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Phys. Scr. T136 (2009) 014027 (7pp) doi:10.1088/0031-8949/2009/T136/014027

Self-organization phenomena

in reaction–diffusion systemswith non-integer order time derivatives

B Y Datsko1 and V V Gafiychuk2

1 Institute for Applied Problems in Mechanics and Mathematics of the National Academy of Sciences,

Naukova 3b, Lviv 79063 Ukraine2 Physics Department, New York City College of Technology, CUNY, 300 Jay, NY, USA

E-mail: b datsko@yahoo.com and vagaf@yahoo.com

Received 27 January 2009

Accepted for publication 28 January 2009

Published 12 October 2009

Online at stacks.iop.org/PhysScr/T136/014027

Abstract

This paper considers a one-dimensional fractional reaction–diffusion model with a different

order of derivative indices. The corresponding linear stability analysis is presented for any

number of fractional derivative indices that are greater than zero but less than two. It was

shown that instability can arise for the Hopf and the Turing modes. The self-organization

phenomena are more diverse when the activator derivative index is greater than the inhibitor

one than when the inhibitor variable index is greater. This analysis was supported by the

results of computer simulations of the fractional nonlinear reaction–diffusion

problem.

PACS number: 02.10.Ud

1. Introduction

Recently, the study of fractional diffusion has attracted much

interest [1–9]. Part of this interest comes from the attempt to

understand the phenomena of fractal and irregular systems [4],

and the relaxation processes [10] in nature. In fact, fractional

derivatives are used for a description of heterogeneous porous

systems [11], tumor growth [12], plasma [13], polymers [14],turbulence [15], disordered semiconductors [16], magnetic

resonance imaging [17], etc.

However, the study of the fractional reaction–diffusion

system (RDS) in general is of interest in other fields, like the

complex system and self-organization [19–25]. The present

paper is devoted to the investigation of self-organization

phenomena, widely studied in the standard systems [26–28],

in the media described by fractional systems. We will show

that in the fractional RDS with different orders of derivatives

we have phenomena that are not possible to find in RDSs with

integer derivatives. We confirm the linear stability analysis

by numerical simulation of a Bonhoeffer–van der Pol type

fractional RDS.

2. Mathematical model

The starting point of our consideration is the coupled

reaction–diffusion equations with indices of different orders

τ αu∂ αu( x , t )

∂t α= l2 ∂ 2u( x , t )

∂ x 2+ W (u, v, A ), (1)

τ βv∂β v( x , t )

∂t β=  L2 ∂ 2v( x , t )

∂ x 2+ Q(u, v, A ), (2)

subject to the Neumann:

∂u/∂ x | x =0,l x = ∂v/∂ x | x =0,l x 

= 0 (3)

boundary conditions and with certain initial conditions. Here,

u( x , t ), v( x , t ) are activator and inhibitor variables, 0  x l x , τ u , τ v, l, L are the characteristic times and lengths of 

the system, correspondingly, A  is an external parameter,

and W (u, v, A ), Q(u, v, A ) are the nonlinear sources of the

system modeling their production rates.

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Phys. Scr. T136 (2009) 014027 B Y Datsko and V V Gafiychuk  

Time derivatives ∂α u( x ,t )

∂ t α, ∂β v( x ,t )

∂t βon the left-hand side

of equations (1) and (2), instead of the standard ones,

are the Caputo fractional derivatives in time of the order

0 < α, β < 2 and, for a certain rational value of  α, are

represented as [29, 30]

∂α

 f (t )∂t α

: s = 1(m − α)

 t 

0

 f (m)

(τ )(t − τ )α+1−m

dτ,

where m − 1 < α < m, m = 1, 2.

The system (1) and (2) with arbitrary rational α and β,

by a certain substitution, can be transformed into the system

of differential equations with the same order of fractional

derivative index. In fact, if the new fractional derivative index

γ  is the greatest common factor of  α (α = pγ ) and β(β =r γ ), m, p ∈N, we obtain the system of  p + r  differential

equations

τ γ u

∂γ u p( x , t )

∂t γ 

=l2 ∂ 2u( x , t )

∂ x 2

+ W (u, v, A ), (4)

τ γ v

∂ γ vr ( x , t )

∂t γ =  L2 ∂2v( x , t )

∂ x 2+ Q(u, v, A ), (5)

where the derivatives on the right-hand side generate

recurrent equations for ui , i = p, p − 1, . . . , 1 and v j , j =r , r − 1, . . . , 1

τ γ u

∂γ ui−1( x , t )

∂t γ = ui ( x , t ), p i > 1, u1 ≡ u, (6)

τ γ v

∂γ v j−1( x , t )

∂t γ = v j ( x , t ), r  j > 1, v1 ≡ v. (7)

This system of equations (4)–(7) is equivalent to the system

(1) and (2) and our analysis is devoted to the investigation of 

its possible solutions for given forms W (u, v, A ), Q(u, v, A ).

We consider the RDS with two specific variables: one of 

them is a variable with positive feedback and the other one

is a variable with negative feedback. Such systems possess

a variety of nonlinear phenomena investigated in the last

decades [26–28].

3. Linear stability analysis

3.1. Standard RDS 

Let us start with α = β = 1, and we obtain a system (1)

and (2) with standard derivatives, where we can analyze its

nullclines

W (u, v, A ) = 0, Q(u, v, A ) = 0. (8)

The simultaneous solution of the system (8) leads to

homogeneous distribution of  u and v. The stability of the

steady-state solutions of the system (1) and (2) corresponding

to homogeneous equilibrium state (u, v) is determined by the

eigenvalue problem for the matrix

F (k ) = a11

(k )/τ u

a12

/τ u

a21/τ v a22(k )/τ v

,

k = πl x 

 j, j = 1, 2, . . . , a11(k ) = a11 − k 2l2, a11 = W u , a12 =W v, a21 = Q

u , a22(k ) = a22 − k 2 L2, a22 = Qv (all derivatives

are taken at homogeneous equilibrium states W  = Q = 0).

For this square matrix, eigenvalues are given by the quadratic

equation and can be determined as

λ1,2 = 12

(tr F ±√ 

tr2 F − 4det F ). (9)

For α

=β

=1 and k 

=0 under the conditions

trF (0) > 0, det F (0) > 0, (10)

we can have a Hopf bifurcation. For k = 0 it is possible to

obtain that at a certain value of  k 0 eigenvalues λ1,2 are real

and one of them is greater than zero (a Turing bifurcation).

The conditions of this instability are

trF  < 0, det F (0) > 0, det F (k 0) < 0. (11)

We can rewrite the inequality (10) as a11 > −a22τ u /τ vaccording to the time frequency oscillation ω =√ 

det F (0)/(τ u τ v ) and the inequality (11) as [26–28]

a11 > −a22(l2/ L2) + 2 

det F (0)(l/ L) (12)

according to wave numbers

k 0 = 4 

det F (0)/√ 

l L. (13)

Instability conditions for these two types of bifurcations are

realized due to positive feedback  (a11 > 0) and at τ u /τ v → 0

and l/ L → 0 they coincide and approach the extremum point

(or points) of W (u, v, A ) = 0.

3.2. Fractional RDS 

Now we will analyze the fractional RDS with arbitrary

rational α, β. By introducing a new parameter γ , we have

a system of  m + p differential equations (4)–(7). In this

case, the linearization of the system in the equilibrium

conditions described by vectors u = (u, 0, . . . , 0) p×1, v =(v, 0, . . . , 0)r ×1 leads to the characteristic equation det( J −λ I ) = 0 of the (r + p) degree polynomial

(−λ)r + p + (−1)r −1 a22(k )

τ βv

(−λ) p + (−1) p+1 a11(k )

τ αu(−λ)r 

+(−1)r + p det F = 0. (14)

In the common case, the solution of such a type of equationcan be obtained numerically. This solution will determine a

stability of the system (4)–(7) [34]. At a small value of  =detF  (this situation takes place when nullclines W (u, v, A ) =0, Q(u, v, A ) = 0 are practically tangent to each other), it is

always possible to find the roots with the value close to zero

λi ≈ (τ βv /a22)1/ p, α < β, i ∈ ¯1, p, (15)

λi ≈ (τ αu /a11)1/r , α > β, i ∈ ¯1, r , (16)

and to determine where they are greater than zero. At

det F 

≈0, the condition det F (0) > 0 can be rewritten

as dv/du|Q=0 > dv/du|W =0, which means that the secondnullcline (Q = 0) has a greater slope than the first one ( W  =0). Here, we would like to consider the roots of several

specific cases, described by third degree polynomials, namely,

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Phys. Scr. T136 (2009) 014027 B Y Datsko and V V Gafiychuk  

(a) (b)

Figure 1. Imaginary (gray lines) and real parts (black lines) of eigenvalues as function u at α = β for k = 0 (a), and k = 1 (b).The other parameters are B = 2, τ αu = 12, τ βv = 1, l2 = 0.1, L2 = 1.

(a) (b)

Figure 2. Imaginary (gray lines) and real parts (black lines) of eigenvalues as a function of  u1 obtained from the solution of equation (19) at α = 2β for k = 0 (a) and k = 1 (b). The otherparameters are B = 1.1, τ αu = 0.1, τ βv = 1, l2 = 0.1, L2 = 1.

α = 2β = 2γ  and β = 2α = 2γ . In this case, the characteristic

equation (14) has the form of the cubical equation

λ3 + λ2b + λc + d = 0, (17)

where

b = −a22(k )/τ βv , c = −a11(k )/τ αu , d = det F (k ) (18)

for α = 2β = 2γ , and

b = −a11(k )/τ u , c = −a22(k )/τ v, d = det F (k ) (19)

for β = 2α = 2γ (0 < γ < 1).

4. Eigenvalue problem analysis

for a Bonhoeffer–van der Pol type RDS

As an example, we consider here a Bonhoeffer–van der

Pol type RDS with cubical nonlinearity (see [24, 25, 27,

28]). In this case, the source term for the activator variable

is nonlinear, W  = u − u3/3 − v, and it is linear for the

inhibitor one, Q = −v + Bu +A . The homogeneous solution

of variables u and v can be obtained from the system W  =Q = 0, and for the determination of  u, we have a cubic

algebraic equation

( B − 1)u + (u3/3) + A = 0. (20)

Calculation of the coefficients ai j : a11 = (1 − u2), a12 =−1, a21 =  B, a22 = −1 at the homogeneous state (20) makes

it possible to investigate the eigenvalues of the system

explicitly. As a result, we can see that at τ αu /τ βv → 0

(a) (b)

Figure 3. Imaginary (gray lines) and real parts (black lines) of eigenvalues as a function of u1 at 2α = β for k = 0 (a) and k = 1(b). The other parameters are B = 1.1, τ αu = 0.1, τ βv = 1, l2 = 0.1, L2 = 1.

(a)

(b)

Figure 4. The oscillatory structures obtained from numericalsimulations of the system (1) and (2). (a) Dynamics of the variableu1 on the time interval (0,10) for α = 1.9, β = 1.75, τ αu = 0.1,τ βv = 1, l2 = 0.1, L2 = 1, A = −10, B = 3. (b) Dynamics of variable u1 on the time interval (0,60) for α

=1.9, β

=1.85,

τ αu = 12.5, τ βv = 1, l2 = 5.0, L2 = 1, A = −0.2, B = 2.

and l/ L → 0 in a standard RDS the instability domain

is determined at |u| < 1. In this case, the simultaneous

conditions of the Hopf (10) and the Turing (11) bifurcations

are realized.

4.1. The case α = β

The linear stability analysis of the system when fractional

derivative indices are equal is considered in [32, 33]. Here,

we would like to single out the special case when the index

is greater than one and a new type of bifurcation arises. Thereal and imaginary parts of eigenvalues for k = 0 obtained

numerically for each particular point u as a solution of 

the equation (17) for certain values B, τ αu , τ βv , l2, L2 are

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Phys. Scr. T136 (2009) 014027 B Y Datsko and V V Gafiychuk  

(a) (b)

(c) (d)

Figure 5. Pattern formation scenario for A = −0.1, α1 = 1.4, α2 = 0.7, l2 = 0.1, L2 = 1.0, B = 1.1, τ βv = 1, l x  = 4π and different valuesof τ αu ((a) τ αu = 0.2, (b) τ αu = 0.25, (c) τ αu = 0.3 and (d) τ αu = 0.35).

presented in figure 1(a). For these parameters, we see thatthe real part of the roots is always less than zero and the

imaginary one on some interval of  u becomes nonzero. In

this case, when the fractional derivative index becomes greater

than some critical value γ 0 = π2

tan−1(Im λ/Re λ): γ > γ 0, we

have homogeneous oscillations. In this case, at the values

of  γ 0 > 1.5 instability conditions take place on the interval

1.8 |u| 4.5.

A similar plot can be presented for the eigenvalues when

wave number k = 0 (for example, k = 1). On figure 1(b),

imaginary parts of the eigenvalues are presented for the same

parameters of the system. As we can see from this figure,

when nullclines have an intersection point on the interval|u| between 4.5 and 6, the system is stable according to

homogeneous oscillations. At a certain value of  α, instability

conditions are possible to realize for k = 1. This means that

at least perturbations with this wave number are unstable.

Moreover, they are unstable for oscillatory fluctuations. This

situation is qualitatively different from the integer RDS,

whether either the Turing (k = 0) or the Hopf bifurcations

(k = 0) take place. This depends on which conditions are

more easy to realize. In the system under consideration, we

can choose the parameter 4.5 < u < 6 when we do not have

a standard Hopf or Turing bifurcation at all. Nevertheless,

we obtained that the conditions for the Hopf bifurcation can

be realized for non-homogeneous perturbations with wave

numbers k = 0 [24, 25]. This phenomenon is inherent to the

system with different orders of fractional derivatives if the

difference between them is not so substantial.

4.2. The case α > β(0 < α, β < 2)

The real and imaginary parts of eigenvalues for k = 0,

depending on parameter u, for α = 2β and parameters typical

for pattern formation in regular systems, are presented in

figure 2(a). We see that one real root is always less than zero.

The roots of two others at |u| > u0 are complex conjugate

roots. At |u| < u0 these roots become real and positive. As

a result of these conditions, the instability in the system takes

place practically for any value of α.

Except homogeneous oscillations, the condition of the

Turing instability becomes true if the ratio l/ L 1. As an

example, the plot of eigenvalues for k 

=1 is presented in

figure 2(b) where at |u| < uk  all the roots are real and one of them corresponds to stationary non-homogeneous structures.

At |u| < uk , we could have non-homogeneous

oscillations of the structures if the conditions of this

instability are softer than conditions of homogeneous

oscillations.

4.3. The case α < β(0 < α, β < 2)

The dependence of eigenvalues as a function of u for 2α = β

and some other parameters are presented in figure 3. Similar to

the case considered above at |u| > u0, two roots are complex

and one is real. At certain values of  u, all three roots become

real, two of them are positive and the system loses its stabilityat any value α.

At the same time, at l/ L 1 and |u| < 1, the system

is unstable according to the Turing instability. The plot of 

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Phys. Scr. T136 (2009) 014027 B Y Datsko and V V Gafiychuk  

(a)

(b)

Figure 6. Pattern formation scenario for A = −0.1, α = 1.0,β = 0.5, l2 = 0.1, L2 = 1.0, τ αu = 0.15, τ βv = 1, l x  = 7.5 anddifferent values of  B ((a) B = 2.0 and (b) B = 1.05).

the roots of the characteristic equation (17) for k 

=1 is

presented in figure 3(b). At |u| < uk  the system has onepositive real root of great value and two complex conjugate

roots. Namely, the first root is responsible for stationary

pattern formation. By increasing α, we can always obtain

homogeneous oscillations that, on the one hand, will lead

to oscillations of the structures, and on the other hand, can

destroy them and lead to homogeneous oscillations.

5. Nonlinear dynamics of Bonhoeffer–van der Poltype RDS

For a numerical simulation of the system (1) and (2) with

corresponding initial and boundary conditions, we used the

finite difference schemes based on the Grünwald–Letnikov

and Riemann–Liouville definition [29–31] the application of 

which is considered in detail in the paper [32].

5.1. The case α β

The results of computer simulation of nonlinear

non-homogeneous oscillations for α β are presented

in figure 4. Realized non-homogeneous oscillations emerge

spontaneously at the given parameters. This agrees with the

theoretical investigation presented in section 4. It should be

noted that the presented phenomena are typical for differentrelationships between the system parameters and are realized

outside the increasing path of nullcline W  = 0 (|u| > 1)

(figure 4(b)) as well as for l > L (figure 4(a)).

(a)

(b)

Figure 7. Pattern formation scenario for (a) α = 1.1, β = 0.55,A = −0.01, B = 1.1 and (b) α = 1.0, α = 0.5, A = −0.01, B = 1.1.The other parameters are τ αu = 0.1, τ βv = 1, l2 = 0.1, L2 = 1.0.

5.2. The case α > β(0 < α, β < 2)

Computer simulation of the system for this case is presented

in figures 5–7. In figure 5(a), we can mostly see the simplepattern formation scenario corresponding to homogeneousoscillations when the homogeneous solution u is close to zero.

By moving out of this point by increasing A , homogeneousoscillations are modulated by non-homogeneous mode.

Computer simulation shows that the interplay between

the Hopf and the Turing bifurcations, which leads tocomplicated dynamics, is typical for a wide spectrum of parameter α from a small one such as α = 0.1 to α = 1.

For example, typical spatio-temporal structures for differentparameters of  τ u are presented in figures 5(c)–(d). Thesespatio-temporal patterns emerge as a result of the interplay

between the Hopf and the Turing bifurcations. In fact,

according to figure 2, eigenvalues for k = 0 and k = 1, thesystem parameters are practically the same in the vicinity of 

the point |u| = 0.By changing the system parameters, for example β,

spatio-temporal patterns (see figure 6) are similar to those

displayed in figures 5(c) and (d) and we have a profile of complex ‘zigzag’ oscillations. It should be noted that the formof these structures is strongly dependent on the fractional

derivative indices as well as other parameters of the systemunder consideration.

At a certain value of  A , where the Hopf and the

Turing instabilities have the same realization conditions, wehave stable space-time oscillations of the form presented in

figures 7(a) and (b). It should be noted that for a successiveincrease of  A  to u 1, we obtain a stable homogeneousdistribution, which for integer derivative indices, is unstableand we have homogeneous oscillations.

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Phys. Scr. T136 (2009) 014027 B Y Datsko and V V Gafiychuk  

(a) (b)

(c) (d)

Figure 8. Pattern formation scenario for A = −0.1, l2 = 0.1, L2 = 1.0, τ αu = 0.15, τ βv = 1, B = 1.1 and different values of derivativeorders: (a) α = 0.4, β = 0.8 and (b) α = 0.75, β = 1.5. Oscillating dissipative structures for (c) α = 0.75, β = 1.5, A = −0.12, l2 = 0.05, L2 = 1.0, τ αu = 0.55, τ βv = 1, B = 1.1 and (d) α = 0.7, β = 1.4, A = −0.1, l2 = 0.05, L2 = 1.0, τ αu = 0.55, τ βv = 1, B = 1.1.

The obtained scenario of pattern formation is typical

for the general case α > β . We can conclude that at

certain parameters the solutions may have a simple form

of homogeneous oscillations or stationary non-homogeneous

structures, and can correspond to spatio-temporal structures

similar to those presented in figures 5–7. In addition to

homogeneous oscillations or stationary structure formation

inherent to the standard system with integer derivatives, the

system considered here with indices α > β possesses more

complicated nonlinear dynamics.

5.3. The case α < β(0 < α, β < 2)

The typical scenario of the structure formation in the case

2α = β is not so diverse as for the case considered above,

and at a wide limit of system parameters we have either

homogeneous oscillation or stationary dissipative structures.

Nevertheless, even homogeneous oscillation can have a

variety of forms depending on the relationship between α

and β (figure 8(a)–(d)). On the other hand, at the parameters

close to realization of the Turing bifurcation we can obtain

the oscillatory non-homogeneous structures period the shape

of which depends on derivative indices.

For α β the diversity of the structure formation

increases. This is due to the fact that the Turing and theHopf bifurcations have independent parameters for their

realization. The Turing bifurcation depends on the ratio of 

the characteristic lengths and is connected with the instability

domain |u| < 1. The Hopf bifurcation is not connected with

the domain |u| < 1 and is realized at a wide spectrum of 

parameters α < β . This makes it possible to find the scenario

of a complicated pattern formation due to the interplay

between these two types of instabilities.

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