THE CHEMICAL BASIS OF MORPHOGENESIS BY REACTION DIFFUSION SYSTEM by W.A.B. Janith( SC/2007/6624 ) Group members S.H. Madarasinghe( SC/2007/6678 ) D.M.S. Thushani( SC/2007/6705 ) Supervisor: Mr. L. W. Somathilake Demostrater: Mr. Anjana Prabhath MMA 3b23 report submitted to the faculty of science University Of Ruhuna in partial fulfillment of the requirements for the degree of Bachelor of Science Department of Mathematics University Of Ruhuna October 2010
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THE CHEMICAL BASIS OF MORPHOGENESIS BY REACTION DIFFUSION SYSTEM
Recent progress in development biology has identified various micro mechanisms so we used mathematical methods is Gray-Scott model and Turing model for make animal various skin patterns just like zebra skin, tiger skin patterns.
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THE CHEMICAL BASIS OF MORPHOGENESIS BYREACTION DIFFUSION SYSTEM
byW.A.B. Janith( SC/2007/6624 )
Group membersS.H. Madarasinghe( SC/2007/6678 )D.M.S. Thushani( SC/2007/6705 )
Supervisor: Mr. L. W. SomathilakeDemostrater: Mr. Anjana Prabhath
MMA 3b23 report submitted to the faculty of scienceUniversity Of Ruhuna
in partial fulfillment of the requirements for the degree of
1.1 Principle and properties of the reaction-diffusion
model
This chapter briefly explains, without recourse to mathematics, how
the reaction-diffusion system can form a periodic structure. Sir Alan Tur-
ing presented a idea that a combination of reaction and diffusion can
generate spatial patterns (Alan Mathison Turing worked from 1952 until
his death in 1954 on mathematical biology, specifically morphogenesis.
He published one paper on the subject called The Chemical Basis of Mor-
phogenesis in 1952,). In the paper, he studied the behavior of a complex
system in which two substances interact with each other and diffuse at
different diffusion rates, which is known as the reaction-diffusion system.
Turing proved mathematically that such system is able to form some char-
acteristic spatio-temporal patterns in the field. One of the most significant
deviations is s formation of a stable periodic pattern. He stated that the
spatial pattern generated by the system might provide positional infor-
1
1.1 Principle and properties of the reaction-diffusion model 2
mation for a developing embryo.
In spite of the importance of the idea in the developmental biology, his
model was not accepted by most experimental biologists. But,At finally
most of those who took over and developed the Turing’s idea were applied
mathematicians and physicists. They proposed various types of model
that developed Turing’s original equation to fit real, naturally occurring
phenomena. Although the equations for each model differ, they all share
the basic requirement of the original model; that is, ’waves’ are made
from the interactions of two putative chemical substances which we refer
to here as the ’activator’ and the ’inhibitor’. Suppose that the activa-
tor enhances the synthesis of itself and another substance-the inhibitor.
In turn, the inhibitor inhibits the synthesis of the activator. The auto-
catalytic property of the activator and the feedback circuit of the inhibitor
make the system oscillate when the catalytic constants are set properly.
Like normal molecules, both substances are expected to diffuse into neigh-
boring cells according to the concentration gradient. The ratio of the dif-
fusion constants of the two substances plays a main role in determining
the behavior of the system.
The most important and interesting phenomenon in this system occurs
when the diffusion of the inhibitor is much faster than that of the acti-
vator. For example, imagine a one-dimensional (1D) cell array in which
the above interactions function (Fig. 1b-g). In the central region, as an
initial condition, the concentration of the activator is set relatively higher
than in other regions (Fig. 1a). Due to the self-catalytic nature of the
activator, the concentration of activator increases at the center, as well
as the concentration of the inhibitor (Fig. 1c). The concentration curve
1.1 Principle and properties of the reaction-diffusion model 3
Figure 1.1 (a) Schematic drawing of a reaction-diffusion system(activator-inhibitor type). The white and black arrows represent activa-tion and inhibition, respectively. (bd): Wave formation from almost randominitial conditions (for details, see text). (eg) Doubling of the waves in thegrowing field (for details, see text). (h) Different patterns generated byan identical RD equation. One of the parameter values in the equation ischanged for each pattern.
becomes steeper for the activator and shallower for the inhibitor, accord-
ing to the ratios of their diffusion constants. In the side regions, then,
the activator concentration decreases because of the high concentration
of inhibitor diffusing from the central region. Eventually, the small initial
concentration difference between the central and side regions are gradually
amplified (Fig. 1d). When the concentration of activator reaches a peak,
the balance of reaction and diffusion stabilizes the slopes of the concentra-
tion. The concentration wave created by this mechanism has an intrinsic
1.1 Principle and properties of the reaction-diffusion model 4
wavelength that is determined by the constants of reaction and diffusion.
Suppose that the 1D field enlarges (i.e. grows) gradually. This process
does not immediately change the number of peaks but changes the gradi-
ent of the slope (Fig. 1e,f ). As the field grows, the concentration of the
inhibitor increases, particularly at the centre, because this region is far-
thest from the low concentration regions. When the field reaches a critical
length, the feedback effect of the inhibitor exceeds the auto-catalysis of
the activator, and the widened peak divides into two peaks of the original
width (Fig. 1g). The shapes of the waves are determined by the values
of the parameters that represent the constants of reaction and diffusion.
The two-dimensional (2D) patterns of the reaction and diffusion wave are
more sensitive to the parameter values. Figure 1h shows the stable 2D
pattern of an reaction and diffusion system calculated with different pa-
rameters. One of the merits of the reaction and diffusion model is that
it can explain each of the clearly different patterns that are often seen in
animal skin. The most important properties of this system are:
(i) that the pattern forms autonomously without any other positional in-
formation
(ii) that the pattern is stable once formed
(iii) That it regenerates when it is artificially disturbed.
Turing and many other theoretical researchers thought that these prop-
erties appropriately explain the marvelous robustness of animal develop-
ment. In Turing’s, as well as many other models, ’diffusion’ is used as a
means of propagating a condition that occurs in a cell.
1.2 About Our programming language 5
1.2 About Our programming language
Figure 1.2
For in our project we used java programming language as Java is a high-level lan-
guage , third generation programming language, like C, Fortran, Smalltalk, Perl, and
many others. We can use Java to write computer applications that crunch numbers,
process words, play games, store data or do any of the thousands of other things
computer software can do. Compared to other programming languages, Java is most
similar to C. However although Java shares much of C’s syntax, it is not C. C , will
certainly help Us to learn Java more quickly. Unlike C++ Java is not a superset of
C. A Java compiler won’t compile C code, and most large C programs need to be
changed substantially before they can become Java programs.
Java is a platform for application development. A platform is a loosely defined
computer industry buzzword that typically means some combination of hardware and
system software that will mostly run all the same software. For instance PowerMacs
running Mac OS 9.2 would be one platform. DEC Alphas running Windows NT
would be another.
object oriented programming is the catch phrase of computer programming in the
1990’s. Although object oriented programming has been around in one form or an-
other since the Simula language was invented in the 1960’s, it’s really begun to take
1.2 About Our programming language 6
hold in modern GUI environments like Windows, Motif and the Mac. In object-
oriented programs data is represented by objects. Objects have two sections, fields
(instance variables) and methods. Fields tell what an object is. Methods tell what
an object does. These fields and methods are closely tied to the object’s real world
characteristics and behavior. When a program is run messages are passed back and
forth between objects. When an object receives a message it responds accordingly as
defined by its methods.
Object oriented programming is alleged to have a number of advantages including:
” Simpler, easier to read programs ” More efficient reuse of code ” Faster time to
market ” More robust, error-free code
Java is Platform Independent, Java was designed to not only be cross-platform in
source form like C, but also in compiled binary form. Since this is frankly impossible
across processor architectures Java is compiled to an intermediate form called byte-
code. A Java program never really executes natively on the host machine. Rather a
special native program called the Java interpreter reads the byte code and executes
the corresponding native machine instructions. Thus to port Java programs to a new
platform all that is needed is to port the interpreter and some of the library routines.
Even the compiler is written in Java. The byte codes are precisely defined, and remain
the same on all platforms.
Chapter 2
Mathematical back ground
2.1 Reaction-Diffusion Equations
Figure 2.1
A system of reaction-diffusion equations is a system of equations of
the form
∂x
∂t= D4 u + f(u,5u, x, t)
Over a region D ⊆ Rn , where u (x,t) is a vector representing the states
(in our model morphogenesis concentrations) of a group of substances at
time t and position X ⊆ Rn A is a matrix of diffusion coefficients, which
in a two species system is typically of the form D=
u 0
0 v
, and 4u
7
2.1 Reaction-Diffusion Equations 8
is the Laplacian differential operator acting on u with respect to x ∈ D
It is the second order spatial rate of change of u. In the most general
case, the inputs in the reaction function f are u,5u the gradient of u with
respect to x, x and t. These partial differential equations are subject to
boundary conditions over Ω ⊆ D and initial conditions. In these equa-
tions the term containing the Laplacian operator is the diffusion term.
Without the function f, (see Fig. 2.1) is the heat equation, one of the first
equations encountered in any partial differential equation course. The
heat equation models the diffusion of heat from regions of higher tem-
perature, or heat concentration, to regions of lower temperature, which
is very similar to chemical diffusion. The function f is called the reac-
tion function because it represents the interactions between particles that
act to increase or decrease the quantities of each species,and may depend
on the concentration of particles themselves (u), the gradient of the con-
centrations with respect to space(5u),and the location of the reaction in
space and time, (x and t). The use of chemical terms is meant merely as
an analogy, as reaction-diffusion equations have found broad application
in areas other than chemistry, such as neurological signal transmission,
Belousov-Zhabotinsky chemical waves, geochemical systems, combustion
theory, and other complex systems.
This study is primarily concerned with functions f that depend only
on the concentrations of the reactants. This idealization is a good approx-
imation for many chemical reactions held at constant temperature, as is
often true for biological reactions. The general system now reduces to
∂x
∂t= D4 u + f(u)
2.1 Reaction-Diffusion Equations 9
This system is augmented by initial conditions, u(x, 0) = h(x) ,and bound-
ary conditions. We will be dealing with two morphogens, that is, the
vector u will be given by:
u(x, t) =
u(x, t)
v(x, t)
J.D.Murray gives a good overview of several reaction-diffusion equations
used to model morphogenesis. The three primary reactions he mentions
are Schnakenbergs reaction, Gierer and Meinhardts activator/inhibitor
model, and Thomas experimental model. Schnakenbergs model has not
found much biological application. It is given in nondimensionalized form
by
ut = γ(b− v2u) + d4 u,
vt = γ(a− v + uv2) +4v
following model which is known as an activator/inhibitor system.
ut = γ(a− bv +u2
v) +4u
vt = γ(u2 − v) + d4 v
The nondimensionalized parameters are the same as above. For this equa-
tion we will call u the activator, since it acts to increase the population of
both chemicals, and v will be the inhibitor, since it decreases the rate of
change over time for each morphogen. For patterns to occur, Gierer and
Meinhardt showed that d À 1, in other words, that the inhibitor must dif-
fuse significantly faster than the activator. As an illustration, consider a
predator/prey system. Think of the prey as the activators and the preda-
tors as the inhibitors. The predators, cheetahs for instance, diffuse faster
2.2 Gray-Scott model 10
than the prey, antelope. Where the antelope gather together they create
an environment where more of their kind can thrive, but the fast moving
cheetahs inhibit their numbers (through digestion) when they stray from
the herd. Also contact between antelope and cheetahs (again through
digestion) activates the production of cheetahs. For the right parame-
ters, activator/inhibitor reaction-diffusion systems form dappled patterns
where activators clump together that can be thought of as analogous to
herds of prey species.
2.2 Gray-Scott model
Figure 2.2 Continuous stirred tank reactor
For the study of a chemical reaction we are going to look at a continu-
ous stirred tank reactor. The reactor contains two chemicals: U and V. In
the tank reactor we have a continuous inlet stream which, in our case, only
contains the chemical U and the product P is continuously drained. The
reactor is well mixed so that there is a uniform concentration of the chem-
icals U and V throughout the reactor. We study the following chemical
2.2 Gray-Scott model 11
reaction:
U + 2V −→ 3V
This is an autocatalytic reaction in which V is called the catalyst or the
activator and U the inhibitor of the reaction. Since we want the catalyst
V to have a finite lifetime,we use a second chemical reaction which is of
the form
V −→ P
P is an inert product. It is assumed for simplicity that the reverse reactions
do not occur (this is a useful simplification when a constant supply of
reactants prevents the attainment of equilibrium). Because V appears on
both sides of the first reaction, it acts as a catalyst for its own production.
The overall behavior of the system is described by the following for-
mula, two equations which describe three sources of increase and decrease
for each of the two chemicals:
∂U
∂t= Du 52 U − UV 2 + F (1− U)
∂V
∂t= Dv 52 V + UV 2 − (F + K)V
These equations were posed by P.Gray and S.K. Scott in 1983, that’s why
it’s called the Gray-Scott model. In this model, the two partial differential
equations are the mass-balance equations for U and V. In the model DU
and DV are the diffusivities, which represent the rate of speed by which U
and V diffuse. For the sake of simplicity we can consider Du, Dv, F and
k to be constants. In computer simulations there are also quantization
constants for time and space (4t and 4x) that are used to break ∂t and
52 into discrete intervals.
2.2 Gray-Scott model 12
The first equation tells how quickly the quantity u increases. There are
three terms. The first term, Du52u is the diffusion term. It specifies that
u will increase in proportion to the Laplacian (a sort of multidimensional
second derivative giving the amount of local variation in the gradient) of U.
When the quantity of U is higher in neighboring areas, u will increase. 52u
will be negative when the surrounding regions have lower concentrations
of U, and in such cases the diffusion term is negative and u decreases.
If we made an equation for u with only the first term, we would have
∂U∂t
= Du 52 U , which is a diffusion-only system equivalent to the heat
equation.
The second term is −uv2. This is the reaction rate. The first reaction
shown above requires one U and two V, such a reaction takes place at
a rate proportional to the concentration of U times the square of the
concentration of V. Also, it converts U into V: the increase in v is equal
to the decrease in u (as shown by the positive uv2 in the second equation).
There is no constant on the reaction terms, but the relative strength of
the other terms can be adjusted through the constants Du, Dv, F and k,
and the choice of the arbitrary time unit implicit in ∂t.
The third term, F(1-u), is the replenishment term. Since the reaction
uses up U and generates V, all of the chemical U will eventually get used
up unless there is a way to replenish it. The replenishment term says that
u will be increased at a rate proportional to the difference between its
current level and 1. As a result, even if the other two terms had no effect,
1 would be the maximum value for u. The constant F is the feed rate and
represents the rate of replenishment. In the systems this equation is mod-
eling, the area where the reaction occurs is physically adjacent to a large
2.2 Gray-Scott model 13
supply of U and separated by something that limits its flow, such as a
semi-permeable membrane; replenishment takes place via diffusion across
the membrane, at a rate proportional to the concentration difference 4[U ]
across the membrane. The value 1 represents the concentration of U in
this supply area, and F corresponds to the permeability of the membrane.
The only significant difference in the v equation is in its third term. The
third term in the v equation is the dimishment term. Without the di-
minishment term, the concentration of V could increase without limit.
In practice, V could be allowed to accumulate for a long time without
interfering with further production of more V, but it naturally diffuses
out of the system through the same (or a similar) process as that which
introduces the new supply of U. The diminishment term is proportional
to the concentration of V that is currently present, and also to the sum
of two constants F and k. F, as above, represents the permeability of
the membrane to U, and k represents the difference between this rate
and that for V. Notice that there is nothing in the equations that states
whether the system exists in a two-dimensional space (like a Petri dish)
or in three dimensions, or even some other number of dimensions. In fact,
any number of dimensions is possible, and the resulting behavior is fairly
similar. The only significant difference is that in higher dimensions, there
are more directions for diffusion to happen in and the first term of the
equation becomes relatively stronger. It is for this reason that phenomena
depending on diffusion for their action (such as gradient-sustained stable
”spots”) occur at higher k values in the 2-D system as compared to the
1-D system, and at yet higher values for the 3-D system
2.3 Euler forward method 14
2.3 Euler forward method
In mathematics and computational science, the Euler method, named
after Leonhard Euler, is a first-order numerical procedure for solving or-
dinary differential equations (ODEs) with a given initial value. It is the
most basic kind of explicit method for numerical integration of ordinary
differential equations.
A method for solving ordinary differential equations using the Euler
formula
From Calculus we know that calculating an integral is equivalent to com-
puting the area under the curve given by a(s) = f(s, y(s)) over the interval
[tk, tk + h] by the (left) rectangular rule
Figure 2.3 Calculate area by left rectangular rule
∫ tk+1
tk
f(s, y(s))ds = hf(tk, y(tk)),
which after substitution and replacing exact values with approximate ones
2.4 Finite difference method 15
(yk ≈ y(tk), yk+1 ≈ y(tk+1)) results in
yk+1 = yk + hf(tk, yk) k = 0, ..., N − 1.
Naturally, y0 is given by the initial condition at t0 in problem . The
formula yk+1 = yk +hf(tk, yk) is called Forward Euler Method or Explicit
Euler Method. Note that the method increments a solution through an
interval while using derivative information from only the beginning of the
interval.
2.4 Finite difference method
2.4.1 Difference methods
We assume throughout our discussion that our mathematical problem
is well posed, that is ,if its solution exists then it is unique and depends
continuously on the given data.
In the finite difference method, we superimpose on the region R of
interest a network or a mash by lines, as follows:
(1) one-dimensional case:
xm = a + mh, m = 0, 1, 2, .....
where h is the mash size in the x-direction.
(2)two-dimensional case:
xl = a + lh1, l = 0, 1, 2, ......
ym = b + mh2, m = 0, 1, 2, ......
2.4 Finite difference method 16
where h1, h2 are the mash sizes in x and y directions respectively. If we
are considering an initial value problem, then we also have the lines
tn = nk, n = 0, 1, 2, .....
where k is the step length in the t-direction. The points of intersection of
the network are called nodes. The network and nodes for boundary value
problem are shown in (see Fig. 2.4) The partial derivatives in the differ-
R
Y
X
0h
k
Figure 2.4 The region R and nodes
ential equation are replaced by suitable difference quotients, converting
the differential equation to a difference equation at each nodal point. We
may call this procedure as the discretization of the differential equation.
The given data is used to modify the difference equation at the nodes near
or on the boundary. The solution of this system of equations gives the
numerical solution of the given initial/boundary value problem.
2.4 Finite difference method 17
2.4.2 Parabolic Partial Differential Equations Two Space Di-
mensions
Parabolic partial differential equations arise in various branches of sci-
ence and engineering,such as fluid dynamics,heat flow, diffusion, elastic
vibration etc. We assume that a steady state solution does not exist, and
one of the independent variables t has the role of time. We also assume
that unique solution exists, the solution being uniquely determined by the
differential equations together with the initial and boundary conditions.
Two Space Dimensions
we can readily extend the one dimensional difference schemes to higher
space dimension especially when the region is rectangular. The tow di-
mensional heat flow equation in the unit square R = [0 ≤ x, y ≤ 1]× [0, T ]
is given by
∂u
∂t=
∂2u
∂x2+
∂2u
∂y2
subject to the initial condition
u(x, y, 0) = f(x, y)
and the boundary conditions
u(0, y, t) = g1(y, t) u(1, y, t) = g2(y, t)
u(x, 0, t) = h1(x, t) u(x, 1, t) = h2(x, t) t > 0
We place a uniform mesh of spacing h on the square region 0 ≤ x, y ≤ 1
with Mh=1. Let k be the step size in the time direction such that t-
nk,n=0,1,.....N where Nk=T. The nodal points are defined by
xl = lh, l = 0, 1, 2, ..., M
2.4 Finite difference method 18
ym = mh, m = 0, 1, 2, ..M
tn = nk, n = 0, 1, 2, ..., N
The solution value u(x, y, t) at the nodal point (l,m,n) is denoted by Unl,m
.Then may be written as
un+1l,m = un
l,m + λ(δ2x + δ2
y)[θun+1l,m + (1− θ)un
l,m]
where unl,m is an approximate value of Un
l,m .For example, the value θ = 0
gives the difference scheme
un+1l,m = un
l,m + λ(δ2x + δ2
y)unl,m
which has order off accuracy (k + h2).
Using the Von Neumann method of stability analysis, We substitute
unl,m = Aξneiθ1lheiθ2mh
in the explicit difference scheme un+1l,m = un
l,m + λ(δ2x + δ2
y)unl,m.The
propagating factor is given by
ξ = 1− 4λ(sin2 φ1 + sin2 φ2)
where φ1 = θ1h/2 and φ2 = θ2h/2.
For stability, we require that |ξ| ≤ 1 and hence
−1 ≤ 1− 4λ(sin2 φ1 + sin2 φ2) ≤ 1
since 0 ≤ sin2 φ1, sin2 φ2 ≤ 1 the stability condition is obtained as
0 < λ ≤ 1/4
Again for θ = 1/2,we write the difference scheme above
un+1l,m = un
l,m + λ(δ2x + δ2
y)[θun+1l,m + (1− θ)un
l,m equation as
un+1l,m = un
l,m +λ
2(δ2
x + δ2y)(u
n+1l,m + un
l,m)
2.4 Finite difference method 19
which is of order k2 + h2 .using the Von Neumann method,we obtain
the propagating factor as
ξ =1− 2λ(sin2 φ1 + sin2 φ2)
1 + 2λ(sin2 φ1 + sin2 φ2)
where φ1 = θ1h/2 and φ2 = θ2h/2. Since 0 ≤ sin2 φ1, sin2 φ2 ≤ 1 and
λ > 0,the condition |ξ| ≤ 1 is always satisfied. Hence the method above
eqn un+1l,m = un
l,m + λ2(δ2
x + δ2y)(u
n+1l,m + un
l,m) is unconditionally stable.
M
1 2 m-1
m
m+1
2M-2
M-1
y
h
h
A:(j-1)(M-1)+m
B:(j-1)(M-1)+m+1
C:j(M-1)+m
D:(j-1)(M-1)+m-1
E:(J-2)(m-1)+m
j A B
C
D
E
Figure 2.5 Application of the implicit method on the level n+1
On each time level, a system of linear algebraic equations is to be
solved.The coefficient matrix of this system is a band matrix whose total
band with is 2M-1 as shown in 2.5 We number the unknowns in the
interior stating from left to right in the x-direction and from bottom to
top in the y-direction. When we apply above eqn un+1l,m = un
l,m + λ2(δ2
x +
δ2y)(u
n+1l,m + un
l,m) at A, the five point A,B,C,D,E (and hence the unknowns
at these nodes ) enter the scheme. The points E and C are M-1 unite
2.4 Finite difference method 20
away from A while B and D are one unit away from A. Hence the total
band width of this system of equations is 2M-1. If h is small, then the
band width is very large and the solution of this system of equations takes
a lot of computer time . To avoid this difficulty we use the Alternating
Direction implicit methods. we can get idea about this differential method
by a example. now we try to find the solution of the two dimension
equation
∂u
∂t=
∂2u
∂x2+
∂2u
∂y2
subject to the initial condition
u(x, y, 0) = sin πx sin πy, 0 ≤ x, y ≤ 1
and the boundary conditions
u = 0, ontheboundaries, t ≥ 1
using the explicit method
un+1l,m = un
l,m + λ(unl−1,m + un
l+1,m + unl,m−1 + un
l,m+1 − 4unl,m)
with h = 1/3 and λ = 1/8. Integrate upto two time levels. Compare the
results with the exact solution
u(x, y, t) = e−π2t sin πx sin πy
The nodal points are given in Figure 2.6 For λ = 1/8,we have
un+1l,m =
1
2un
l,m +1
8(un
l−1,m + unl+1,m + un
l,m−1 + unl,m+1)
the initial and boundary conditions become
u0l,m = sin(πl/3)sin(πm/3), l, m = 0, 1, 2, 3
2.5 Periodic Boundary Conditions 21
u0
0,0U
0
3,0
u2
0,0
u2
3,0
u2
3,3
u2
0,3
u0
3,3
Zeroth level
First level
Second level
Figure 2.6 Representation of nodal points in Example
unl,0 = un
0,m = un3,m = un
l,3 = 0, l,m = 0, 1, 2, 3, andn = 0, 1, 2, ...
We get for n=0
u1l,m =
1
2u0
l,m +1
8(u0
l−1,m + u0l+1,m + u0
l,m−1 + u0l,m+1)
so now we can calculate every unl,m for l,m = 0, 1, 2, ...
2.5 Periodic Boundary Conditions
In mathematical models and computer simulations, periodic bound-
ary conditions (PBC) are a set of boundary conditions that are often
used to simulate a large system by modeling a small part that is far from
its edge. Periodic boundary conditions resemble the topologies of some
video games; a unit cell or simulation box of geometry suitable for perfect
three-dimensional tiling is defined, and when an object passes through
2.5 Periodic Boundary Conditions 22
one face of the unit cell, it reappears on the opposite face with the same
velocity. The simulation is of an infinite perfect tiling of the system. In
topological terms, the space can be thought of as being mapped onto a
four-dimensional torus. The tiled copies of the unit cell are called im-
ages, of which there are infinitely many. During the simulation, only
the properties of the unit cell need be recorded and propagated. The
minimum-image convention is a common form of PBC particle bookkeep-
ing in which each individual particle in the simulation interacts with the
closest image of the remaining particles in the system. An example occurs
in molecular dynamics, where PBC are usually applied to simulate bulk
gasses, liquids, crystals or mixtures. A common application uses PBCs to
simulate solvated macromolecules in a bath of explicit solvent.
Practical implementation:
To implement periodic boundary conditions in practice, at least two steps
are needed. he first is to make an object which leaves the simulation cell
on one side enter back on the other. This is we used operation,
If (periodicx) then
if (x < 0) x=x+width
if (x >= width) x=x-width
endif
Chapter 3
Method Of Solution
According to Gray-Scott model there two equation as follows
∂U
∂t= Du 52 U − UV 2 + F (1− U)
∂V
∂t= Dv 52 V + UV 2 − (F + K)V
where U,V are input reactants and P is product of this reaction.
U + 2V −→ 3V
V −→ P
3.1 Problem formulation
we will assume that Ω ≡ (0, L)× (0, L) is an open square representing the
square reactor where the chemical reaction takes place, ∂Ω is its boundary
and is its outer normal. Then initial-boundary value problem for the
Gray-Scott model then we solve is a system of two partial differential
equations with initial condition and periodic boundary conditions there
23
3.2 Numerical scheme 24
initial condition are
U(x, y, 0) =
0.5 if 13≤ x < 2
3, 5
7≤ x < 6
7and 1
3≤ y < 2
3, 3
5≤ y < 4
5
1 elsewhere,
V (x, y, 0) =
0.25 if 13≤ x < 2
3, 5
7≤ x < 6
7and 1
3≤ y < 2
3, 3
5≤ y < 4
5
0 elsewhere,
this model boundary condition is periodic boundary conditions
u(0, y, t) = u(1, y, t) u(x, 0, t) = u(x, 1, t)
v(0, y, t) = v(1, y, t) v(x, 0, t) = v(x, 1, t)
3.2 Numerical scheme
In this two-dimensional numerical experiments the following choices
for the model parameters are made: Du = 2e−5 Dv = 1e−5 and f =
0.02 k = 0.059 but through the program can be changed k and f
value. we can get range on the domain [0, 1] [0, 1] where L =1 in above
defined. We choose the same time step, and tolerances as in 1D on a
spatial mesh of 100 100 mesh points with Dirichlet boundary conditions
and diffusion coefficients
∇u = (∂
∂xi +
∂
∂yj)u =
∂u
∂xi +
∂u
∂yj
∇2u = ∇(∇u) = (∂
∂xi +
∂
∂yj)(
∂u
∂xi +
∂u
∂yj)
=∂2u
∂x2+
∂2u
∂y2
In practice, we represent the concentration functions as two-dimensional
arrays of discrete samples. To evaluate U and V we approximate the space
3.2 Numerical scheme 25
derivative ∇2U by a finite difference. The second finite difference in the
x direction is
∂2u
∂x2≈ ui+1,j + ui−1,j − 2ui,j
h2
The second finite difference in the y direction is
∂2v
∂y2≈ ui,j+1 + ui,j−1 − 2ui,j
h2
where the i and j are array subscripts, and h is the distance between
adjacent samples.therefore we can get h value as h = LN
= 1100
= 0.01
where L is domain length, N is mash size. Taking the corresponding
second difference in the y direction, and summing, gives