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 .
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