Mathematical Theory and Applications of the Kinetic Theory ... · PDF fileMathematical Theory...

Mathematical Theory and Applications of the

Kinetic Theory of Active Particles

Nicola Bellomo and Livio Gibellihttp://staff.polito.it/nicola.bellomo

EVACUATE FPT7 - April 2013 - March 2017.

Porous Media Modelling in Biological Processes

Dundee, Scotland, August 27-28, 2015

Mathematical Theory and Applications of the Kinetic Theory of Active Particles

Theory and Application of the Kinetic Theory of ActiveParticles

Theory → Applications → Mathematical problems

Mathematical theory of active particles

Social

systems

Crowd

dynamics

Immune

competition

Analytical and computational problems


Lecture 1 - Mathematical Theory

Quest toward a mathematical structure

• B.N., Knopoff D., and Soler J., On the difficult interplay between life, “complexity”,

and mathematical sciences,Math. Models Methods Appl. Sci., 23 (10) (2013),

1861–1913.



Strategy

1. Understanding the links between the dynamics of living systems and their

complexity features;

2. Understanding the multiscale features of living systems;

3. Derivation a general mathematical structure, consistent with the aforesaid

features, with the aim of offering the conceptual frameworktoward the derivation

of specific models;

4. Derivation of specific models corresponding to well defined classes of systems by

implementing the structure by suitable micro-scale modelsof individual-based

interactions;

5. Validation by quantitative comparison empirical data and assessment of their

ability to depict qualitatively emerging behaviors.



Five Common Features and Sources of Complexity

1. Ability to express a strategy:Living entities are capable to develop specific

strategiesandorganization abilitiesthat depend on the state of the surrounding

environment.

2. Heterogeneity:The ability to express a strategy is not the same for all entities as

expression of heterogeneous behaviorsis a common feature of a great part of living

systems.

3. Learning ability: Living systems receive inputs from their environments and have

the ability to learn from past experience.

4. Interactions: Interactions are nonlinearly additive and involve immediate

neighbors, but in some cases also distant entities.

5. Darwinian mutations and selection:All living systems are evolutionary, as birth

processes can generate entities more fitted to the environment, who in turn generate

new entities again more fitted to the outer environment.



Representation of a large system of interacting entities called active particles

• The description of the overall state of the system is delivered by theone-particle

distribution function

fi = fi(t,x,v, u) = fi(t,w) : [0, T ]× Ω×Dv ×Du → IR +,

such thatfi(t,x,v, u) dx dv du = fi(t,w) dw denotes the number of active particles

whose state, at timet, is in the interval[w,w + dw] of thei-th functional subsystem

(still to be defined).

•w = x,v, u is an element of thespace of the microscopic states.

• x andv represent themechanical variables.

• u is theactivity variable which models the strategy expressed in each

functional subsystem.



Stochastic Games

Interactions by evolutionary stochastic games:Living entities, at each interaction,

play a gamewith an output that depends on their strategy often related to surviving and

adaptation abilities. The output of the game generally is not deterministic even when a

causality principle is identified.

1. Competitive (dissent):When one of the interacting particle increases its status by

taking advantage of the other, obliging the latter to decrease it. Therefore the

competition brings advantage to only one of the two.

2. Cooperative (consensus):When the interacting particles exchange their status,

one by increasing it and the other one by decreasing it. Therefore, the interacting

active particles show a trend to share their micro-state.

3. Learning: One of the two modifies, independently from the other, the micro-state,

in the sense that it learns by reducing the distance between them.

4. Hiding-chasing: One of the two attempts to increase the overall distance fromthe

other, which attempts to reduce it.



Pictorial illustration, where black and grey bullets denote, respectively, the pre- and

post-interaction states.

F F F

F C FF

T

T

FF FF FF T

i + 1

i

i − 1



Interactions among individuals need not have an additive linear character. As a

consequence, the global impact of a given number of entities(field entities) over a

single one (test entity) cannot be assumed to merely consistin the linear superposition

of the actions exerted individually by single field entities. This nonlinear feature

represents a serious conceptual difficultyto the derivation, and subsequent analysis,

of mathematical models for that type of systems.



• J. Hofbauer and K. Sigmund, Evolutionary game dynamics,Bull. Am.

Math. Society, 40 479-519, (2003).

Evolutionary game theory deals with entire population of players, all

programmed to use the same strategy (or type of behaviour). Strategies with

higher payoff will spread within the population, this can be achieved by

learning, by copying or inheriting strategies, or even by infection. The payoffs

depend on the actions of the coplayers and hence on the frequencies of the

strategies within the population. Since these frequencies change according to

the payoffs, this leads to a feedback loop. The dynamics of this feedback loop

is the object of evolutionary game theory.

E. Mayr,Populations, Species, and Evolution, Harward University Press, (1970).

E. Mayr,What Evolution Is , Basic Books, New York, (2001).

Mathematical Theory and Applications of the Kinetic Theory of Active Particles–


Interactions by evolutionary stochastic games:

• Functional subsystems:The overall system is subdivided into several populations,

calledfunctional subsystems, of entities calledactive particles, which express the

same strategy.

• Testparticles of thei-th functional subsystem with microscopic state, at timet,

delivered by the variable(x,v, u) := w, whose distribution function is

fi = fi(t,x,v, u) = fi(t,w). The test particle is assumed to be representative of the

whole system.

• Field particles of thek-th functional subsystem with microscopic state, at timet,

defined by the variable(x∗,v∗, u∗) := w∗, whose distribution function is

fk = fk(t,x∗,v∗, u∗) = fk(t,w

∗).

• Candidateparticles, of theh-th functional subsystem, with microscopic state, at

time t, defined by the variable(x∗,v∗, u∗) := w∗, whose distribution function is

fh = fh(t,x∗,v∗, u∗) = fh(t,w∗).



Mathematical Structures: Models with Space Dynamics

H.1. Candidate or test particles inx, interact with the field particles in the interaction

domainx∗ ∈ Ω. Interactions are weighted by theinteraction rate ηhk[f ]

supposed to depend on the local distribution function in theposition of the field

particles.

H.2. A candidate particle modifies its state according to the probability density:

Cihk[f ](v∗ → v, u∗ → u|w∗,w), which denotes the probability density that a

candidate particles of theh-subsystems with statew∗ = x∗,v∗, u∗ reaches the state

x = x∗,v, u in thei-th subsystem after an interaction with the field particles of the

k-subsystems with statew∗ = x∗,v∗, u∗.

H.3. A candidate particle, inx, can proliferate, due to encounters with field particles in

x∗, with rateµhkPihk[f ], which denotes the proliferation rate into the functional

subsystemi, due the encounter of particles belonging the functional subsystemsh and

k. Destructive events occur only within the same functional subsystem with rate

µikDik[f ].



Balance within the space of microscopic states and Structures

Variation rate of the number of active particles

= Inlet flux rate caused by conservative interactions

+Inlet flux rate caused by proliferative interactions

−Outlet flux rate caused by destructive interactions

−Outlet flux rate caused by conservative interactions,

where the inlet flux includes the dynamics of mutations.

This balance relation corresponds to the following structure:

(∂t + v · ∂x) fi(t,x,v, u) =(

JCi − J

Li + J

Pi − J

Di

)

[f ](t,x,v, u),

where the various termsJi can be formally expressed, consistently with the definition

of η, µ, C, P , andD.

•Michail Gromov, In a Search for a Structure, Part 1: On Entropy,Preprint,

(2013), http://www.ihes.fr/ gromov/.



Mathematical Structures

JCi =

n∑

h,k=1

∫

Ω×D2u×D2

v

ηhk[f ](w∗,w∗) Cihk[f ](v∗ → v, u∗ → u|w∗,w

∗, u∗)

× fh(t,x,v∗, u∗)fk(t,x∗,v

∗, u

∗) dx∗dv∗ dv

∗du∗ du

∗,

JLi =

n∑

k=1

fi(t,x,v)

∫

Ω×Du×Dv

ηik[f ](w∗,w∗) fk(t,x

∗,v

∗, u

∗) dx∗dv

∗du

∗,

JPi =

n∑

h,k=1

∫

Ω×D2u×Dv

µhk[f ](w∗,w∗)Pi

hk[f ](u∗, u∗)

× fh(t,x,v, u∗)fk(t,x∗,v

∗, u

∗) dx∗dv

∗du∗ du

∗,

JDi =

n∑

k=1

fi(t,x,v)

∫

Ω×Du×Dv

µik[f ](w∗,w∗)Dik[f ](u∗, u

∗)

× fk(t,x∗,v

∗, u

∗) dx∗dv

∗du

∗.


Lecture 2 - Crowd Dynamics

Modeling crowd dynamics

• B.N., Bellouquid A., and Knopoff D., From the micro-scale tocollective crowd

dynamics,SIAM Multiscale Model. Simul., 11 (2013), 943–963.

• B.N. and Bellouquid A., On multiscale models of pedestrian crowds - From

mesoscopic to macroscopic,Comm. Math. Sci., 13 (2015), 1649–1664.

• B.N. and Gibelli L., Toward a Mathematical Theory of Behavioural-Social

Dynamics for Pedestrian Crowds,Math. Models Methods Appl. Sci., 25 (2015).



Why a crowd is a social, hence complex, system?

• Ability to express a strategy:Walkers are capable to develop specific strategies,

which depend on their own state and on that of the entities in their surrounding

environment.

• Heterogeneity and hierarchy:The ability to express a strategy is

heterogeneously distributed and includes, in addition to different walking abilities,

also different objectives.

• Nonlinear interactions: Interactions are nonlinearly additive and involve

immediate neighbors, but also distant individuals.

• Social communication and learning ability: Walkers have the ability to learn

from past experience. Therefore, their strategic ability evolves in time due to

inputs received from outside induced by the tendency to adaptation.

• Influence of environmental conditions:The dynamics is remarkably affected by

the quality of environment.



Mesoscale (kinetic type) representation

• The system can be subdivided into different groups, calledfunctional subsystems, of

of persons, calledactive particles.

• The approach of the so-calledbehavioral crowd dynamicsintroduces an additional

microscopic variableu ∈ [0, 1], which models the heterogeneous ability to express a

strategy specific for each functional subsystem.

• The overall state of the system is described by theone-particle distribution function

fi = fi(t,x,v, u) such thatfi(t,x,v, u) dx dv du denotes the number of active

particles whose state, at timet, is in the elementary volume of thespace of the

microscopic states. Local density and flux are computed as follows:

ρi(t,x) =

∫

Dv×Du

fi(t,x,v, u) dv du ,

qi(t,x) =

∫

Dv×Du

v fi(t,x,v, u) dv du ,



Step 1: Modeling the decision process of adjustment of the velocity directionThree types of stimuli contribute to modify the walking direction:

1. desire to reach a well defined target,ν(τ)i ;

2. attraction toward the mean stream,ν(s);

3. attempt to avoid overcrowded areas,ν(v).

The preferred direction is defined by

ν(p)i =

(1− ρ)ν(τ)i + ρ ((1− β)ν(v) + βν

(s))∥

∥

∥(1− ρ)ν(τ)i + ρ ((1− β)ν(v) + βν

(s))∥

∥

∥

where the direction of vacuum and stream are given by

ν(v) = −

∇xρ

|∇ρ|, ν

(s) =V

|V |

whereβ ∈ [0, 1] a parameter which models the sensitivity to the stream with respect to

the search of vacuum.



Step 2: Adjustment of the speed to the new density conditions

Two cases can be distinguished:

• The walker’s speed is greater or equal than the mean speed. The walker either

maintains its speed, if for instance there is enough space toovertake the leading

walker, or decelerate to a speed which is as much lower as the higher is the density. It

is reasonable to assume that the probability to decelerate increases with the congestion

of the space and with the badness of the environmental conditions.

• The walker’s speed is lower than the mean speed. The walker either maintains its

speed or accelerate to a speed which is as much higher as the lower is the density, the

higher is the gap between the mean speed and the preferred speed and the goodness are

the environmental conditions.

C(v∗ → v) =

p+δ(

v − V(+))

+ (1− p+) δ (v − v∗)

H (V − v∗)

p−δ(

v − V(−))

+ (1− p−) δ (v − v∗)

H (v∗ − V )

wherep+ = αβ(1− ρ) andp− = (1− α)(1− β)ρ.



Step 3: Adjustment of the velocity in the presence of boundaries

Walkers whose distance from the wall,d, is within a specified cutoff,dw, modify their

velocityv to a new velocityvr, by reducing the normal component linearly with the

distance from the wall but keeping the speed constant, that is

v(r) =

d

dw(v · n)n+ sign(v · τ )

[

v2 −

d2

d2w(v · n)2

]1/2

τ

wheren andt are the normal and tangent to the solid wall.

Interaction domain: Interactions occur within an interaction domain related tothe

local velocity and visibility. Theperceived density ρaθ along the directionθ is:

ρaθ = ρ

aθ [ρ] = ρ+

∂θρ√

1 + (∂θρ)2

[

(1− ρ)H(∂θρ) + ρH(−∂θρ)]

,

where∂θ denotes the derivative along the directionθ, whileH(·) is the heaviside

functionH(· ≥ 0) = 1, andH(· < 0) = 0.

∂θρ→∞⇒ ρa → 1 , ∂θρ = 0⇒ ρ

a = ρ , ∂θρ→ −∞⇒ ρa → 0.



Individuals walking in a corridor with opposite directions

Ly

Lx

exitexit

wall

wall

• The kinetic model of pedestrian crowds is applied to the problem of two groups of

people walking in opposite directions.

• The segregation of walkers into lanes of uniform walking direction is

quantitatively assess by computing the band index

YB(t) =1

LxLy

∫ Ly

0

∫ Lx

0

∣

∣ρ1(t,x)− ρ2(t,x)∣

∣

ρ1(t,x) + ρ2(t,x)dx dy



Pedestrians walking in a corridor with opposite directions


Lecture 3 - Darwinian Dynamics and Immune Competition

Modeling Darwinian Mutations and Selections and Immune Competition

Bellouquid A., De Angelis E., and Knopoff D., From the modeling of the immune

hallmarks of cancer to a black swan in biology,Math. Mod. Meth. Appl. Sci., 23,

(2013), 949–978.

B.N. et al., On the Interplay between Mathematics and Biology - Hallmarks Looking

for a New System Biology,Physics of Life Review, 12 (2015), 85-90.

R.A. Weinberg, The Biology of Cancer, Garland Sciences - Taylor and Francis, New

York, (2007). Mathematical Theory and Applications of the Kinetic Theory of Active Particles–


Immune competition

• Malignant progression: Mutations, self-sufficiency in growth signals, insensitivity

to anti-growth signals, evading apoptosis, limitless replicative potential, evading

immune system attack, and tissue invasion and metastasis, incorporate some aspects of

genetic mutation, gene expression, and evolutionary selection.

• Immune defence:Immune cellslearn the presence of carriers of a pathology and

attempt to deplete them. It is a complex process, where cellsfrom theinnate immunity

improve their action by learning the so-calledacquired immunity.

• Darwinian selection: Can potentially initiate in each birth process, where mutations

bring new genetic variants into populations and natural selection then screens them. In

some cases, new cell phenotypes can originate from random mistakes during

replication.



Decomposition into functional subsystemsThe model considers two types of active

particles, namely epithelial and cancer cells, which move from the differentiate state to

various levels of progression, and immune cells characterized by different values of

activation.

↓ i = 1 Epithelial cells i = 5 Innate immune cells ↓

↓ i = 2 First hallmark ← i = 6 Acquired immunity 1 ↓

↓ i = 3 Secondhallmark ← i = 7 Acquired immunity 2 ↓

i = 4 Third hallmark ← i = 8 Acquired immunity 3

Table Functional subsystems

- i = 2 corresponds to the ability to thrive in a chronically inflamed

micro-environment;

- i = 3 to the ability to evade the immune recognition;

- i = 4 to the ability to suppress the immune reaction.



Mathematical structure

d

dtfij(t) = Jij [f ](t) = Cij [f ](t) + Pij [f ](t)−Dij [f ](t) + Lij [f ](t)

=

n∑

k=1

m∑

p=1

m∑

q=1

ηik[f ]Bpqik (j)[f ] fip fkq − fij

n∑

k=1

m∑

q=1

ηik[f ] fkq

=

n∑

h=1

n∑

k=1

m∑

p=1

m∑

q=1

ηhk[f ]µpqhk(ij) fhp fkq − fij

n∑

k=1

m∑

q=1

ηik[f ] νjqik fkq,

+ λ (f0ij − fij),

for i = 1, ..., 8 andj = 1, ..., m, and it is assumed that the activity variable attains

values in the following discrete set:Iu = 0 = u1, ..., uj , ..., um = 1. The overall

state of the system is described by the distribution function

fij = fij(t), i = 1, ..., 8 , j = 1, ...,m,

where the indexi labels each subsystem,j labels the level of the activity variable, and

fij(t) represents the number of active particles from functional subsystemi that, at

time t, have the stateuj .Mathematical Theory and Applications of the Kinetic Theory of Active Particles–


Interactions

• Conservative interactions:Cells modify their activity within the same functional

subsystem. A candidateh-particle with stateup can experiment a conservative

interaction with a fieldk-particle. The output of the interaction can be in the

contiguous statesup−1, up or up+1.

• Net proliferative events: Can generate, although with small probability, a

daughter cell that presents genetic modifications with respect to the mother cell. A

candidateh-particle (mother cell) can generate, by interacting with afield

k-particle, a daughter cell, belonging either to the same functional subsystem with

same state, or eventually to the following functional subsystem with the lowest

activity value.

• Destructive events:The immune system has the ability to suppress a cancer cell.

A h-candidate particle with stateup, interacting with a fieldk-particle with state

uq can undergo a destructive action which occurs within the same state of the

candidate particle.



Interactions

Figure 1: A h-candidate particleP (mother cell) by interaction with a k-field particleF

proliferates giving a daughter cellT , belonging either to the same functional

subsystem with same state (identical daughter), or to the following functional

subsystem with the lowest activity value (mutated daughter). Candidate parti-

cleC can experiment a conservative interaction with the field particleF , with

an output in the same functional subsystem. Finally, candidate particleD can

be subject a destructive action which occurs within the samestate.



Qualitative analysis and simulations

• The objective of the qualitative and computational analysis consists in understanding

if the immune system, possibly thank also to therapeutical actions, has the ability to

suppress cells of the last hallmark.

• Existence of solutions for arbitrary large times has been proved, while simulations

have shown the whole panorama of the competition depending on a critical parameter

that separate the situations where the immune system gains from those where it looses.

It is the ratio between the mutation rates of the immune cellsversus cancer cells, both

corresponding to the last mutation.

• Simulations show different trajectories are obtained for the number density of tumor

cells corresponding to increasing values of the ratio between the said parameter. The

first trajectory shows that for low values of the parameter the model predicts a rapid

growth of cancer cells due to the lack of contrast of the immune system. However, for

increasing values of the parameter the trajectory shows a trend to an asymptotic value

corresponding to a certain equilibrium. This asymptotic value decreases for increasing

value of the parameter up to when the defence is strong enoughto deplete the presence

of tumor cells.



Simulations

t

n

123

tn

123

Figure 2: Mutations and evolution in competition with the immune system with the

effect of therapeutical actions


Lecture 4 - New Trends in Behavioral Economy and Soci-ology

• B.N., Herrero M.A., Tosin A., On the Dynamics of Social Conflicts Looking for the

Black Swan,Kinet. Relat. Models, 6(3), 459–479, (2013).

• Ajmone Marsan G., B. N., and Tosin A.,Complex Systems and Society –Modeling and Simulations, Springer Briefs, Springer, New York, (2013).

• B.N., Colasuonno F., Knopoff D., and Soler J., From a SystemsTheory of Sociology

to Modeling the Onset and Evolution of Criminality,Networks Heter. Media, 10,

421-441, (2015).

• N.N. Taleb,The Black Swan: The Impact of the Highly Improbable, 2007.Mathematical Theory and Applications of the Kinetic Theory of Active Particles–

Lecture 4 - New Trends in Behavioral Sociology

Preliminary Reasonings

• The dynamics of social and economic systems are necessarilybased on individual

behaviors, by which single subjects express, either consciously or unconsciously, a

particular strategy, which is heterogeneously distributed. The latter is often based not

only on their own individual purposes, but also on those theyattribute to other agents.

• In the last few years, a radical philosophical change has been undertaken in social

and economic disciplines. An interplay among Economics, Psychology, and Sociology

has taken place, thanks to a new cognitive approach no longergrounded on the

traditional assumption of rational socio-economic behavior. Starting from the concept

of bounded rationality, the idea of Economics as a subject highly affected by individual

(rational or irrational) behaviors, reactions, and interactions has begun to impose itself.

• A key experimental feature of such systems is that interaction among heterogeneous

individuals often produces unexpected outcomes, which were absent at the individual

level, and are commonly termed emergent behaviors. Mathematical sciences can

significantly contribute to a deeper understanding of the relationships between

individual behaviors and the collective social outcomes they spontaneously generate.



Modeling strategy - Modeling on the Onset and Dynamics of Criminality

• The overall system is partitioned into functional subsystems, whose elements,

called active particles, have the ability to collectively develop a common strategy;

• The strategy is heterogeneously distributed among the components and

corresponds to an individual state, defined activity, of theactive particles;

• The state of each functional subsystem is defined by a probability distribution over

the activity variable;

• Active particles interact within the same functional subsystem as well as with

particles of other subsystems, and with agents from the outer environment;

• Interactions generally are nonlinearly additive and are modeled as stochastic

games, meaning that the outcome of a single interaction event can be known only

in probability;

• The evolution of the probability distribution is obtained by a balance of particles

within elementary volumes of the space of microscopic states, the inflow and

outflow of particles being related to the aforementioned interactions.



Functional subsystems, representation, and structure

i = 1 Normal citizens, whose microscopic state is identified by their wealth, which

constitutes the attraction for the eventual perpetration of criminal acts.

i = 2 Criminals, whose microscopic state is given by their criminal ability, namely

their ability to succeed in the perpetration of illegal acts.

i = 3 Detectives who chase criminals according to their individual ability.

Functional subsystem Micro-state

i = 1, citizens u ∈ D1, wealth

i = 2, criminals u ∈ D2, criminal ability

i = 3, detectives u ∈ D3, experience/prestige



∂tfi(t, u) = Ji[f ](t, u) =

=

3∑

h,k=1

∫

Dh

∫

Dk

ηhk(u∗, u∗)Bi

hk(u∗ → u|u∗, u∗)fh(t, u∗)fk(t, u

∗) du∗ du∗

−fi(t, u)

3∑

k=1

∫

Dk

ηik(u, u∗) fk(t, u

∗) du∗

+

∫

Di

µi(u∗,Ei)Mi(u∗ → u|u∗,Ei)fi(t, u∗)du∗

−µi(u,Ei)fi(t, u),

wherefi : [0, T ]×Di → IR +, i = 1, 2, 3. Moreoverηhk(u∗, u∗) and

µh(u∗,Eh) are, respectively, the encounter rate of individual based interactions and

that between a candidateh-particle and the mean activity. Moreover,

Bihk(u∗ → u|u∗, u

∗) andMh(u∗ → u|u∗,Eh) are, respectively, the probability

density for the state transition of individual based interactions and that between a

candidateh-particle and the mean activity.



Interaction Qualitative description η

Closer social states

1 ↔ 1© tend to interact η11(u∗, u∗) = η0 (1− |u∗ − u∗|)

more frequently

Experienced lawbreakers

2 ↔ 2© are more expected to η22(u∗, u∗) = η0(u∗ + u∗)

expose themselves

2 ↔ 3© Experienced detectives η23(u∗, u∗) = η0

(

(1− u∗) + u∗)

are more likely tohunt

3 ↔ 2© less experienced criminals η32(u∗, u∗) = η0

(

u∗ + (1− u∗))

Table 1: Non-trivial interactions between ah-particle (represented by a square) with

stateu∗ and ak- particle (represented by a circle) with stateu∗.Mathematical Theory and Applications of the Kinetic Theory of Active Particles–


Interaction Qualitative description µ

Experienced criminals

2 ↔ E2 are more expected µ2(u∗,E2) = µ0|u∗ − E2|

to expose themselves

Detectives interact with

3 ↔ E3 with the mean value through µ3(u∗,E3) = µ0|u∗ − E3|

the mean micro-state distance

Table 2: Non-trivial interactions between ah-candidate particle (represented by a

square) with activityu∗ and the mean activity valueEh.



Parameters involved in the table of games

αT Susceptibility of citizens to become criminals

αB Susceptibility of criminals to reach back the state of normal citizen

β Learning dynamics among criminals

γ Motivation/efficacy of security forces to catch criminals

λ Learning dynamics among detectives



Case Studies and Trends

Case 1 - Role of the mean wealth

Increasing number of criminals

Decreasing mean wealth of the society=⇒

Increasing criminal ability

Case 2 - Role of the shape of wealth distribution

Equal distribution =⇒ Slow growth in the number

of criminals

Poor society

Unequal distribution =⇒ Fast growth in the number

of criminals

Equal distribution =⇒ Fast decrease in the number

of criminals

Rich society

Unequal distribution =⇒ Slow decrease in the number

of criminals


Lecture 5 - Derivation of Chemotaxis Models

Derivation of Chemotaxis Models



The classical Keller and Segel model

∂tn = divx(Dn(n, S)∇xn− χ(n, S)n∇xS) +H(n, S),

∂tS = DS∆S +K(n, S),

wheren = n(t,x) is the cell (or organism) density at positionx and timet, and

S = S(t,x) is the density of the chemoattractant. The termsDS andDn are the

diffusivity of the nchemoattractant and of the cells, respectively. In a more general

framework in which diffusions are not isotropic,DS andDn could be positive definite

matrices.

E.F. Keller and L.A. Segel, Model for chemotaxis,J. Theor. Biol., 30, (1971),

225-234.

N. Bellomo, A. Bellouquid, Y. Tao, and M. Winkler, On the Theory of Keller-Segel

Models, an Overview with Perspectives,Math. Mod. Meth. Appl. Sci., 25 (2015),

1663-1763.



Flux limited Keller and Segel models:A nonlinear limited flux that allows a more

realistic dynamics: finite speed of propagationc, 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.

∂tn = divx

nDn(n, S)√

n2 +D2

n(n,S)

c2|n∇x|2

∇xn−nχ(n, S)

√

1 + |∇xS|2∇xS

+H1(n, S),

∂tS = divx(DS · ∇xS) +H2(n, S).

A challenging problem consists in the derivation of the model from the underlyingdescription at the cellular scale and, possibly, a revisionof the model itself to

avoid unrealistic blow up description of phenomena.



Parabolic-Parabolic Scaling

(

∂t + v · ∇x

)

f1 = ν1 L1[f1] + η1 G1[f , f ] + η1 µ1 I1[f , f ],

(

∂t + v · ∇x

)

f2 = ν2 L2[f2] + η2 G2[f , f ] + η2 µ2 I2[f , f ] ,

Li[fi] =

∫

Dv

(

Ti(v∗ → v)fi(t,x,v

∗, u)− Ti(v→ v

∗)fi(t,x,v, u))

dv∗,

whereTi(v∗ → v) is, for theith subsystem, the probability kernel for the new

velocityv ∈ Dv assuming that the previous velocity wasv∗.

(

ε∂t + v · ∇x

)

fε1 =

1

εpL1[f

ε1 ] + ε

qC1[f

ε, f

ε] + εq+r1P1[f

ε, f

ε],

(

ε∂t + v · ∇x

)

fε2 =

1

εL2[f

ε1 ](f

ε2 ) + ε

qG2[fε, f

ε] + εq+r2I2[f

ε, f

ε].

p, q ≥ 1 , r1, r2 ≥ 0, andε is a small parameter that is allowed to tend to zero.



Assumptions

Assumption H.1. We assume that the turning operatorL2[f2] is decomposed as

L2[f2] = L02[f2] + εL1

2[f1][f2], whereLi2, for i ∈ 0, 1, is given by

Li2[f2] =

∫

Dv

(

Ti2(v,v

∗)f2(t,x,v∗, u)− T

i2(v

∗,v)f2(t,x,v, u)

)

dv∗.

with T 12 ≡ T 1

2 [f1] depending onf1 andT 02 independent onf1.

Assumption H.2. We also assume that the turning operatorsL1 andL2 satisfy, for all

g, the following conditions:∫

Dv

L1[g]dv =

∫

Dv

L02[g]dv =

∫

Dv

L12[f1][g]dv = 0.



Assumption H.3. There exists a bounded velocity distributionMi(v) > 0, for

i ∈ 1, 2, independent oft,x, such that the detailed balance

T1(v,v∗)M1(v

∗) = T1(v∗,v)M1(v)

T02 (v,v

∗)M2(v∗) = T

02 (v

∗,v)M2(v)

Moreover, the flow produced by these equilibrium distributions vanishes, andMi are

normalized, i.e.∫

Dv

vMi(v)dv = 0 and∫

Dv

Mi(v)dv = 1.

Assumption H.4. The kernelsT1(v,v∗) andT 0

2 (v,v∗) are bounded, and there exist

constantsσi > 0, i = 1, 2 such that for all(v,v∗) ∈ Dv ×Dv, x ∈ Ω:

T1(v,v∗) ≥ σ1M1(v), T

02 (v,v

∗) ≥ σ2M2(v),



Derivation of Keller Segel Models

LettingL1 = L1 andL2 = L02, the above assumptions yields the following:

Lemma

i) For f ∈ L2, the equationLi[g] = f , for i ∈ 1, 2, has a unique solution

g ∈ L2

(

Dv,dv

Mi

)

,

which satisfies∫

Dv

g(v) dv = 0 if and only if∫

Dv

f(v) dv = 0.

ii) The operatorLi is self-adjoint in the spaceL2

(

Dv,dv

Mi

)

.

iii) There exists a unique functionθi(v) verifying Li[θi(v)] = vMi(v), i = 1, 2.

iv) The kernel ofLi is N(Li) = vect(Mi(v)), i=1,2.



Derivation of Keller Segel Models

Theorem Let fεi (t,x,v, u) be a sequence of solutions to the scaled kinetic system,

which verifies Assumptions (H.1.–H.4.) such thatfεi converges a.e. in

[0,∞)×Dx ×Dv ×Du to a functionf0i asε goes to zero and

supt≥0

∫

Dx

∫

Dv

∫

Du

|fεi (t,x,v, u)|

mdu dv dx ≤ C <∞

for some positive constantsC > 0 andm > 2. Moreover, we assume that the

probability kernelsBij are bounded functions and that the weight functionswij have

finite integrals. It follows that the asymptotic limitsf0i is such thatn, S are the weak

solutions of the following equation (that depends on the values ofp, q, r1 andr2)

∂tS − δp,1 divx (DS · ∇xS) = δq,1 G1(n, S) + δq,1δr1,0 I1(n, S),

∂tn+ divx (nα(S)−Dn · ∇xn) = δq,1 G2(n, S) + δq,1δr2,0 I2(n, S),



whereδa,b stands for the Kronecker delta andDn, DS andα(S) are given by

DS = −

∫

Dv

v ⊗ θ1(v)dv, Dn = −

∫

Dv

v ⊗ θ2(v)dv (1)

and

α(S) = −

∫

Dv

θ2(v)

M2(v)L1

2[M1S](M2)(v)dv, (2)

and where, fori = 1, 2, Gi andIi are given by the following:

Gi(n, S)(t,x, u) =

∫

Dv

Gi

[(

M1S

M2n

)

,

(

M1S

M2n

)]

dv

and

Ii(n, S)(t,x, u) =

∫

Dv

Ii

[(

M1S

M2n

)

,

(

M1S

M2n

)]

dv.


The End

Thank You!


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