All interesting domain of research. Their importance isn’t

All
collective motion like bacterial colonies, insects swarm, fish schools and bird
flocks are made up of self-propelled individual organisms and move by following
the interaction of their closed neighbors and exhibit a fascinating behavior,
self-propelled particles (SPP) model are known as self-driven particles.  In recent years, studying the collective
behavior in biological systems of SPP model is an ongoing challenge and become
an interesting domain of research. Their importance isn’t solely restricted to
biological system but they have a significant impact on space flight control,
in robotics of multiagent systems and controlling traffic flow etc.  In this work we thoroughly study the impact
by varying the radius on the collective behavior of SPP models .They produced a
range of new phenomenon that can be relevant for experimental systems. It will
be fruitful in future for further study to demonstrate the similarities and
differences between standard minimal fixed radiuses and varying radius. Inspite
of these differences between these processes, we can draw some analogies that
go beyond their visual appearance.

 

 

 

 

 

 

 

Fig (a) Demonstrating wingless locust’s marching in the field

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Fig
(b) a colony of ants displaying a complex social organization.

Fig
(c) Demonstrating a fascinating three dimensional motion of golden ray fishes

Fig
(d) Demonstrating the collective motion pattern in fish schools individual’s
interaction giving a dynamical structure.

.,

 

 

Fig
(e) Starlings displaying a magnanimous performance of collective behavior
during flight and also avoiding from the predator bird close to the central,
finger like structure

Fig
(f) a gregarious herd of zebras creating an optical illusion for predators.

Fig
(g ) a huge crowd of people ordering into traffic lanes during crossing the
pedestrian bridge and signifying a captivating    collective behavior

Fig
(h) Sheep are exhibiting a coherent behavior 
4

 

Assembled flock of birds, locust plagues manifest the captivating
role in our surroundings. Human mob also exhibits the fascinating act infact in
all classes shows charming motion 1 .Assembled role of schools of fish, swarm
of bees and locust’s plagues confiscated the vision, also the elegant facet during
motion is the functioning of locust’s plagues, bee’s swarms, schools of fish
has empirical utilizations. They show the well-furnished scenario because they
not only interact but also save from calamitous situations 2,3 . They can
save from highly disastrous situations in over mobbed humans in public places
like play grounds, shopping malls or in concerts by executing after a deep
analysis of their motion 1 . One of the most
fascinating features of the collective motion is the emergence of ordered motion
patterns from simple short range inter-individual interactions. Recently,
explorations of the principles that govern the formation of natural group
motion patterns have been highly beneficial for understanding the complex
physical and biological collective motions 4 .Collective motion in natural
systems is a well-studied phenomena that spans many spatial and temporal time
scales 5,6

Literature Review

                            In 1987  Reynolds 7 
gave attention to the individual 
behavior of each organism in the system 
and presented the first computational model which is  widely-accepted interaction theory for
natural biological flocks /  swarms,
which comprises three heuristic rules, their model shares features with an
earlier simulation carried out by Aoki (1982) 8 ,  that fish must poses three behavioral
attitude in order to produce the schools structures displayed in nature who used
the following rules (similar to those assumed by Reynolds) : (a)
avoidance/Separation: avoid collisions with nearby group mates;  (b) parallel orientation movements
/Alignment; match the velocity of group mates with moderate distances; and (c)
approach/Cohesion: stay close to remote group mates. The speed and direction of
the individuals were considered to be stochastic, but the direction of the
units was related to the location and heading of the neighbors. In this
pioneering paper of the field it was already declared that collective motion can
occur without a leader and the individuals having information regarding the
movement of the entire school. In 1995  a
1st computational model was presented  in 
order to establish a quantitative interpretation of the behavior of huge
flocks in the presence of perturbations, this statistical physics type of
approach to flocking was introduced  by
Vicsek et al. 9 and worked on the collective behavior of all organisms or
particles rather than individual particles

Self-propelled particles (SPP) or
the Vicsek model (SPP) model is fundamentally the   most popular instrument which utilizes to
describe the modeling and the collective movement of large groups of organisms.   A system displaying collective motion that
is made of units

• That is of same kind.

• moving almost constant absolute
velocity and able to change their direction.

• moving within a limited
interaction range by changing their direction of motion, in a way involving an
effective alignment.

• subject a perturbation of varying
magnitude.

Self-propelled
particles ( SPP) or IBM models has been suggested after the background study
of   twenty five years ,and analyzed in
one dimensional 10  or higher
dimensional 9,11,12, ,with or without an autonomous agents 13-15 ,with or
without interior or exterior perturbation and also integrate the living
organisms co-operation principle 7,8,11,16-18 . Biological SPP or   IBM 
models can be compared with other SPP models like robot to understand
the complex functioning .By simulating 
with original SPP models, different experiments performed to formulate
their features19and contrast with the natural swarm for progressing the
features 20. The goal is to portrait the deep analysis on SPP models by
taking archetypical examples of fundamental types to exhibits their drawbacks
and benefits. Acquire innovative results by altering the parameters relate to
the phase transitions of the model. By pondering  the outcomes of SPP models when autonomous
agents exists and  permit to alter in the
consequence of organization  of particle
in order to find steady scheme 20 to present the comprehensive outcomes for
SPP or IBM models .by  adopting  those representatives models  which include over all living circumstances
and schemes except some zones like traffic 21,22 and human mob scheming or
modeling 23,24. The 1 dimensional fundamental SPP models which was
articulated by 10 and then  scrutinize
in  the modifications of  locusts swarming functioning. Vicsek et al.
9  proposed a basic higher dimensional
SPP models to understand the complex coordinate cooperative behavior by
suggesting the modest collaborating principle of two dimensional. Modifications
can be   inquired during motion when
entities move coherently and move in same directions .They acknowledge the
characteristics of a paradigmatic model of 12. For discrete and continuous
articulation and the living model 11. By suggesting three distinct categories
of autonomous agents in three  archetypical
SPP models to scrutinize the effects of autonomous agents on the entire
entities, in last explore the functioning of SPP models by changing the
parameters as an outcome of each entity by updating the autonomous agents. And
wish to summarize the latest rank of SPP models in more areas for different
platform.

Czirok
et al  10 presented an amenable SPP
model for one dimension by taking a domain of 
size L with periodic boundary conditions having particles N .the ith
particle have xi  positions
and have ui speed drives in common position .The velocity of each
particle is updated by  considering the
average of its own velocity and its closest particles .closest particles are
the particles which are at a distance of r.r is the interaction radius and it
is a fixed quantity from each individual .The particles log in accidentally in
this situation with different velocities .Particle’s location and heading of
velocity can be emerge by using two principles.Czirok also suggests the
duration t .Vi denotes the dimensional velocity of the individuals.

Noise  parameter ? can be, measured by taking the
square root of the duration t, it is favorable to take normal distributions, to
find the current average velocity , add all the individuals by summing two or
greater than two consistence distributions gives the unstandardized,
non-differentiable, triangular distributions while adding two or more non
consistence distributions give standardize distribution. For getting accessible
distribution implement central limit theorem by adding constant random variables.

In
living system it’s a typical situation to find the mode of the disturbance.
That’s why use distributions to get the actual disturbance in a small duration,
Czirok model shows the complicated evolution of the particles .Particles move
in the same pattern when there is no external interaction. Otherwise they
become scattered from class. Classes not depend on time period and its position
changes with time to time. After introducing disturbance in random particles
it’s possible that they turned back for one dimensional .Czirok et al 10
introduced a   continuum analogue for
discrete particles. He observed the complicated dimensionless average behavior
on several locations after 1000 time steps, by taking each time step is equal
to 1.by keeping the same time interval and showed that at t> 4000 entities
show an ordered movement but they change the orientation at t?3000 to t?5000
after this they move as previous pattern.

Czirok
and Vicsek used continuity analogue (1999) and showed two behavior in the
absence of disturbance ?= 0 and observed that all the particles flow in the
common headings and move smoothly, that’s why the model is not so fascinating
in one dimension, so talk about a model in which  disturbance is  allowed for this sake  apply the current Velocity. After ignoring
the close individuals and previous Velocities launching a weighting parameter ?
. When ? = ? the entities are fully independent from other neighbors whereas
when ? = 0 the current velocity will not depend on the previous one. When ? = 1
the entity and its close entities have same weight.

In
? =0 case it may be possible that it has no close entity .in this situation the
velocity update formulation will not work and it will move with its own
velocity .Now  we observe the heading of
individuals by taking two parameters r and ?. We ponder after sometime step of
Czirok models in the absence and presence of disturbance, the individuals show
less alignment when they aware from? own Velocities .They show greater
alignment when they concentrate on other neighbors .by altering parametric
values after 50 time steps of weighted Czirok indicates that at ?=0 and  ?=2. 
When (perturbation) ? =2 the entities show ordered motion and reaching
towards unity and at zero figure they shows fluctuations. When perturbation is
allowed or not .the individuals show less alignment when they are aware from
their own velocities i.e. when ?>1 or 
r =0 .They show greater alignment when they concentrate on other
neighbors ? 1. Transitions can be observed by changing the
values of the parameters r and ?. One dimensional SPP models have bounded
complexity .Eftimie et al 25,26 suggests continuity .He proposed distinct
continuity categories in IBM model by applying hyperbolic PDEs and presented
different outcomes .Levine et al 27 presents one dimensional model in which
attraction and repulsion forces are involved .individuals are moving fastly to
being categorized  by keeping the domain
constant after increasing the individual’s quantity he observed that the attractive
forces are weaker than the individuals repulsive forces that’s why the size of
the class is increasing by managing the 
interaction radius .Czirok 
depends on physically periodic boundary conditions but Levine introduced
the attractive or repulsive forces that’s why the Czirok model is not as
attractive as Levine.

Name of author

year

 

I . Aoki

1982

Showed that fish poses three behavioral rules
separation , alignment and cohesion to make schools. 8

Reynolds

1987

Presented the first computational model for
natural flocks.7

Tamas Vicsek

1995

Presented the concept of SPP and worked on
collective behavior  of particles in
higher dimension rather than individuals in the presence of perturbation.9

Andras Czriok

1999

Analyzed the one dimensional model of self
propelled particles.10

Levine

2000

Present the 1 dimensional model by introducing
the attraction and repulsion forces and observed that repulsion forces are
stronger than attraction forces.27

Lain . D . Couzin

2002

Designed a 3 dimensional self organizing  model of group formation.28

L.Spector

2005

 Compared
the SPP model with other SPPmodel like robots to understand the complex
functioning of SPP model.19

S.-H.Lee

2006

.Studied the 
movement in escaping behavior of prey flock in response to a single
predator attack.29 

 Wood and
Ackland

2007

Got  the
progressive outcomes by changing the parameters and  displayed 
the benefits and drawbacks of SPP models.20

Jaime A,Pimentel
 

2008

showed
that the phase transition can be continuous or discontinuous it depends on
the way in which noise is innovate into the system 34.

Yu-Jian

2011

Worked on 3 dimensional model and showed that an
optimal view angle exists , particles move generally in a 3d space.30

 Chepizhko,
F. Peruani

2015

 Worked on
heterogeneous medium and showed that they change the behavior of collective
motion in  the presence of static or
dynamic obstacles also display  the
behavior of motion by varying the obstacles density and turning speed.31

 A.P. Solon

2015

Present a theoretical comparison of run and
tumble and active motion32. 

Israr Ahmed

2017

Work on SPP in the presence of moving obstacles
and predict the numerical evidence on the effect of noise, avoidance radius
and density.33

 
 
 

 

 

 

 

 

Methodology;

To
investigate the collective behavior of the self-propelled particles vicsek
model will be used. He showed a dynamic phase transition from disordered to an
ordered phase by taking a square shape domain of size L with individual
velocities vi can be find at each time step and current position by
the equation 

Xi (t + 1) = xi (t) + vi(t)
? t                     eq (1)                                                                                  

Periodic
boundary conditions having N particles placed in it, take time step t = 1 and
place r=1 interaction radius between each particle at initial stage when t=0
particles elicit completely random uncorrelated behavior in random directions
but they have common absolute velocity and direction determine by the angle 

? (t + I) = (?(t))r + ? ?                            eq (2)

Where
represents
the average direction of the velocities of particle i and its local;
neighborhood The average direction was given by the angle
arctan(sin (?(t))r /(cos (?(t))r .

In
Eq. (2) ?? is the magnitude of the perturbation selected from interval -?/2,
?/2.Used a constant distance between two updating by altering the parameters
perturbation ? and density ? and obtained a normalized average velocity using
equation

                                                             

FORTRAN
programming will be used for simulation purpose .for visualization gnu plot
will be used.

 

 

 

 

 

 

 

 

 

         Aim and Objective:

·        
The aim of this research is to model the collective behavior of
SPP in homogeneous medium that will be helpful in a highly multidisciplinary
field of collective behavior of SPP in a homogeneous medium.

·        
To investigate the impact of varying radius on the collective
motion of self-propelled particles.

·        
To investigate the effect of time on the collective behavior of
self-propelled particles.

·        
To explore the simulations results and contrast with previous
experimental results for fixed radius, so that the readers with a variety of
background could get the basics and a broader, more detailed clear picture of
the field.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Reference 

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 9 Vicsek, T., Czirók, A., Ben-Jacob, E.,
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Iain D. Couzin , Jens Krausew, Richard Jamesz, Graeme
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29 Journal of Theoretical Biology 240 (2006)
250–259 Dynamics of prey-flock escaping behavior in response to predator’s
attack S.-H. Lee  , H.K. Pak  , T.-S. Chon

30Optimal view angle in the three-dimensional
self-propelled particle model Yu-Jian Li , Su Wang  ,Zhong-Lin Han , Bao-Mei Tian  , Zhen-Dong Xi and Bing-Hong Wang March 2011
EPL, 93 (2011) 68003 doi: 10.1209/0295-5075/93/68003

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 32 A.P.
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 33 Ahmed et al / Materials Today:
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34 Intrinsic and extrinsic noise effects on
phase transitions of network models with applications to swarming systems Jaime A. Pimentel, Maximino
Aldana, Cristián Huepe, and Hernán Larralde Phys. Rev. E 77, 061138 – Published 27 June 2008 .

 

 

 

 

 

 

 

 

 

 

Impact of Varying Radius

On the Collective Behavior of

Self-Propelled Particles

BY

Bushra
Malik

A
research Proposal submitted in partial fulfillment for the approval of the
thesis

MASTER
OF PHILOSOSPHY IN

MATHEMATICS

Roll # MAT-2016-34

Under
the supervision of

DrIsrarMemon

at
the

SHAH ABDUL LATIF
UNIVERSITY, KHAIRPUR