Marine bacteria – virus interaction in a chemostat Amsterdam – 24 th January 2008 Jean-Christophe Poggiale Laboratoire de Microbiologie, de Géochimie et d’Ecologie Marines (UMR CNRS 6117) Université de la Méditerranée Centre d’Océanologie de Marseille Marseille Cedex 09 France

? Andreasen, Iwasa, Levin, 1987, 1989 Amsterdam – 24 th January 2008 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 » in Structured Population Models in Biology and Epidemiology Series: Lecture Notes in Mathematics Subseries: Mathematical Biosciences Subseries, Vol Magal, Pierre; Ruan, Shigui (Eds.), 345 p.Lecture Notes in MathematicsMathematical Biosciences Subseries Singular perturbation theory Discrete systems Delayed and partial differential equations Applications to population dynamics models Amsterdam – 24 th January 2008 Aggregation of variables

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

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

?? Time scales and singular perturbation theory Aggregation of variables Amsterdam – 24 th January 2008

Time scales and singular perturbation theory Aggregation of variables

The fundamental theorems : normal hyperbolicity theory Def. : The invariant manifold M 0 is normally hyperbolic if the linearization of the previous system at each point of M 0 has exactly k 2 eigenvalues on the imaginary axis. Amsterdam – 24 th January 2008 Aggregation of variables

The fundamental theorems : normal hyperbolicity theory Theorem (Fenichel, 1971) : if is small enough, there exists a manifold M 1 close and diffeomorphic to M 0. Moreover, it is locally invariant under the flow, and differentiable. Theorem (Fenichel, 1971) : « the dynamics in the vicinity of the invariant manifold is close to the dynamics restricted on the manifolds ». Amsterdam – 24 th January 2008 Aggregation of variables

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 Alcalà de Henares - April 2005 Intuitive ideas used everywhere (quasi-steady state assumption, adiabatic assumptions, time scale separation…)

Marine bacteria – virus interaction in a chemostat Amsterdam – 24 th January state variables : S, I and V If the «burst coefficient » increases then oscillations appear

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

Model description R Resistant D S Susceptible D V Virus D I Infected D C Carbon substrate D C 0 Reservoir D Amsterdam – 24 th January 2008

Susceptibles Infected Resistant

Virus Carbon substrate Amsterdam – 24 th January 2008

The Model Amsterdam – 24 th January 2008

General property : it is a dissipative system (Compact) Result: the vector field defined by the model satisfies the following properties: - K + is positively invariant -  is positively invariant - all trajectories initiated in K + has its  –limit in  Amsterdam – 24 th January 2008

VariablesUnit S10 6 cell/ml V I C R Unité de temps : 10 heures ParametersUnitsValues K10 6 cells/ml0.3 C0C cells/ml3 11 1/time2.5  2 1/time1 – 4 kml/time0.1  virus/lysis10 – 60 D1/time0.2 – 1 1/time5 Variables and parameters From Middelboe, 2000 Amsterdam – 24 th January 2008

Comparison versus experimental data From Middelboe, 2000 Amsterdam – 24 th January 2008

Fast Slow Time scales Amsterdam – 24 th January 2008

Fast dynamics Two fast variables and three slow variables Amsterdam – 24 th January 2008

The complete model FAST SLOW Amsterdam – 24 th January 2008

Fast dynamics While H>0, E 1 is hyperbolically stable While H>0, E 2 is a saddle point Equilibria Amsterdam – 24 th January 2008

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 : Slow dynamics (GSP Theory) Normal hyperbolicity condition Amsterdam – 24 th January 2008

Comparison between complete and agregated model Amsterdam – 24 th January 2008

Comparison between complete and agregated model

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

Normally stable invariant manifold Normally unstable invariant manifold Amsterdam – 24 th January 2008 Loss of Normal Hyperbolicity

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

Fast Lotka-Volterra Model Singular perturbation theory Amsterdam – 24 th January 2008

Asymptotic expansion of the invariant manifold with respect to the small parameter Amsterdam – 24 th January 2008 Singular perturbation theory

Centre perturbation Letbe the duale form of the vector field defined by the previous system: Poincaré map s y x Amsterdam – 24 th January 2008 Singular perturbation theory

s y Displacement map x x P(x) Amsterdam – 24 th January 2008 Letbe the duale form of the vector field defined by the previous system: Centre perturbation Singular perturbation theory

Poincaré lemma: Application : Stockes theorem: Centre perturbation Amsterdam – 24 th January 2008

Simulations

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 (C 0 ;D) exhibits a transcritical curve. If the ingestion rate of resistant is lower than that of susceptible, oscillations appear Amsterdam – 24 th January 2008

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