Marine bacteria – virus interaction in a chemostat

1. Marine bacteria – virus interaction in a chemostat Jean-Christophe Poggiale Laboratoire de Microbiologie, de Géochimie et d’Ecologie Marines (UMR CNRS 6117) Université de la Méditerranée Centre d’Océanologie de Marseille 13288 Marseille Cedex 09 France Jean-christophe.poggiale@univmed.fr http://www.com.univ-mrs.fr/~poggiale/ Amsterdam – 24thJanuary 2008

2. Aggregation of variables Andreasen, Iwasa, Levin, 1987, 1989 ? Amsterdam – 24thJanuary 2008

3. Aggregation of variables P. Auger, R. Bravo de la Parra, J.-C. Poggiale, E. Sanchez, and T. Nguyen-Huu, 2008, « Aggregation of Variables and Applications to Population Dynamics » inStructured Population Models in Biology and Epidemiology Series: Lecture Notes in MathematicsSubseries: Mathematical Biosciences Subseries , Vol. 1936 Magal, Pierre; Ruan, Shigui (Eds.) , 345 p. Singular perturbation theory Discrete systems Delayed and partial differential equations Applications to population dynamics models Amsterdam – 24thJanuary 2008

4. Aggregation of variables Individuals to populations : growth Mechanistic model (IBM) Population model Individual parameters Population parameters Amsterdam – 24thJanuary 2008

5. Aggregation of variables Individuals to populations : growth Comparison between DEB model and logistic equation Comparison between DEB model and a Substrate-Structure model Amsterdam – 24thJanuary 2008

6. ? ? Aggregation of variables Time scales and singular perturbation theory Amsterdam – 24thJanuary 2008

7. Aggregation of variables Time scales and singular perturbation theory Amsterdam – 24thJanuary 2008

8. Aggregation of variables The fundamental theorems : normal hyperbolicity theory Def. : The invariant manifold M0 is normally hyperbolic if the linearization of the previous system at each point of M0 has exactly k2 eigenvalues on the imaginary axis. Amsterdam – 24thJanuary 2008

9. Theorem (Fenichel, 1971) : if is small enough, there exists a manifold M1 close and diffeomorphic to M0. Moreover, it is locally invariant under the flow, and differentiable. Aggregation of variables The fundamental theorems : normal hyperbolicity theory Theorem (Fenichel, 1971) : « the dynamics in the vicinity of the invariant manifold is close to the dynamics restricted on the manifolds ». Amsterdam – 24thJanuary 2008

10. Alcalà de Henares - April 2005 Geometrical Singular Perturbation theory The fundamental theorems : normal hyperbolicity theory • Simple criteria for the normal hyperbolicity in concrete cases (Sakamoto, 1991) • Good behavior of the trajectories of the differential system in the vicinity of the perturbed invariant manifold. • Reduction of the dimension • Powerful method to analyze the bifurcations for the reduced system and link them with the bifurcations of the complete system • Intuitive ideas used everywhere (quasi-steady state assumption, adiabatic assumptions, time scale separation…)

11. Marine bacteria – virus interaction in a chemostat • 3 state variables : S, I and V • If the «burst coefficient » increases then oscillations appear Amsterdam – 24thJanuary 2008

12. Pseudoalteromonas sp. hbmmd.hboi.edu/ jpegs2/L261.jpg An experiment in a chemostat Amsterdam – 24thJanuary 2008

13. D I Infected D D S Susceptible V Virus R Resistant D C Carbon substrate C0Reservoir D D Model description Amsterdam – 24thJanuary 2008

14. Susceptibles Resistant Infected Amsterdam – 24thJanuary 2008

15. Virus Carbon substrate Amsterdam – 24thJanuary 2008

16. The Model Amsterdam – 24thJanuary 2008

17. General property : itis a dissipative system (Compact) Result: the vector field defined by the model satisfies the following properties: - K+ is positively invariant - W is positively invariant - all trajectories initiated in K+ has its w–limit in W Amsterdam – 24thJanuary 2008

18. Variables and parameters Unité de temps : 10 heures From Middelboe, 2000 Amsterdam – 24thJanuary 2008

19. Comparison versus experimental data From Middelboe, 2000 Amsterdam – 24thJanuary 2008

20. Fast Slow Time scales Amsterdam – 24thJanuary 2008

21. Two fast variables and three slow variables Fastdynamics Amsterdam – 24thJanuary 2008

22. The complete model FAST SLOW Amsterdam – 24thJanuary 2008

23. While H>0, E1 is hyperbolically stable Equilibria While H>0, E2 is a saddle point Fastdynamics Amsterdam – 24thJanuary 2008

24. Slow dynamics (GSP Theory) The Geometrical Singular Perturbation theory (e.g. Fenichel, 1971, Sakamoto, 1990, Tychonov, ) allows to conclude that the previous complete model can be reduced to the following 3D system, under the normal hyperbolicity condition : Normal hyperbolicity condition Amsterdam – 24thJanuary 2008

25. Comparisonbetweencomplete and agregated model Amsterdam – 24thJanuary 2008

26. Comparisonbetweencomplete and agregated model Amsterdam – 24thJanuary 2008

27. S,I ? H {(0;0)}x{H}X{(C;R)} C,R Loss of Normal Hyperbolicity The reduced system well approximates the complete one Amsterdam – 24thJanuary 2008

28. Normally stable invariant manifold Normally unstable invariant manifold Loss of Normal Hyperbolicity Amsterdam – 24thJanuary 2008

29. Two ODE’s systems (H. Thieme, 1992) Loss of Normal Hyperbolicity Blow-up Fast Slow Amsterdam – 24thJanuary 2008

30. Singular perturbation theory Fast Lotka-Volterra Model Amsterdam – 24thJanuary 2008

31. Singular perturbation theory Asymptotic expansion of the invariant manifold with respect to the small parameter Amsterdam – 24thJanuary 2008

32. Singular perturbation theory Centre perturbation Let be the duale form of the vector field defined by the previous system: Poincaré map x y s Amsterdam – 24thJanuary 2008

33. Singular perturbation theory Centre perturbation Let be the duale form of the vector field defined by the previous system: Displacement map x y P(x) x s Amsterdam – 24thJanuary 2008

34. Poincaré lemma: Application : Stockes theorem: Centre perturbation Amsterdam – 24thJanuary 2008

35. Simulations Amsterdam – 24thJanuary 2008

36. Summary • If the maximum ingestion rate of resistant population is larger to that of susceptible, the initial 5D system reduces to 2+1 equations. • In this case, the bifurcation diagram in the plan (C0;D) exhibits a transcritical curve. • If the ingestion rate of resistant is lower than that of susceptible, oscillations appear Amsterdam – 24thJanuary 2008

37. CONCLUSIONS • Different time scales induced by the virus efficiency • The resistant population affects Beretta and Kuang conclusions. Amsterdam – 24thJanuary 2008