Jean-Christophe POGGIALE
Aix-Marseille UniversitéMediterranean Institute of Oceanography
U.M.R. C.N.R.S. 7249
Case 901 – Campus de Luminy – 13288 Marseille CEDEX [email protected]
Leicester – Feb. 2013
Mathematical formulation of ecological processes : a problem of scaleMathematical 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 scalesI-2) Some mathematical methodsI-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
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Example
A tri-trophic food chains model
22
11
1
ydcyez
ddz
zyd
cyxb
axeyddy
yxb
axKxrx
ddx
where 1
Deng and Hines (2002) de Feo and Rinaldi (1998)Muratori and Rinaldi (1992)
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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)
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Slow-Fast vector fields
yxgddy
yxfddx
,
,
0
0,,y
yxfx yxx * Top – down
,,
,,yxgyyxfx
,,,,
,*
*
yGyyxgy
yxx
Bottom – up
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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
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Example
How to deal with this multi-time scales dynamical system ?
11
1
xbaxey
ddy
yxb
axKxrx
ddx
0
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?
?
Example
How to deal with this multi-time scales dynamical system ?
11
1
xbaxey
ddy
yxb
axKxrx
ddx
0
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Example
How to deal with this multi-time scales dynamical system ?
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Geometrical Singular Perturbation theory (GSP)
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Geometrical Singular Perturbation theory
The fundamental theorems : normal hyperbolicity theory
,,
,,yxgyyxfx
2
1
k
k
IRy
IRx
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.
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Geometrical Singular Perturbation theory
The fundamental theorems : normal hyperbolicity theory
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.
Theorem (Fenichel, 1971) : « the dynamics in the vicinity of the invariant manifold is close to the dynamics restricted on the manifolds ».
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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
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Geometrical Singular Perturbation theory
Why do we need theorems ?
• Intuitive ideas used everywhere (quasi-steady state assumption, …)
• Complexity of the involved mathematical techniques
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Geometrical Singular Perturbation theory
Why do we need theorems ?
11
222121212
1111212121
NebPddP
NrNmNmddN
ParNNmNmddN
21 NNN
11
112121
11112112121
NebPddP
PNaNrNrrddN
ParNNmmNmddN
Slow-fast system
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Geometrical Singular Perturbation theory
Why do we need theorems ?
PeaNPdtdP
aNPrNdtdN
11
112121
11112112121
NebPddP
PNaNrNrrddN
ParNNmmNmddN
2112
21*2
2112
12*1 mm
NmNmmNmN
NNr
NNrr
*2
2
*1
1 NNaa
*1
1t
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Geometrical Singular Perturbation theory
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Geometrical Singular Perturbation theory
Why do we need theorems ?
2112
2121
*2
*1
1 ;mmN
ParrNNPN
PNNPeaPeaNPdtdP
ParrPNNaNPrNdtdN
;
;
11
1211
Fenichel theorem
where
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Geometrical Singular Perturbation theory
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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
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Loss of normal hyperbolicity
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Loss of normal hyperbolicity
A simple example
xyddyddx
x
y
Unstable
Solve the system :
0xx
2
00 2exp xyy
Dynamical bifurcation
? ? ?
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Loss of normal hyperbolicity
A simple example
0xx
2
00 2exp xyy
20
0
0 0
2 ln
;
Kxy
y K t x x
0 :n bifurcatio theofDelay x
xK
0x
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Loss of normal hyperbolicity
The saddle – node bifurcation
0 if * xxxy(Stable slow manifold)
x
y
0
2yxddyddx
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Loss of normal hyperbolicity
The saddle – node bifurcation
2yxddyddx
0 if * xxxy
(Stable slow manifold)
3/2
2
3/21
3/23/1
3/2*
21
0*
000
if 3)
if 2)
if 1)
:such that and , constants someexist there then ;; if :1990) ,. (Jung Theorem
ctx-Lty
ctxty
txtxyty
ccLxyxyxalet
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Loss of normal hyperbolicity
The saddle – node bifurcation
3/2
3/1
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Loss of normal hyperbolicity
The pitchfork bifurcation
3yxyddyddx
.0 toclose timeaafter axis then thisleavesand 0;0point theuntil axis thisfollows axis,- thetogoesy trajector then the,0)0( and 00 if :Result
xx
yx
x
y
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The pitchfork bifurcation
Loss of normal hyperbolicity
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Loss of normal hyperbolicity
A general geometrical method
Dumortier and Roussarie, 1996, 2003, and now others...
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Loss of normal hyperbolicity
A general geometrical method
.hyperbolicnormally not is manifold the whereon point a be 0Let 0M
1 and ,with
;;;,,;;;0:
:ation transforma is A
2
1
2
11
1
n
ii
ki
ki
nn
nn
xrxrx
xxrxxIRS
i
up blowing
.;0on field vector a is ~ field vector The * nSXX
.space phase in the 0 of odneighborho aon 0 smallfor
of dynamics theprovides ;0on ~ ofstudy The 0
ε
n XrSX
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Loss of normal hyperbolicity
Example
yxa
xydtdy
xaxyxx
dtdx
1
Blow up
ayvxu
:Let
a
y
x
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Loss of normal hyperbolicity
Example
u
v
0;0
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Loss of normal hyperbolicity
Example
u
v
1u1v
0;0
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Loss of normal hyperbolicity
Example
Ovrara
vdtvd
vrarar
dtdr
u
1
1
: 1chart In the
r
va1
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Loss of normal hyperbolicity
Example
u
v y
x
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What’s about noise effects?
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What’s about noise effects ?
The general model
tttttt
tttttt
dWyxGdtyxgdy
dWyxFdtyxfdx
,,1,',
0 when bounded is '
process Wiener a is 0ttW
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What’s about noise effects ?
The generic case : normal hyperbolicity
xyyyxM *;
: stableally asymptoticuniformly manifold slow a has system the,0'for : Hypothesis
theorem.Fenichel the viaobtained manifold theLet M
time.longlly exponentiaan during in stays in enteringctory each trajesuch that of
odneighborho a exists There : 2003) Gentz, and (Berglund TheoremBBMB
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What’s about noise effects ?
The generic case : normal hyperbolicity
0M
M
B
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What’s about noise effects ?
The saddle node bifurcation
tt
t
dWyxdy
dx
211
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What’s about noise effects ?
The saddle node bifurcation
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What’s about noise effects ?
The pitchfork bifurcation
tt
t
dWyxydy
dx
311
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What’s about noise effects ?
The pitchfork bifurcation
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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
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Introduction
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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?
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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. 2013
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. 2013
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. 2013
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. 2013
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. 2013
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. 2013
Introduction
STRUCTURE SENSITIVITY
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Structure sensitivity
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Sensitivity to function g ? gR : Reference model = MR
gP : Perturbed model = MP
Structure sensitivity
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Structure sensitivity
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Structure sensitivity
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Structure sensitivity
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Structure sensitivity
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Ref. formulation= Holling Ref. formulation= Ivlev
Structure sensitivity
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0 5 10 15 20 25 30 35 40 45 500
0.5
1
1.5
2
2.5
3
3.5
4
Taux
d’a
bsor
ptio
n de
Si (
d-
1 )
Concentration de Si (mol.l-1)
Structure sensitivity
Zooplancton
Phytoplancton
Nutriments
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Structure sensitivity
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Structure sensitivity
0
0.5
1
1.5
2
2.5
30
0.5
1
1.5
2
2.5
3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
S2 concentration
S1 concentration
j S2
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PROCESS FORMULATION :FUNCTIONAL RESPONSE
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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.
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Functional response
Small scale : experiments
Large scale : integrate spatial variability and individuals displacement (behavior, …)
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Functional response
Small scale : experiments
Large scale : integrate spatial variability and individuals displacement (behavior, …)
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Functional response
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Functional response
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Functional response
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11 1
11
bxaxey
ddy
ybxax
Kxrx
ddx
Holling idea:
Searching Handling
Functional response
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11 1
11
bxaxey
ddy
ybxax
Kxrx
ddx
Holling idea:
Searching Handling
Functional response
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11 1
11
bxaxey
ddy
ybxax
Kxrx
ddx
Holling idea:
Searching Handling
x is assumed constant at this scale of description
Functional response
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bxaxxg
1
Holling type II (Disc equation – Holling 1959)
Searching Handling
x
Functional response
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bxaxxg
1
Holling type II (Disc equation – Holling 1959)
Searching Handling
x
Functional response
or yyy
ooro
rrorr
r
yyxyddy
yeaxyyxyddy
axyxxrddx
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bxaxxg
1
Holling type II (Disc equation – Holling 1959)
Searching Handling
x
ooro
rrorr
r
yyxyddy
yeaxyyxyddy
axyxxrddx
or yyy yx
yr
x
axxg
1
Functional response
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Functional response
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Functional response
2
1
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Functional response
2
1
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Functional response
2
1
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Functional response
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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
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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.
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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.
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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?
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SHIFTS BETWEEN MODELS : LOSS OF NORMAL HYPERBOLICITY
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Slow-Fast vector fields
yxgddy
yxfddx
,
,
0
0,,y
yxfx yxx * Top – down
,,
,,yxgyyxfx
,,,,
,*
*
yGyyxgy
yxx
Bottom – up
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Fenichel theorem (Geometrical Singular Perturbation Theory)
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Fenichel theorem (Geometrical Singular Perturbation Theory)
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Fenichel theorem (Geometrical Singular Perturbation Theory)
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Fenichel theorem (Geometrical Singular Perturbation Theory)
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• 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
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