Mathematical Models and Methods for Complex...
Transcript of Mathematical Models and Methods for Complex...
Mathematical Models and Methods for Complex
Systems
Nicola [email protected]
Department of Mathematics
Politecnico di Torino
http://calvino.polito.it/fismat/poli/
Mathematical Models and Methods for Complex Systems– p. 1/40
Homepage: http://calvino.polito.it/fismat/poli
Mathematical Models and Methods for Complex Systems– p. 2/40
Topics
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Mathematical Models and Methods for Complex Systems– p. 3/40
Unraveling Complex (Living) Systems
APRIL 2011 UNRAVELING COMPLEX SYSTEMS
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 complexsystems 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 inthe 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.
Mathematical Models and Methods for Complex Systems– p. 4/40
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. Foundations of Mathematical Modeling and Scaling
T.3. Representation, Nonlinear Stochastic Games, Pathways, Networks and
Mathematical Tools
Mathematical Models and Methods for Complex Systems– p. 5/40
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. BarabasiLinked. 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).
Mathematical Models and Methods for Complex Systems– p. 6/40
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. Dogbe, 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.Mathematical Models and Methods for Complex Systems– p. 7/40
Lecture 1. - Common Features of Living Systems
• E. Kant 1790, daCritique 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 interactingmolecules. Cells exist far
from thermal equilibrium by harvesting energy from their environment.
Mathematical Models and Methods for Complex Systems– p. 8/40
Lecture 1. - Common Features of Complex Living Systems
E. Schrodinger, 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 2003In 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., 1960The 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 e una proprieta 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 primitivodovra spiegare l’emergere
per selezione delle dinamiche proprie del singolo vivente.
Mathematical Models and Methods for Complex Systems– p. 9/40
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.
Mathematical Models and Methods for Complex Systems– p. 10/40
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
Mathematical Models and Methods for Complex Systems– p. 11/40
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 adilemma, 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.
Mathematical Models and Methods for Complex Systems– p. 12/40
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
specificstrategies andorganization abilitiesthat depend on the state of the
surrounding environment. These can be expressed without the application of any
external organizing principle. In general, they typicallyoperate
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:Heterogeneitycharacterizes 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 learningMathematical Models and Methods for Complex Systems– p. 13/40
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 entitiesplay a game at each
interaction with an output that is technically related to their strategyoften 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 cangenerate 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.Mathematical Models and Methods for Complex Systems– p. 14/40
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 maybe 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.
Mathematical Models and Methods for Complex Systems– p. 15/40
Lecture 1. - Common Features of Complex Living Systems
MULTISCALE REPRESENTATION OF TUMOUR GROWTH: gene interactions (stochastic
games), cells (kinetic theory), tissues (continuum mechanics), mixed (hybrid models).
Mathematical Models and Methods for Complex Systems– p. 16/40
Lecture 1. - Common Features of Complex Living Systems
EVOLUTION OF CELLULAR PHENOTYPE
Mathematical Models and Methods for Complex Systems– p. 17/40
Lecture 2. - Modeling and Scaling
Part 1. - Toward a Mathematical Theory of Complex Systems
T.1. Common Features of Living (Complex) Systems
T.2. Foundations of Mathematical Modeling and Scaling
T.3. Representation, Nonlinear Stochastic Games, Pathways, Networks and
Mathematical Tools
Mathematical Models and Methods for Complex Systems– p. 18/40
Lecture 2. - Modeling and Scaling
Introduction to Modeling and Scaling: This Lecture refers to Chapter 1
of the e-book:
• N. Bellomo, E. De Angelis, M. DelitalaLecture Notes on Mathematical Modelling
From Applied Sciences to Complex SystemsSIMAI e-Lecture Notes, vol. 8, 2010.
Systems belonging to the real world can be observed and phenomenologically studied
in order to reach a knowledge, as deep as possible, of the inner structure and of their
behavior. For such a purpose, experiments often need to be organized, and suitable
experimental devices need to be designed. These may be, in some cases, quite
sophisticated inventions of the human brain. When, throughthe collection and
organization of the experimental data, the systematic observation is developed into the
analysis, interpretation, and prediction of the behavior of the observed system, then a
mathematical model is produced.
In past times, mathematical modeling was essentially related to mathematical physics.
Such a science provided, mainly through the last century, all fundamental models of
the modern mechanical and technological sciences. Mathematical modeling is
nowadays a discipline that plays a crucial role in all applied sciences, from natural to
technological ones.Mathematical Models and Methods for Complex Systems– p. 19/40
Lecture 2. - Modeling and Scaling
Some Definitions
Definition 2.1. Independent variables: That istime t ∈ [0, T ] ⊆ IR + and
space x = x, y, z ∈ D ⊆ IR 3, are theindependent variables. A physical
system can be observed in a time interval[0, T ] ⊆ IR + and in a volumeD ⊆ IR 3. In
order to define the physical volume occupied by the observed system, a fixed system of
orthogonal axesO(xyz) with unit vectorsi, j, k can be used. Thepast is represented
by the negative values of the time,t ∈ IR −; thefuture by positive valuest ∈ IR +. A
system can be observed for positive times, however, the mathematical model can refer
to negative times. In this caset ∈ IR .
Definition 2.2. State variable: Thestate variable is a vector valued variable,
generally a function of the independent variables,u = u(t,x) : [0, T ] ×D 7−→ IR n,
whereu = u1, . . . , ui, . . . , un, such thatu is the set of the variables in charge of
describing (in the mathematical model) the physical state of the observed system. The
state variable is also calleddependent variable. It can take values either
for all values or for discrete values of theindependent variables.
Mathematical Models and Methods for Complex Systems– p. 20/40
Lecture 2. - Modeling and Scaling
Definition 2.3. Parameters: Theparameters are physical quantities that
characterize the physical system to be modeled. These quantities can be defined either
in a dimensional or in a dimensionless form and can be obtained either by direct
measurements on the system itself or by comparisons betweenthe predictions of the
model and the behavior of the real system. In the latter, the definition of
identification parameters is also used.
Definition 2.4. Mathematical model: A mathematical model is an
equation, or a set of equations, whose solution (that is the solution of the related
mathematical problems) provides the time-space evolutionof thestate variable,
that is the physical behavior, in the framework of the mathematical model, of the
related physical system. The equation that defines the mathematical model can also be
called thestate equation. The structure of the state equation is not, and cannot be
fixeda priori. It depends upon the adopted modeling technique.
Mathematical Models and Methods for Complex Systems– p. 21/40
Lecture 2. - Modeling and Scaling
The classifications
• Analysis of the structure of the state variable;
• Analysis of the type of the state equation;
• Analysis of the structure of the evolution equation;
• Analysis of the characteristics of the parameters.
Classification by state variableThis type of classification is based upon the analysis
of the structure of the state variable and, in particular, ofits arguments (the
independent variables).
Definition 2.5. Dynamic and static models A mathematical model is
dynamic if the state variableu depends on the time variablet. Otherwise the
mathematical model isstatic.
Definition 2.6. Discrete and continuous models A mathematical model is
discrete if the state variable does not depend on the space variables.Otherwise the
mathematical model iscontinuous.
Mathematical Models and Methods for Complex Systems– p. 22/40
Lecture 2. - Modeling and Scaling
Definition 2.7. Algebraic model: A mathematical model isalgebraic if the
state equation is an algebraic equation over the state variable and the set of parameters
r:
f(u; t,x, r) = 0.
Definition 2.8. Models and ODE: A mathematical model is expressed in terms
of ordinary differential equations if the state variable satisfies an ordinary differential
equation of the typedu
dt= f(t, u; r).
Definition 2.9. Models and PDE: A mathematical model is expressed in terms
of partial differential equations if the state equation involves partial derivatives of the
state variable with respect to the independent variables.
∂u
∂t= f
(t, x, u,
∂u
∂x,∂2x
∂x2; r
).
Remark If the state variable is a multidimensional vector, then theabove equations are
replaced by systems of equations: One for each componentui of the state variableu.
Mathematical Models and Methods for Complex Systems– p. 23/40
Lecture 2. - Modeling and Scaling
Classification by structure of the state equation
Definition 2.9. Linear and nonlinear models: A mathematical model is
linear if the state equation can be written in the formLu = 0, whereL is a linear
operator that satisfies the following conditions:
L(λu) = λL(u) ,
whereλ is a scalar and
L(u1 + u2) = Lu1 + Lu2.
A mathematical model isnonlinear if the state equation can be written as
Nu = 0 ,
whereN is a nonlinear operator.
Mathematical Models and Methods for Complex Systems– p. 24/40
Lecture 2. - Modeling and Scaling
Consistency of Models and Good Position of Problems
• Definition of the consistency properties of a mathematical model.
• Definition of the correct formulation of a mathematical problem.
• Well-posedness of a mathematical problem.
Definition 2.10.Consistency: A mathematical model defined by a system of
PDEs isconsistent if the number of equations is equal to the dimension of the
state-dependent variable.
• Elliptic equations describe systems in the equilibrium or steady state. They can be
seen as equations describing, for instance, the final state reached by a physical system
described by a parabolic equation after the transient term has died out.
Mathematical Models and Methods for Complex Systems– p. 25/40
Lecture 2. - Modeling and Scaling
• Hyperbolic equations are evolution equations that describe wave-like phenomena.
The solution of a hyperbolic initial value problem cannot besmoother than the initial
data. On the other hand, it can actually develop, as time goesby, singularities even
from smooth initial data, which then characterize the wholeevolution and are
propagated along special curves calledcharacteristics. In particular, if the
solution has initially a compact support, for instance,
u(t = 0, x) = 0 , for |x| > M0 ,
then
∀t , ∃M(t) such that u(t, x) = 0 for |x| > M(t) .
The rate at which the support expands can be interpreted as thespeed of
propagation of the effect modelled by the hyperbolic equation.
Mathematical Models and Methods for Complex Systems– p. 26/40
Lecture 2. - Modeling and Scaling
• Parabolic equations are evolution equations that describediffusion-like phenomena.
In general, the solution of a parabolic problem is infinitelydifferentiable with respect
to both space and time for anyx andt > 0, even if the initial data are not continuous.
In other words, parabolic systems have asmoothing action and singularities can
neither develop nor be maintained. Moreover, even if the initial data have compact
support, at anyt > 0 the initial condition is felt everywhere, that is
∀t and ∀M , ∃x with |x| > M such that u(t, x) 6= 0 .
This is usually indicated by saying that parabolic systems have aninfinite speed
of propagation.
Mathematical Models and Methods for Complex Systems– p. 27/40
Lecture 3. - Mathematical Tools
Part 1. - Toward a Mathematical Theory of Complex Systems
T.1. Common Features of Living (Complex) Systems
T.2. Foundations of Mathematical Modeling and Scaling
T.3. Representation, Nonlinear Stochastic Games, Pathways, Networks and
Mathematical Tools
Mathematical Models and Methods for Complex Systems– p. 28/40
Lecture 3. - Mathematical Tools
Toward a representation of complex systems
Let us consider a system constituted by a large number of interacting living entities,
calledactive particles. Their microscopic state includes, in addition to
geometrical and mechanical variables, also an additional variable, calledactivity,
which is heterogeneously distributed. Moreover, considera system of networks and
nodes such that the role of the space variable is not relevantin 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 byby
decomposing the system into suitable functional subsystems,
where afunctional subsystem is a collection of active particles, which have the
ability to express cooperatively the sameactivity regarded as a scalar variable.
Mathematical Models and Methods for Complex Systems– p. 29/40
Lecture 3. - 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 indexi, which denotes the functional subsystem within each node.
fi = fi(t, u) , i = 1, . . . , n,
such thatfi(t, u) du denotes the number of active particles whose state, at timet, 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.
Mathematical Models and Methods for Complex Systems– p. 30/40
Lecture 3. - Representation, Nonlinear Stochastic Games,Pathways, Mathematical Tools
Nonlinear Stochastic Games
Active particles interact with a certainencounter rate and play acollective
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. Amongthem:
• Distribution function conditioning the encounter rate also in connection to
hiding and learning dynamics;
• Nonlinearity induced bytopological distribution of the interacting
entities.
• Distribution function conditioning the output of the games;
Mathematical Models and Methods for Complex Systems– p. 31/40
Lecture 3. - Representation, Nonlinear Stochastic Games,Pathways, and Mathematical Tools
Nonlinear Stochastic GamesInteraction involves:
• Test particles with microscopic stateu, at the timet: fij = fij(t, u).
• Field particles with microscopic state,u∗ at the timet: fij = fij(t, u∗).
• Candidate particles with microscopic state,u∗ at the timet: fij = fij(t, u∗).
Rule:
i) Thecandidate particle interacts withfield particles and acquires, in probability,
the state of thetest 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.
Mathematical Models and Methods for Complex Systems– p. 32/40
Lecture 3. - Representation, Nonlinear Stochastic Games,Pathways, and Mathematical Tools
Nonlinear Stochastic Games
Interactions within the action domain
C ΩT
F F
F
F
F
Figure 1: – Active particles interact with other particles in their action domain
Mathematical Models and Methods for Complex Systems– p. 33/40
Lecture 3. - 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
F C FF
T
T
FF FF FF T
i + 1
i
i − 1
Figure 2: – Active particles during proliferation move fromone functional subsystem
to the other through pathways.
Mathematical Models and Methods for Complex Systems– p. 34/40
Lecture 3. - Representation, Nonlinear Stochastic Games,Pathways, and Mathematical Tools
Cooperative/Competitive Games
Cooperative behaviorThe active particle with stateh < k improves its state by
gaining from the particlek , which cooperates loosing part of its state.
Competitive behavior The active particle with stateh < k decreases its state by
contributing to the particlek, which due to competition increases part of its state.
Mathematical Models and Methods for Complex Systems– p. 35/40
Lecture 3. - 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 = η0hk ψhk(f) ψhk(f) = e
−αhk(1 + |u∗ − u∗|)(1 + ||fh − fk||) ,
whereη0hk andαhk are positive constants, and
||fh − fk||(t) =
∫
R+
|fh(t, u) − fk(t, u)| du.
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)) , E
p(t) =
∫
R+
upfi(t, u) du ,
for p = 1, 2, ....
Proliferation rate The termµihk is supposed to depend on the state of the interacting
pairs and low order moments.
Mathematical Models and Methods for Complex Systems– p. 36/40
Lecture 3. - 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:
∂tfi(t, u) + Fi(t) ∂ufi(t, u) = Ci[f ](t, u) + Pi[f ](t, u)
where
Ci[f ] =
n∑
j=1
η0ij
∫
Du×Du
ψij(f)Bij (u∗ → u|u∗, u∗
, f)
− fi(t, u)n∑
j=1
η0ij
∫
Du
ψij(f) fj(t, u∗) du∗
.
and
Pi[f ] =
n∑
h=1
n∑
k=1
η0hk
∫
Du
∫
Du
ψhk(f)µihk(u∗, u
∗
, f)fh(t, u∗)fk(t, u∗) du∗
.
Mathematical Models and Methods for Complex Systems– p. 37/40
Lecture 3. - Common Features of Complex Living Systems,and Mathematical Tools
Models with Space Structure
H.1. Candidate or test particles inx, interact with the field particles inx∗ ∈ Ω located
in the interaction domainΩ. Interactions are weighted by theinteraction rate
ηhk[ρ](x∗) supposed to depend on the local density in the position of thefield
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 theh-subsystems with statev∗, u∗ reaches the statev, u after an
interaction with the field particlesk-subsystems with statev∗, u∗.
H.3. Candidate particle, inx, can proliferate, due to encounters with field particles in
x∗, with rateµihk, which denotes the proliferation rate into the functional subsystemi,
due the encounter of particles belonging the functional subsystemsh andk.
Destructive events can occur only within the same functional subsystem.
Mathematical Models and Methods for Complex Systems– p. 38/40
Lecture 3. - Representation, Nonlinear Stochastic Games,Pathways, and Mathematical Tools
(∂t + v · ∂x) fi(t,x,v, u) =
[ n∑
j=1
Cij [f ] +
n∑
h=1
n∑
k=1
Sihk[f ]
](t,x,v, u) ,
where
Cij =
∫
Ω×D2u×D2
v
ηij [ρ](x∗)Ahk(v∗ → v, u∗ → u|v∗,v
∗
, u∗, u∗
, f)
× fi(t,x,v∗, u∗)fj(t,x∗
,v∗
, u∗) dv∗ dv
∗
du∗ du∗
dx∗
,
− fi(t,x,v)
∫
Ω×Du×Dv
ηij [ρ](x∗) fj(t,x
∗
,v∗
, u∗) dv∗
du∗
dx∗
Sihk =
∫
Ω×D2u×Dv
ηhk[ρ](x∗)µihk(u∗, u
∗|f)
× fh(t,x,v, u∗)fk(t,x∗
,v∗
, u∗) dv∗
du∗ du∗
dx∗
.
Mathematical Models and Methods for Complex Systems– p. 39/40
Lecture 3. - 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∗) ,
whereA, B, andC are, for positive definedf , 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 whereTi(v∗ → v) is, for theith subsystem, the probability kernel for the new
velocityv ∈ Dv assuming that the previous velocity wasv∗.
Mathematical Models and Methods for Complex Systems– p. 40/40