ISSN 0130–9420. Ìàò. ìåòîäè òà ô³ç.-ìåõ. ïîëÿ. 2008. – 51, ¹ 3. – Ñ. 193-201. 193

UDK 517.519: 517.96 B. Y. Datsko, V. V. Gafiychuk MATHEMATICAL MODELING OF FRACTIONAL REACTION-DIFFUSION SYSTEMS WITH DIFFERENT ORDER TIME DERIVATIVES

The linear stability analysis is studied for a two-component fractional reaction-diffusion system with different derivative indices. Two different cases are considered when an activator index is larger than an inhibitor one and when an inhibitor variable index is larger than an activator one. General analysis is con-firmed by computer simulation of the system with cubic nonlinearity. It is shown that the systems with a higher activator variable index lead to a much more complicated spatio-temporal dynamics.

Introduction. Since implementation of fractional derivatives to the reac-

tion-diffusion systems (RDS), the investigation of new phenomena in these systems has been a popular field of research. The matter is that most of stan-dard reaction-diffusion systems, describing phenomena in complex heteroge-neous living systems, are based on purely qualitative features. In this way, many well-known mathematical models, such as Oregonator, Brusselator, Gierer – Meinhard, Gray – Scott models (see, for example, a review [13]), were written phenomenologically to explain specific properties in complex systems and revolutionized our understanding of pattern formation phenome-na. Recent experiments show that the models in real systems are probably better described by fractional equations [14, 15].

Among the applications of time fractional RDS one can find the descrip-tion of transport of fission cells during a tumor growth, as well as transport of a substance across a thin membranes [12]. Heterogeneous porous systems are often more effectively described by a medium of reaction diffusion type with fractional derivatives [20]. A charge carrier transport in disordered semicon-ductors, due to non-Gaussian processes and multiple trapping, can be much better described by fractional derivatives [19].

It should be noted that at the present time an experimental media for investigating phenomena in reaction-diffusion systems with derivatives of fractional order can be created synthetically, with the help of circuits and modern solid state technology [2]. In this case we can design the layered solid state distributive media, the corresponding layers of which have to be endo-wed with the properties inherent to the fractional order controllers [16]. As a result, each layer can be described by fractional differential equations and can even have its own fractional index.

Mathematical model. The starting point of our consideration is the frac-tional reaction-diffusion system [1, 3–11] with indices of different order

21

21 11 1 221

( , ) ( , )( , , )

n x t n x tW n n A

xt

α

α

∂ ∂τ = +

∂∂ , (1)

22

22 22 1 222

( , ) ( , )( , , )

n x t n x tL Q n n A

xt

α

α

∂ ∂τ = +

∂∂, (2)

subject to Neumann

0

0, 1,2i i

x x x

n ni

x x= =

∂ ∂= = =

∂ ∂ , (3)

boundary conditions and with a certain initial conditions. Here 1( , )n x t ,

2 ( , )n x t are activator and inhibitor variables, 0 xx≤ ≤ , 1 2, , , Lτ τ are the

characteristic times and lengths of the system, correspondingly, A is an external parameter.

194

Time derivatives ( , )i

i

i

n x t

t

α

α

∂

∂ on the left hand side of equations (1), (2)

instead of standard ones are the Caputo fractional derivatives in time of the order 0 2< α < and are represented as [17, 18]

( )

10

( )1( ) :( ) ( )

t mi

i m

nn t d

mt t

α

α α + −

τ∂ = τΓ − α∂ − τ∫ ,

where 1 , 1,2m m m− < α < = . It should be noted that equations (1), (2) at 1α = correspond to standard integer derivatives.

Linear stability analysis. We consider an RDS with two variables: one of them is a variable with positive feedback and the second one is a variable with a negative one. Namely, these systems possess a variety of nonlinear phenomena investigated in the last decades [13]. Positive and negative feed-backs require a special form of nonlinearities. For example, the source term which corresponds to an activator variable must be nonmonotonous and the second one can be monotonous. For our consideration, it is good to analyze nullclines of the system (1), (2): 1 2( , , ) 0W W n n A= = , 1 2( , , ) 0Q Q n n A= = . Si-

multaneous solution of the system 0W Q= = leads to homogeneous distribu-

tion of 1n and 2n . The eigenvalues 21,2

1 tr tr 4det2

F F Fλ = ± −( ) on the li-

nearized right hand side of the system (1), (2) play an important role in the

system evolution. Here, the matrix 11 12

1 1

21 222 2

1 1

1 1

( )( )

( )

a k aF k

a a k

τ τ=

τ τ, is determined

by 2 211 11( )a k a k= − , 11 1na W′= , 12 2na W′= , 21 1na Q′= , 2 2

22 22( )a k a k L= − ,

22 2na Q′= (all derivatives are taken at homogeneous equilibrium states

0W Q= = ), , 1,2,x

k j jπ= =

.

In standard RDS (see, for example [13]) we have two types of bifurcati-ons: for 0k = at conditions

tr (0) 0, det (0) 0F F> > , (4)

we have a Hopf bifurcation, and for 0k ≠ at

0tr 0, det (0) 0, det ( ) 0F F F k< > < , (5)

we have a Turing one. These two types of bifurcations could be realized only if 11 0a > (positive feedback).

For fractional RDS the situation differs. First of all, the conditions of the Hopf bifurcation are qualitatively different. The instability conditions for

1 2α = α are determined by a new parameter 02 Arg ( )iα = λπ

, which can be

represented as [3–5]

0Re 0 Re 0

Im Imarctan 2 arctan2 Re 2 Reλ > λ<

π λ π λα = −λ λ

. (6)

In the case of the fractional derivative index, the Hopf bifurcation can be not connected with the condition 11 0a > [3, 5] and is determined as 0α > α .

In the case of different indices the conditions of the Hopf bifurcation are softer and the system can be unstable in a wide spectrum of parameters [6].

195

The Turing bifurcation conditions are the same as for the standard sys-tem. At the same time, here we reveal a new type of instability for 0k ≠ . In fact, having critical value of α from (6) for homogenous perturbation 0k = , we can analyze if there is a condition where the Hopf bifurcation for 0k = is not realized but at the same time the conditions of the Hopf bifurcation for

0k ≠ become true. In particular, this situation is considered in [3, 5], and the conditions are represented as:

2 20 0tr (0) 0, 4det (0) tr (0), 4det ( ) tr ( )F F F F k F k< < > . (7)

In this case, the instability conditions are determined by condition (6) for a gi-ven wave number 0k ≠ .

Fractional RDS with arbitrary rational 1 2,α α . With arbitrary rational

1α and 2α (for example 1 2 )α > α by certain substitution, the system can be transformed to the differential equations with fractional derivative index being the «greatest common factor» α of the values 1 2, mα = α α = α simul-

taneously, ,m ∈ . Due to the properties of the Caputo derivatives [17, 18], we obtain the system of m + equations

13

( , )n x tn

t

α

α

∂=

∂,

21

( , )m

n x tn

t

α

+α

∂=

∂,

1

( , ), 3, 1 1, 1i

i

n x tn i m m m

t

α

+α

∂= ∈ − + + −

∂ [ ] [ ] ,

2

2 12 1 22

( , ) ( , )( , , )mn x t n x t

L Q n n At x

α

α

∂ ∂τ = +

∂ ∂,

2

2 21 1 22

( , ) ( , )( , , )mn x t n x t

W n n At x

α+

α

∂ ∂τ = +

∂ ∂ . (8)

The Jacobian, on the right hand side of the system, is

11 12

1 1

21 222 2

0 1 0 0 0 0 00 0 0 0 0 1 00 0 1 0 0 0

1

0 0 0 0 1 0 0 001 1( ) 0 0 0 0 0

0 0 0 0 0 1 00 0 0 0 0 0 1

1 1( ) 0 0 0 0

a k a

a a k

− λ− λ

− λ− λ

− λ=− λ

τ τ

− λ− λ

− λτ τ

,

which is equivalent to the next characteristic equation

122

2

1( ) ( 1) ( ) ( )m ma k+ −−λ + − −λ +τ

111

1

1( 1) ( ) ( ) ( 1) det 0m ma k F+ ++ − −λ + − =τ

. (9)

The solution of such type of equation can be obtained numerically. At a small value of detFε = , it is always possible to find the roots with the value close to zero:

196

1

22 1 2

22

1 ( ) , , 1, ,m

i ai m λ = τ ε + Ο ε α < α ∈

/,

1

21 1 2

11

1 ( ) , , 1, ,i ai λ = τ ε + Ο ε α > α ∈

/, (10)

and to determine where they are greater then zero entailing the instability. This situation takes place when nullclines are practically tangent to each

other. At det 0F ≈ condition det (0) 0F > can be rewritten as 2 2

1 10 0Q W

dn dndn dn= =

> ,

which means that the second nullcline ( 0)Q = has a greater slope then the first one ( 0)W = .

Case 1 22 2 , 0 1, 2, 1mα = α = α < α < = = . The characteristic equation (9) has the form

3 2 0b c dλ + λ + λ + = , (11) where

22 112 1

1 1( ) , ( ) , det ( )b a k c a k d F k= − = − = −τ τ . (12)

As a result, eigenvalues of the equation (11) can be represented by Car-dano formulas: 1 3A B bλ = + − / ,

2 ( ) 2 3( ) 2 3A B i A B bλ = − + + − −/ / / ,

3 ( ) 2 3( ) 2 3A B i A B bλ = − + − − −/ / / , (13)

where 3 2A q= − + ∆/ , 3 2B q= − − ∆/ , 2 3( 2) ( 3)q p∆ = +/ / , 2 3p c b= − / , 33 2 27q d bc b= − +/ / .

Let us analyze eigenvalues of the equation (11). If the value of 0∆ < , than all roots are real. In this case, if at least one of the roots is positive, than the system will be unstable and will lead to oscillations for practically any va-lue of α. We can see that at 3A B b+ > / , the first root leads to instability and the second two can be less than zero. In the opposite situation, the first root is negative and the second root leads to instability. So, as a result, the system can always be unstable either to the first eigenvalue or to one of the remai-ning two. A detailed analysis of the eigenvalues for specific nonlinearities is given in the next section. Here we would like to conclude just the general properties of the system.

If 0∆ > , than ,A B and consequently λ1 are real and the roots 2 3,λ λ

are complex. In this case, if 3A B b+ > / than the system is unstable according

to any value of α . At 3A B b+ < / , the first root is real and negative and we

find that the system can be unstable for a certain value of α , 0α > α :

0( ) ( )2( ) arctan 3

2 ( ) 3 ( ) ( )A k B k

kb k A k B k

−α =π + +/

. (14)

As a result of this instability, we can expect nonlinear oscillations which can be homogeneous for 0k = if 0 0(0) ( )kα < α and nonhomogeneous, 0k ≠ ,

if 0 0( ) (0)kα < α .

It should be noted that for 0k ≠ , the conditions of instability according to this wave number, are stricter than the conditions of emerging homogene-ous oscillations. This means that homogeneous oscillations arise at smaller va-lues of α0 then inhomogeneous ones. So, as the conditions of the Turing ins-

tability do not depend on α0 and can be easily realized in the system, we can

197

expect complicated dynamics of pattern formation when simultaneous conditi-ons of instability become true.

Case 1 22 2 , 0 1, 1, 2mα = α = α < α < = = . In this case, characteristic equation is also represented by the equation (11) where the values b and c interchange their values

11 221 2

1 1( ) , ( )b a k c a k= − = −τ τ . (15)

A similar speculation as the one above shows that the system can be uns-table practically for all values of α .

Here we explore numerically the system dynamics when the parameters of the derivative indices are different and change from small values to the value of two. We employ a two-component reaction-diffusion system (1), (2) for specific source terms.

Computer simulation. As an example, here we consider that the source

term for activator variable is nonlinear, 31 1 2W n n n= − − , and it is linear for

the inhibitor one, 2 1Q n n A= − + β + [13]. The homogeneous solution of vari-

ables 1n and 2n can be obtained from the system of equations 0W Q= = ,

and for the determination of 1n , we have a cubic algebraic equation

31 1( 1) 3 0n n Aβ − + + =/ . (16)

The calculation of the coefficients ija : 211 1 12 211 , 1, a n a a= − = − = β , 22a =

1= − at homogeneous state (16) makes it possible to explicitly investigate the

eigenvalues of the system. As a result, we can see that at 1 2 0τ τ →/ and

0L →/ in a standard RDS, the instability domain is realized at 1 1n < ,

11 1a > . In this case, the simultaneous conditions of the Hopf (4) and the Tu-ring (5) bifurcations are realized.

Case 1 2 1, 0 2α = α < α < . This situation was analyzed in the papers [3, 5,

7–9]. Here we would like to analyze instability conditions for 0k ≠ (7) which lead to a new pattern formation phenomena [3, 5]. In this case the instability conditions (7) are

0 0( ) 2 arctan2

k Tπα > α = − , (17)

where 20 04det ( ) tr ( ) 1T F k F k= −/ . This condition can be satisfied for a cer-

tain optimal value of 0k k= :

1 2

0 12 21 2 2

2 1

1 112k a a

Lτ τ

= − −

/

. (18)

Having obtained (18), we can estimate the marginal value of α0 (17),

where

1 2

12 21 1 22 2

2 111 2 22 1 11 2 22 12 2

2 1

( 4 )

( )

a aT

La a a a

L

− τ τ=

τ + ττ − τ − τ − τ

τ − τ

/

.

As a result, we can obtain the new mechanism [3, 5] of pattern formation, which leads to oscillatory inhomogeneous structures.

Case 1 2 1 2, 0 , 2α > α < α α < . The investigation of the eigenvalues for

0k = , depending on parameter 1n , is presented on Fig. 1a for the case L/

1 21, 1τ τ < / . (The eigenvalues, as a function of 1n , are obtained from the

numerical solution of Eq. (11) for 1 22α = α , 21 21.1, 0.1, 1, 0.1β = τ = τ = = ,

198

2 1L = .) We see that one real root is always less than zero. The others two

roots at 01 1n n> are complex conjugate roots. At 0

1 1n n< these roots be-

come real and positive. As a result of these conditions, instability in the sys-tem takes place practically for any small values of α .

Along with homogeneous oscillations, a condition of the Turing instability becomes true if the ratio of 1L / . As an example, a plot of the eigenvalues

for 1k = (all other parameters are the same) is presented on Fig. 1b, where at

1 1kn n< all of the roots are real and one of them corresponds to stationary

inhomogeneous structures.

-4

-2

0

2

4

-4 -2 0 2 4 n1

Re λIm λ

01n− 0

1n

)a

-4

-2

0

2

4

-4 -2 0 2 4 n1

Re λIm λ

1kn− 1

kn

)b

Fig.1

At 1 1kn n< , we can have inhomogeneous oscillations of the structures if

the conditions of this instability are softer than the conditions of homogeneous oscillations. However, this domain is very narrow in parameters, and practi-cally for all parameters we have 0 0 ( )kα < α . Computer simulation of the evo-lutionary dynamics of the system is presented on Fig. 2 and Fig. 3. On Fig. 2 we present a certain pattern formation scenario for 0.1A = − , 1 1.0α = , 2α =

0.5= , 1 0.1= , 2 1= , 1 0.15τ = , 2 1τ = , 7.5x = and different values of β (Fig. 2a for 2.0β = and Fig. 2b for 1.05β = ). In this situation we can observe

the complex evolution of variable 1n in the form of inhomogeneous «zigzag»

oscillations when the homogeneous solution 1n is close to zero. At that point, by decreasing the value of β , the structures increase amplitude of their «zig-zag» oscillations (Fig. 2b).

Fig. 2

At a small value of 1 2τ τ/ we revealed space temporal oscillations in the

instability domain. The scenario of dynamics for variable 1n for 0.1A = − ,

1 1.4α = , 2 0.7α = , 1 0.1= , 2 1= , 21.1, 1β = τ = , 4x = π and 1 0.2τ = is

presented on Fig. 3a. By increasing 1 2τ τ/ , homogenous oscillations are slightly

modulated by inhomogeneous mode. At a certain value of 1 0.25τ = the Turing instability comes into play, and we get very complicated patterns oscillating in space and time (Fig. 3b). A successive increase of 1 2τ τ/ leads to

more regular spatio-temporal structures (Fig. 3c for 1 0.3τ = , and Fig. 3d for

1 0.35τ = ) and eventually to inhomogeneous stationary structures.

199

Fig. 3

The obtained scenario of pattern formation is typical for the general case of 1 2α > α . We can find that at certain parameters the solutions can have a simple form of homogenous oscillations or stationary inhomogeneous structu-res and can correspond to spatio-temporal structures similar to those presen-ted on Fig. 2, 3. In addition to homogenous oscillations or stationary structure formation, inherent to standard system with integer derivative, the system considered here with indices 1 2α > α possesses more complicated nonlinear

dynamics. We can conclude that for 1 2α >≈ α the possible solutions are even more diverse.

Case 1 2α < α , 1 20 , 2< α α < . We start analyzing case 1 2α < α by assu-

ming 1 22α = α . The dependence of eigenvalues as a function of 1n for L /

1 21, 1τ τ < / is given on Fig. 4 for 2 21 21.1, 0.1, 1, 0.1, 1Lβ = τ = τ = = =

( 0k = on Fig. 4a and 2k = on Fig. 4b). Similar to the case considered above,

at 01 1n n> two roots are complex and one is real. At a certain parameter of

01n , all three roots become real, two of them are positive and the system loses

its stability at any value α . We can see that in the vicinity of 1 0, 1n ≈ β ≈ , the roots correspond to analytical solutions (10).

-4

-2

0

2

4

6

8

10

-3 -2 -1 0 1 2 3 n1

Re λIm λ

01n− 0

1n

)a

-4

-2

0

2

4

6

8

10

-3 -2 -1 0 1 2 3 n1

Re λIm λ

11kn =− 1

1kn =

)b

Fig. 4

At the same time, at 1L / and 1 1n < the system is unstable accor-

ding to the Turing instability. The plot of the characteristic roots for 1k = is

presented on Fig. 4b. At 1 1kn n< the system has one positive real root of lar-

ge value and two complex conjugate roots. Namely the first root is responsible

200

for the stationary pattern formation. By increasing α we can always obtain homogeneous oscillations which, on one hand, will lead to oscillations of the structures and, on the other hand, can destroy them and lead to homogenous oscillations.

A typical scenario of the structure formation in the case 1 22α = α is not so diverse as for the case considered above, and at a wide limit of system pa-rameters we have either homogenous oscillations or stationary dissipative structures. In a certain way, this is similar to the case that we have for inte-ger derivative indices. Nonlinear dynamics corresponding to these roots is rep-resented on Fig. 5. For example, at certain parameters 1 20.55, 1.1α = α = , 1τ =

2 220.4, 1, 0.1, 1.0, 0.01L A= τ = = = = − , 1.1β = we get sufficiently smooth

homogenous oscillations (Fig. 5b). A decrease of α ( 1 20.45, 0.9α = α = ) leads to interplay between structures which arise from the Hopf and the Turing bifurcations (Fig. 5c). A successive decrease of α ( 1 20.5, 1.0α = α = ) leads to stable inhomogeneous structures (Fig. 5a).

Fig. 5

For 1 2α <≈ α the diversity of the structure formation increases and we can find the solutions similar to those presented on Fig. 2, 3. This is due to the fact that the Turing and the Hopf bifurcations have independent parameters for their realization. The Turing bifurcation depends on the ratio of the cha-racteristic lengths and is connected with the instability domain 1 1n < . The

Hopf bifurcation is not connected with the domain 1 1n < and is realized at

wide spectrum of parameters 1 2α < α . This makes it possible to find a scena-rio of complicated pattern formation due to interplay between these two types of instabilities.

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МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ДРОБОВИХ СИСТЕМ РЕАКЦІЇ-ДИФУЗІЇ З ЧАСОВИМИ ПОХІДНИМИ РІЗНОГО ПОРЯДКУ Ïðîâåäåíî ë³í³éíèé àíàë³ç ñò³éêîñò³ äâîêîìïîíåíòíî¿ ñèñòåìè ðåàêö³¿-äèôó糿 äðîáîâîãî ïîðÿäêó ç ð³çíèìè ³íäåêñàìè ïîõ³äíèõ äëÿ êîæíîãî ç ð³âíÿíü ñèñòåìè. Ðîçãëÿíóòî äâà âèïàäêè: êîëè ïîõ³äíà â ð³âíÿíí³ äëÿ àêòèâàòîðà º á³ëüøîþ, í³æ â ð³âíÿíí³ äëÿ ³íã³á³òîðà, ³ íàâïàêè, êîëè ïîðÿäîê ïîõ³äíî¿ äëÿ ð³âíÿííÿ ç äîäàò-íèì çâîðîòíèì çâ’ÿçêîì º ìåíøèì, í³æ äëÿ ð³âíÿííÿ ç â³ä’ºìíèì çâîðîòíèì çâ’ÿçêîì. Çàãàëüíèé àíàë³ç ï³äòâåðäæåíî çà äîïîìîãîþ êîìï’þòåðíîãî ìîäåëþâàí-íÿ ñèñòåìè ç êóá³÷íîþ íåë³í³éí³ñòþ. Ïîêàçàíî, ùî ñèñòåìè ç âèùèì ³íäåêñîì ïî-õ³äíî¿ â ð³âíÿíí³ äëÿ àêòèâàòîðà ìàþòü ñóòòºâî ñêëàäí³øó ïðîñòîðîâî-÷àñîâó äèíàì³êó, í³æ ñèñòåìè ç âèùèì ³íäåêñîì ïîõ³äíî¿ äëÿ ³íã³á³òîðà. МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ДРОБНЫХ СИСТЕМ РЕАКЦИИ-ДИФФУЗИИ С ВРЕМЕННЫМИ ПРОИЗВОДНЫМИ РАЗНОГО ПОРЯДКА Ïðîâåäåíî ëèíåéíîå èññëåäîâàíèå óñòîé÷èâîñòè äâóõêîìïîíåíòíîé ñèñòåìû ðå-àêöèè-äèôôóçèè äðîáíîãî ïîðÿäêà ñ ðàçëè÷íûìè èíäåêñàìè ïðîèçâîäíûõ äëÿ êàæäîãî èç óðàâíåíèé ñèñòåìû. Ðàññìîòðåíû äâà ñëó÷àÿ: êîãäà ïðîèçâîäíàÿ â óðàâíåíèè äëÿ àêòèâàòîðà áîëüøå, ÷åì â óðàâíåíèè äëÿ èíãèáèòîðà, è íàîáîðîò, êîãäà ïîðÿäîê ïðîèçâîäíîé äëÿ óðàâíåíèÿ ñ ïîëîæèòåëüíîé îáðàòíîé ñâÿçüþ ìåíüøå, ÷åì äëÿ óðàâíåíèÿ ñ îòðèöàòåëüíîé îáðàòíîé ñâÿçüþ. Îáùèé àíàëèç ïîäòâåðæäåí ñ ïîìîùüþ êîìïüþòåðíîãî ìîäåëèðîâàíèÿ ñèñòåìû ñ êóáè÷åñêîé íåëèíåéíîñòüþ. Ïîêàçàíî, ÷òî ñèñòåìû, èìåþùèå áîëüøèé èíäåêñ ïðîèçâîäíîé â óðàâíåíèè äëÿ èíãèáèòîðà, îáëàäàþò ñóùåñòâåííî áîëåå ñëîæíîé ïðîñòðàíñò-âåííî-âðåìåííîé äèíàìèêîé, ÷åì ñèñòåìû ñ âûñøèì èíäåêñîì ïðîèçâîäíîé äëÿ èíãèáèòîðà. Pidstryhach Inst. of Appl. Problems Received of Mech. and Math. of NASU, L’viv 07.03.08