Jean-Christophe POGGIALE Aix-Marseille Université Mediterranean Institute of Oceanography U.M.R. C.N.R.S. 7249

1. Mathematical formulation of ecological processes : a problem of scale Mathematical approach and ecological consequences Jean-Christophe POGGIALE Aix-Marseille Université Mediterranean Institute of Oceanography U.M.R. C.N.R.S. 7249 Case 901 – Campus de Luminy – 13288 Marseille CEDEX 09 jean-christophe.poggiale@univ-amu.fr Leicester – Feb. 2013

2. Outline I – Mathematicalsystemswithseveral time scales : singular perturbation theory and itsgeometricalframework I-1) An examplewiththree time scales I-2) Somemathematicalmethods I-3) Somecomments on theirusefulness : limits and extensions II – Mathematicalmodelling – Processe formulation - Structure sensitivity III – Several formulations for one process : a dynamical system approach V – Conclusion Leicester – Feb. 2013

3. where Example A tri-trophic food chains model Deng and Hines (2002) de Feo and Rinaldi (1998) Muratori and Rinaldi (1992) Leicester – Feb. 2013

4. Example Some results • Under some technical assumptions, there exists a saddle-focus in the positive orthant. • Under some technical assumptions, there exists a singular homoclinic orbit. • This orbit is a Shilnikov orbit. • This orbit is contained in a chaotic attractor These results are obtained “by hand”, by using ideas of the Geometrical Singular Perturbation theory (GSP) Leicester – Feb. 2013

5. Slow-Fast vector fields Top – down Bottom – up Leicester – Feb. 2013

6. Example How to deal with this multi-time scales dynamical system ? 1) Neglect the slow dynamics 2) Analyze the remaining systems (when slow variables are assumed to be constant) 3) Eliminate the fast variables in the slow dynamics and reduce the dimension 4) Compare the complete and reduced dynamics Leicester – Feb. 2013

7. Example How to deal with this multi-time scales dynamical system ? Leicester – Feb. 2013

8. ? ? Example How to deal with this multi-time scales dynamical system ? Leicester – Feb. 2013

9. Example How to deal with this multi-time scales dynamical system ? Leicester – Feb. 2013

10. Geometrical Singular Perturbation theory (GSP) Leicester – Feb. 2013

11. Geometrical Singular Perturbation theory 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. Leicester – Feb. 2013

12. 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. Geometrical Singular Perturbation theory 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 ». Leicester – Feb. 2013

13. 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 Leicester – Feb. 2013

14. Geometrical Singular Perturbation theory Why do we need theorems ? • Intuitive ideas used everywhere (quasi-steady state assumption, …) • Complexity of the involved mathematical techniques Leicester – Feb. 2013

15. Geometrical Singular Perturbation theory Why do we need theorems ? Slow-fast system Leicester – Feb. 2013

16. Geometrical Singular Perturbation theory Why do we need theorems ? Leicester – Feb. 2013

17. Geometrical Singular Perturbation theory Leicester – Feb. 2013

18. Geometrical Singular Perturbation theory Why do we need theorems ? Fenichel theorem where Leicester – Feb. 2013

19. Geometrical Singular Perturbation theory Leicester – Feb. 2013

20. Geometrical Singular Perturbation theory Why do we need theorems ? • Theorems help to solve more complex cases where intuition is wrong • Theorems provide tools to analyse bifurcations on the slow manifold • Theorems provide a global theory which allows us to extend the results to non hyperbolic cases Leicester – Feb. 2013

21. Loss of normal hyperbolicity Leicester – Feb. 2013

22. ? ? ? Solve the system : Loss of normal hyperbolicity A simple example y Unstable x Dynamical bifurcation Leicester – Feb. 2013

23. Loss of normal hyperbolicity A simple example x Leicester – Feb. 2013

24. Loss of normal hyperbolicity The saddle – node bifurcation (Stable slow manifold) Leicester – Feb. 2013

25. Loss of normal hyperbolicity The saddle – node bifurcation (Stable slow manifold) Leicester – Feb. 2013

26. Loss of normal hyperbolicity The saddle – node bifurcation Leicester – Feb. 2013

27. Loss of normal hyperbolicity The pitchfork bifurcation Leicester – Feb. 2013

28. Loss of normal hyperbolicity The pitchfork bifurcation Leicester – Feb. 2013

29. Loss of normal hyperbolicity A general geometrical method Dumortier and Roussarie, 1996, 2003, and now others... Leicester – Feb. 2013

30. Loss of normal hyperbolicity A general geometrical method Leicester – Feb. 2013

31. Loss of normal hyperbolicity Example Blow up Leicester – Feb. 2013

32. Loss of normal hyperbolicity Example Leicester – Feb. 2013

33. Loss of normal hyperbolicity Example Leicester – Feb. 2013

34. Loss of normal hyperbolicity Example Leicester – Feb. 2013

35. Loss of normal hyperbolicity Example Leicester – Feb. 2013

36. What’s about noise effects? Leicester – Feb. 2013

37. What’s about noise effects ? The general model Leicester – Feb. 2013

38. What’s about noise effects ? The generic case : normal hyperbolicity Leicester – Feb. 2013

39. What’s about noise effects ? The generic case : normal hyperbolicity Leicester – Feb. 2013

40. What’s about noise effects ? The saddle node bifurcation Leicester – Feb. 2013

41. What’s about noise effects ? The saddle node bifurcation Leicester – Feb. 2013

42. What’s about noise effects ? The pitchfork bifurcation Leicester – Feb. 2013

43. What’s about noise effects ? The pitchfork bifurcation Leicester – Feb. 2013

44. Conclusions • GSP theory provides a rigorous way to build and analyze aggregated models • It has recently been completed for the analysis of non normally hyperbolic manifolds • This advance permits to deal with systems having more than one aggregated model • Noise can have important effectswhenitsvariousis large enough • Various applications : Food webs models, gene networks, chemical reactors Leicester – Feb. 2013

45. Introduction Leicester – Feb. 2013

46. Introduction Leicester – Feb. 2013

47. Introduction • Complex systems dynamics (high number of entities interacting in nonlinear way, networks, loops and feed-back loops, etc.) - Ecosystems • Response of the complete network to a given perturbation (contamination, exploitation, global warming, …) on a particular part of the system? (amplified, damped, how and why?) • processes intensities and variations; • the whole system dynamics; • from individuals to communities and back; • MODELLING: • How does the formulation of a process in a complex system affect the whole system dynamics? How to measure the impacts of a perturbation? Leicester – Feb. 2013

48. Introduction Information - Data Bioenergetics – Genetic properties – Metabolism – Physiology - Behaviours Individuals Functional groups Activities – Genetic and Metabolic expressions Biotic interactions – Trophic webs Communities Environmental forcing – – Energy assessments – Human activities Ecosystems Complexity Leicester – Feb. 2013

49. Introduction How canweused data got in laboratoryexperiments to fieldmodels? How canwetakebenefit of the large amount of data obtainedatsmallscales to understand global system functioning? Can welinkdifferent data sets obtainedatdifferentscales? For a givenprocess in a complex system, whatis the effect of itsmathematical formulation on the wholedynamics? Doesitmatter if itiswellquantitativelyvalidated? For a givenprocess, weoften use functionseven if we know thatitis a badrepresentation, becauseitissimpler : isthere a simple alternative? For an givenecosystem, manymodelscanbedeveloped. How to choose? One of themcanbevalidduring a given time periodwhileanotherwillbe efficient for anotherperiod : how do we know the sequence of the models to use? Leicester – Feb. 2013

50. Introduction How canweused data got in laboratoryexperiments to fieldmodels? How canwetakebenefit of the large amount of data obtainedatsmallscales to understand global system functioning? Can welinkdifferent data sets obtainedatdifferentscales? For a givenprocess in a complex system, whatis the effect of itsmathematical formulation on the wholedynamics? Doesitmatter if itiswellquantitativelyvalidated? For a givenprocess, weoften use functionseven if we know thatitis a badrepresentation, becauseitissimpler : isthere a simple alternative? For an givenecosystem, manymodelscanbedeveloped. How to choose? One of themcanbevalidduring a given time periodwhileanotherwillbe efficient for anotherperiod : how do we know the sequence of the models to use? Leicester – Feb. 2013