A propos de quelques modèles à compartiments

Transcription

1 A propos de quelques modèles à compartiments Ecole-chercheur sur les modèles ressources-consommateurs UMR MISTEA et UMR Eco&Sols septembre 2015

2 Le chémostat avec beaucoup d espèces D = 0.4h 1 T dble = log 2 D 1.73h

3 Le chémostat avec beaucoup d espèces

4 Le chémostat avec beaucoup d espèces

5 Le chémostat avec beaucoup d espèces

6 Le chémostat avec beaucoup d espèces ṡ = n µ j (s)x j + D(s in s) j=1 ẋ i = µ i (s)x i Dx i (i = 1 n) Simulations numériques avec n > 100 : tirage aléatoire des fonctions µ i ( ) dans une famille de fonctions de Monod tirage aléatoire de la composition initiale en biomasse

7 biomasse totale Le chémostat avec beaucoup d espèces Soit b = n i=1 x i et p i = x i b { ṡ = µ(s, p)b + D(sin s) ḃ = µ(s, p)b Db and ṗ i = (µ i (s) µ(s, p)) p i n with µ(s, p) = p i µ i (s) i=1 0 0 substrat

8 Séparation en deux groupes sous ensemble de m souches efficaces µ souches moins efficaces D λ λj

9 Dynamique rapide n ṡ = µ(s) x µ j (s)x j + D(s in s) x = µ(s) x D x j=m+1 ẋ j = µ j (s)x j Dx j (j = m + 1 n) Principe d Exclusion Compétitive : lim t + s(t) = s lim t + x(t) = x = s in s lim t + x j(t) = 0 (j = m + 1 n)

10 La dynamique lente Pour i = 1 m, on écrit µ i (s) = µ(s) + ɛν i (s) m ṗ i = ɛ ν i (s ) p j ν j (s ) p i j=1 Solution explicite : p i (t) = p i(0)e A i ɛt m p j (0)e A jɛt j=1 avec A j = ν j (s )

11 Propriétés de la dynamique lente Toute proportion t p i (t) est - soit croissante jusqu à un temps T i puis décroissante, - soit croissante pour tout t, - soit décroissante pour tout t. Lorsque ε 0 alors T i + pour toutes les souches sauf une. L entropie réactionnelle E(t) := j µ j ( s)p j (t) est croissante jusqu à D. see R. Dochain Harmand. Long run coexistence in the chemostat with multiple species, J. Theor. Bio. 2009

12 Remarque Le Principe d Exclusion Compétitive est un résultat asymptotique.

13 About modelling flocculation and bacteria attachment at a macroscopic level Results obtained in the ANR DISCO project - May 2013

14 Content Extensions of the classical chemostat model Dynamics with mono-specific flocs Dynamics with multi-specific flocs Some results about overyielding Concluding remarks and perspectives

15 Main objective Introduce in the simple chemostat model : ṡ = µ(s)x + D(s in s) (substrate concentration) ẋ = µ(s)x Dx (biomass concentration) a consideration of planktonic bacteria and attached bacteria : good access to substrate less access to substrate

16 Attachement and detachment processes

17 A simple modelling The classical chemostat model : ṡ = µ(s)x + D(s in s) ẋ = µ(s)x Dx An extension with planktonic bacteria (of concentration v) and attached bacteria (of concentration w) with x = v + w : where a( )v : b( )w : ṡ = µ v (s)v µ w (s)w + D(s in s) v = µ v (s)v D v v a( )v+b( )w ẇ = µ w (s)w D w w+a( )v b( )w attachement detachement

18 Examples adaptive nutrient uptake. a( ) = a(s), b( ) = b(s) Tang Sitomer Jackson 97 wall attachement. a( ) = a, b( ) = b Pilyguin Waltman 99 intestine( model. ) a( ) = a 1 v v max, b( ) = b+µ v (s)(1 G( v v max )) with G( ) Freter 83 flocs. a( ) = av, b( ) = b Haegeman Rapaport 08

19 Main assumptions ṡ = µ v (s)v µ w (s)w + D(s in s) v = µ v (s)v D v v a( )v+b( )w ẇ = µ w (s)w D w w+a( )v b( )w Assumption A1. µ v (s) µ w (s), s 0 Assumption A2. a( ) = 1 ε α( ) and b( ) = 1 β( ) with ε small ε quasi-stationnary approximation : α(s, v, w)v = β(s, v, w)w v + w = x

20 Reduced dynamics Assumption A3. α(s, v, w)v = β(s, v, w)w can be solved as v = p(s, x)x w = (1 p(s, x))x with p( ) smooth Define µ(s, x) = p(s, x)µ v (s)+(1 p(s, x))µ w (s). Model 1 : { ṡ = µ(s, x)x + D(sin s) ẋ = µ(s, x)x Dx if D v = D w = D Model 2 : { ṡ = µ(s, x)x + d(s, x)(sin s) ẋ = µ(s, x)x d(s, x)x if D v D w with d(s, x) = p(s, x)d v +(1 p(s, x))d w

21 Examples flocs of two individuals : α( ) = av, β( ) = b p(x) = undifferentiated flocs : a/bx α( ) = a(v + w), β( ) = b p(x) = a/bx

22 The Competitive Exclusion Principle ṡ = n µ j (s)x j + D(s in s) j=1 ẋ i = µ i (s)x i Dx i i = 1 n Proposition (Hsu Hubbell Waltman 77...) Under the conditions 0 < µ 1 1 µ 1 1 (D) < s in (D) < µ 1 2 any solution with x 1 (0) > 0 satisfies (D) µ 1(D) lim (s(t), x 1(t),, x n (t)) = (µ 1 t + 1 (D), s in µ 1 1 (D), 0,, 0) n

23 Chemostat model with density dependent growths ṡ = n µ j (s, x j )x j + D(s in s) j=1 ẋ i = µ i (s, x i )x i Dx i i = 1 n Proposition (Lobry Mazenc Rapaport 05...) Under the conditions µ i (, ) increasing w.r.t. s, decreasing w.r.t. x i µ i (s in, 0) > D µ i (s, + ) = 0, s 0 there exists an unique positive equilibrium, that is globally exponentially stable.

24 Sketch of proof in 2D Example of two species on the attractive manifold x 1 + x 2 = s in s X 2 X 2 X 2 X 2 X =0 2 X =0 2 X =0 1 X =0 1 0 X 1 without density dependency X 1 0 X 1 with density dependency X 1

25 Chemostat model with multi-specific flocs ṡ = n µ j (s, x 1,, x n )x j + D(s in s) j=1 ẋ i = µ i (s, x 1,, x n )x i Dx i i = 1 n Typically a i ( ) = 1 α ij x j and b i ( ) = 1 ε ε β i j p i (x 1,, x n ) = β i β i + j α ijx j Lobry Harmand 06 : Under the conditions µ i ( ) increasing in s, decreasing in each x j x i = 0 µ i (s in, ) > D x i = + µ i ( ) = 0 simulations show the existence of asymptotically stable positive equilibrium, but the theory is not ready at the moment...

26 The case of density-dependent dilution rate { µ(s, x )x = D(s in s ) µ(s, x ) = d(x ) = { s = g(x ) := s in x d(x) D s = f (x ) or? s = f (x ) : p(x ) [µ v (s ) D v ] +(1 p(x )) [µ w (s ) D w ] = 0 }{{}}{{} = 0 for s = λ v = 0 for s = λ w µ v µ v D v µ w D v µ w Dw Dw λ w λv λ v > λ w λv λ w λ v < λ w

27 Multiplicity of equilibria Proposition (Fekih-Salem Harmand Lobry Rapaport Sari 12) λ v < λ w λ v > λ w (f ) (f ) λ v < s in λ v > s in λ v < s in λ v > s in!(s, x ) no positive even nb. uneven nb. G.A.S. equ. of equ. alter. of equ. alter. stab. and unstab. stab. and unstab. wash-out eq. repulsive wash-out eq. attractive

28 Example µ v (s) = 2s 1 + s, µ w(s) = 1.5s s D u = D = E, D 1 = 0.5, α = 4, β = 1, s in = 0.9 intersection of the null-clines phase portrait

29 The multiple species case ṡ = n µ j (s, x)x j + D(s in s) j=1 ẋ i = µ i (s, x)x i d i (x)x i i = 1 n Define λ v = max λ v,i i and λ w = min i λ w,i. Assumptions. λ v,i < λ w,i, d i (x i) > xi µ i (s, x i ), λ v < min(λ w, s in ). Proposition (Fekih-Salem Harmand Lobry Rapaport Sari 12) Dynamics admits an unique positive equilibrium E if and only if i µ i ( λ v, gi 1 ( λ v ))gi 1 ( λ v ) < D(s in λ v ) When E exists, it is locally exponentially stable.

30 Ongoing work Consider a species with µ v ( ) non monotonic (Haldane) µ w ( ) monotonic (Monod) Possible behaviors : i. no positive equilibrium ii. one positive equilibrium (L.A.S) iii. bi-stability - Add a species with monotonic growth, that does not aggregate : Hopf bifurcation

31 About niches and overyielding Consider two species and a spatial structure α Q S in S in (1 α) Q D* µ 2 µ 1 V 1 = r V V 2 = (1 r) V S 1 S 2 + S S out

32 Bioconversion overyielding S in S in S in S in αq (1 α)q αq (1 α)q S 1 S 2 S 1 S S out S out S in S in S in S in αq (1 α)q αq (1 α)q S 1 S 2 S 1 S S out Which is the best configuration? S out

33 Steady state overyielding Assumption : µ( ) is increasing. µ Let λ(d) = 1 (D) if D µ(s in ) s in if D > µ(s in ) ( ) ( ) α 1 α Define F (α, r) := αλ r D + (1 α)λ 1 r D Definition : There is overyielding exactly when G(α, r) := αλ 1 ( α r D ) +(1 α)λ 2 ( 1 α 1 r D ) < min(f 1 (α, r), F 2 (α, r))

34 Steady state overyielding C = {(α, r) [0, 1] 2 (α/r)d µ(s in ) and ((1 α)/(1 r)) D µ(s in )} T 1 = {(α, r) [0, 1] 2 (α/r)d > µ(s in )} T 2 = {(α, r) [0, 1] 2 ((1 α)/(1 r)) D > µ(s in ))} Proposition. If µ( ) is concave then the restriction of F on C is convex. Proposition. Define T in (D) = λ(d) + Dλ (D). One has min F = λ(d) = F (α, α), α [0, 1], for s in T in (D) [0,1] 2 min F (α, 0) < λ(d), for s in < T in (D) α

35 Steady state overyielding Assume that there exists D such that λ 1 (D ) = λ 2 (D ). Proposition. When D = D and α (0, 1), there exist configurations close to (α, α) that present overyielding. Proposition. When D D, there exists (α, r) with α r D < D < 1 α 1 r D < min(µ 1(s in ), µ 2 (s in )) that corresponds to overyelding.

36 Example (large s in ) 10 r= r= r= r= F F F F1 F2 G α 4 3 F 2 G α 4 3 F 1 G α 4 3 F 1 G α

37 Example (small s in ) 5.0 r= r= r= r= F F 1 F 1 F 2 G α F 2 G α F 1 G α F 2 F 1 G α

38 Concluding remarks and perspectives Macroscopic models of flocs with same dilution rate lead to density-dependent growth rate for the overall biomass. Macroscopic models of flocs with different dilution rate lead in addition to density-dependent dilution rate for the overall biomass. Richness of possible behaviors with possibly multiple positive equilibria, bi-stability, limit cycle... How to infer the right attachment and detachment terms? We would like to see if all these predictions could occur on more realistic (and more complex) models as well as in real flocs or biofilms...

39 References Haegeman, B., C. Lobry and J. Harmand (2007) Modeling bacteria flocculation as density-dependent growth, AIChE Journal, Vol. 53(2), pp Haegeman, B. and Rapaport, A. (2008) How flocculation can explain coexistence in the chemostat, Journal of Biological Dynamics, Vol. 2, No. 1, pp Fekih-Salem R., Harmand J., Lobry C., Rapaport, A. and Sari T. (2012) Extensions of the chemostat model with flocculation, J. of Mathematical Analysis and Applications, Vol. 397, pp Fekih-Salem R., Rapaport, A. and Sari T. (2013) Effect of the flocculation on the coexistence of microbial species and apparition of limit cycles with non monotonic growth rates, submitted.