a S b a X K a K px qs a S c a S a a X K a X px qs βi a KS px qs a X K a K c a Id a 33 I βs σ KS px qs Id a 33 I βs σ KS px qs a cs a S a 33 I a S σ K q a a a S a a 33 a I a X K a a S a a 33 I σ K a q. a Inegraing equaion 8from o,we obain V V M = a a 33 a a Xθ K dθ a a a S θ dθ I θ dθ σ K q M, 9 σ Xθ KSθ pxθqsθ a σ SθXθ db θ a pxθqsθ is a real-valued coninuous local maringale. Thus M, M = C σ q a σ a q σ Xθ KSθ pxθqsθ <. Applying he srong law of large numbers, we obain lim M =. a σ SθXθ dθ a pxθqsθ

10 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page of 9 Dividing equaion 9by and aking he limi superior, we have hus a Xθ K a a a S θ a a 33 a I θ dθ σ K q, a Xθ K S θi θ dθ σ K q W. Corollary 3. From Theorem 3., when σ =,we have LV a X K a a a S a a 33 a I, hus when R <and K = b a c a hold, he equilibrium poin E K,, of sysem is globally asympoically sable. Remark 3. From Theorem 3., if he inerference inensiy is sufficienly small, he soluion of model will flucuae around he equilibrium poin E K,,.Moreover,he flucuaion inensiy is relaed wih he disurbance inensiy: he flucuaion inensiy is posiively correlaed wih he value of σ. 3. Asympoic behaviors around he equilibrium poin E of sysem When R >andr <,sysem has an equilibrium poin E X, S,, bu i is no he equilibrium poin of sysem. In his subsecion, we sudy he asympoic behaviors of sysem arounde X, S,. Theorem 3. Le X,S,I be he soluion of model wih iniial value X,S, I R 3. If R >,R <and a q > a p, hen we have U = σ X q Xθ X Sθ S I θ dθ U W, a px σ S a qs p σ S C and { W = min a a p q, a a px, a } a 33 px. a qs a qs Proof Noing ha X, S, is he equilibrium poin of sysem, hus b a X a S px qs =, c a a X S px qs =.

11 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page of 9 Define VX, S, I= X X X ln X a px S S S ln S a px X a qs S a qs I := V a px a qs V a px a qs V 3. Applying Iô s formula o sochasic differenial sysem yields dv = LV d σ X XS px qs db, Similarly, LV =X X b a X a S px qs =X X b a X X a X σ XS px qs a S px qs psx X S S px =X X a X Xa px qs px qs = a X X σ XS px qs. σ XS px qs σ XS px qs a psx X px qs px qs a pxs SX X px qs px qs dv = LV d σ S SX px qs db σ S SIdB, LV =S S c a S a X px qs βi =S S c a S S a S σ SX px qs σ S a X px qs βi X X qs qxs S =S S a S Sa px qs px qs βi σ SX px qs σ S I I σ SX px qs σ S I = a S S qsx XS S a px qs px qs a qxs S px qs px qs βis S σ SX px qs σ S I.

12 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page of 9 Also, we have dv 3 = I d a 33 I βs d σ SI db. Hence dv = LV d σ X XS px qs db a px σ S SX a qs px qs db σ S SIdB σsi db, LV = LV a px a qs LV a px a qs LV 3 = a X X a psx X px qs px qs a pxs SX X px qs px qs σ XS px qs a px a qs a S S a qsx XS S px qs px qs qxs S a px qs px qs βis S σ SX px qs σ S I d a 33 I βs a a p q Since βs < d,hus a px σ βs di a qs a px a qs LV a a p q σ X q a px a qs X X a a px a qs XS px qs σ SX px qs σ S I. X X a a px a qs σ S p σ S C. Inegraing boh sides of equaion fromo yields V V a a p Xθ X q I S S a a 33 px I a qs S S a a 33 px I a qs a a px a a 33 px Sθ S a qs a qs σ X q a px σ S a qs p σ S C I θ dθ M M 3,

13 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 3 of 9 and M = σ Xθ XSθ pxθqsθ a px a qs a σ pxsiθ M 3 = db θ a qs are real-valued coninuous local maringales. Thus and = C M, M σ q M 3, M 3 σ Xθ XSθ pxθqsθ a px a4 σ px a p qs < σ Sθ SXθ pxθqsθ db θ σ Sθ SXθ dθ a qs pxθqsθ a σ pxsiθ = dθ a qs a C σ S px <. qs Applying he srong law of large numbers, we have lim M i =i =,3. Dividing equaion by and aking he limi superior, we have Thus a a p q a a 33 px I θ a qs Xθ X a a px Sθ S dθ σ X q a qs a px a qs σ S p σ S C. Xθ X Sθ S I θ dθ U W. Corollary 3. From Theorem 3., when σ = σ = σ =,we have LV a a p q X X a a px a qs S S a a 33 px I, a qs hus when a q > a p, R >and R <hold, he equilibrium poin E X, S,of sysem is globally asympoically sable.

14 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 4 of 9 Remark 3. From Theorem 3., if he inerference inensiy is sufficienly small, he soluion of model will flucuaes around he equilibrium poin E X, S,.Moreover,he flucuaion inensiy is relaed wih he disurbance inensiy: he flucuaion inensiy is posiively correlaed wih he value of σ, σ and σ. 3.3 Asympoic behaviors around he posiive equilibrium poin E 3 of sysem When R >,sysem has a posiive equilibrium poin E 3 X, S, I,buiisnohe equilibrium poin of model. Now, we explore he asympoic behaviors of sysem around E 3 X, S, I. X is a emporally homogeneous Markov process in E l, which is given by he sochasic differenial equaion dx =bx d k σ m x db m, m= E l R l represens a l-dimensional Euclidean space. The diffusion marix of Xisgivenby x= a i,j x, a i,j x= k σm i xσ m j x. m= Assumpion Assume ha here is a bounded domain U E l wih regular boundary, saisfying he following condiions: In he domain U and some of is neighbors, he minimum eigenvalue of he diffusion marix Ax is nonzero. When x E l \U, he mean ime τ a which a pah saring from x o he se U is limied, and sup x H E x τ < for every compac subse H E l. Lemma When Assumpion 3. holds, he Markov process X has a saionary disribuion μ wih densiy in E l. Le f x be a funcion inegrable wih respec o he measure μ, x E l, hen, for any Borel se B E l, we have lim P, x, B=μB and { P x lim T T T f x } d = f xμdx =. E l Theorem 3.3 Le X,S,I be he soluion of model wih iniial value X,S, I R 3. If a q > a pandr >hold, hen Xθ X Sθ S Iθ I dθ U 3 W 3, U 3 = σ X a px σ S C σ S I q a qs p

15 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 5 of 9 and { W 3 = min a a p q, a a px a qs, a a 33 px }. a qs Proof Noing ha X, S, I is he equilibrium poin of sysem, hus b a X a S px qs =, c a S a X px qs βi =, βs d a 33 I =. Define VX, S, I= X X X ln XX a px S S S ln SS a qs a px I I I ln I a qs I := V a px a qs V a px a qs V 3. Applying Iô s formula o he sochasic differenial sysem yields Similarly, dv = LV d σ X X S px qs db, LV = X X b a X a S px qs = X X b a X X a X σ X S px qs a S px qs σ X S px qs = X X a X X a ps X X S S px px qs px qs σ X S px qs = a X X a ps X X px qs px qs σ X S a px S S X X px qs px qs. dv = LV d σ S S X px qs db σ S S IdB, px qs

16 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 6 of 9 LV = S S c a S = S S c a S S a S σ S X px qs σ S I a X px qs βi σ S X px qs σ S I a X px qs β I I βi = S S a S S a X X qs qx S S px qs px qs Also, we have σ S X px qs σ S I β I I = a S S a qs X X S S px qs px qs β I I S S a qx S S px qs px qs σ S X px qs σ S I. dv 3 = LV 3 d σ I I SdB, LV 3 = I I d a 33 I βs σ I S = I I d a 33 I I a 33 I β S S βs σ I S = a 33 I I β S S I I σ I S. Then we have dv = LV d σ X X S px qs db a px σ S S X a qs px qs db σ S S IdB σ I I SdB, LV = LV a px a qs LV a px a qs LV 3 = a X X a ps X X px qs px qs σ X S a px S S X X px qs px qs a px a qs px qs a S S a33 I I qs X X S S a px qs px qs a qx S S px qs px qs σ S X px qs σ S I σ I S

17 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 7 of 9 a a p q σ X a px q a qs I is easy o see ha, for any { φ < min a a p q he ellipsoid X X a a px a qs σ S C σ S I. p X, a a px a qs a a p X X a a px S S q a qs a a 33 px I I φ = a qs S S a a 33 px I I a qs S, a a 33 px I }, a qs lies enirely in R 3.LeU o be any neighborhood of he ellipsoid wih Ū E 3 = R 3,hus for any x U\E l,wehavelv M M is a posiive consan. Therefore, condiion in Assumpion 3. is saisfied. Moreover, here exiss a G = min{σ x, σ x, σ 3 x 3,x, x, x 3 U} >suchha 3 3 a ik xa jk x ξ i ξ j = σ x ξ σ x ξ σ 3 x 3 ξ 3 G ξ i,j= k= for all x Ū, ξ R 3, which means condiion in Assumpion 3. is saisfied. Therefore, he sochasic model has a unique saionary disribuion μ, i also has he ergodic propery. Inegraing equaion fromo on boh sides yields V V a a p Xθ X a a px Sθ S q a qs a a 33 px Iθ I dθ a qs σ X q a px a qs σ S C σ S I p M 4 M 5, 3 and M 4 = M 5 = σ Xθ X Sθ pxθqsθ a px a qs σ Sθ S Xθ db θ pxθqsθ a px σ Sθ S Iθσ Iθ I Sθ db a qs θ are real-valued coninuous local maringales.

18 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 8 of 9 Thus and = C M 4, M 4 a px σ Sθ S Xθ a qs σ q a4 σ px a p qs M 5, M 5 = pxθqsθ σ Xθ X Sθ dθ pxθqsθ < σ S Iθ σ I Sθ dθ C σ S I <. Applying he srong law of large numbers, we have lim M i =i =4,5. Dividing equaion 3by and aking he limi superior, we have hus a a p Xθ X a a px Sθ S q a qs dθ a a 33 px Iθ I a qs σ X a px q a qs σ S C σ S I, p Xθ X Sθ S Iθ I U 3 dθ. 4 W 3 Corollary 3.3 From Theorem 3.3, when σ = σ = σ =,we have LV a a p q. X X a a px a qs S S a a 33 px I I a qs Thus when a q > a pandr >hold, he posiive equilibrium poin E 3 X, S, I of sysem is globally asympoically sable. Remark 3.3 From Theorem 3.3, if he inerference inensiy is sufficienly small, he soluion of model will flucuaes around he equilibrium poin E 3 X, S, I. Moreover, he flucuaion inensiy is relaed wih he disurbance inensiy: he flucuaion inensiy is posiively correlaed wih he value of σ, σ and σ. Remark 3.4 If he condiions in Theorem 3.3 arehold, hen he soluion of model has a unique saionary disribuion, i also has he ergodic propery.

19 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 9 of 9 4 Persisence in mean and exincion When we consider a biological populaion sysem, persisence in mean and exincion are wo very imporan properies. In his secion, we invesigae he persisence in mean and exincion of sysem. Since here is no equilibrium poin in sysem, we canno deermine he persisence of sysem by proving he sabiliy of he equilibrium poin as a deerminisic sysem. Definiion 4. 5 The definiion of persisence in mean and exincion are given as follows: The species X is said o be in exincion if lim X=. The species X is said o be in persisence in mean if lim X >. Lemma Le X C,, R. If here exis T >,λ >,λ, n i, when T, we have ln X λ λ Xs ds j n i B i= a.s., hen X λ λ a.s., if λ ; lim X=a.s., if λ <. If here exis T >,λ >,λ >,n i, when T, we have ln X λ λ Xs ds hen X λ λ a.s. j n i B i= a.s., 4. Persisence in mean Theorem 4. Le X,S,I be he soluion of model wih iniial value X,S, I R 3. Model has persisence in mean if condiions a q > a p, R >,and } ϱ = max{σ, σ, σ } < min {X W 3, S W 3, I W 3 U U U hold, ha is, lim inf Xθ dθ >, lim inf Sθ dθ >, lim inf Iθ dθ >, U = X q a px S a qs p C S I, U 3 and W 3 are defined in Theorem 3.3.

20 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page of 9 Proof Applying equaion 4inTheorem3.3we have Xθ X U 3 Sθ S U 3 W 3, Iθ I U 3 W 3. W 3, 5 Applying he inequaliy a ab a a b o X, we have X X X X X. Therefore U 3 = σ X a px σ S C σ q a qs p S X ϱ q a px a qs = ϱ U. S I p C S I When ϱ < X W3 U,wehave lim inf Xθ dθ X X U 3 W 3 X X σ U W 3 X >. Xθ X dθ X Similarly, when ϱ < S W3 U,wehave lim inf Sθ dθ S S U 3 W 3 S S σ U W 3 S >. Sθ S dθ S When ϱ < I W3 U,wehave lim inf Iθ dθ I Iθ I dθ I I U 3 W 3 I

21 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page of 9 I σ U W 3 I >. Remark 4. From Theorem 4.,whenR >,a q > a p and he inensiy of random disurbance is sufficienly small, sysem will persisence in mean. This shows ha biological populaions can resis a small environmenal disurbance o mainain he original persisence. 4. Exincion Exincion and persisence in mean are closely relaed, so we also concern ourselves wih he siuaion of populaion exincion. In his subsecion, we poin ou he condiions of predaor exincion. Theorem 4. Le X,S,I be he soluion of model wih iniial value X,S, I R 3. If one of he following condiions holds: σ > max{ a c, a p}, R = a pc σ p c <,σ a p, hen lim X= b, a lim S=, lim I=. Proof Applying Iô s formula o he second equaion of sochasic differenial sysem yields d ln S= c σ I σ X px qs a a X S px qs βi d σ X px qs db σ IdB c a S σ X px qs a σ a σ d σ X px qs db σ IdB. 6 Case I. When σ > max{ a c, a p}, inequaliy6 akes is maximum value on he inerval, p a a,sowehave σ d ln S c a S a σ X σ d px qs db σidb. Inegraing 6fromo and dividing i by,wege S ln S a σ c a S σ Xθ pxθqsθ db θ Applying Lemma 4.,weobain lim S=. σ Iθ db θ.

22 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page of 9 Case II. When R = a pc σ p c <andσ a p,inequaliy6 akes is maximum value on he inerval, p a p,sowehave a d ln S p σ p c a σ X S d px qs db σidb. Inegraing 6from o and dividing i by,weobain S ln S a p σ p c a S σ Xθ pxθqsθ db θ σ Iθ db θ a = c cp σ cp σ Iθ db θ = c R a S σ Iθ db θ. Applying Lemma 4.,weobain a S σ Xθ pxθqsθ db θ σ Xθ pxθqsθ db θ lim S=. Applying Iô s formula o he hird equaion of sochasic differenial sysem, one has d ln I= d σ S a 33 I βs d σ SdB. Since lim S =, here is an arbirarily small consan ε >suchhawhen > T,we have σ S βs < ε,hus ln I= d a 33 I βs σ S d σ SdB ε d a 33 I d σ SdB. 7 Inegraing equaion 7from o and dividing i by yields I ln I ε d a 33 I σ Sθ db θ. Applying Lemma 4. and he arbirariness of ε,we obain lim I=.

23 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 3 of 9 Similarly, d ln X= b a X a S px qs σ S d px qs σ S px qs db. Since lim S =, here is an arbirarily small consan ε >suchhawhen > T, S we have pxqs < ε,hus d ln X b a X a ε σ ε d σ S px qs db. Inegraing he above equaion from o and dividing i by,onehas X ln X b a ε σ ε a X σ Sθ pxθqsθ db θ. Applying Lemma 4. and he arbirariness of ε,we obain lim X b. 8 a On he oher hand, d ln X= b a X b a X d a S px qs σ S d px qs σ S px qs db σ S px qs db. 9 Inegraing equaion 9from o and dividing i by,we have X ln X b a X σ Sθ pxθqsθ db θ. Applying Lemma 4.,weobain lim X b. a From 8and, we have lim X= b. a Remark 4. From Theorem 4., if he inensiy of random disurbance is sufficienly large or R <andσ a p, he predaor populaion will be exinc. 5 Conclusions and numerical simulaions This paper invesigaes a sochasic infeced predaor-prey model wih Beddingon- DeAngelis funcional response. The exisence of a global posiive soluion of model is firs proved, hen we show he sochasically ulimae boundedness of he soluion. In addiion, by using he Lyapunov mehod and Iô s formula, we sudy he asympoic properies

24 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 4 of 9 Figure Time sequence diagram and phase porrai of model. a The deerminisic model; b he Brownian moion model wih σ = σ = σ =.5;c phase porrai: he red is corresponding o he deerminisicmodel, whilehe blue represens he sochasic model. and saionary disribuion of he soluion of model around he equilibrium poins of is deerminisic. A las, we discuss he persisence in mean and exincion of model. The biological significance of he resul shows ha he exernal environmen disurbance may have a cerain effec on he sabiliy of he biological sysem: he populaion s abiliy o adap o he environmen is limied. If he disurbance in he environmen is small enough, he sabiliy of he populaion will no be desroyed; if large disurbances occur in he environmen, i may lead o he exincion of species. We nex give some numerical simulaions o suppor our resuls. We consider he following discree equaions: X n = X n X n b a X n a S n px n qs n σ S n X n px n qs n W k, S n = S n S n c a S n a X n px n qs n βi n σ X n S n px n qs n W k σ S n I n W k, I n = I n I n βs n d a 33 I n σ S n I n W k, =., W ik W k W k obeys he Gaussian disribuion N,. In Figure, wechoosex =, S =, I =, b =,c =.4,d =.,a =.8,a =.55, a =.,a =.,a 33 =.,β =.,p =,q =, and sep size =.. Under his condiion, E =K,,=.5,,, R =.76, K = b a =.5 c a =. The numerical simulaion of Figure is consisen wih our conclusionin Theorem 3.. In Figure,wechooseX =, S =, I =, b =.6,c =.,d =.3,a =.8,a =.6, a =.8,a =.,a 33 =.4,β =.,p =,q =, and sep size =.. Under his condiion, E =X, S,=.6,.4,, a q =.8>a p =., R =5>, R =.53<. In Figure a, we choose σ = σ = σ =.,hus Xθ X Sθ S I θ dθ U W =.8648.

25 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 5 of 9 Figure Time sequence diagram and phase porrai of model. a-c are a Brownian moion model wih σ = σ = σ =.,.,.4, respecively. d-f are phase porrais of a-c, respecively. The red is correspondingo hedeerminisicmodel, whilehe blue represens he sochasic model. In Figure b, we choose σ = σ = σ =.,hus Xθ X Sθ S I θ dθ U W = In Figure c, we choose σ = σ = σ =.4,hus Xθ X Sθ S I θ dθ U W = Figure shows ha he soluion of model flucuaes around he equilibrium E.6,.4,. In addiion, he flucuaion inensiy is relaed wih he disurbance inensiy: wih he increase of σ, σ, σ, he flucuaion inensiy is also increasing. These all mee he condiions of Theorem 3.. In Figure 3, wechoosex =, S =, I =, b =,c =.,d =.,a =.5,a =.3, a =,a =.,a 33 =.,β =.5,p =,q =, and sep size =.. Under his condiion, E 3 = X, S, I =.985,.53,.87, a q =.5>a p =.3, R =5.>. In Figure 3a, we choose σ = σ = σ =.3,hus Xθ X Sθ S Iθ I dθ U 3 W 3 = 8.3.

26 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 6 of 9 Figure 3 Time sequence diagram and phase porrai of model. a The Brownian moion model wih σ = σ = σ =.3,.6,., respecively. d-f are phase porrais of a-c, respecively. The red is correspondingo hedeerminisicmodel, whilehe blue represens he sochasic model. In Figure 3b, we choose σ = σ = σ =.6,hus Xθ X Sθ S Iθ I dθ U 3 W 3 =.453. In Figure 3c, we choose σ = σ = σ =.,hus Xθ X Sθ S Iθ I dθ U 3 W 3 = Figure 3 shows ha he soluion of model flucuaes around E 3.985,.53,.87. In addiion, he flucuaion inensiy is relaed wih he disurbance inensiy: wih he increase of σ, σ and σ, he flucuaion inensiy is also increasing. These all mee he condiions of Theorem 3.. In Figure 4, wechoosex =, S =, I =, b =,c =.,d =.,a =.5,a =.3, a =,a =.,a 33 =.,β =.5,p =,q =, and sep size =..Figure4 shows ha he soluion of model flucuaes up and down in a small neighborhood. According o he densiy funcions in Figure 4b-d, we see ha here is a saionary disribuion. This is in line wih our conclusions. In Figure 5, wechoosex =, S =, I =, b =,c =.,d =.,a =.5,a =.3, a =.,a =.,a 33 =.,β =.5,p =.5,q =.5, and sep size =.. In his condiion, E 3 = X, S, I =.8793,.4339,.5847, a q =.5>a p =.5, R =3.4>.

27 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 7 of 9 Figure 4 Time sequence diagram and densiy funcion of model wihσ = σ = σ =..atime sequence diagram; b-d he densiy funcions of X, S, I, respecively. Figure 5 Persisence in mean and exincion of model. a The deerminisicmodel; b persisence in mean of model ; c exincion of model. In Figure 5b, we choose σ = σ = σ =.6.Inhiscase, { ϱ = max{σ, σ, σ } =.6<min X W 3 U, S W 3, I U W 3 U } =.64, which saisfies he condiions in Theorem 4.. Figure5b shows ha X, S, I have persisence in mean, his is in line wih our conclusion in Theorem 4.. In Figure 5c, we choose σ = σ =.5and { a σ =.>max, } a p =.3, c

28 Feng e al. Journal of Inequaliies and Applicaions 6 6:37 Page 8 of 9 which saisfies he condiions in Theorem 4..Figure5c shows ha S, I are exinc and lim X= b =, a his is in line wih our conclusion in Theorem 4.. To sum up, we have he following conclusions: I. Asympoic behaviors When R <and Ka < c, he soluion of model is flucuaing around E. Therefore, he inensiy of he flucuaion is posiively correlaed wih σ. When R >,R <and a q > a p, he soluion of model isflucuaing around E. Therefore, he inensiy of he flucuaion is posiively correlaed wih σ, σ and σ. 3 When R >and a q > a p, he soluion of model is flucuaing around E 3. Therefore, he inensiy of he flucuaion is posiively correlaed wih σ, σ and σ. When he inerference inensiy is sufficien small, he soluion of model has a unique saionary disribuion, i also has he ergodic propery. II. Persisence in mean and exincion When R >,a q > a p and ϱ = max{σ, σ, σ } < min{x W3 U, S W3 U, I W3 U }, he soluion of model can have persisence in mean. When σ > max{ a c, a p} or R <and σ a p, he predaor of model canbeexinc. Compeing ineress The auhors declare ha hey have no compeing ineress. Auhors conribuions All auhors conribued equally o he wriing of his paper. All auhors read and approved he final manuscrip. Auhor deails College of Mahemaics and Sysems Science, Shandong Universiy of Science and Technology, Qingdao, 6659, P.R. China. Key Laboraory of Jiangxi Province for Numerical Simulaion and Emulaion Techniques, Gannan Normal Universiy, Ganzhou, 34, P.R. China. 3 Sae Key Laboraory of Mining Disaser Prevenion and Conrol Co-founded by Shandong Province and he Minisry of Science and Technology, Shandong Universiy of Science and Technology, Qingdao, 6659, P.R. China. Acknowledgemens The auhors would like o hank Dr. Tonghua Zhang, who helped hem during he wriing of his paper. This work was suppored by Naional Naural Science Foundaion of China 373, 533, 564, he SDUST Research Fund 4TDJH, Shandong Provincial Naural Science Foundaion, China ZR5AQ, BS5SF, Join Innovaive Cener for Safe And Effecive Mining Technology and Equipmen of Coal Resources, he Open Foundaion of he Key Laboraory of Jiangxi Province for Numerical Simulaion and Emulaion Techniques, Gannan Normal Universiy, China, SDUST Innovaion Fund for Graduae Sudens No. SDKDYC75. Received: 3 June 6 Acceped: 9 November 6 References. Gao, SJ, Chen, LS, Nieo, JJ, Torres, A: Analysis of a delayed epidemic model wih pulse vaccinaion and sauraion incidence. Vaccine 4, Meng, XZ, Jiao, JJ, Chen, LS: The dynamics of an age srucured predaor-prey model wih disurbing pulse and ime delays. Nonlinear Anal., Real World Appl. 9, Tong, YL: Relaionship beween sochasic inequaliies and some classical mahemaical inequaliies. J. Inequal. Appl. 997, Zu, L, Jiang, DQ, O Regan, D: Condiions for persisence and ergodiciy of a sochasic Loka-Volerra predaor-prey model wih regime swiching. Commun. Nonlinear Sci. Numer. Simul. 9, Meng, XZ, Zhao, SN, Feng, T, Zhang, TH: Dynamics of a novel nonlinear sochasic SIS epidemic model wih double epidemic hypohesis. J. Mah. Anal. Appl. 433, 7-4 6

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