Some aspects concerning the dynamics of stochastic chemostats

Some aspec s conce ning he dynamics o s ochas ic chemos a s Tom´ as Ca aballo, Ma ´ ıa J. Ga ido-A ienza and Ja ie L´ opez-de-la-C uz Abs ac In his pape we s udy a simple chemos a model in luenced by whi e noise which makes his kind o models mo e ealis ic. We use he heo y o andom a ac o s and, o ha end, we i s pe o m a change o a iable using he O ns ein- Uhlenbeck p ocess, ans o ming ou s ochas ic model in o a sys em o di e en ial equa ions wi h andom coe icien s. A e p o ing ha his andom sys em possesses a unique solu ion o any ini ial alue, we analyze he exis ence o andom a ac o s. Finally we illus a e ou esul s wi h some nume ical simula ions. 1 In oduc ion Modeling chemos a s is a eally in e es ing and impo an p oblem wi h special in- e es in ma hema ical biology, since hey can be used o s udy ecombinan p ob- lems in gene ically al e ed mic oo ganisms [13, 14], was e wa e ea men [10, 18] and play an impo an ole in heo e ical ecology [2, 9, 12, 17, 22, 23, 24, 26]. De i a ion and analysis o chemos a models a e well documen ed in [19, 20, 25] and e e ences he ein. Two s anda d assump ions o simple chemos a models a e as ollows: (1) he a ailabili y o he nu ien and i s supply a e a e ixed and (2) he endency o he Tom´ as Ca aballo Dp o. Ecuaciones Di e enciales y An´ alisis Num´ e ico, Uni e sidad de Se illa, Apdo. de Co eos 1160, 41080-Se illa, Spain. e-mail: [email p o ec ed] Ma ´ ıa J. Ga ido A ienza Dp o. Ecuaciones Di e enciales y An´ alisis Num´ e ico, Uni e sidad de Se illa, Apdo. de Co eos 1160, 41080-Se illa, Spain. e-mail: [email p o ec ed] Ja ie L´ opez de la C uz Dp o. Ecuaciones Di e enciales y An´ alisis Num´ e ico, Uni e sidad de Se illa, Apdo. de Co eos 1160, 41080-Se illa, Spain. e-mail: [email p o ec ed] 1 2 T. Ca aballo, M.J. Ga ido-A ienza, J. L´ opez-de-la-C uz mic oo ganisms o adhe e o su aces is no aken in o accoun . Howe e , hese a e e y s ong es ic ions as he eal wo ld is non-au onomous and s ochas ic, and his jus i ies he analysis o s ochas ic chemos a models. Le us i s conside one o he simples chemos a models, dS d = (S0−S)D−mSx a+S,(1) dx d =xmS a+S−D,(2) whe e S( )and x( )deno e concen a ions o he nu ien and he mic obial biomass, espec i ely; S0deno es he olume ic dilu ion a e, ais he hal -sa u a ion con- s an , Dis he dilu ion a e and mis he maximal consump ion a e o he nu ien and also he maximal speci ic g ow h a e o mic oo ganisms. We no ice ha all pa- ame e s a e posi i e and we use a unc ion Holling ype-II as unc ional esponse o he mic oo ganism desc ibing how he nu ien is consumed by he species (see [21] o mo e de ails and biological explana ions abou his model). Howe e , we can conside a mo e ealis ic model by in oducing a whi e noise in one o he pa ame e s, he e o e we eplace he dilu ion a e Dby D+α˙ W( ), whe e W( )is a whi e noise, i.e., is a B ownian mo ion, and α≥0 ep esen s he in ensi y o noise. Then, sys em (1)-(2) is eplaced by he ollowing sys em o s ochas ic di e en ial equa ions dS =(S0−S)D−mSx a+Sd +α(S0−S)dW( ),(3) dx =xmS a+S−Dd −αxdW( ).(4) Sys em (3)-(4) has been analyzed in [27] by using he classic echniques om s ochas ic analysis and some s abili y esul s a e p o ided he e. Howe e , as in ou opinion he e a e some unclea poin s in he analysis ca ied ou in [27], ou aim in his pape is o use an al e na i e app oach o his p oblem, speci ically he heo y o andom dynamical sys ems, which will allow us o pa ially imp o e he esul s in [27]. In addi ion, we will p o ide some esul s which hold wi h p obabili y one while hose om [27] a e said o hold in p obabili y. Sys em (3)-(4) is unde s ood in he I ˆ o sense. Then we i s conside i s equi alen S a ono ich o mula ion which is gi en by dS =(S0−S)D+α2 2−mSx a+Sd +α(S0−S)◦dW( ),(5) S ochas ic chemos a s 3 dx =xmS a+S−D+α2 2d −αx◦dW( ).(6) In Sec ion 2 we ecall some basic esul s on andom dynamical sys ems. In Sec ion 3 we s a wi h he s udy o equilib ia and we p o e a esul ela ed o he exis ence and uniqueness o global solu ion o (5)-(6), by using he so-called O ns ein-Uhlenbeck p ocess. Then, we de ine a andom dynamical sys em and p o e he exis ence o a andom a ac o o sys em (5)-(6) gi ing an explici exp ession o i . Finally, in Sec ion 3.5 we show some nume ical simula ions wi h di e en alues o αand we can see wha happens when αinc eases. 2 Random dynamical sys ems In his sec ion we p esen some basic esul s ela ed o andom dynamical sys ems (RDSs) and andom a ac o s which will be necessa y o ou analysis. Fo mo e de ailed in o ma ion abou RDSs and hei impo ance, see [1]. Le (X,k·kX)be a sepa able Banach space and le (Ω,F,P)be a p obabili y space whe e Fis he σ−algeb a o measu able subse s o Ω(called “e en s”) and Pis he p obabili y measu e. To connec he s a e ωin he p obabili y space Ωa ime 0 wi h i s s a e a e a ime o elapses, we de ine a low θ={θ } ∈Ron Ω wi h each θ being a mapping θ :Ω→Ω ha sa is ies (1) θ0=IdΩ, (2) θs◦θ =θs+ o all s, ∈R, (3) he mapping ( ,ω)7→ θ ωis measu able, (4) he p obabili y measu e Pis p ese ed by θ , i.e., θ P=P. This se -up es ablishes a ime-dependen amily θ ha acks he noise, and (Ω,F,P,θ) is called a me ic dynamical sys em [1]. De ini ion 1. A s ochas ic p ocess {ϕ( ,ω)} ≥0,ω∈Ωis said o be a con inuous RDS o e (Ω,F,P,{θ } ∈R)wi h s a e space Xi ϕ:[0,+∞)×Ω×X→Xis (B[0,+∞)×F×B(X),B(X))- measu able, and o each ω∈Ω, (i) he mapping ϕ( ,ω):X→X,x7→ ϕ( ,ω)xis con inuous o e e y ≥0, (ii) ϕ(0,ω)is he iden i y ope a o on X, (iii) (cocycle p ope y) ϕ( +s,ω) = ϕ( ,θsω)ϕ(s,ω) o all s, ≥0. De ini ion 2. Le (Ω,F,P)be a p obabili y space. A andom se Kis a measu able subse o X×Ωwi h espec o he p oduc σ−algeb a B(X)×F. The ω−sec ion o a andom se Kis de ined by K(ω) = {x:(x,ω)∈K},ω∈Ω. 4 T. Ca aballo, M.J. Ga ido-A ienza, J. L´ opez-de-la-C uz In he case ha a se K⊂X×Ωhas closed o compac ω−sec ions i is a andom se as soon as he mapping ω7→ d(x,K(ω)) is measu able ( om Ω o [0,∞)) o e e y x∈X, see [8]. Then Kwill be said o be a closed o a compac , espec i ely, andom se . I will be assumed ha closed andom se s sa is y K(ω)6=/0 o all o a leas o P−almos all ω∈Ω. Rema k 1. I should be no ed ha in he li e a u e e y o en andom se s a e de ined p o ided ha ω7→ d(x,K(ω)) is measu able o e e y x∈X. Ob iously his is sa is ied, o ins ance, when K(ω) = N o all ω, whe e Nis some non-measu able subse o X, and also when K= (U×F)∪(U×Fc) o some open se U⊂Xand F/∈F. In bo h cases ω7→ d(x,K(ω)) is cons an , hence measu able, o e e y x∈X. Howe e , bo h cases gi e K⊂X×Ωwhich is no an elemen o he p oduc σ−algeb a B(X)×F. De ini ion 3. A bounded andom se K(ω)⊂Xis said o be empe ed wi h espec o {θ } ∈Ri o a.e. ω∈Ω, lim →∞e−β sup x∈K(θ− ω) kxkX=0, o all β>0; a andom a iable ω7→ (ω)∈Ris said o be empe ed wi h espec o {θ } ∈Ri o a.e. ω∈Ω, lim →∞e−β sup ∈R | (θ− ω)|=0, o all β>0. In wha ollows we use D(X) o deno e he se o all empe ed andom se s o X. De ini ion 4. A andom se B(ω)⊂Xis called a andom abso bing se in D(X)i o any D∈D(X)and a.e. ω∈Ω, he e exis s TD(ω)>0 such ha ϕ( ,θ− ω)D(θ− ω)⊂B(ω),∀ ≥TD(ω). De ini ion 5. Le {ϕ( ,ω)} ≥0,ω∈Ωbe an RDS o e (Ω,F,P,{θ } ∈R)wi h s a e space Xand le A(ω)(⊂X)be a andom se . Then A={A(ω)}ω∈Ωis called a global andom D−a ac o (o pullback D−a ac o ) o {ϕ( ,ω)} ≥0,ω∈Ωi (i) (compac ness) A(ω)is a compac se o X o any ω∈Ω; (ii) (in a iance) o any ω∈Ωand all ≥0, i holds ϕ( ,ω)A(ω) = A(θ ω); (iii) (a ac ing p ope y) o any D∈D(X)and a.e. ω∈Ω, lim →∞dis X(ϕ( ,θ− ω)D(θ− ω),A(ω)) = 0, whe e dis X(G,H) = sup g∈G in h∈Hkg−hkX S ochas ic chemos a s 5 is he Hausdo semi-me ic o G,H⊆X. P oposi ion 1. [6, 11] Le B ∈D(X)be a closed abso bing se o he con inuous andom dynamical sys em {ϕ( ,ω)} ≥0,ω∈Ω ha sa is ies he asymp o ic compac - ness condi ion o a.e.ω∈Ω, i.e., each sequence xn∈ϕ( n,θ− nω)B(θ− nω)has a con e gen subsequence in X when n→∞. Then ϕhas a unique global andom a ac o A={A(ω)}ω∈Ωwi h componen subse s A(ω) = τ≥TB(ω)[ ≥τ ϕ( ,θ− ω)B(θ− ω). I he pullback abso bing se is posi i ely in a ian , i.e., ϕ( ,ω)B(ω)⊂B(θ ω) o all ≥0, hen A(ω) = ≥0 ϕ( ,θ− ω)B(θ− ω). Rema k 2. When he s a e space X=Rdas in his pape , he asymp o ic compac - ness ollows i ially. No e ha he andom a ac o is pa h-wise a ac ing in he pullback sense, bu does no need o be pa h-wise a ac ing in he o wa d sense, al- hough i is o wa d a ac ing in p obabili y, due o some possible la ge de ia ions, see e.g. [1]. The nex esul ensu es when wo andom dynamical sys ems a e conjuga ed (see also [3, 4]). Lemma 1. Le ϕube a andom dynamical sys em on X. Suppose ha he mapping T:Ω×X→X possesses he ollowing p ope ies: o ixed ω∈Ω, T(ω,·)is a homeomo phism on X, and o x ∈X, he mappings T (·,x), T −1(·,x)a e measu - able. Then he mapping ( ,ω,x)→ϕ ( ,ω)x:=T−1(θ ω,ϕu( ,ω)T(ω,x)) is a (conjuga ed) andom dynamical sys em. 3 Random chemos a In his sec ion we will in es iga e he s ochas ic sys em (5)-(6). To his end, we i s ans o m i in o di e en ial equa ions wi h andom coe icien s and wi hou whi e noise. Le Wbe a wo sided Wiene p ocess. Kolmogo o ’s heo em ensu es ha Whas a con inuous e sion, ha we will deno e by ω, whose canonical in e p e a ion is as ollows: le Ωbe de ined by Ω={ω∈C(R,R):ω(0) = 0}=C0(R,R), 6 T. Ca aballo, M.J. Ga ido-A ienza, J. L´ opez-de-la-C uz Fbe he Bo el σ−algeb a on Ωgene a ed by he compac open opology (see [1] o de ails) and P he co esponding Wiene measu e on F. We conside he Wiene shi low gi en by θ ω(·) = ω(·+ )−ω( ), ∈R, hen (Ω,F,P,{θ } ∈R)is a me ic dynamical sys em. Now le us in oduce he ollowing O ns ein-Uhlenbeck p ocess on (Ω,F,P,{θ } ∈R) z∗(θ ω) = − 0 Z −∞ esθ ω(s)ds, ∈R,ω∈Ω, which sol es he ollowing Lange in equa ion [1, 5] dz +zd =dω( ), ∈R. P oposi ion 2. ([1, 5]) The e exis s a θ -in a ian se e Ω∈Fo Ωo ull Pmeasu e such ha o ω∈e Ω,we ha e (i) he andom a iable |z∗(ω)|is empe ed. (ii) he mapping ( ,ω)→z∗(θ ω) = − 0 Z −∞ esω( +s)ds+ω( ) is a s a iona y solu ion o (7) wi h con inuous ajec o ies; (iii) in addi ion, o any ω∈˜ Ω: lim →±∞ |z∗(θ ω)| =0; lim →±∞ 1 Z 0 z∗(θsω)ds =0; lim →±∞ 1 Z 0 |z∗(θsω)|ds =E[z∗]<∞. In wha ollows we will conside he es ic ion o he Wiene shi θ o he se ˜ Ω, and we es ic acco dingly he me ic dynamical sys em o his se , ha is also a me ic dynamical sys em, see [4]. Fo simplici y, we will s ill deno e he es ic ed me ic dynamical sys em by he old symbols (Ω,F,P,{θ } ∈R). S ochas ic chemos a s 7 3.1 S ochas ic chemos a becomes a andom chemos a In wha ollows we use he O ns ein-Uhlenbeck p ocess o ans o m (5)-(6) in o a andom sys em. Le us no e ha analyzing he equilib ia we ob ain ha he only one is he axial equilib ium (S0,0)and hen we de ine wo new a iables σand κby σ( )=(S( )−S0)eαz∗(θ ω),(7) κ( ) = x( )eαz∗(θ ω).(8) Fo he sake o simplici y we will w i e z∗ins ead o z∗(θ ω), and σand κin- s ead o σ( )and κ( ). On he one hand, by di e en ia ion, we ha e dσ=eαz∗dS +(S−S0)eαz∗αdz∗ =(S0−S)D+α2 2−mSx a+Sd +α(S0−S)◦dW( )eαz∗ +(S−S0)eαz∗α{−z∗d +dW( )} = (S0−S)D+α2 2eαz∗d −mSx a+Seαz∗d +α(S0−S)eαz∗◦dW( ) −(S−S0)αeαz∗z∗d +(S−S0)eαz∗α◦dW ( ) =−D+α2 2σ−mSκ a+S−ασz∗d ="−D+α2 2σ−m(S0+σe−αz∗) a+S0+σe−αz∗κ−ασz∗#d . On he o he hand, dκ=eαz∗dx +xeαz∗αdz∗ =xmS a+S−D+α2 2d −αx◦dW( )eαz∗+αxeαz∗[−z∗d +dW( )] =xmS a+Seαz∗d +x−D+α2 2eαz∗d −αxeαz∗◦dW( ) −αxz∗eαz∗d +αxeαz∗◦dW( ) 8 T. Ca aballo, M.J. Ga ido-A ienza, J. L´ opez-de-la-C uz ="m(S0+σe−αz∗) a+S0+σe−αz∗κ−D−α2 2κ−αz∗κ#d . Thus, we ha e ob ained he ollowing andom sys em dσ d =−(¯ D+αz∗)σ−m(S0+σe−αz∗) a+S0+σe−αz∗κ,(9) dκ d =−(e D+αz∗)κ+m(S0+σe−αz∗) a+S0+σe−αz∗κ,(10) whe e ¯ D:=D+α2 2and e D:=D−α2 2. 3.2 Random chemos a gene a es an RDS Nex we p o e ha he andom chemos a sys em (9)-(10) gene a es an RDS. F om now on, we deno e X:={(x,y)∈R2:x∈R,y≥0}, he uppe hal -plane. Lemma 2. Assume ha D≥α2 2,˜ λ:=˜ Da m−˜ D≥S0.(11) Then o any ω∈Ωand any ini ial alue u0:= (σ0,κ0)∈X, whe e σ0:=σ(0) and κ0:=κ(0), sys em (9)-(10) possesses a unique global solu ion u(·;ω,u0):= (σ(·;ω,u0),κ(·;ω,u0)) ∈C1([0,+∞),X)wi h u(0;ω,u0) = u0. Mo eo e he so- lu ion mapping gene a es a andom dynamical sys em ϕu:R+×Ω×X→X de ined as ϕu( ,ω)u0=u( ;ω,u0),∀ ∈R+,u0∈X,ω∈Ω. P oo . Obse e ha we can ew i e one o he e ms in he p e ious equa ions as m(S0+σe−αz∗) a+S0+σe−αz∗κ=m(S0+σe−αz∗+a−a) a+S0+σe−αz∗κ=mκ−maκ a+S0+σe−αz∗ and he e o e sys em (9)-(10) u ns in o dσ d =−(¯ D+αz∗)σ−mκ+ma a+S0+σe−αz∗κ,(12) dκ d =−(e D+αz∗)κ+mκ−ma a+S0+σe−αz∗κ.(13) S ochas ic chemos a s 9 Deno ing u(·;ω,u0):= (σ(·;ω,u0),κ(·;ω,u0)), sys em (12)-(13) can be ew i - en as du d =L(θ ω)·u+F(u,θ ω), whe e L(θ ω) = −(¯ D+αz∗)−m 0−(e D+αz∗)+ m and F:X×[0,+∞)−→ R2is gi en by F(ξ,θ ω) =    ma a+S0+ξ1e−αz∗ξ2 −ma a+S0+ξ1e−αz∗ξ2  , whe e ξ= (ξ1,ξ2)∈X. Since z∗(θ ω)is con inuous, Lgene a es an e olu ion sys em on R2. Mo eo e , we no ice ha ∂ ∂ξ2±am a+S0+ξ1e−αz∗ξ2=±am a+S0+ξ1e−αz∗ and ∂ ∂ξ1±am a+S0+ξ1e−αz∗ξ2=∓ame−αz∗ (a+S0+ξ1e−αz∗)2ξ2 so F(·,θ ω)∈C(X×[0,+∞);R2)and is con inuously di e en iable wi h espec o he a iables (ξ1,ξ2), which implies ha i is locally Lipschi z wi h espec o (ξ1,ξ2)∈X. The e o e, hanks o classical esul s om he heo y o o dina y di e en ial equa ions, sys em (12)-(13) possesses a unique local solu ion. Le us check now ha in ac his solu ion is a global one. In o de o do ha , we spli ou analysis in o wo di e en cases: i s , we assume σ( )≥0 o all ≥0. Thus, om (9)-(10) d d (σ+κ) = −¯ Dσ−αz∗σ−e Dκ−αz∗κ 16 T. Ca aballo, M.J. Ga ido-A ienza, J. L´ opez-de-la-C uz 3.4 Exis ence o he andom a ac o o he s ochas ic chemos a sys em We ha e p o ed ha he sys em (9)-(10) has a unique global solu ion u( ;ω,u0) which emains in X o all u0∈Xand gene a es he RDS ϕu. Now, we de ine a mapping T:Ω×X−→ X as ollows T(ω,ζ) = T(ω,(ζ1,ζ2)) = T1(ω,ζ1) T2(ω,ζ2)=(ζ1−S0)eαz∗(ω) ζ2eαz∗(ω) whose in e se is gi en by T−1(ω,ζ) = S0+ζ1e−αz∗(ω) ζ2e−αz∗(ω). We know ha ( )=(S( ),x( )) and u( )=(σ( ),κ( )) a e ela ed by (7)-(8). Since Tis a homeomo phism, hanks o Lemma 1 we ob ain a conjuga ed RDS gi en by ϕ ( ,ω) 0:=T−1(θ ω,ϕu( ,ω)T(ω, 0)) =T−1θ ω,ϕu( ,ω)(S(0)−S0)eαz∗(ω) x(0)eαz∗(ω) =T−1(θ ω,ϕu( ,ω)u0) =T−1(θ ω,u( ;ω,u0)) =S0+σ( )e−αz∗(θ ω) κ( )e−αz∗(θ ω) = ( ;ω, 0) which means ha ϕ is an RDS o ou o iginal s ochas ic sys em (5)-(6). Mo eo e , he global andom a ac o o he andom sys em (9)-(10) A={A(ω)}ω∈Ω={(0,0)} becomes A={e A(ω)}ω∈Ω={(S0,0)}, he global andom a ac o o he s ochas ic sys em (5)-(6). S ochas ic chemos a s 17 3.5 Nume ical simula ions and inal commen s To con i m he esul s abo e, in his sec ion we show some nume ical simula ions o (3)-(4). We use he Eule -Ma uyama me hod [15] conside ing an ini ial alue (S0,x0) = (5,10),S0=1, D=3, a=0.6, m=3 and he ollowing nume ical scheme: Sj=Sj−1+ (xj−1,Sj−1)∆ +g(xj−1,Sj−1)·(W(τj)−W(τj−1)), xj=xj−1+e (xj−1,Sj−1)∆ +e g(xj−1,Sj−1)·(W(τj)−W(τj−1)), whe e we de ine unc ions ,g,e and e gas (xj−1,Sj−1) = (S0−Sj−1)D−mSj−1xj−1 a+Sj−1, g(xj−1,Sj−1) = α(S0−Sj−1), e (xj−1,Sj−1) = xj−1mSj−1 a+Sj−1 −D, e g(xj−1,Sj−1) = αxj−1, and W(τj)−W(τj−1) = jR ∑ k=jR−R+1 dWk, whe e Ris a nonnega i e in ege numbe and dWka e N(0,1)−dis ibu ed inde- penden andom a iables which can be gene a ed nume ically by pseudo andom numbe gene a o s. F om now on, he ed lines in he pic u es ep esen he s ochas ic solu ions o sys em (3)-(4) and he blue ones he de e minis ic solu ions o he same sys em. By he p e ious sec ions, we know ha sys em (3)-(4) possesses a andom a ac- o gi en by ˜ A={(S0,0)}as long as (11) is sa is ied. Fo he ollowing di e en alues o αwe ob ain he ollowing alues o ˜ λ: (a) Case α=0.1: eλ:=e Da m−e D=359.4≥1=S0. 18 T. Ca aballo, M.J. Ga ido-A ienza, J. L´ opez-de-la-C uz (b) Case α=0.5: eλ:=e Da m−e D=13.8≥1=S0. (c) Case α=1: eλ:=e Da m−e D=3≥1=S0. (d) Case α=1.5: eλ:=e Da m−e D=1≥1=S0. Summing up, in all he abo e cases eλ≥S0and D≥α2 2hold, hence he solu ions o sys em (3)-(4) o he p e ious alues o he pa ame e s go o (S0,0)=(1,0), he andom a ac o . The ollowing pic u es show wha we expec ed om he heo y and nume ical compu ing and we also can obse e wha happens when he in ensi y o noise in- c eases. S( ) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x( ) 0 2 4 6 8 10 12 Phase plane S( ) 0123456 x( ) 0 2 4 6 8 10 12 Phase plane Fig. 1 α=0.1 on he le and α=0.5 on he igh S( ) -10123456 x( ) 0 2 4 6 8 10 12 14 16 Phase plane S( ) -10123456 x( ) 0 2 4 6 8 10 12 Phase plane Fig. 2 α=1 on he le and α=1.5 on he igh S ochas ic chemos a s 19 Howe e , he nex pic u es show wha happens i eλ<S0holds ue. In his case D=1.5 ins ead o D=3 as in he p e ious cases. Fig. 3 α=0.1 on he le and α=0.5 on he igh Fig. 4 α=0.7 on he le and α=0.9 on he igh Rema k 3. We would like o men ion ha he ac ha he subs a e S(o i s co e- sponding σ) may ake nega i e alues does no p oduce any ma hema ical incon- sis ence in ou analysis, in o he wo ds, ou ma hema ical analysis is accu a e o handle he ma hema ical p oblem. Howe e , om a biological poin o iew, his may e lec some oubles and sugges s ha ei he he ac o pe u bing he dilu ion a e wi h an addi i e noise may no be a ealis ic si ua ion, o ha we should y o use a some kind o swi ching sys em o model ou eal chemos a in such a way ha when he dilu ion may be nega i e we use a di e en equa ion o model he sys em. This will lead us o a di e en analysis in some subsequen pape s by conside ing a di e en kind o andomness o s ochas ici y in his pa ame e o designing a di e - en model o ou p oblem. 20 T. Ca aballo, M.J. Ga ido-A ienza, J. L´ opez-de-la-C uz On he o he hand, i could also be conside ed a noisy e m in each equa ion o he de e minis ic model in he same ashion as in he pape by Imho and Walche [16], which ensu es he posi i i y o bo h he nu ien and biomass, al hough does no p ese e he wash ou equilib ium om he de e minis ic o he s ochas ic model. We a e cu en ly in e es ed on his kind o chemos a models and we will analyze hem in u u e pape s. Acknowledgemen s: Pa ially suppo ed by FEDER and Minis e io de Econom´ ıa y Compe i i i- dad unde g an MTM2015-63723-P and Jun a de Andaluc´ ıa unde P oyec o de Excelencia P12- FQM-1492. We also would like o hank Alain Rapapo and S e anie Sonne o he nice discus- sions ha we had wi h hem du ing he inal w i ing o he pape . Thanks o hei help ul sugges- ions we we e able o imp o e he p elimina y e sion o his pape . Finally, we a e eally g a e ul o he e e ee o he kind commen s and use ul sugges ions which helped us o make he cu en pape . Re e ences 1. L. A nold, Random Dynamical Sys ems, Sp inge -Ve lag, Be lin, 1998. 2. H. R. Bungay and M. L. Bungay, Mic obial in e ac ions in con inuous cul u e, Ad ances in Applied Mic obiology,10 (1968) 269–290. 3. T. Ca aballo, P. E. Kloeden and B. Schmal uß, Exponen ially S able S a iona y Solu ions o S ochas ic E olu ion Equa ions and Thei Pe u ba ion, Applied Ma hema ics & Op imiza ion, 50 (2004) 183–207. 4. T. Ca aballo, M.J. Ga ido-A ienza, B. Schmal uß and J. Vale o, Asymp o ic Beha iou o a S ochas ic Semilinea Dissipa i e Func ional Equa ion Wi hou Uniqueness o Solu ions, Disc e e and Con inuous Dynamical Sys ems Se ies B, ol. 14 2(2010), 439–455. 5. T. Ca aballo, K. Lu, A ac o s o s ochas ic la ice dynamical sys ems wi h a mul iplica i e noise, F on . Ma h. China, 3 (2008), no. 3, 317–335. 6. T. Ca aballo, G. Lukaszewicz and J. Real. Pullback a ac o s o asymp o ically compac nonau onomous dynamical sys ems. Nonlinea Analysis TMA 6 (2006), 484–498. 7. H. C auel and F. Flandoli, A ac o s o andom dynamical sys ems, P obab. Theo y Rela ed Fields 100 (1994), 365–393. 8. H. C auel, Random P obabili y Measu es on Polish Spaces. Taylo & F ancis, London and New Yo k (2002). 9. A. Cunningham and R. M. Nisbe , T ansien s and oscilla ions in con inuous cul u es, Ma he- ma ics in Mic obiology, 77–103, Academic P ess, London. 1983. 10. G. D’ans, P. V. Koko o ic and D. Go lieb, A nonlinea egula o p oblem o a model o biological was e ea men , IEEE T ansac ions on Au oma ic Con ol AC-16 (1971), 341–347. 11. F. Flandoli and B. Schmal uß. Random a ac o s o he 3D s ochas ic Na ie -S okes equa ion wi h mul iplica i e noise. S ochas ics S ochas ics Rep., 59 (1996), no. 1-2, 21–45. 12. A. G. F ed ickson and G. S ephanopoulos, Mic obial compe i ion, Science,213 (1981), no. 4511, 972–979. 13. R. F e e , Mechanisms ha con ol he mic o lo a in he la ge in es ine, in Human In es inal mic o lo a in Heal h and Disease, 33–54, D. J. Hen ges, ed., Academic P ess, New Yo k, 1983. 14. R. F e e , An unde s anding o coloniza ion o he la ge in es ine equi es ma hema ical anal- ysis, Mic oecology and The apy,16 (1986) 147–155. 15. D. J. Higham, An algo i hmic in oduc ion o nume ical simula ion o s ochas ic di e en ial equa ions, SIAM Re iew, ol. 43, 3, (2001), 525–546. 16. L. Imho and S. Walche , Exclusion and pe sis ence in de e minis ic and s ochas ic chemos a models, J. Di e en ial Equa ions, 217 (2005), 26–53 S ochas ic chemos a s 21 17. H. W. Jannash and R. T. Ma eles, Expe imen al bac e ial ecology s udies in con inuous cul u e, Ad ances in Mic obial Physiology 11 (1974) 165–212. 18. J. W. M. La Ri ie e, Mic obial ecology o liquid was e, Ad ances in Mic obial Ecology 1 (1977), 215–259. 19. H. L. Smi h, Mono one Dynamical Sys ems: an In oduc ion o he Theo y o Compe i i e and Coope a i e Sys ems, Ma hema ical Su eys and Monog aphs 41. Ame ican Ma hema ical Socie y, P o idence, RI (1995). 20. H. L. Smi h and P. Wal man, The Theo y o he Chemos a : Dynamics o Mic obial Compe i- ion, Camb idge Uni e si y P ess, Camb idge, UK (1995). 21. V. S ee Ha i Rao and P. Raja Sekha a Rao, Dynamic Models and Con ol o Biological Sys- ems, Sp inge -Ve lag, Heidelbe g (2009). 22. P. A. Taylo and J. L. Williams, Theo e ical s udies on he coexis ence o compe ing species unde con unous low condi ions, Canadian Jou nal o Mic obiology 21 (1975) 90–98. 23. H. Veldcamp, Ecological s udies wi h he chemos a , Ad ances in Mic obial Ecology,1 (1977), 59–95. 24. P. Wal man, Compe i ion Models in Popula ion Biology, CBMS-NSF Regional Con e ence Se- ies in Applied Ma hema ics 45. Socie y o Indus ial and Applied Ma hema ics, Philadelphia (1983). 25. P. Wal man, Coexis ence in chemos a -like model, Rocky Moun ain Jou nal o Ma hema ics 20 (1990), 777–807. 26. P. Wal man, S. P. Hubbel and S. B. Hsu, Theo e ical and expe imen al in es iga ions o mi- c obial compe i ion in con inuous cul u e, Modeling and Di e en ial Equa ions in Biology (Con ., sou he n Illinois Uni . Ca bonadle, III., 1978), pp. 107–152. Lec u e No es in Pu e and Appl. Ma h., 58, Dekke , New Yo k (1980). 27. C. Xu, S. Yuan and T. Zhang, Asymp o ic Beha iou o a Chemos a Model wi h S ochas ic Pe u ba ion on he Dilu ion Ra e, Hindawi Publishing Co po a ion. Abs ac and Applied Analysis (2013).