Toward a Mathematical Theory of Complex
Systems
Nicola Bellomo
[email protected]
Department of Mathematics
Politecnico di Torino
http://calvino.polito.it/fismat/poli/
BCAM - Bilbao - October 2011
Toward a Mathematical Theory of Complex Systems – p. 1/6
Homepage: http://calvino.polito.it/fismat/poli
Toward a Mathematical Theory of Complex Systems – p. 2/6
Topics
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Toward a Mathematical Theory of Complex Systems – p. 3/6
Unraveling Complex (Living) Systems
A PRIL 2011 U NRAVELING C OMPLEX S YSTEMS
The American Mathematical Society, the American Statistical Association, the
Mathematical Association of America, and the Society for Industrial and Applied
Mathematics announce that the theme of Mathematics Awareness Month 2011 is
”Unraveling Complex Systems.”
From www.mathaware.org
We live in a complex world. Many familiar examples of complex systems may be found
in very different entities at very different scales: from power grids, to transportation
systems, from financial markets, to the Internet, and even in the underlying
environment to cells in our bodies. Mathematics and statistics can guide us in
unveiling, defining and understanding these systems,in order to enhance their
reliability and improve their performance.
Toward a Mathematical Theory of Complex Systems – p. 4/6
Lecture 1. - Common Features of Living Systems
Part 1. - Toward a Mathematical Theory of Complex Systems
T.1. Common Features of Living (Complex) Systems
T.2. Representation, Nonlinear Stochastic Games, Pathways, Networks and
Mathematical Tools
Part 2 - Applications
A.1 On the Derivation of Flux Limited Keller Segel Models
A.2 A Mathematical Model for the in-vitro Dynamics of Cancer Hepatocytes under
Therapeutic Actions
A.3 On the Modeling the Collective Dynamics of Vehicles
Toward a Mathematical Theory of Complex Systems – p. 5/6
Lecture T.1. - Common Features of Living Systems
A Personal Bibliographic Search
• E. Mayr, The philosophical foundation of Darwinism, Proc. American Philosophical
Society, 145 (2001) 488-495.
• A.L. Barabasi Linked. The New Science of Networks, (Perseus Publishing,
Cambridge Massachusetts, 2002.)
• F. Schweitzer, Brownian Agents and Active Particles, (Springer, Berlin, 2003).
• M. Talagrand, Spin Glasses, a Challenge to Mathematicians, Springer, (2003).
• M.A. Nowak and K. Sigmund, Evolutionary dynamics of biological games, Science,
303 (2004) 793–799.
• M.A. Nowak, Evolutionary Dynamics, Princeton Univ. Press, (2006).
• F.C. Santos, J.M. Pacheco, and T. Lenaerts, Evolutionary dynamics of social
dilemmas in structured heterogeneous populations, PNAS, 103(9), 3490-3494, (2006).
• N.N. Taleb, The Black Swan: The Impact of the Highly Improbable, (2007).
• K. Sigmund, The Calculus of Selfishness, Princeton Univ. Press, (2011).
Toward a Mathematical Theory of Complex Systems – p. 6/6
Lecture T.1. - Common Features of Living Systems
A Personal Quest Through Complexity
• N.Bellomo, Modelling Complex Living Systems. A Kinetic Theory and
Stochastic Game Approach, (Birkhauser-Springer, Boston, 2008).
• N.Bellomo and M. Delitala, From the mathematical kinetic, and stochastic game
theory to modelling mutations, onset, progression and immune competition of cancer
cells, Phys. Life Rev., 5, (2008), 183-206.
• N. Bellomo, A. Bellouquid, J. Nieto J., and J. Soler, Multiscale biological tissue
models and flux-limited chemotaxis from binary mixtures of multicellular growing
systems, Math. Models Methods Appl. Sci., 10 (2010) 1179-1207.
• N. Bellomo and B. Carbonaro, Towards a mathematical theory of living systems
focusing on developmental biology and evolution: a review and perspectives, Physics
of Life Reviews, 8 (2011) 1-18.
• N. Bellomo and C. Dogbé, On The Modelling of traffic and crowds - A survey of
models, speculations, and perspectives, SIAM Review, 53(3) (2011) 409-463.
• N. Bellomo and A. Bellouquid, On the modeling of crowd dynamics looking at the
beautiful shapes of swarms, Networks Heterog. Media, 3, (2011) 383-399.
Toward a Mathematical Theory of Complex Systems – p. 7/6
Lecture 1. - Common Features of Living Systems
• E. Kant 1790, da Critique de la raison pure, Traduction Fracaise, Press Univ. de
France, 1967,
Living Systems: Special structures organized and with the ability to chase a purpose.
Hartwell - Nobel Laureate 2001, Nature 1999
• Biological systems are very different from the physical or chemical systems analyzed
by statistical mechanics or hydrodynamics. Statistical mechanics typically deals with
systems containing many copies of a few interacting components, whereas cells
contain from millions to a few copies of each of thousands of different components,
each with very specific interactions.
• Although living systems obey the laws of physics and chemistry, the notion of
function or purpose differentiate biology from other natural sciences. Organisms exist
to reproduce, whereas, outside religious belief rocks and stars have no purpose.
Selection for function has produced the living cell, with a unique set of properties
which distinguish it from inanimate systems of interacting molecules. Cells exist far
from thermal equilibrium by harvesting energy from their environment.
Toward a Mathematical Theory of Complex Systems – p. 8/6
Lecture 1. - Common Features of Complex Living Systems
E. Schrödinger, P. Dirac - 1933, What is Life?, I living systems have the ability to
extract entropy to keep their own at low levels.
R. May, Science 2003 In the physical sciences, mathematical theory and experimental
investigation have always marched together. Mathematics has been less intrusive in
the life sciences, possibly because they have been until recently descriptive, lacking the
invariance principles and fundamental natural constants of physics.
E.P. Wiegner , Comm. Pure Appl. Math., 1960 The miracle of the appropriateness of
the language of mathematics for the formulation of the laws of physics is a wonderful
gift which we neither understand nor deserve.
G. Jona Lasinio , La Matematica come Linguaggio delle Scienze della Natura Preprint, La vita è una proprietà emergente della materia? - La vita rappresenta una
fase avanzata di un processo evolutivo e selettivo. Mi pare difficile spiegare il vivente
ignorando la sua dimensione storica. La dinamica delle popolazioni, di cui esiste una
teoria matematica ancora in uno stato abbastanza primitivo dovrà spiegare l’emergere
per selezione delle dinamiche proprie del singolo vivente.
Toward a Mathematical Theory of Complex Systems – p. 9/6
Lecture 1. - Common Features of Living Systems
N.B. H. Berestycki, F. Brezzi, and J.P. Nadal, Mathematics and Complexity in Life
and Human Sciences, Mathematical Models and Methods in Applied Sciences, 2010.
• The study of complex systems, namely systems of many individuals interacting in a
non-linear manner, has received in recent years a remarkable increase of interest
among applied mathematicians, physicists as well as researchers in various other
fields as economy or social sciences.
• Their collective overall behavior is determined by the dynamics of their interactions.
On the other hand, a traditional modeling of individual dynamics does not lead in a
straightforward way to a mathematical description of collective emerging behaviors.
• In particular it is very difficult to understand and model these systems based on the
sole description of the dynamics and interactions of a few individual entities localized
in space and time.
Toward a Mathematical Theory of Complex Systems – p. 10/6
Lecture 1. - Common Features of Living Systems
Five Key Questions
1. How far is the state-of-the-art from the development of a biological-mathematical
theory of living systems and how an appropriate understanding of multiscale issues can
contribute to this ambitious objective? Application of methods of the inert matter to
living systems is highly misleading. Moreover multiscale approaches should be
interpreted in the framework of an evolutionary dynamics rather that within a static
framework.
2. Can mathematics contribute to reduce the complexity of living systems by splitting
it into suitable subsystems? Toward a new system biology
3. Can mathematics offer suitable tools to constrain into equations the common
complexity features all living systems? Not at present, but looking ahead to new
mathematical approaches should contribute to this objective
Toward a Mathematical Theory of Complex Systems – p. 11/6
Lecture 1. - Common Features of Living Systems
Five Key Questions
4. Should a conceivable mathematical theory show common featuress in all field of
applications? Although a theory should be linked to a specific class of systems, all
theories should have common features.
The last question is also a dilemma
5. Should mathematics attempt to reproduce experiments by equations whose
parameters are identified on the basis of empirical data, or develop new structures,
hopefully a new theory able to capture the complexity of biological phenomena and
subsequently to base experiments on theoretical foundations? This last question
witnesses the presence of a dilemma, which occasionally is the object of intellectual
conflicts within the scientific community. However, we are inclined to assert the
second perspective, since we firmly believe that it can also give a contribution to
further substantial developments of mathematical sciences.
Toward a Mathematical Theory of Complex Systems – p. 12/6
Lecture 1. - Common Features of Living Systems
Five Common Features and Sources of Complexity
1. Ability to express a strategy: Living entities are capable to develop
specific strategies and organization abilities that depend on the state of the
surrounding environment. These can be expressed without the application of any
external organizing principle. In general, they typically operate
out-of-equilibrium. For example, a constant struggle with the environment is
developed to remain in a particular out-of-equilibrium state, namely stay alive.
2. Heterogeneity: The ability to express a strategy is not the same for all
entities: Heterogeneity characterizes a great part of living systems, namely, the
characteristics of interacting entities can even differ from an entity to another
belonging to the same structure. In developmental biology, this is due to different
phenotype expressions generated by the same genotype.
3. Learning ability: Living systems receive inputs from their environments and
have the ability to learn from past experience. Therefore their strategic ability and the
characteristics of interactions among living entities evolve in time. Societies can
induce a collective strategy toward individual learning
Toward a Mathematical Theory of Complex Systems – p. 13/6
Lecture 1. - Common Features of Living Systems
4. Interactions: Interactions nonlinearly additive and involve immediate
neighbors, but in some cases also distant particles. Indeed, living systems have the
ability to communicate and may possibly choose different observation paths. In some
cases, the topological distribution of a fixed number of neighbors can play a prominent
role in the development of the strategy and interactions. Interactions modify their state
according to the strategy they develop. Living entities play a game at each
interaction with an output that is technically related to their strategy often related
to surviving and adaptation ability. Individual interactions in swarms can depend on
the number of interacting entities rather that on their distance.
5. Darwinian selection and time as a key variable: All living
systems are evolutionary. For instance birth processes can generate individuals more
fitted to the environment, who in turn generate new individuals again more fitted to the
outer environment. Neglecting this aspect means that the time scale of observation and
modeling of the system itself is not long enough to observe evolutionary events. Such
a time scale can be very short for cellular systems and very long for vertebrates.
Micro-Darwinian occurs at small scales, while Darwinian evolution is generally
interpreted at large scales.
Toward a Mathematical Theory of Complex Systems – p. 14/6
Lecture 1. - Common Features of Living Systems
Technical Complexity
• Large number of components: Complexity in living systems is induced by
a large number of variables, which are needed to describe their overall state. Therefore,
the number of equations needed for the modeling approach may be too large to be
practically treated. Reduction of complexity is the first step of the modeling approach.
• Multiscale aspects: The study of complex living systems always needs a
multiscale approach. For instance, the functions expressed by a cell are determined by
the dynamics at the molecular (genetic) level. Moreover, the structure of macroscopic
tissues depends on such a dynamics.
• Time varying role of the environment: The environment surrounding
a living system evolves in time, in several cases also due to the interaction with the
inner system. Therefore the output of this interaction evolves in time. One of the
several implications is that the number of components of a living system evolves in
time.
Toward a Mathematical Theory of Complex Systems – p. 15/6
Lecture 1. - Common Features of Complex Living Systems
M ULTISCALE REPRESENTATION OF TUMOUR GROWTH: gene interactions (stochastic
games), cells (kinetic theory), tissues (continuum mechanics), mixed (hybrid models).
Toward a Mathematical Theory of Complex Systems – p. 16/6
Lecture 1. - Common Features of Complex Living Systems
E VOLUTION OF CELLULAR PHENOTYPE
Toward a Mathematical Theory of Complex Systems – p. 17/6
Lecture 2. - Common Features of Living Systems
Part 1. - Toward a Mathematical Theory of Complex Systems
T.1. Common Features of Living (Complex) Systems
T.2. Representation, Nonlinear Stochastic Games, Pathways, Networks, and
Mathematical Tools
Part 2 - Applications
A.1 On the Derivation of Flux Limited Keller Segel Models
A.2 A Mathematical Model for the in-vitro Dynamics of Cancer Hepatocytes under
Therapeutic Actions
A.3 On the Modeling the Collective Dynamics of Vehicles
Toward a Mathematical Theory of Complex Systems – p. 18/6
Lecture 2. - Representation, Nonlinear Stochastic Games,
and Mathematical Tools
Toward a representation of complex systems
Let us consider a system constituted by a large number of interacting living entities,
called active particles. Their microscopic state includes, in addition to
geometrical and mechanical variables, also an additional variable, called activity,
which is heterogeneously distributed. Moreover, consider a system of networks and
nodes such that the role of the space variable is not relevant in the nodes, while
transition from one node to the other has to be properly modeled.
1. Living systems are constituted different types of active particles, which
express several different functions. This source of complexity can be reduced by by
decomposing the system into suitable functional subsystems,
where a functional subsystem is a collection of active particles, which have the
ability to express cooperatively the same activity regarded as a scalar variable.
Toward a Mathematical Theory of Complex Systems – p. 19/6
Lecture 2. - Representation, Nonlinear Stochastic Games,
Pathways, and Mathematical Tools
2. The description of the overall state of the system within each node is delivered by a
probability distribution function over such microscopic state, labeled by
the index i, which denotes the functional subsystem within each node.
fi = fi (t, u) ,
i = 1, . . . , n,
such that fi (t, u) du denotes the number of active particles whose state, at time t, is in
the interval [u, u + du]. If the number of active particles is constant in time, then the
distribution function can be normalized with respect to such a number and
consequently is aprobability density. The physical meaning of the activity
variable differs for each subscript.
3. The overall number of functional subsystems is given by the sum of all of them in
the nodes: fij = fij (t, u) ,
i = 1, . . . , n, j = 1, . . . , m, where functional
subsystems differ if are localized in different nodes although they express the same
activity variable.
Toward a Mathematical Theory of Complex Systems – p. 20/6
Lecture 2. - Representation, Nonlinear Stochastic Games,
Pathways
Nonlinear Stochastic Games
Active particles interact with a certain encounter rate and play a collective
game at each interaction, which can occur within each node or through the network.
The modeling consists in computing, in probability, the output of the interaction.
Various sources of nonlinearities can be considered. Among them:
• Distribution function conditioning the encounter rate also in connection to
hiding and learning dynamics;
• Nonlinearity induced by topological distribution of the interacting
entities.
• Distribution function conditioning the output of the games;
Toward a Mathematical Theory of Complex Systems – p. 21/6
Lecture 2. - Representation, Nonlinear Stochastic Games,
Pathways, and Mathematical Tools
Nonlinear Stochastic Games Interaction involves:
• Test particles with microscopic state u, at the time t: fij = fij (t, u).
• Field particles with microscopic state, u∗ at the time t: fij = fij (t, u∗ ).
• Candidate particles with microscopic state, u∗ at the time t: fij = fij (t, u∗ ).
Rule:
i) The candidate particle interacts with field particles and acquires, in probability,
the state of the test particle. Test particles interact with field particles and lose their
state.
ii) Interactions can: modify the microscopic state of particles; generate proliferation or
destruction of particles in their microscopic state; and can also generate a particle in a
new functional subsystem.
Loss of determinism: Modeling of systems of the inert matter is developed
within the framework of deterministic causality principles, which does not any longer
holds in the case of the living matter.
Toward a Mathematical Theory of Complex Systems – p. 22/6
Lecture 2. - Representation, Nonlinear Stochastic Games,
Pathways, and Mathematical Tools
Nonlinear Stochastic Games
Interactions within the action domain
F
F
F
C
F
T
Ω
Figure 1: – Active particles interact with other particles in their action domain
Toward a Mathematical Theory of Complex Systems – p. 23/6
Lecture 2. - Representation, Nonlinear Stochastic Games,
Pathways, and Mathematical Tools
Nonlinear Stochastic Games
Interactions with modification of activity and transition Generation of particles into
a new functional subsystem occurs through pathways. Different paths can be chosen
according to the dynamics at the lower scale.
F
F
F
T
F
C
F
T
F
F
F
i+1
i
T
i−1
Figure 2: – Active particles during proliferation move from one functional subsystem
to the other through pathways.
Toward a Mathematical Theory of Complex Systems – p. 24/6
Lecture 2. - Representation, Nonlinear Stochastic Games,
Pathways, and Mathematical Tools
Cooperative/Competitive Games
Cooperative behavior The active particle with state h < k improves its state by
gaining from the particle k , which cooperates loosing part of its state.
Competitive behavior The active particle with state h < k decreases its state by
contributing to the particle k, which due to competition increases part of its state.
Toward a Mathematical Theory of Complex Systems – p. 25/6
Lecture 2. - Common Features of Complex Living Systems,
and Mathematical Tools
Encounter rate ηij : ηij is supposed to decay with the distance between the
interacting active particles, which depends both on their state and on the distance of
their functional subsystems.
ηhk =
0
ηhk
ψhk (f )
∗
−α
hk (1 + |u∗ − u |)(1 + ||fh − fk ||)
ψhk (f ) = e
,
0
where ηhk
and αhk are positive constants, and
!
||fh − fk ||(t) =
|fh (t, u) − fk (t, u)| du.
R+
Transition probability density: Bhk is supposed to depend on the state of the
interacting pairs and low order moments:
!
Bhk (u∗ → u|u∗ , u∗ , Ep [fk ](t)) , Ep (t) =
up fi (t, u) du ,
R+
for p = 1, 2, ....
Proliferation rate The term µihk is supposed to depend on the state of the interacting
pairs and low order moments.
Toward a Mathematical Theory of Complex Systems – p. 26/6
Lecture 2. - Representation, Nonlinear Stochastic Games,
Pathways, and Mathematical Tools
Mathematical Structures
The evolution equation follows from a balance equation for net flow of particles in the
elementary volume of the space of the microscopic state by transport and interactions:
∂t fi (t, u) + Fi (t) ∂u fi (t, u) = Ci [f ](t, u) + Pi [f ](t, u)
where
Ci [f ]
=
n
"
0
ηij
j=1
−
fi (t, u)
!
ψij (f )Bij (u∗ → u|u∗ , u∗ , f )
Du ×Du
n
"
j=1
0
ηij
!
ψij (f ) fj (t, u∗ ) du∗ .
Du
and
Pi [f ] =
n "
n
"
h=1 k=1
0
ηhk
!
Du
!
ψhk (f ) µihk (u∗ , u∗ , f )fh (t, u∗ )fk (t, u∗ ) du∗ .
Du
Toward a Mathematical Theory of Complex Systems – p. 27/6
Lecture 2. - Common Features of Complex Living Systems,
and Mathematical Tools
Models with Space Structure
H.1. Candidate or test particles in x, interact with the field particles in x∗ ∈ Ω located
in the interaction domain Ω. Interactions are weighted by the interaction rate
ηhk [ρ](x∗ ) supposed to depend on the local density in the position of the field
particles.
H.2. The candidate particle modifies its state according to the term defined as follows:
Ahk (v∗ → v, u∗ → u|v∗ , v∗ , u∗ , u∗ , f ), which denotes the probability density that a
candidate particles of the h-subsystems with state v∗ , u∗ reaches the state v, u after an
interaction with the field particles k-subsystems with state v∗ , u∗ .
H.3. Candidate particle, in x, can proliferate, due to encounters with field particles in
x∗ , with rate µihk , which denotes the proliferation rate into the functional subsystem i,
due the encounter of particles belonging the functional subsystems h and k.
Destructive events can occur only within the same functional subsystem.
Toward a Mathematical Theory of Complex Systems – p. 28/6
Lecture 2. - Representation, Nonlinear Stochastic Games,
Pathways, and Mathematical Tools
(∂t + v · ∂x ) fi (t, x, v, u) =
#"
n
j=1
where
Cij
=
×
−
!
Cij [f ] +
n "
n
"
h=1 k=1
$
i
Shk
[f ] (t, x, v, u) ,
ηij [ρ](x∗ ) Ahk (v∗ → v, u∗ → u|v∗ , v∗ , u∗ , u∗ , f )
2 ×D 2
Ω×Du
v
fi (t, x, v∗ , u∗ )fj (t, x∗ , v∗ , u∗ ) dv∗ dv∗ du∗ du∗ dx∗ ,
!
fi (t, x, v)
ηij [ρ](x∗ ) fj (t, x∗ , v∗ , u∗ ) dv∗ du∗ dx∗
Ω×Du ×Dv
i
Shk
=
×
!
ηhk [ρ](x∗ ) µihk (u∗ , u∗ |f )
2 ×D
Ω×Du
v
fh (t, x, v, u∗ )fk (t, x∗ , v∗ , u∗ ) dv∗ du∗ du∗ dx∗ .
Toward a Mathematical Theory of Complex Systems – p. 29/6
Lecture 2. - Representation, Nonlinear Stochastic Games,
Pathways, and Mathematical Tools
Conjecture 1. Interactions modify the activity variable according to topological
stochastic games, however independently on the distribution of the velocity variable,
while modification of the velocity of the interacting particles depends also on the
activity variable.
Ahk (·) = Bhk (u∗ → u, |u∗ , u∗ ) × Chk (v∗ → v |v∗ , v∗ , u∗ , u∗ ) ,
where A, B, and C are, for positive defined f , probability densities.
Conjecture 2. Stochastic velocity perturbation
%
&
∂t + v · ∇x fi = νi Li [fi ] + ηi Ci [f ] + ηi µi Si [f ],
where
Li [fi ] =
!
Dv
'
∗
∗
∗
(
Ti (v → v)fi (t, x, v , u) − Ti (v → v )fi (t, x, v, u) dv∗ ,
and where Ti (v∗ → v) is, for the ith subsystem, the probability kernel for the new
velocity v ∈ Dv assuming that the previous velocity was v∗ .
Toward a Mathematical Theory of Complex Systems – p. 30/6
Lecture 3. - Application I
Keller and Segel model: The mathematical approach to study chemotaxis was
boosted by Keller and Segel. They introduced a model to study the aggregation of
Dictyostelium discoideum due to an attractive chemical substance. The model consists
in an advection-diffusion system of two coupled parabolic equations:



 ∂t n = divx (Dn (n, S)∇x n − χ(n, S) n ∇x S) + H(n, S),



∂t S = DS ∆S + K(n, S),
where n = n(t, x) is the cell (or organism) density at position x and time t, and
S = S(t, x) is the density of the chemo-attractant. The positive definite terms DS and
Dn are the diffusivity of the chemo-attractant and of the cells, respectively, while
χ ≥ 0 is the chemotactic sensitivity. In a more general framework in which diffusions
are not isotropic, DS and Dn could be positive definite matrices.
• T. H ILLEN AND K.J. PAINTER, A users guide to PDE models for chemotaxis, J.
Math. Biol., 58 (2009), 183-217
Toward a Mathematical Theory of Complex Systems – p. 31/6
Lecture 3. - Application I
It is not completely clear how the term divx (χn∇x S) induces per se the optimal
movement of the cells towards the pathway determined by the chemoattractant. This
term could be modified in a fashion that the flux density of particles is optimized along
the trajectory induced by the chemoattractant, namely by minimizing the functional
!
!
χ(n, S) n dS = χ(n, s) n 1 + |∇x S|2 dx
with respect to S, where dS is the measure of the curve defined by S. This approach
provides an alternative term in the corresponding Euler-Lagrange equation of type
.
/
∇x S
divx χ(n, S) n .
2
1 + |∇x S|
It does not seem realistic to think that cells or bacteria move simply by (linear
Fokker-Planck) diffusion, divx (Dn ∇x n). Other possibilities to modify this approach
based on incorporating real phenomena related with cell or bacteria motion (cilium
activation or elasticity properties of the membrane, among others) can be considered.
Toward a Mathematical Theory of Complex Systems – p. 32/6
Lecture 3. - Application I
A nonlinear limited flux that allows a more realistic dynamics: finite speed of
propagation c, preservation of fronts in the evolution, or formation of biological
patterns. The model collects two of the innovating improved terms consisting in the
choice of a flux limited and in the optimal transport.




n∇x n
∇x S





2
∂
n
=
div
D
(n,
S)
−
nχ(n,
S)
t
x
n

2
2

1 + |∇x S|

2 + Dn (n,S) |∇ n|2

n
x
2

c


+H1 (n, S),











∂t S = divx (DS · ∇x S) + H2 (n, S).
A challenging problem consists in the derivation of the model from the underlying
description at the cellular scale and, possibly, a revision of the model itself to avoid
unrealistic blow up description of phenomena.
This needs a specific characterization of the perturbation term.
Toward a Mathematical Theory of Complex Systems – p. 33/6
Lecture 3. - Application I
•N.B., A. B ELLOUQUID , J. N IETO , AND J. S OLER, Complexity and Mathematical
Tools Toward the Modelling of Multicellular Growing Systems, Math. Models Methods
Appl. Sci., 20 1179-1207, (2010).
•N.B., A. B ELLOUQUID , J. N IETO , AND J. S OLER, On the asymptotic theory from
microscopic to macroscopic tissue models: an overview with perspectives, to be
published.
 %
&


 ∂t + v · ∇x f1 = ν1 L1 [f1 ] + η1 G1 [f , f ] + η1 µ1 I1 [f , f ],


 %
Li [fi ] =
!
&
∂t + v · ∇x f2 = ν2 L2 [f2 ] + η2 G2 [f , f ] + η2 µ2 I2 [f , f ] ,
Dv
'
∗
∗
∗
(
Ti (v → v)fi (t, x, v , u) − Ti (v → v )fi (t, x, v, u) dv∗ ,
where Ti (v∗ → v) is, for the ith subsystem, the probability kernel for the new
velocity v ∈ Dv assuming that the previous velocity was v∗ .
Toward a Mathematical Theory of Complex Systems – p. 34/6
Lecture 3. - Application I
Moreover:Gi [f , f ] = Gi1 [fi , f1 ] + Gi2 [fi , f2 ],
!
Gij =
wij (x, x∗ )Bij (u∗ → u|u∗ , u∗ ) fi (t, x, v, u∗ )fj (t, x∗ , v, u∗ ) dx∗ du∗ du∗
Γ
! !
− fi (t, x, v, u)
wij (x, x∗ ) fj (t, x∗ , v, u∗ ) dx∗ du∗ ,
Du
Ii [f , f ] =
2
"
Dx
fi (t, x, v, u)
j=1
!
Du
!
wij (x, x∗ ) fj (t, x∗ , v, u∗ ) dx∗ du∗ .
Dx
with wij (x, ·) being a nonnegative weight supported in some subset Ω ⊆ Dx .
Parabolic-Parabolic Scaling
%
%
&
=
1
ε
q
ε ε
q+r1
ε ε
L
[f
]
+
ε
C
[f
,
f
]
+
ε
P
[f
, f ],
1
1
1
1
εp
&
=
1
L2 [f1ε ](f2ε ) + εq G2 [f ε , f ε ] + εq+r2 I2 [f ε , f ε ].
ε
ε∂t + v · ∇x f1ε
ε∂t + v · ∇x f2ε
p, q ≥ 1 ,
r1 , r2 ≥ 0, and ε is a small parameter that is allowed to tend to zero.
Toward a Mathematical Theory of Complex Systems – p. 35/6
Lecture 3. - Application I
Assumptions
Assumption H.1. We assume that the turning operator L2 [f2 ] is decomposed as
L2 [f2 ] = L02 [f2 ] + ε L12 [f1 ][f2 ], where Li2 , for i ∈ {0, 1}, is given by
! '
(
i
i
∗
∗
i
∗
L2 [f2 ] =
T2 (v, v )f2 (t, x, v , u) − T2 (v , v)f2 (t, x, v, u) dv∗ .
Dv
with T21 ≡ T21 [f1 ] depending on f1 and T20 independent on f1 .
Assumption H.2. We also assume that the turning operators L1 and L2 satisfy, for all
g, the following conditions:
!
!
!
L1 [g]dv =
L02 [g]dv =
L12 [f1 ][g]dv = 0.
Dv
Dv
Dv
Toward a Mathematical Theory of Complex Systems – p. 36/6
Lecture 3. - Application I
Assumption H.3. There exists a bounded velocity distribution Mi (v) > 0, for
i ∈ {1, 2}, independent of t, x, such that the detailed balance
T1 (v, v∗ )M1 (v∗ ) = T1 (v∗ , v)M1 (v)
T20 (v, v∗ )M2 (v∗ ) = T20 (v∗ , v)M2 (v)
Moreover, the flow produced by these equilibrium distributions vanishes, and Mi are
5
5
normalized, i.e. Dv v Mi (v)dv = 0 and Dv Mi (v)dv = 1.
Assumption H.4. The kernels T1 (v, v∗ ) and T20 (v, v∗ ) are bounded, and there exist
constants σi > 0, i = 1, 2 such that for all (v, v∗ ) ∈ Dv × Dv , x ∈ Ω:
T1 (v, v∗ ) ≥ σ1 M1 (v),
T20 (v, v∗ ) ≥ σ2 M2 (v),
Assumption H.5. The turning operator L2 [f1ε ] = L02 + εL12 [f1ε ] satisfies:
!
!
!
L1 [g]dv =
L02 [g]dv =
L12 [f1 ](g)dv = 0.
Dv
Dv
Dv
Toward a Mathematical Theory of Complex Systems – p. 37/6
Lecture 3. - Application I
Letting L1 = L1 and L2 = L02 , the above assumptions yields the following:
Lemma
i) For f ∈ L2 , the equation Li [g] = f , for i ∈ {1, 2}, has a unique solution
6
7
dv
,
g ∈ L2 Dv ,
Mi
which satisfies
!
g(v) dv = 0 if and only if
Dv
!
6
f (v) dv = 0.
Dv
ii) The operator Li is self-adjoint in the space L2 Dv ,
7
dv
.
Mi
iii) There exists a unique function θi (v) verifying Li [θi (v)] = v Mi (v), i = 1, 2.
iv) The kernel of Li is N (Li ) = vect(Mi (v)), i=1,2.
Toward a Mathematical Theory of Complex Systems – p. 38/6
Lecture 3. - Application I
The relaxation kernels presented in the Section together with the choice
T21 [f1 ] = K f1 (v, v∗ ) · ∇x
M1
f1
,
M1
where K f1 (v, v∗ ) is a vector valued function, leads to the model
M1
L12 [M1 S](M2 ) = h(v, S) · ∇x S,
h(v, S) =
!
Dv
'
∗
∗
∗
(
KS (v, v )M2 (v ) − KS (v , v)M2 (v) dv∗ .
Finally, α(n, S) = χ(n, S) · ∇x S, where the chemotactic sensitivity χ(n, S) is given
by the matrix
!
1
χ(n, S) =
v ⊗ h(v, S)dv.
σ2 Dv
Therefore, the drift term divx (n α(S)) that appears in the macroscopic case, becomes:
divx (n α(n, S)) = divx (n χ(n, S) · ∇x S) ,
which gives a Keller-Segel type model.
Toward a Mathematical Theory of Complex Systems – p. 39/6
Lecture 3. - Application I
Theorem Let fiε (t, x, v, u) be a sequence of solutions to the scaled kinetic system,
which verifies Assumptions (H.1.–H.5.) such that fiε converges a.e. in
[0, ∞) × Dx × Dv × Du to a function fi0 as ε goes to zero and
! ! !
sup
|fiε (t, x, v, u)|m du dv dx ≤ C < ∞
(1)
t≥0
Dx
Dv
Du
for some positive constants C > 0 and m > 2. Moreover, we assume that the
probability kernels Bij are bounded functions and that the weight functions wij have
finite integrals. It follows that the asymptotic limits fi0 have the form (??)-(??) where
n, S are the weak solutions of the following equation (that depends on the values of p,
q, r1 and r2 )
∂t S − δp,1 divx (DS · ∇x S)
=
δq,1 G1 (n, S) + δq,1 δr1 ,0 I1 (n, S),
∂t n + divx (n α(S) − Dn · ∇x n)
=
δq,1 G2 (n, S) + δq,1 δr2 ,0 I2 (n, S),
Toward a Mathematical Theory of Complex Systems – p. 40/6
Lecture 3. - Application I
where δa,b stands for the Kronecker delta and Dn , DS and α(S) are given by
!
!
DS = −
v ⊗ θ1 (v)dv,
Dn = −
v ⊗ θ2 (v)dv
Dv
and
α(S) = −
(2)
Dv
!
Dv
θ2 (v) 1
L2 [M1 S](M2 )(v)dv.
M2 (v)
(3)
The asymptotic quadratic terms of converge, for i = 1, 2, to the following functionals:
8.
/ .
/9
!
M1 S
M1 S
Gi (n, S)(t, x, u) =
Gi
,
dv
M2 n
M2 n
Dv
and
Ii (n, S)(t, x, u) =
!
Ii
Dv
8.
/ .
M1 S
M1 S
,
M2 n
M2 n
/9
dv.
Therefore, we can derive macroscopic models by taking limits in the scaled equation.
To pass to the limit it is sufficient assuming pointwise convergence together with a
global Lm bound of fiε .
Toward a Mathematical Theory of Complex Systems – p. 41/6
Lecture 1. - Common Features of Living Systems
Part 1. - Common Features of Living Systems, Stochastic Games, Nonlinear
Interactions, and Learning
Part 2 - Applications
A.1 On the Derivation of Flux Limited Keller Segel Models
A.2 A Mathematical Model for the in-vitro Dynamics of Cancer Hepatocytes
under Therapeutic Actions
A.3 On the Modeling of the Collective Dynamics of Vehicles
Toward a Mathematical Theory of Complex Systems – p. 42/6
Toward a Mathematical Theory of Complex Systems – p. 43/6
Part 2 - Application II
• M. D ELITALA AND T. L ORENZI, A mathematical model for the in-vitro dynamics
of cancer hepatocytes under therapeutic actions, work in progress.
Hepatocellular carcinoma is regarded, among cancer pathologies, as one of the major
malignant diseases in the world because of high incidence, extremely poor prognosis
and absence of symptoms during the early stages.
A human model of hepatocellular carcinoma is being developed by Mikulits and
coworkers (Mikulits et al.2010), which describes the dynamics of malignant
hepatocytes expressing epithelial and mesenchymal phenotypes.
The human model contributes to select the functional sybsystems and the activity
vaariables.
• W. M IKULITS , ET AL ., A Human Model of Epithelial to Mesenchymal Transition to
Monitor Drug Efficacy in Hepatocellular Carcinoma Progression, Mol. Cancer Ther.
10 (2011) 850-860
Toward a Mathematical Theory of Complex Systems – p. 44/6
Part 2 - Application II
Modeling Approach
Consider an in-vitro sample composed of:
• epithelial and mesenchymal malignant hepatocytes with heterogeneous
genotypic-phenotypic profiles.
• Cells proliferate due to the interactions with cytokines regulating proliferation, that
are produced by the cells themselves. The sample is exposed to the therapeutic actions
of Cytotoxic Agents, for short CAs, which lead cells to die almost independently from
their genotypic-phenotypic expression, and Targeted Therapeutic Agents, for short
TTAs, which can be addressed to act over those cells that are characterized by peculiar
genotypic-phenotypic profiles.
Cancer cells expressing the epithelial and mesenchymal phenotypes are modeled as
active particles divided into two functional subsystems, which are labeled by indexes
i = 1 and i = 2, respectively. Cytokines constitute the third functional subsystem.
Toward a Mathematical Theory of Complex Systems – p. 45/6
Part 2 - Application II
Modeling Approach
The genotypic-phenotypic profile of each cell is modeled by the activity variable
u ∈ Du ⊂ R, which identifies the cellular microscopic state. We assume u to be
normalized with respect to a suitable reference value; therefore, Du := [0, 1]. The
state of functional subsystems i = 1, 2 is identified by the distribution functions:
f1 = f1 (t, u) : [0, T ] × Du → R+ ,
f2 = f2 (t, u) : [0, T ] × Du → R+ ,
and the related number densities are computed as follows:
!
!
n1 (t) =
f1 (t, u)du, n2 (t) =
f2 (t, u)du.
Du
Du
Cytokines responsible for cell proliferation are supposed to be grouped into an
additional subsystem (i = 3). The microscopic state of cytokines is not characterized
by any variable and the state of subsystem i = 3 is modeled by the number density:
n3 = n3 (t) : [0, T ] → R+ .
Toward a Mathematical Theory of Complex Systems – p. 46/6
Part 2 - Application II
Modeling approach
Cytotoxic and targeted therapeutic agents are modeled as active particles of the outer
environment (i.e. external agents) that generate two additional functional subsystems
labeled, respectively, by indexes j = 1 and j = 2.
The microscopic state of the external agents is identified by the activity variable
w ∈ Dw := [0, 1]. In subsystem j = 1, w describes the intensity of the expressed
anti-cancer function, while, in subsystem j = 2, the activity variable is related to the
genotypic-phenotypic profile of cancer cells that can be mainly recognized, and then
attacked, by the curing agents. The state of functional subsystems j = 1, 2 is identified
by functions:
g1 = g1 (t, w) : [0, T ] × Dw → R+ ,
g2 = g2 (t, w) : [0, T ] × Dw → R+ ,
which are supposed to be given functions of their arguments.
Toward a Mathematical Theory of Complex Systems – p. 47/6
Part 2 - Application II
∂t fi (t, u)
=
−
−
dn3 (t)
dt
=
2 !
"
Aik (u, u∗ ; ,)fk (t, u∗ )du∗ − fi (t, u) + η K κ(u)n3 (t)fi (t, u)
:
;<
=
k=1 Du
:
;<
=
cell proliferation
EMT, mutations and renewal
!
η K µi (u)fi (t, u)(n1 (t) + n2 (t)) − η C fi (t, u)
µC (w∗ )g1 (t, w∗ )dw∗
:
;<
=
Dw
:
;<
=
cell-cell competition
destruction due to CAs
!
µT fi (t, u)
η T (u, w∗ )g2 (t, w∗ )dw∗
Dw
:
;<
=
destruction due to TTAs
!
µK (n1 (t) + n2 (t)) − η K n3 (t)
κ(u∗ )(f1 (t, u∗ ) + f2 (t, u∗ ))du∗ ,
:
;<
=
Du
:
;<
=
secretion of cytokines
consumption of cytokines
Toward a Mathematical Theory of Complex Systems – p. 48/6
Part 2 - Application II
Biological Phenomena
Model
Probabability epithelial to mesenchymal transition
A21 (u, u∗ ; ,), A12 (u, u∗ ; ,)
Probability for mutations
A11 (u, u∗ ; ,), A22 (u, u∗ ; ,)
Average variation in the genotypic-phenotypic profile
Cell-cytokine interaction rate
,
ηK
Probability for cell proliferation
κ(u)
Probability for secretion of cytokines
µK
Cell-cell interaction rate
ηK
Probability for destruction due to cell-cell competition
Cell-CA interaction rate
Probability for destruction due to CAs
Cell-TA interaction rate
Probability for destruction due to TTAs
µ1 (u), µ2 (u)
ηC
µC (w∗ )
η T (u, w∗ )
µT
Toward a Mathematical Theory of Complex Systems – p. 49/6
Part 2 - Application II
Testing biological consistency of the model
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
n1
0.2
0
0
n2
5
10
15
20
25
t
30
35
40
45
50
The figure on the right (Mikulits et al., 2011) shows a possible trend for the densities
of epithelial and mesenchymal cancer cells obtained by means of laboratory
experiments. Such a trend can be viewed as the expression of an emerging behavior
resulting from the interactions among cells.
The qualitative behaviors of the curves on the left are in agreement with the ones of the
curves on the right. This suggests that the present model is able to reproduce such an
emerging behavior.
Toward a Mathematical Theory of Complex Systems – p. 50/6
Part 2 - Application II
f
f
1
2
100
100
80
0.6
80
0.4
0.4
60
0.2
40
0
0
20
0.5
u
1 0
60
0.2
40
0
0
t
20
0.5
u
t
1 0
These figures show the dynamics of f1 (t, u) and f2 (t, u) provided by numerical
simulations. They highlight a branching process and show that f1 and f2 concentrate,
along time, around the points where the probability of cell proliferation κ(u) attains its
maximum.
Toward a Mathematical Theory of Complex Systems – p. 51/6
Part 2 - Application II
Numerical simulations have been developed under the following assumptions:
•
mutations lead to small variations in the genotypic-phenotypic profile (i.e. the
average size of mutations ε is small);
•
there are some genotypic-phenotypic profiles that endow cells with a probability
for proliferation higher than the other ones (i.e. the probability for cell
proliferation κ = κ(u) is not constant but it has a maximum reached at points
u = 0.25 and u = 0.75).
Intra-tumor heterogeneity is due to the presence of cells with different
geno-phenotypic profiles. These results support the idea that a strong reduction in
intra-tumor heterogeneity occurs, if mutations cause small variations, due to the fact
that only cells endowed with strong proliferative abilities can survive inside the
sample. In fact, the above figures show, in the limit of small mutations, the fixation of
highly proliferative clones only. The same results imply, on the one side, that weakly
proliferative mutants die out and, on the other side, that the genotypic-phenotypic
profiles endowing cells with strong proliferative abilities are, under an evolutionary
game perspective, evolutionary stable strategies.
Toward a Mathematical Theory of Complex Systems – p. 52/6
Part 2 - Application III
Part 1. - Common Features of Living Systems, Stochastic Games, Nonlinear
Interactions, and Learning
Part 2 - Applications
A.1 On the Derivation of Flux Limited Keller Segel Models
A.2 A Mathematical Model for the in-vitro Dynamics of Cancer Hepatocytes under
Therapeutic Actions
A.3 On the Modeling of the Collective Dynamics of Vehicles
Toward a Mathematical Theory of Complex Systems – p. 53/6
Part 2 - Application III
Bibliography
• Prigogine I. and Herman R., Kinetic theory of Vehicular Traffic,
Elsevier, New York, (1971).
• D. Helbing, Traffic and related self-driven systems, Review Modern Physics, 73,
(2001).
• N. Bellomo, and C. Dogbè, On The Modeling of Traffic and Crowds a
Survey of Models, Speculations, and Perspectives, SIAM Review, 53(3) (2011).
• N. Bellomo and A. Bellouquid, Global Solution to The Cauchy Problem
For Discrete Velocity Models of Vehicular Traffic, J. Differential Equations, (2012), In
press.
• A. Bellouquid, E. De Angelis, and L. Fermo, Lights, Shades and
Perspectives of the Kinetic Theory Approach to Vehicular Traffic Modeling,
Submitted, (2011).
Toward a Mathematical Theory of Complex Systems – p. 54/6
Part 2 - Application III
F IVE K EY FEATURES OF V EHICULAR DYNAMICS AS A L IVING S YSTEM
I N ONLINEAR INTERACTIONS : Vehicles on roads should be regarded as complex
living systems which interact in a nonlinear manner. Interactions follow specific
strategies generated both by communications, one’s strategy, and interpretation of that
of the others.
II H ETEROGENEOUS EXPRESSION OF STRATEGIC ABILITY: The individual behavior
of the driver-vehicle system, regarded as a micro-system, is heterogeneously
distributed.
III G RANULAR DYNAMICS : The dynamics shows the behavior of granular matter
with aggregation and vacuum phenomena. Namely, the continuity assumption of the
distribution function in kinetic theory cannot be straightforwardly taken.
IV I NFLUENCE OF THE ENVIRONMENTAL CONDITIONS : The dynamics is
remarkably affected by the quality of environment, including weather conditions and
quality of the road.
V PARAMETERS : The parameters should be related to specific observable phenomena
and their identification should be pursued either by using existing experimental data or
by experiments to be properly designed.
Toward a Mathematical Theory of Complex Systems – p. 55/6
Part 2 - Application III
Daganzo’s Critical Analysis - Vehicular Traffic
– Shock waves and particle flows in fluid dynamics refer to thousands of particles,
while only a few vehicles are involved by traffic jams;
– A vehicle is not a particle but a system linking driver and mechanics, so that one has
to take into account the reaction of the driver, who may be aggressive, timid, prompt
etc. This criticism also applies to kinetic type models;
– Increasing the complexity of the model increases the number of parameters to be
identified.
C. Daganzo, Requiem for second order fluid approximations of traffic flow, Transp.
Research B, 29B (1995), pp. 277-286.
Toward a Mathematical Theory of Complex Systems – p. 56/6
Part 2 - Application III
On the Use and Abuse of Empirical Data
• Empirical data must be used to design and validate models. Specifically:
Quantitative data: should be used to verify if the model has the ability to
reproduce quantitatively them.
Qualitative data on emerging behaviors: should be used to verify if the
model has the ability to reproduce qualitatively them.
Data on individual behaviors: should be used to design models at the
kinetic scale by a detailed modeling of interactions at the microscopic scale.
• The use of quantitative data as a direct input to derive models is
an abuse as, at least in principles, a careful modeling of microscopic interactions
should provide the above result. Models should depict empirical data
without forcing them into the model itself.
Toward a Mathematical Theory of Complex Systems – p. 57/6
Part 2 - Application III
Toward a Mathematical Theory of Complex Systems – p. 58/6
Part 2 - Application III
Discrete states models versus granular flow
n "
m
"
f (t, x, v, u) =
fij (t, x)δ(v − vi )δ(u − uj ) ,
i=1 j=1
Iv = {v0 = 0 , . . . , vn = 1},
and
Iu = {u0 = 0 , . . . , um = 1},
with fij (t, x) = f (t, x, vi , uj ).
ρ(t, x) =
n "
m
"
i=1 j=1
fij (t, x). q(t, x) =
n "
m
"
vi fij (t, x).
i=1 j=1
Reference Physical Quantities: nM is the maximum density of vehicles
corresponding to bumper-to-bumper traffic jam; VM is the maximum admissible mean
velocity which can be reached, in average, by vehicles running in free flow conditions;
a limit velocity can be defined as follows: V" = (1 + µ)VM ,
µ > 0; t is the
dimensionless time variable obtained referring the real time tr to a suitable critical
time Tc = ./VM ; x is the dimensionless space variable obtained dividing the real
space xr by the length . of the lane.
Toward a Mathematical Theory of Complex Systems – p. 59/6
Part 2 - Application III
Nonlinear Interactions by Stochastic Games
∂t fi + vi ∂x fi
=
!
n
"
h,k=1
−
x+ξ
η[f ](t, y)Aihk [ρ; u, α]fh (t, x)fk (t, y)w(x, y) dy
x
fi (t, x)
n !
"
h=1
Aihk [ρ; u, α] ≥ 0,
n
"
x+ξ
η[f ](t, y)fh (t, y)w(x, y) dy ,
x
Aihk [ρ] = 1,
∀ h, k = 1, . . . , n,
ρ(t, x) ∈ [0, 1).
i=1
w(x, y) ≥ 0,
!
x+ξ
w(x, y) dy = 1 ,
x
Toward a Mathematical Theory of Complex Systems – p. 60/6
Part 2 - Application III
• PHASE I: 0 ≤ ρ ≤ ρc . The candidate vehicle has the tendency to keep the
maximum velocity:

 1,
i = n;
i
Ahp =
 0,
otherwise.
• PHASE II: ρc < ρ ≤ ρs .
[A] Interaction with faster vehicles
When vh < vp the candidate vehicle maintains its current speed or increase the
velocity with a probability that depends, not only by the road conditions and by the
density, but also by the distance between the velocity classes involved.



1 − αuk (ρs + ρc − ρ),
i = h;



−1

(i
−
h)

 αuk (ρs + ρc − ρ) p
,
i = h + 1, ..., p;
i
"
Ahp =
−1
(i
−
h)




i=h+1



 0,
otherwise.
Toward a Mathematical Theory of Complex Systems – p. 61/6
Part 2 - Application III
[B] Interaction with slower vehicles
Aihp =


αuk (ρs + ρc − ρ),





 [1 − αuk (ρs + ρc − ρ)]







0,
i = h;
h−i
h−1
"
,
i = p, ..., h − 1;
(h − i)
i=p
otherwise.
• [C] Interaction with equally fast vehicles h = p
Aihp


(1 − α uk )(1 − ρs − ρc + ρ) h−1h − i ,



"



(h − i)




i=1

α uk + (1 − 2α uk )(ρs + ρc − ρ),
=



(i − h)−1


α uk (ρs + ρc − ρ) I
,


"


−1

(i
−
h)


i = 1, ...h − 1;
i = h;
i = h + 1, ..., n.
i=h+1
Toward a Mathematical Theory of Complex Systems – p. 62/6
Part 2 - Application III
When h = 1 or h = n the candidate vehicle cannot brake or accelerate, respectively
due to the lack or further lower or higher velocity classes. Thus, we merge the
deceleration or the acceleration into the tendency to preserve the current velocity:


i = 1;

 1 − α uk (ρs + ρc − ρ),
Ai11 =
,
α uk (ρs + ρc − ρ),
i = 2;



0,
otherwise
AiII =



 α uk (1 − ρs − ρc + ρ),



i = n − 1;
1 − α uk (1 − ρs − ρc + ρ),
i = n;
0,
otherwise.
• PHASE III: ρs < ρ < 1. In this case the density is very high and so regardless of
road conditions, the candidate vehicle has a tendency to stop. Then we have:

 1,
i = 1;
j
Ahp =
 0,
otherwise.
Toward a Mathematical Theory of Complex Systems – p. 63/6
Part 2 - Application III
1
0.5
0.45
0.9
α=0.95
0.4
0.8
0.35
0.7
0.3
0.6
0.25
0.5
α=0.95
ξ
q
α=0.7
α=0.5
α=0.7
0.2
0.4
0.15
0.3
0.1
0.2
α=0.5
α=0.3
0.05
0.1
α=0.3
0
0
0.2
0.4
0.6
0.8
1
0
0
0.2
0.4
ρ
0.6
0.8
1
ρ
Fundamental and velocity diagrams
Toward a Mathematical Theory of Complex Systems – p. 64/6
Part 2 - Application III
Mean velocity and variance
Toward a Mathematical Theory of Complex Systems – p. 65/6
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
ρ
ρ
Part 2 - Application III
0.4
0.4
t=0
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
t=1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.7
0.8
0.9
1
x
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
ρ
ρ
x
0.4
0.4
0.3
t=2.19
0.3
t=1.9
0.2
0.2
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
x
0.5
0.6
x
Fast and slow wave interactions
Toward a Mathematical Theory of Complex Systems – p. 66/6
Part 2 - Application III
1
1
0.9
0.9
0.8
0.7
0.7
0.6
0.6
ρ
0.5
t=0
0.4
0.5
0.4
0.3
t=1.1
0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
1
0
0
x
0.2
0.4
0.6
0.8
1
x
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
ρ
ρ
ρ
0.8
0.4
0.4
t=1.14
0.3
0.3
0.2
0.2
0.1
0
0
t=1.40
0.1
0.2
0.4
0.6
x
0.8
1
0
0
0.2
0.4
0.6
0.8
1
x
Toward a Mathematical Theory of Complex Systems – p. 67/6