Jean-Christophe POGGIALE Aix-Marseille Université Mediterranean Institute of Oceanography U.M.R. C.N.R.S Case 901 – Campus de Luminy – Marseille CEDEX 09 Leicester – Feb Mathematical formulation of ecological processes : a problem of scale Mathematical approach and ecological consequences
Outline I – Mathematical systems with several time scales : singular perturbation theory and its geometrical framework I-1) An example with three time scales I-2) Some mathematical methods I-3) Some comments on their usefulness : limits and extensions II – Mathematical modelling – Processe formulation - Structure sensitivity III – Several formulations for one process : a dynamical system approach V – Conclusion Leicester – Feb. 2013
Example A tri-trophic food chains model where Deng and Hines (2002) de Feo and Rinaldi (1998) Muratori and Rinaldi (1992) Leicester – Feb. 2013
Example Some results Under some technical assumptions, there exists a singular homoclinic orbit. Under some technical assumptions, there exists a saddle-focus in the positive orthant. 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
Slow-Fast vector fields Top – down Bottom – up Leicester – Feb. 2013
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
Example How to deal with this multi-time scales dynamical system ? Leicester – Feb. 2013
??Example How to deal with this multi-time scales dynamical system ? Leicester – Feb. 2013
Example How to deal with this multi-time scales dynamical system ? Leicester – Feb. 2013
Geometrical Singular Perturbation theory (GSP) Leicester – Feb. 2013
Geometrical Singular Perturbation theory 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. Leicester – Feb. 2013
Geometrical Singular Perturbation theory 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 ». Leicester – Feb. 2013
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
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
Geometrical Singular Perturbation theory Why do we need theorems ? Slow-fast system Leicester – Feb. 2013
Geometrical Singular Perturbation theory Why do we need theorems ? Leicester – Feb. 2013
Geometrical Singular Perturbation theory Leicester – Feb. 2013
Geometrical Singular Perturbation theory Why do we need theorems ? Fenichel theorem where Leicester – Feb. 2013
Geometrical Singular Perturbation theory Leicester – Feb. 2013
Geometrical Singular Perturbation theory Why do we need theorems ? Theorems help to solve more complex cases where intuition is wrong Theorems provide a global theory which allows us to extend the results to non hyperbolic cases Theorems provide tools to analyse bifurcations on the slow manifold Leicester – Feb. 2013
Loss of normal hyperbolicity Leicester – Feb. 2013
Loss of normal hyperbolicity A simple example x y Unstable Solve the system : Dynamical bifurcation ??? Leicester – Feb. 2013
Loss of normal hyperbolicity A simple example x Leicester – Feb. 2013
Loss of normal hyperbolicity The saddle – node bifurcation (Stable slow manifold) Leicester – Feb. 2013
Loss of normal hyperbolicity The saddle – node bifurcation (Stable slow manifold) Leicester – Feb. 2013
Loss of normal hyperbolicity The saddle – node bifurcation Leicester – Feb. 2013
Loss of normal hyperbolicity The pitchfork bifurcation Leicester – Feb. 2013
The pitchfork bifurcation Loss of normal hyperbolicity Leicester – Feb. 2013
Loss of normal hyperbolicity A general geometrical method Dumortier and Roussarie, 1996, 2003, and now others... Leicester – Feb. 2013
Loss of normal hyperbolicity A general geometrical method Leicester – Feb. 2013
Loss of normal hyperbolicity Example Blow up Leicester – Feb. 2013
Loss of normal hyperbolicity Example Leicester – Feb. 2013
Loss of normal hyperbolicity Example Leicester – Feb. 2013
Loss of normal hyperbolicity Example Leicester – Feb. 2013
Loss of normal hyperbolicity Example Leicester – Feb. 2013
What’s about noise effects? Leicester – Feb. 2013
What’s about noise effects ? The general model Leicester – Feb. 2013
What’s about noise effects ? The generic case : normal hyperbolicity Leicester – Feb. 2013
What’s about noise effects ? The generic case : normal hyperbolicity Leicester – Feb. 2013
What’s about noise effects ? The saddle node bifurcation Leicester – Feb. 2013
What’s about noise effects ? The saddle node bifurcation Leicester – Feb. 2013
What’s about noise effects ? The pitchfork bifurcation Leicester – Feb. 2013
What’s about noise effects ? The pitchfork bifurcation Leicester – Feb. 2013
Conclusions It has recently been completed for the analysis of non normally hyperbolic manifolds Noise can have important effects when its various is large enough Various applications : Food webs models, gene networks, chemical reactors GSP theory provides a rigorous way to build and analyze aggregated models This advance permits to deal with systems having more than one aggregated model Leicester – Feb. 2013
Introduction
Leicester – Feb 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 Introduction
Individuals Functional groups Communities Ecosystems Bioenergetics – Genetic properties – Metabolism – Physiology - Behaviours Activities – Genetic and Metabolic expressions Biotic interactions – Trophic webs Environmental forcing – – Energy assessments – Human activities Complexity Information - Data Leicester – Feb Introduction
How can we used data got in laboratory experiments to field models? How can we take benefit of the large amount of data obtained at small scales to understand global system functioning? Can we link different data sets obtained at different scales? For a given process in a complex system, what is the effect of its mathematical formulation on the whole dynamics? Does it matter if it is well quantitatively validated? For a given process, we often use functions even if we know that it is a bad representation, because it is simpler : is there a simple alternative? For an given ecosystem, many models can be developed. How to choose? One of them can be valid during a given time period while another will be efficient for another period : how do we know the sequence of the models to use? Leicester – Feb Introduction
How can we used data got in laboratory experiments to field models? How can we take benefit of the large amount of data obtained at small scales to understand global system functioning? Can we link different data sets obtained at different scales? For a given process in a complex system, what is the effect of its mathematical formulation on the whole dynamics? Does it matter if it is well quantitatively validated? For a given process, we often use functions even if we know that it is a bad representation, because it is simpler : is there a simple alternative? For an given ecosystem, many models can be developed. How to choose? One of them can be valid during a given time period while another will be efficient for another period : how do we know the sequence of the models to use? Leicester – Feb Introduction
How can we used data got in laboratory experiments to field models? How can we take benefit of the large amount of data obtained at small scales to understand global system functioning? Can we link different data sets obtained at different scales? For a given process in a complex system, what is the effect of its mathematical formulation on the whole dynamics? Does it matter if it is well quantitatively validated? For a given process, we often use functions even if we know that it is a bad representation, because it is simpler : is there a simple alternative? For an given ecosystem, many models can be developed. How to choose? One of them can be valid during a given time period while another will be efficient for another period : how do we know the sequence of the models to use? Leicester – Feb Introduction
How can we used data got in laboratory experiments to field models? How can we take benefit of the large amount of data obtained at small scales to understand global system functioning? Can we link different data sets obtained at different scales? For a given process in a complex system, what is the effect of its mathematical formulation on the whole dynamics? Does it matter if it is well quantitatively validated? For a given process, we often use functions even if we know that it is a bad representation, because it is simpler : is there a simple alternative? For an given ecosystem, many models can be developed. How to choose? One of them can be valid during a given time period while another will be efficient for another period : how do we know the sequence of the models to use? Leicester – Feb Introduction
How can we used data got in laboratory experiments to field models? How can we take benefit of the large amount of data obtained at small scales to understand global system functioning? Can we link different data sets obtained at different scales? For a given process in a complex system, what is the effect of its mathematical formulation on the whole dynamics? Does it matter if it is well quantitatively validated? For a given process, we often use functions even if we know that it is a bad representation, because it is simpler : is there a simple alternative? For an given ecosystem, many models can be developed. How to choose? One of them can be valid during a given time period while another will be efficient for another period : how do we know the sequence of the models to use? Leicester – Feb Introduction
STRUCTURE SENSITIVITY Leicester – Feb. 2013
Structure sensitivity Leicester – Feb. 2013
Sensitivity to function g ? g R : Reference model = M R g P : Perturbed model = M P Structure sensitivity Leicester – Feb. 2013
Structure sensitivity Leicester – Feb. 2013
Structure sensitivity Leicester – Feb. 2013
Structure sensitivity Leicester – Feb. 2013
Structure sensitivity Leicester – Feb Ref. formulation= HollingRef. formulation= Ivlev
Structure sensitivity Leicester – Feb. 2013
Taux d’absorption de Si (d -1 ) Concentration de Si ( mol.l -1 ) Structure sensitivity Zooplancton Phytoplancton Nutriments Leicester – Feb. 2013
Structure sensitivity Leicester – Feb. 2013
Structure sensitivity Leicester – Feb. 2013
PROCESS FORMULATION : FUNCTIONAL RESPONSE PROCESS FORMULATION : FUNCTIONAL RESPONSE Leicester – Feb. 2013
Functional response Process which describes the biomass flux from a trophic level to another one : the functional response aims to describe this process at the population level. However, it results from many individual properties : behavior (interference between predators, optimal foraging, ideal free distribution, etc.) physiology (satiation, starvation, etc.) And population properties as well: population densities (density-dependence effects) populations distribution (encounter rates, etc.) How should we formulate the functional response? At which scale? Leicester – Feb. 2013
Functional response Process which describes the biomass flux from a trophic level to another one : the functional response aims to describe this process at the population level. However, it results from many individual properties : behavior (interference between predators, optimal foraging, ideal free distribution, etc.) physiology (satiation, starvation, etc.) And population properties as well: population densities (density-dependence effects) populations distribution (encounter rates, etc.) How should we formulate the functional response? At which scale? Current ecosystem models are sensitive to the functional response formulation. Leicester – Feb. 2013
Functional response Small scale : experiments Large scale : integrate spatial variability and individuals displacement (behavior, …) Leicester – Feb. 2013
Functional response Small scale : experiments Large scale : integrate spatial variability and individuals displacement (behavior, …) Leicester – Feb. 2013
Functional response Leicester – Feb. 2013
Functional response Leicester – Feb. 2013
Functional response Leicester – Feb. 2013
Holling idea: Searching Handling Functional response Leicester – Feb. 2013
Holling idea: Searching Handling Functional response Leicester – Feb. 2013
Holling idea: Searching Handling x is assumed constant at this scale of description Functional response Leicester – Feb. 2013
Holling type II (Disc equation – Holling 1959) Searching Handling Functional response Leicester – Feb. 2013
Holling type II (Disc equation – Holling 1959) Searching Handling Functional response Leicester – Feb. 2013
Holling type II (Disc equation – Holling 1959) Searching Handling Functional response Leicester – Feb. 2013
Functional response Leicester – Feb. 2013
Functional response 2 1 Leicester – Feb. 2013
Functional response 2 1 Leicester – Feb. 2013
Functional response 2 1 Leicester – Feb. 2013
Functional response Leicester – Feb. 2013
Functional response Local Holling type II functional responses Global Holling type III functional response <0 >0 => Conditions can be found to get the criterion for Holling Type III functional responses Leicester – Feb. 2013
Dynamical consequences 1 – On each patch separately, the parameter values are such that periodic solutions occur. 2 – With density-dependent migration rates of the predator satisfying the above mentioned criterion for Holling Type III functional response, the system is stabilized 3 – With constant migration rates, taking extreme values observed in the situation described in 2, the system exhibits periodic fluctuations : the stabilization results from the change of functional response type. Leicester – Feb. 2013
Dynamical consequences Global type III FR can emerge from local type II functional responses associated to density-dependent displacements The Holling Type III functional response leads to stabilization The stability actually results from the type (type II functional responses lead to periodic fluctuations even if they are quantitatively close to the type III FR) The Type III results from density-dependence : the effect of density- dependent migration rates on the global functional response can be understood explicitely. Leicester – Feb. 2013
Dynamical consequences Functional response in the field : a set of functions instead of one function? Shifts between models Multi-stability of the fast dynamics Changes of fast attractors : bifurcation in the fast part of the system induced by the slow dynamics We use functions to represent FR at a global scale even if we know that it is a bad representation, because it is simpler : is there a simple alternative? Leicester – Feb. 2013
SHIFTS BETWEEN MODELS : LOSS OF NORMAL HYPERBOLICITY Leicester – Feb. 2013
Slow-Fast vector fields Top – down Bottom – up Leicester – Feb. 2013
Fenichel theorem (Geometrical Singular Perturbation Theory) Leicester – Feb. 2013
Fenichel theorem (Geometrical Singular Perturbation Theory) Leicester – Feb. 2013
Fenichel theorem (Geometrical Singular Perturbation Theory) Leicester – Feb. 2013
Fenichel theorem (Geometrical Singular Perturbation Theory) Leicester – Feb. 2013
The jump between two formulations can be described by the Geometrical Singular Perturbation Theory : follow the trajectories of the full system around the points where normal hyperbolicity is lost («Blow up techniques ») Conclusions (2/2) Instead of one function to formulate one process at large scale, several functions can be used. Multiple equilibria in the fast dynamics can provide a mechanism for this multiple representation in large scale models. Bifurcations of the fast dynamics induced by slow dynamics lead to shifts in the fast variables. This leads to several mathematical expressions of the fast equilibrium with respect to slow variables : each of them provides a mathematical formulation at large scales Leicester – Feb. 2013
Thanks to my collaborators… Roger ARDITI Julien ARINO Ovide ARINO Pierre AUGER Rafael BRAVO de la PARRA François CARLOTTI Flora CORDOLEANI Marie EICHINGER Frédérique FRANCOIS Mathias GAUDUCHON Franck GILBERT Bas KOOIJMAN Bob KOOI Horst MALCHOW Claude MANTE Marcos MARVA Andrei MOROZOV David NERINI Tri NGUYEN HUU Robert ROUSSARIE Eva SANCHEZ Richard SEMPERE Georges STORA Caroline TOLLA … and thanks for your attention! Leicester – Feb. 2013