Random dynamical systems to analyze different ways of modeling stochastic chemostats

Random dynamical systems to analyze different ways of modeling stochastic chemostats Javier L´opez-de-la-Cruz Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla - Sevilla (Spain) joint work with T. Caraballo, M. J. Garrido-Atienza (PhD advisors) and A. Rapaport (UMR INRA-SupAgro 0729 MISTEA, Montpellier, France) XVI Conferencia Espa˜nola de Biometr´ıa (Sevilla) - September 13th, 2017 Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 1 / 55 1Introduction 2Our models 3Preliminaries on stochastic processes 4Preliminaries on the theory of random dynamical systems Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 2 / 55 What is a chemostat? Laboratory device used for growing microorganisms in a cultured environment Standard assumptions for simple chemostats: Availability of nutrient and its supply rate are both fixed Wall growth is not taken into account Feed Bottle Collection Vessel Culture Vessel Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 3 / 55 What is a chemostat? However... VERY STRONG RESTRICTIONS! since real world is non-autonomous and STOCHASTIC more realistic situation: WALL GROWTH It encourages us to study STOCHASTIC CHEMOSTATS WITH WALL GROWTH Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 4 / 55 Importance of chemostat models 1They play an important role in ecological studies 2They are used as models of wastewater treatment processes 3They can be considered as starting point for other several models: Problems of genetically altered organisms Antibiotic production models Fermentation models: WINE,BEER...!!!!!! Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 5 / 55 1Introduction 2Our models 3Preliminaries on stochastic processes 4Preliminaries on the theory of random dynamical systems Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 6 / 55 The simplest chemostat model dS dt = (S0−S)D−mSx a+S, dx dt =−Dx +mSx a+S, where S(t) : concentration of the nutrient x(t) : concentration of the microbial biomass S0>0 : input concentration a>0 : half-saturation constant D>0 : dilution rate m>0 : maximal consumption rate of the nutrient and maximal specific growth rate of microorganisms Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 7 / 55 Setting up the simplest chemostat model dS dt = (S0−S)D−mSx a+SEquation for the nutrient dx dt =−Dx +mSx a+SEquation for the specie Feed Bottle Collection Vessel Culture Vessel S0 Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 8 / 55 Setting up the simplest chemostat model dS dt = (S0−S)D−mSx a+SEquation for the nutrient dx dt =−Dx +mSx a+SEquation for the specie Feed Bottle Collection Vessel Culture Vessel S0 D Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 9 / 55 Chemostat with wall growth dS dt =D(S0−S)−mS a+Sx1−mS a+Sx2+bνx1, dx1 dt =−(ν+D)x1+cS a+Sx1−r1x1+r2x2, dx2 dt =−νx2+cS a+Sx2+r1x1−r2x2 S(t),x1(t),x2(t) : concentrations of the nutrient and the two different microorganisms b∈(0,1) : fraction of dead biomass which is recycled ν > 0 : collective death rate coefficient r1,r2>0 : rates at which the species stick on to and shear off the walls, respectively 0<c≤m: growth rate coefficient of the consumer species Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 13 / 55 1Introduction 2Our models 3Preliminaries on stochastic processes 4Preliminaries on the theory of random dynamical systems Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 14 / 55 The Wiener process Definition Astandard Wiener process is a family of random variables W(t)(·) : ω∈Ω7→ W(t)(ω)∈R,t≥0, s.t. P−almost surely W(0)=0 continuous (but NOT bounded variation) paths: t∈R+7→ W(t)(ω)∈R independent increments: for 0 <t1<···<tn, W(t1),W(t2)−W(t1),...,W(tn)−W(tn−1) are independent random variables stationarity: joint distribution of {W(t1+t),...,W(tk+t)}does NOT depend on t W(t)−W(s), 0 ≤s≤t, is a Gaussian variable with mean 0 and variance t−s Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 15 / 55 The Wiener process -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -4 -3 -2 -1 0 1 2 3 4 5Noise Paths Figure: Realizations of the standard Wiener process Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 16 / 55 The Wiener process W: Wiener process Kolmogorov’s theorem ensures that Whas a continuous version, ω, whose canonical interpretation is: Ω := {ω∈ C(R,R) : ω(0) = 0} F: Borel σ−algebra on Ω P: Wiener measure on F We consider the Wiener shift flow θtω(·) = ω(·+t)−ω(t),t∈R, then (Ω,F,P,{θt}t∈R) is a metric dynamical system Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 17 / 55 The Ornstein-Uhlenbeck (OU) process The OU process on (Ω,F,P,{θt}t∈R) is defined as z∗(θtω) = −Z0 −∞ esθtω(s)ds,t∈R, ω ∈Ω, which solves the Langevin equation dz +zdt =dω(t),t∈R(1) T. Caraballo, P. E. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Applied Mathematics & Optimization, vol. 50, no. 3, (2004) 183–207. Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 18 / 55 The Ornstein-Uhlenbeck (OU) process Proposition [T. Caraballo, P. E. Kloeden and B. Schmalfuß, 2004] There exists a θt-invariant set e Ω∈ F of Ω of full Pmeasure such that for ω∈e Ω,we have (i) the random variable |z∗(ω)|is tempered. (ii) the mapping (t, ω)→z∗(θtω) = − 0 R −∞ esω(t+s)ds+ω(t) is a stationary solution of (1) with continuous trajectories; (iii) in addition, for any ω∈˜ Ω: lim t→±∞ |z∗(θtω)| t= 0; lim t→±∞ 1 tZt 0 z∗(θsω)ds = 0; lim t→±∞ 1 tZt 0 |z∗(θsω)|ds =E[z∗]<∞. Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 19 / 55 1Introduction 2Our models 3Preliminaries on stochastic processes 4Preliminaries on the theory of random dynamical systems Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 20 / 55 Random dynamical systems (RDSs) (X,k · kX) separable Banach space Definition An RDS on Xconsists of two ingredients: (a) a metric dynamical system (Ω,F,P,{θt}t∈R), where (Ω,F,P) is a probability space and the family of mappings θt: Ω →Ω satisfies (1) θ0= IdΩ, (2) θs◦θt=θs+tfor all s,t∈R, (3) the mapping (t, ω)7→ θtωis measurable, (4) the probability measure Pis preserved by θt, i.e., θtP=P and (b) a mapping ϕ: [0,∞)×Ω×X→Xwhich is (B[0,∞)× F × B(X),B(X))- measurable, such that for each ω∈Ω, (i) mapping ϕ(t, ω) : X→X,x7→ ϕ(t, ω)xis cont. for every t≥0, (ii) ϕ(0, ω) is the identity operator on X, (iii) (cocycle property) ϕ(t+s, ω) = ϕ(t, θsω)ϕ(s, ω) for all s,t≥0. Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 21 / 55 Random dynamical systems (RDSs) Definition Arandom set K is a measurable subset of X×Ω w.r.t. σ−algebra B(X)× F. Moreover Kwill be said a closed (compact) random set if K(ω) = {x: (x, ω)∈K}, ω ∈Ω, is closed (compact) for P−a.e. ω∈Ω. Definition A bounded random set K(ω)⊂Xis said to be tempered with respect to {θt}t∈Rif for a.e. ω∈Ω, lim t→∞ e−βtsup x∈K(θ−tω) kxkX= 0,for all β > 0; a random variable ω7→ r(ω)∈Ris said to be tempered with respect to {θt}t∈Rif for a.e. ω∈Ω, lim t→∞ e−βtsup t∈R |r(θ−tω)|= 0,for all β > 0. Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 22 / 55 Perturbing the input flows Biological process: particles of dirt inside the pumps Feed Bottle Collection Vessel Culture Vessel bbbb bbb b b bbb bbbbbbb bbbb bb Biologists claim that in the real life The flow passing through the pumps IS NOT CONSTANT The best approach: STOCHASTIC/RANDOM FLUCTUATIONS Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 29 / 55 How can we perturb the input flows? We will analyze two different ways of modeling perturbed input flows in the chemostat model Classical way: by using the standard Wiener process D D+α˙ω(t) SEVERAL INCONNVENIENTS!! New idea: by using DIRECTLY the Ornstein-Uhlenbeck process D D+αz∗(θtω) EXCELLENT APPROACH TO THE REAL LIFE!! Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 30 / 55 Classical way: the Wiener process We consider the chemostat model dS dt = (S0−S)D−mSx a+S, dx dt =−Dx +mSx a+S, and introduce the following perturbation D D+α˙ω(t) then we obtain the following stochastic model understood in Itˆo’s sense dS =(S0−S)D−mSx a+Sdt +α(S0−S)dω(t), dx =−Dx +mSx a+Sdt −αxdω(t). Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 31 / 55 Classical way: the Wiener process Itˆo vs Stratonovich conversion dXt=a(t,Xt)dt +b(t,Xt)dWt, dXt=a(t,Xt)−1 2b(t,Xt)b′(t,Xt)dt +b(t,Xt)◦dWt Then, we obtain the following stochastic chemostat model dS =(S0−S)¯ D−mSx a+Sdt +α(S0−S)◦dω(t),(2) dx =−¯ Dx +mSx a+Sdt −αx◦dω(t),(3) in Stratonovich sense, where ¯ D:= D+α2 2.(4) Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 32 / 55 Classical way: the Wiener process We define the following variable change σ(t) = (S(t)−S0)eαz∗(θtω),(5) κ(t) = x(t)eαz∗(θtω).(6) Then, by differentiation, we have the following RANDOM chemostat model given by dσ dt =−(¯ D+αz∗)σ−m(S0+σe−αz∗(θtω) a+S0+σe−αz∗(θtω),(7) dκ dt =−(¯ D+αz∗)κ+m(S0+σe−αz∗(θtω) a+S0+σe−αz∗(θtω).(8) Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 33 / 55 Classical way: the Wiener process Now, we define X:= {(x,y)∈R2:x∈R,y≥0} Theorem (Caraballo et al, 2017) For any ω∈Ω and any initial value u0:= (σ0, κ0)∈ X, where σ0:= σ(0) and κ0:= κ(0), system (7)-(8) possesses a unique global solution u(·; 0, ω, u0) := (σ(·; 0, ω, u0), κ(·; 0, ω, u0)) ∈ C1([0,+∞),X) with u(0; 0, ω, u0) = u0. Moreover, the solution mapping generates a RDS ϕu:R+×Ω× X → X defined as ϕu(t, ω)u0:= u(t; 0, ω, u0),for all t∈R+,u0∈ X, ω ∈Ω, the value at time tof the solution of system (7)-(8) with initial value u0at time zero. Key for the proof: classical results from the theory of ODEs. REMARK: some state variable can take negative values. Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 34 / 55 Classical way: the Wiener process Theorem (T. Caraballo et al, 2017) There exists a tempered compact random absorbing set for the RDS {ϕu(t, ω)}t≥0,ω∈Ω. σ κ −(a+sin)eαz∗(ω)−ε ε −ε B2 ε(ω) B1 ε(ω) Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 35 / 55 Classical way: the Wiener process Thanks to Proposition 25, we deduce that the RDS generated by the system (7)-(8) possesses a unique random pullback attractor given by A={A(ω)}ω∈Ω⊂Bε(ω),for any ε > 0. Thus, σ κ −(a+sin)eαz∗(ω) B0(ω) A={A(ω)}ω∈Ω⊂B0(ω). Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 36 / 55 Classical way: the Wiener process Analysis of the internal structure of the random pullback attractor Proposition (T. Caraballo et al, 2017) The pullback random attractor of system (7)-(8) consists of a singleton components given by A(ω) = {(0,0)} as long as ¯ D> µ(S0) (9) EXTINCTION OF THE MICROORGANISMS!! Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 37 / 55 Classical way: the Wiener process Define now a mapping T: Ω × X −→ X as T(ω, ζ) = (ζ1−S0)eαz∗(ω), ζ2eαz∗(ω) T−1(ω, ζ) = S0+ζ1e−αz∗(ω), ζ2e−αz∗(ω) Denoting v(t) = (S(t),x(t)) and u(t) = (σ(t), κ(t)), since Tis a homeomorphism, thanks to Lemma 26 we obtain a conjugated RDS given by ϕv(t, ω)v0:= v(t; 0, ω, v0) Hence, ϕvis an RDS for our original stochastic system (2)-(3) Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 38 / 55 New idea: the OU process A new framework is required The new set of events: Ωβ:= ω∈Ω : b1≤D+z∗ β,ν(θtω)≤b2,for all t∈R, The new σ−algebra Fβ: Fβ:= {A∩Ωβ,A∈ F} The new probability measure Pβ:Fβ−→ [0,1]: Pβ(Fβ) := P(Fβ) P(Ωβ),for all Fβ∈ Fβ, We can make use of the theory of RDS since (Ωβ,Fβ,Pβ,{θt}t∈R)≡(Ω,F,P,{θt}t∈R) Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 45 / 55 New idea: the OU process Theorem (T. Caraballo et al, 2017) For any ω∈Ωβand any initial value v0:= (S0,x0)∈ X, where S0:= S(0) and x0:= x(0), system (10)-(11) possesses a unique global solution v(·; 0, ω, v0) := (S(·; 0, ω, v0),x(·; 0, ω, v0)) ∈ C1([0,+∞),X) with v(0; 0, ω, v0) = v0. Moreover the solution mapping generates an RDS ϕv:R+×Ωβ× X → X defined by ϕv(t, ω)v0:= v(t; 0, ω, v0),for all t∈R+,v0∈ X, ω ∈Ωβ. Theorem (T. Caraballo et al, 2017) There exists a tempered compact random absorbing set B0(ω)∈ E(X) of the RDS {ϕv(t, ω)}t≥0, ω∈Ω. The absorbing set: B0(ω) := {(S,x)∈ X :S+x=S0}. Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 46 / 55 New idea: the OU process Proposition (T. Caraballo et al, 2017) Assuming that the following condition D> µ(S0) holds, the pullback random attractor of the chemostat model (10)-(11) is reduced to a singleton component which is given by A(ω) = {(S0,0)}. In this case the microorganisms become EXTINCT! Theorem (T. Caraballo et al, 2017) Assume µ−1(b2)<S0 holds true. Then, there exists a strictly positive tempered absorbing set B(ω)∈ E(X) of the RDS {ϕv(t, ω)}t≥0,ω∈Ω. Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 47 / 55 New idea: the Ornstein-Uhlenbeck process S0−µ−1(b1) S0−µ−1(b2) µ−1(b1)µ−1(b2)S0 S0 S x B(ω) B0(ω) Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 48 / 55 New idea: the Ornstein-Uhlenbeck process Figure: Value of parameters: D= 2, S0= 4, a= 0.6, m= 5, α= 0.5, β= 1, ν= 0.7, S(0) = 2 and x(0) = 5 Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 49 / 55 New idea: the Ornstein-Uhlenbeck process Figure: Value of parameters: D= 3.5, S0= 2, a= 0.8, m= 0.5, α= 0.5, β= 1, ν= 0.7, S(0) = 2.5 and x(0) = 5 Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 50 / 55 Comparison: Wiener vs OU Figure: Value of parameters: S0= 2, D= 3.5, a= 0.8, m= 0.5, α= 0.8, σ= 0.8, x(0) = 5, S(0) = 2.5, β= 2 (red) and β= 0.5 (green) Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 51 / 55 Comparison: Wiener vs OU Figure: Value of parameters: S0= 4, D= 2, a= 0.6, m= 5, α= 0.15, σ= 0.8, x(0) = 5, S(0) = 2, β= 2 (red) and β= 0.5 (green) Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 52 / 55 Conclusions and future works Classical way: D D+α˙ω(t) Some state variables can take negative values... Perturbations D+α˙ωare not realistic New way: D D+αz∗ β,ν(θtω) Perturbations can be controlled by taking a suitable O-U Random pullback attractor strictly contained in the positive cone Hence... PERSISTENCE OF THE MICROORGANISM CAN BE GUARANTEED!!!! Future works Chemostat model with wall growth. Generalize the new idea to perturb other models. Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 53 / 55 ¡¡¡¡Muchas gracias por vuestra atenci´on!!!! Thank you very much for your attention!!!! Universidad de Sevilla Javier L´opez de la Cruz Dpto. E.D.A.N. 54 / 55