System of quasilinear equations of reaction diffusion

IJRET: International Journal of Research in Engineering and Technology

eISSN: 2319-1163 | pISSN: 2321-7308

SYSTEM OF QUASILINEAR EQUATIONS OF REACTION-DIFFUSION TASKS OF KOLMOGOROV-FISHER TYPE BIOLOGICAL POPULATION TASK IN TWO-DIMENSIONAL CASE D.K.Muhamediyeva1 1

Junior researcher of the Centre for development of software products and hardware-software complexes, Tashkent, Uzbekistan

Abstract In this paper considered parabolic system of two quasilinear reaction-diffusion equations of Kolmogorov-Fisher type of biological population task relation for the two-dimensional case and localization of the wave solutions of the reaction - diffusion systems with double nonlinearity. Cross-diffusion means that spatial move of one object, described by one of the variables is due to the diffusion of another object, described by another variable.Considered spatial analogue of the Volterra-Lotka competition system with nonlinear power dependence of the diffusion coefficient from the population density.

Keywords: Population, system, differential equation, cross-diffusion, double nonlinearity, wave solution.

--------------------------------------------------------------------***-----------------------------------------------------------------1. INTRODUCTION Let’s consider following system of two equations in partial derivatives in two-dimensional case:

u1  2 u1  2 u1 u    Q1 (u1 , u 2 ) 2  f (u1 , u 2 )  D11  D  h11 12 2 2 t x1  x1 x1 x 2

 u     h12  Q1 (u1 , u 2 ) 2 x 2  x 2 

 , 

u 2  2u2  2u2 u  u       Q2 (u1 , u 2 ) 1   h22  Q2 (u1 , u 2 ) 1 .  g (u1 , u 2 )  D21  D  h21 22 2 2 t x1  x1  x 2  x 2  x1 x 2 At h11  h12  h21  h22  0 the mathematical model (1) is a system type reaction-diffusion with diffusion coefficient D11  0, D12  0, D21  0, D22  0 (at least one

Dij  0 ). In the case when at least one of the coefficients (mark can be any), the system (1) is a cross-diffusion. To the linear cross-diffusion corresponds Qij (u , v )  const for

i, j  1,2 i=1,2; to the linear cross-diffusionQij (u , v)  const at least one of i and j.

Cross-diffusion means that spatial move one object, described one of the variables is due to the diffusion of another object, described by another variable.At the population level simplest example is a parasite (the first object, located within the "host" (the second object) moves through the diffusion of the owner). The term "selfdiffusion" (diffuse, direct diffusion, ordinary diffusion) moves individuals at the expense of the diffusion flow from areas of high concentration, particularly in the area of low concentration. The term "cross-diffusion" means moving/thread one species/ substances due to the presence

(1)

of the gradient other individuals/ substances. В The value of a cross-diffusion coefficient can be positive, negative or equal to zero. The positive coefficient of cross-diffuse indicates that the movement of individuals takes place in the direction of low concentrations of other species occurs in the direction of the high concentration of other types of individuals/ substances. In the nature system with crossdiffusion quite common and play a significant role especially in biophysical and biomedical systems.

Equation (1) is a generalization of the simple diffusion model for the logistic model of population growth [1-16] of Malthus type ( f1 (u1 , u2 )  u1 , f1 (u1 , u2 )  u2 ,

f 2 (u1 , u2 )  u1 , f 2 (u1 , u2 )  u2 ), Ferhulst type ( f1 (u1 , u2 )  u1 (1  u2 ) , f1 (u1 , u2 )  u2 (1  u1 ) , f 2 (u1 , u2 )  u1 (1  u2 ) , f 2 (u1 , u2 )  u2 (1  u1 ) ), and Allee type

( 

f1 (u1 , u2 )  u2 (1  u1 2 ) ,



f1 (u1 , u2 )  u1 (1  u2 1 ) , 

f 2 (u1 , u 2 )  u1 (1  u 2 1 ) ,



f 2 (u1 , u 2 )  u 2 (1  u1 2 ) , 1  1,  2  1) for the case

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of

nonlinear diffusion. In case, when 1  1,  2  1 , it can be regarded as the equation of nonlinear filtration, thermal conductivity at simultaneous influence of the source and the absorption capacity of which

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double

is equalrespectively to

 u1 ,u2 1 , u 2 ,u1 2 . 

2. LOCALIZATION OF WAVE SOLUTIONS OF REACTION - DIFFUSION SYSTEMS WITH DOUBLE NONLINEARITY

Let’s consider the spatial analogue of Volterra-Lotka competition system with nonlinear power dependence of the diffusion coefficient from population density. In the case of the simplest Volterra’s competitive interactions between populations can be constructed numerically, and in some cases analytically heterogeneous in space solutions[19].

Let’s consider in the domain Q={(t,x): 0< t < ,xR2} parabolic system of two quasilinear equations of reactiondiffusion of Kolmogorov-Fisher typebiological population task

p2 p2  u u1    u1  u u   m1 1 u1 m1 1 u1  1  D11u 2  D12 u 2  l1 (t ) 1  l 2 (t ) 1  k1 (t )u1 1  u 21 ,     x1 x1 x1 x 2 x 2 x 2 x1 x 2  t     (2)  p2 p2  u 2 u u 2    u u 2  u u    D21u1m2 1 2  D2 u1m2 1 2  l1 (t ) 2  l 2 (t ) 2  k 2 (t )u 2 1  u1 2 ,  x1  x1 x1  x 2  x 2 x 2  x1 x 2  t    







u1 t  0  u10 ( x) , u2

t 0

 u20 ( x) ,

which describes the process of biological populations in nonlinear two-component environment, it’s diffusion m 1

coefficient is equal D11u 2 1 m2 1 21 1

D u

u 2 x1

u1 x1

p 2

,

m2 1 22 1

D u

p 2 m 1

, D12 u 2 1

u 2 x 2

u1 x 2

,

andconvective

m1 , m2 , p, 1 ,  2 u1  u1 (t , x1 , x2 )  0 ,

transfer with speed l i (t ) , where positive

real

numbers,

u (t , x)  f ( ),   ct  x

p 2

p 2



u (t , x)  u t , x1 , x 2 ,

x

x1 2  x2 2 ,

, Where constant с- is a wave speed Let's build self-similar system of equations (2) - more simple for research system of equations.

u2  u2 (t , x1 , x2 )  0 -search solutions.

Self-similar system of equations will construct by the method of nonlinear splitting [15].

The Cauchy problem and boundary problems for the system (1) in one-dimensional and multidimensional cases investigated by many authors [15-21]

Replacement in (2)

The aim of this work is the investigation of qualitative properties of solutions of the task (2) on the basis of selfsimilar analysis and numerical solutions by using the methods of modern computer technologies, research of methods of linearization to the convergence of iterative process with subsequent visualization. Obtained the estimates of solutions and emerging in this case free boundary, that gives the chance to choose the appropriate initial approximation [15] for each value of the numeric parameters.

It is known that the nonlinear wave equations have solutions in the form of diffusion waves. Under the wave is understood self-similar solution of the equation (2)

t

u1 (t , x1 , x 2 )  e



 k1 ( ) d 0

v1 ( (t ),1 , 2 ) ,

t

1  x1   l1 ( )d

,

0 t

u 2 (t , x1 , x 2 )  e



 k 2 ( ) d 0

v 2 ( (t ),1 , 2 ) ,

t

 2  x 2   l 2 ( )d

,

0

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Lead (2) to the form: p2 p 2  v v1  v1      m1 1 v1 m1 1 v1 1   D11v 2  D12 v 2  k1 (t )e ( 2 p ) k1  ( 1  m1 1) k 2 t v1v 21 , 1 1   2   2  2    1     (3)  p 2 p2      v 2 v v 2  v v 2       D21v1m2 1 2  D22 v1m2 1 2  k 2 (t )e (  2  m2 1) k1  ( 2 p ) k 2 t v1 2 v 2 ,                    1  1 1  2  2 2  

v1 If k1 ( p  (m1  1))  k2 ( p  (m2 choosing

t 0

 1)) ,

 v10 (1 , 2 ) , v2

then

by

t 0

 v20 (1 , 2 ) .

 (t ) 

e[( m1 1) k 2  ( p  2 ) k1 ]t e[( m2 1) k1  ( p  2 ) k 2 ]t  (m1  1)k2  ( p  2)k1 (m2  1)k1  ( p  2)k2

,we get the following system of equations: p2 p2  v v1  v1      m1 1 v1 m1 1 v1 1   D11v 2  D12 v 2  a1 (t ) b1 v1v 21 , 1 1   2   2  2    1     (4)  p 2 p2     v 2  v  v  v  v    2  2   D v m2 1 2  D21v1m2 1 2   a 2 t  b2 v1 2 v 2 ,  22 1     1 1 1  2  2  2      

where

a1  k1 ( p  2)k1  (m1 1)k2  1 , b1  b

b a2  k2 (m2 1)k1  ( p  2)k2  2 , b2 

(2  p )k1  ( 1  m1  1)k 2 , ( p  2)k1  (m1  1)k 2

(  2  m2  1)k1  (2  p )k 2 . (m2  1)k1  ( p  2)k 2

If bi  0 , and ai (t )  const , i  1,2 , then the system has the form: p 2 p2  v v1  v1      m1 1 v1 m1 1 v1 1   D11v 2  D12 v 2  a1 v1v 21 , 1 1   2   2  2    1      p2 p 2  v 2 v 2  v 2      m2 1 v 2 m2 1 v 2  D v  D v  a 2 v1 2 v 2 .  21 1 22 1     1 1 1  2  2  2      

 d 1 1  d   a1 1 2 ,    d 2   a   2 , 2 1 2   d

The Cauchy problem for system (4) in the case when b1  b2  0 studied in [16,19] and prove the existence of wave global solutions and blow-up solutions.

Below we will describe one way of obtaining self-similar system for the system of equations (4). It consists in the following. Firstly we find first the solution of a system of ordinary differential equations

In the form

 1 ( )  c1 (  T0 )  , 2 ( )  c2 (  T0 )  1

2

, T0  0

, Where

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с1  1 ,  1 

1

2

,

с2  1 ,  2 

1

1

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v1 (t ,1 , 2 )  v1 (t ) w1 ( ,1 , 2 ), .

And then the solution of system (3)-(4) is sought in the form

(5)

v 2 (t ,1 , 2 )  v 2 (t ) w2 ( ,1 , 2 ), and

  (t )

chosen as

1  1[  1 ( p  2)  2 ( m1 1)] , if 1  [ 1 ( p  2)   2 (m1  1)]  0, 1  [ ( p  2)   (m  1)] (T   ) 1 2 1     1 ( )   v1( p 2) (t )v2( m1 1) (t )dt  ln(T   ), if 1  [ 1 ( p  2)   2 ( m1  1)]  0, 0 (T   ), if p  2 и m1  1,  

if

 1 ( p  2)   2 (m1  1)   2 ( p  2)   1 (m2  1) .

Then for

wi ( , x), i  1,2

we get the following

system of equations p 2 p 2  w w1  w1      m1 1 w1 m1 1 w1 1   D11 w2  D12 w2   1 ( w1 w21  w1 ) 1  1 1   2   2  2         , (6) p 2 p2     w2  w  w  w  w    2  2   D w m2 1 2  D21 w1m2 1 2    2 ( w2 w1 2  w2 )  22 1     1 1 1  2  2  2      

Where

1   (1  [ ( p  2)   (m  1)]) , 1   1 2 1  с  (1[ 1 ( p2) 2 ( m11)] ) ,  11

1   (1  [ ( p  2)   (m  1)]) , 2   2 1 2  с  (1[ 2 ( p  2) 1 ( m2 1)]) ,  2 1 Representation of system (2) in the form (5) suggests that, when

if 1  [ 1 ( p  2)   2 (m1  1)  0, (7)

if 1  [ 1 ( p  2)   2 (m1  1)  0,

if 1  [ 2 ( p  2)   1 (m2  1)]  0, if 1  [ 2 ( p  2)   1 (m2  1)]  0.   ,i  0

and

p2 p 2  w      w  w  w w1   m  1 m  1 1 1  1 D w 1 D w 1  1  11 2 12 2 1 1   2   2  2    1      p 2 p 2  w2 w2    w2    m2 1 w2 m2 1 w2  D21 w1  D22 w1  1  1 1   2   2  2       

(8)

Therefore, the solution of system (1) with condition (5) tends to the solution of the system (8).

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IJRET: International Journal of Research in Engineering and Technology If 1  [ 1 ( p  2)   2 (m1

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 1)  0 , wave solution of system (6) has the form

1 2   2 2

wi ( (t ),1 , 2 )  yi ( ) ,   c   ,  

i  1,2 ,

where c- wave velocity, and taking into account that the equation for wi ( ,1 , 2 ) without the younger members always has a self-similar solution if

1  [ 1 ( p  2)   2 (m1  1)  0 we get the system p2 d dy1 dy m1 1 dy1 )  c 1   1 ( y1  y1 y 21 )  0,  ( y2 d d d  d  p2 dy2 dy2 d m2 1 dy2 ( y )  c   2 ( y 2  y 2 y1 2 )  0.

1  d d d d 

After integration (8) we get the system of nonlinear differential equations of the first order

 m 1 dy p 2 dy 1 1  cy1  0,  y2 1 d  d    p 2 (9)  m2 1 dy2 dy2  cy2  0.  y1 d d  

t

u 2 (t , x1 , x 2 )  e

0

v 2 ( (t ),1 , 2 )

We have t

u1 (t , x1 , x 2 )  Ae



 k1 ( ) d 0

(c (t )   ) 1 ,

t

u 2 (t , x1 , x 2 )  Be

The system (9) has an approximate solution in the form

y1  A(a   ) 1 , y2  B(a   ) 2 ,



 k 2 ( ) d



 k 2 ( ) d 0

(c (t )   ) 2 , c  0 .

Due to the fact that t

[b (t )   li ( )d  xi ]  0 ,

Where

0

1 

( p  1)( p  (m1  1)) , ( p  2) 2  (m1  1)(m2  1)

If t

2 

xi  [b (t )   li ( )d  xi ]  0 , t  0 ,

( p  1)( p  (m2  1)) . ( p  2) 2  (m1  1)(m2  1)

and the coefficients A and B are determined from the solution of a system of nonlinear algebraic equations

( 1 ) p 1 A p 1B m1 1  c ,

0

Then

u1 (t , x)  0 , u2 (t , x)  0 , t

( 2 ) p 1 Am2 1B p 1  c .

Then by taking into account expressions

xi  [b (t )   li ( )d  xi ]  0 , t  0, i  1,2 .

0

Therefore, localization condition of solutions of the system (2) are conditions

t

u1 (t , x1 , x 2 )  e



 k1 ( ) d 0

v1 ( (t ),1 , 2 ) ,

å

 l ( y)dy  0 ,  (t )   for t  0, i

i  1,2 . (10)

0

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Condition (10) is the prerequisite for the emergence of a new effect - localizationof wave solutions of (2). If condition (10) unfulfilled, then there is the phenomenon of the finite velocity of propagation of a perturbation, i.e.

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As is known, the main problem for the numerical solving of nonlinear tasks, is suitable choosing initial approximation and the linearization method of the system (2). Let’s consider the function:



t

ui (t , x)  0 at x  b(t ) ,  (t )   e 



( m1  p 3) k1 ( y ) dy 0

v10 (t , x1 , x2 )   1 (t )  a   1 , 

d

0

v20 (t , x1 , x2 )   2 (t )  a   2 , 

, and the front of arbitrarily far away, with increasing time, since  (t )   at t   .

The study of qualitative properties of the system (2) allowed to perform numerical experiment based on the values included in the system of numeric parameters. For this purpose, as the initial approximation was used asymptotic solutions. To numerical solving the task for the linearization of system (2) has been used linearization methods of Newton and Picard. To build self-similar system of equationsof biological populations used the method of nonlinear splitting [16, 19].

3. NUMERICAL EXPERIMENT Let’s construct a uniform grid to the numericalsolvingof the task (2)

 h  xi  ih, h  0, i  0,1,..., n, hn  l, and the temporary grit

 h  t j  jh1 , 1

h1  0, j  0,1,..., n, m  T 

Where  1 (t )  e above functions,

 1 (t )

kt

and

 2 (t )  e kt 2 (t )

defined

Notation (a)  means (a)   max(0, a) . These functions have the property of finite speed of propagation of perturbations[16,19]. Therefore, in the numerical solving of the task (1) - (2) at 1   1 as an initial approximation proposed the function vi 0 (t , x1 , x 2 ) , i  1,2 .

Created on input language MathCad program allows you to visually traced the evolution of the process for different values of the parameters and data.

Numerical calculations show that in the case of arbitrary values   0,   0 qualitative properties of solutions do not change. Below listed results of numerical experiments for different values of parameters (Fig.1)

.

Replace task (2) by implicit difference scheme and receive





differential task with the error O h  h1 .

2

Parameter values Results of numerical experiment

m1  0.3 , m2  0.7, p  2.1 1  5, k1  2  2  7, k 2  3 eps  103

time1( FRAM E  1)  time2( FRAM E  4)  time1( FRAM E  8)  time2( FRAM E  10)

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m1  2.2 , m2  2.2, p  2.5 1  1, k1  7  2  1, k 2  2 eps  103

m1  0.2 , m2  0.2, p  2.8 1  0.5, k1  2  2  0.7, k 2  3 time1( FRAM E  1)  time2( FRAM E  4)  time1( FRAM E  10)  time2( FRAM E  17) eps  103

Fig.1 Results of numerical experiment

4. CONCLUSIONS It can be expected that further theoretical and experimental [10] studies of excitable systems with cross-diffusion will make a time1study ( FRAME  20)  time2 (of FRAME significant contribution to the of phenomena self-  20) [11] organization in all of nonlinear systems from the micro and astrophysical systems to public social systems.

[12]

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«ApproximateSelfsimilarApproachtuSolve Quasilinear Parabolic Equation» Experimentation, Modeling and Computation in Flow Turbulence and Combustion vol. 2. p. 19- 26. Арипов М. Метод эталонных уравнений для решения нелинейных краевых задач Ташкент, Фан, 1988, 137 p. Белотелов Н.В., Лобанов A.И.Популяционные модели с нелинейной диффузией. // Математическое моделирование. –М.; 1997, №12, pp. 43-56. В. Вольтерра. Математическая теория борьбы за существование -М.: Наука, 1976, 288 p.

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Гаузе Г.Ф. О процессах уничтожения одного вида другим в популяциях инфузорий // Зоологический журнал,1934, т.13, №1.

Aripov M., Muhammadiev J. Asymptotic behaviour of automodel solutions for one system of quasilinear equations of parabolic type. Buletin StiintificUniversitatea din Pitesti, Seria Matematica si Informatica. N 3. 1999. pg. 19-40 Aripov M.M. Muhamediyeva D.K. To the numerical modeling of self-similar solutions of reactiondiffusion system of the one task of biological population of Kolmogorov-Fisher type. International Journal of Engineering and Technology. Vol-02, Iss11, Nov-2013. India. 2013. Арипов М.М. Мухамедиева Д.К. Подходы к решению одной задачи биологической популяции. Вопросы вычислительной и прикладной математики. -Ташкент. 2013. Вып.129.-pp.22-31.

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