hopf bifurcation and turing instability of 2-d lengyel–epstein system with reaction–diffusion...

Applied Mathematics and Computation 219 (2013) 9229–9244

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Hopf bifurcation and Turing instability of 2-D Lengyel–Epsteinsystem with reaction–diffusion terms

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.03.071

⇑ Corresponding author.E-mail address: [email protected] (H. Zhao).

Ling Wang, Hongyong Zhao ⇑Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China

a r t i c l e i n f o

Keywords:StabilityReaction–diffusionLengyel–EpsteinHopf bifurcation Turing instability

a b s t r a c t

In this paper, we study 2-D Lengye l–Epstein (L–E) reaction–diffusion system with homogeneous Neumann boundary condition. By employing stability theory, Hopf bifurcation theorem and Turing’s theory, we present some sufficient conditions ensuring the equilibrium point of system to be stable and derive conditions on the parameters so that spatial homogenous Hopf bifurcation and Turing instabili ty occur. These conditions obtained have impo r-tant leading significance in applications of L–E system. Finally, we show some numerical examples to verity the theoretical analysis.

� 2013 Elsevier Inc. All rights reserved.

1. Introductio n

Reaction diffusion system is one of the basic systems which describe the movement of nature. Various of dynamical char- acteristics of reaction–diffusion systems in physics, chemistry, biology and ecology have become recently a subject of intense research activity (see [1–3]). In 1952, Alan Turing described patterns and forms about biological systems in his paper [4], and put forward the Turing principle of spatial patterns, showing diffusion could lead to instability. This kind of instability is usu- ally called Turing instabilit y or diffusion-dri ven instability. As is well known, in chemistry, the chlorite–iodide–malonic acid reaction or so called CIMA reaction is a typical example to indicate diffusion-driven instability mechanism. After De Kepper [5], Lengyel and Epstein [6,7] had established the CIMA system, they discovered that although there were five variables in the reaction, in fact, three of them in the reaction process were almost unchanged . Thus, it is able to simplify the original system to a two-dimensi onal model, which we call Lengyel–Epstein system. The correspondi ng dimensionle ss reaction–diffusion system takes the following form

ut ¼ @2u@x2 þ a� u� 4uv

1þu2 ; x 2 X; t > 0;

v t ¼ rd @2v@x2 þ rb u� uv

1þu2

� �; x 2 X; t > 0;

@u@c ¼ @v

@c ¼ 0; x 2 @X; t > 0;

uðx;0Þ ¼ u0ðxÞ; vðx;0Þ ¼ v0ðxÞ; x 2 X;

8>>>>><>>>>>:ð1Þ

where X is a bounded domain in Rn; ðn P 1Þ, with sufficiently smooth boundary @X; X is assumed to be closed, thus reflex-ive Neumann boundary condition is imposed; uðx; tÞ and vðx; tÞ are normalized dimensionle ss concentrations of the two reactants iodide ðI�Þ and chlorite ðCLO�2 Þ respectively; a and b are the parameters related to the concentratio n of acid, d is diffusion coefficient ratio, r > 1 is adjustable parameter which is bound up with starch concentr ation. a; b; r and d arepositive.

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9230 L. Wang, H. Zhao / Applied Mathematics and Computation 219 (2013) 9229–9244

Over the past decades, since the pioneering work of Lengyel and Epstein [6,7], a great deal of research interest of Lengyel–Epstein system has been aroused [8–12]. Recently, the authors [13] discussed the global bifurcation structure of the set of the non-constan t steady states for a 1-D Lengyel–Epstein model. Researchers in [14], by constructing a proper Lyapunov func- tion, showed that when the feed concentratio n was small enough, the constant equilibrium solution of Lengyel–Epstein reaction–diffusion system was globally asymptotically stable. Du and Wang [15] gave the existence of multiple spatially non-homog eneous periodic solutions though all the parameters of the system were spatially homogeneous , meanwhile ifparameter l ¼ 1, the stability results were similar to those of [16].

In [17], the authors considered 2-D L–E system as follows

ut ¼ @2u@x2 þ @2u

@y2 þ a� u� 4uv1þu2 ; ðx; yÞ 2 X�X; t > 0;

v t ¼ rd @2v@x2 þ @2v

@y2

� �þ rb u� uv

1þu2

� �; ðx; yÞ 2 X�X; t > 0;

@u@c ¼ @v

@c ¼ 0; x 2 @X; y 2 @X; t > 0;

uðx; y;0Þ ¼ u0ðx; yÞ; vðx; y; 0Þ ¼ v0ðx; yÞ; ðx; yÞ 2 X�X:

8>>>>><>>>>>:ð2Þ

However, stability and Turing instability of system (2) are only studied in [17] as an numerical example. To the best of our knowledge, there are no theory results about stability, Hopf bifurcation and Turing instability for system (2) in the literature so far, which still remains open and challenging.

Motivated by the above discussions, in this paper, we will discuss further dynamical behaviors of system (2) by employing stability theory, Hopf bifurcation theorem and Turing instability theory. We extend or improve the previously known results.

2. Stability and Hopf bifurcation analysis

Let N be the nonnegati ve integer set and Nþ be the positive integer set.In this paper, we consider system (2) with X ¼ ð0; lpÞ; l 2 Rþ. The unique constant steady state of system (2) is

ðu�;v�Þ ¼ ða;1þ a2Þ, here a ¼ a5.

To simplify the discussions, we translate system (2) by the formatio n u ¼ u� u�; v ¼ v � v� and for simplicity, denoting u; v again by u; v respectively . Furthermore, we incorporate explicitly the length lp into the equations by the transforma- tion x ¼ lp~x; y ¼ lp~y, which changes the domain X�X ¼ ð0; lpÞ � ð0; lpÞ to the unit square eX � eX ¼ ð0;1Þ � ð0;1Þ, denoting eX � eX ¼ ð0;1Þ � ð0;1Þ; x; y again by C; x; y respectively. Thus, model (2) becomes

ut ¼ 1l2p2ð@2u@x2 þ @2u

@y2Þ þ 4a� u� 4ðuþaÞðvþ1þa2Þ1þðuþaÞ2

; ðx; yÞ 2 C; t > 0;

v t ¼ rdl2p2

@2v@x2 þ @2v

@y2

� �þ rb uþ a� ðuþaÞðvþ1þa2Þ

1þðuþaÞ2

� �; ðx; yÞ 2 C; t > 0;

uxðx; y; tÞ ¼ vxðx; y; tÞ ¼ 0; ðx; yÞ 2 @C; t > 0;uyðx; y; tÞ ¼ vyðx; y; tÞ ¼ 0; ðx; yÞ 2 @C; t > 0;uðx; y;0Þ ¼ u0ðx; yÞ; vðx; y; 0Þ ¼ v0ðx; yÞ; ðx; yÞ 2 C:

8>>>>>>>><>>>>>>>>:ð3Þ

We obtain the linear operator M of system (3) at ðu;vÞ ¼ ð0;0Þ is

M ¼1

l2p2@2

@x2 þ @2

@y2

� �þ 3a2�5

1þa2 � 4a1þa2

2a2rb1þa2

rdl2p2

@2

@x2 þ @2

@y2

� �� arb

1þa2

0B@1CA: ð4Þ

Rewrite system (3) to

dudtdvdt

!¼ M

u

v

� �þ

h1ðu;vÞh2ðu;vÞ

� �; ð5Þ

whereh1ðu;vÞ ¼ 4að3�a2Þ

ð1þa2Þ2u2 þ 4ða2�1Þ

ð1þa2Þ2uv þ Oðjuj3; juj2jvjÞ; h2ðu;vÞ ¼ rb

4 h1ðu;vÞ.

It is well known that the eigenvalue problem

�u00 ¼ lu; ðx; yÞ 2 eX � eX;uxð0; y; tÞ ¼ uyðx;0; tÞ ¼ uxð1; y; tÞ ¼ uyðx;1; tÞ ¼ 0

(ð6Þ

has eigenvalues lnm ¼ ðn2 þm2Þp2; ðn;m 2 NÞ, with correspondi ng eigenfunctio ns unm ¼ ðcosnpxÞðcosmpyÞ.Now we define X ¼

P1n;m¼0 � Xn;m, where

Xn;m :¼c1

c2

� �cosðnpxÞ � cosðmpyÞ; c1; c2 2 R

� �; n; m 2 N; ð7Þ

L. Wang, H. Zhao / Applied Mathematics and Computation 219 (2013) 9229–9244 9231

Mn;m :¼ MjXn;m, where

Mn;m ¼� n2þm2

l2þ 3a2�5

1þa2 � 4a1þa2

2a2rb1þa2 �rd n2þm2

l2� arb

1þa2

0@ 1A: ð8Þ

It follows from the analysis above that eigenvalues of M are given by the eigenvalues of Mn;m for n; m 2 N. The characteristic equation of Mn;m is given by

k2 � kTnm þ Dnm ¼ 0; n; m 2 N; ð9Þ

where

Tnm :¼ trðMn;mÞ ¼3a2 � 51þ a2 �

arb1þ a2 �

n2 þm2

l2 ð1þ rdÞ;

Dnm :¼ detðMn;mÞ ¼ðn2 þm2Þ2

l4rdþ arb

1þ a2 � rd3a2 � 51þ a2

� n2 þm2

l2 þ 5arb1þ a2 :

Denote

b� :¼ 3a2 � 5ar ; b1 :¼ 13a2dþ 5d� 4ad

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10ða2 þ 1Þ

pa

; b2 :¼ 13a2dþ 5dþ 4adffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10ða2 þ 1Þ

pa

;

H :¼ n2 þm2

l2; H1 :¼

� arb1þa2 � rd 3a2�5

1þa2

� ��

ffiffiffiffiffiffiDHp

2rd; H2 :¼

� arb1þa2 � rd 3a2�5

1þa2

� �þ

ffiffiffiffiffiffiDHp

2rd:

Then, Dnm can be expressed as the following form

FðHÞ :¼ rdH2 þ arb1þ a2 � rd

3a2 � 51þ a2

� H þ 5arb

1þ a2 : ð10Þ

Obviously , the discriminant of FðHÞ ¼ 0 takes the form

DHðbÞ ¼a2r2b2 � 26a3r2bd� 10ar2bdþ r2d2ð3a2 � 5Þ2

ð1þ a2Þ2:¼ f ðbÞð1þ a2Þ2

: ð11Þ

Lemma 1. f ðbÞ ¼ 0 has two positive roots b1 and b2.

Lemma 2. If DH > 0, then FðHÞ ¼ 0 has two roots H1 and H2.

Lemma 3. If b > b�, then for any n; m 2 N, we have Tnm < 0.

Proof. From b > b�, we have T00 ¼ 3a2�51þa2 � arb

1þa2 < 0. It is easy to derive that for any n; m 2 N; Tnm < 0. This completes the proof. h

Theorem 1. As b > b�, the equilibrium point ð0;0Þ of system (3) is locally asymptot ically stable, if ðA1Þ and one of ðA2Þ and ðA3Þhold. Here

(A1) a2 > 53,

(A2) 0 < d 6 ab3a2�5,

(A3) max f ab3a2�5 ;

1rg < d < ab

13a2þ5�4affiffiffiffiffiffiffiffiffiffiffiffiffiffi10ða2þ1Þp .

Proof. Following from b > b� and Lemma 3, we obtain Tnm < 0, for any n; m 2 N.

(i) When ðA1Þ and ðA2Þ hold. By simple calculation, we have

arb1þ a2 �

rdð3a2 � 5Þ1þ a2 P 0:


Thus Dnm > 0 for any n; m 2 N. Combining Dnm > 0 with Tnm < 0 ðn; m 2 NÞ leads to that all eigenvalues of Mn;m have negative real parts. Therefore, the equilibrium point ð0;0Þ of system (3) is locally asymptotically stable.

(ii) When ðA1Þ and ðA3Þ hold. From ðA1Þ, we obtain

0 < 13a2 þ 5� 4affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10ða2 þ 1Þ

q< 3a2 � 5 < 13a2 þ 5þ 4a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10ða2 þ 1Þ

q;

which implies

13a2dþ 5d� 4adffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10ða2 þ 1Þ

pa

¼ b1 <dð3a2 � 5Þ

að12Þ

and

ab

13a2 þ 5þ 4affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10ða2 þ 1Þ

p <ab

3a2 � 5: ð13Þ

Furthermore, following from ðA3Þ and b > b�, we have

b1 < b <dð3a2 � 5Þ

a; b� <

dð3a2 � 5Þa

: ð14Þ

By (13) and ðA3Þ, we obtain

b < b2: ð15Þ

It follows from b > b� and (12)–(15) that

maxfb�; b1g < b < mindð3a2 � 5Þ

a; b2

� �: ð16Þ

According to Lemma 1 and (16), we have f ðbÞ < 0, and so DH < 0. Noting that rd > 0, thus we obtain FðHÞ > 0, namely, for any n; m 2 N; Dnm > 0. The rest proof is similar to the first case, we omit it here. h

Theorem 2. As b > b�, the equilibrium point ð0;0Þ of system (3) is locally asymptotically stable, If ðA1Þ; ðA4Þ and ðA5Þ hold. Here

ðA4Þ d > max ab13a2þ5�4a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi10ða2þ1Þp ; 3a2�5

rð13a2þ5�4affiffiffiffiffiffiffiffiffiffiffiffiffiffi10ða2þ1Þp

Þ

� �,

ðA5Þ n2þm2

l2< H1 or n2þm2

l2> H2; for any n;m 2 N.

Proof. By ðA1Þ we have

0 < 13a2 þ 5� 4affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10ða2 þ 1Þ

q< 3a2 � 5;

which implies

ab3a2 � 5

<ab

13a2 þ 5� 4affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10ða2 þ 1Þ

p : ð17Þ

From ðA4Þ and (17), it is easy to derive that

b < min b1;dð3a2 � 5Þ

a

� �ð18Þ

and

b� < min b1;dð3a2 � 5Þ

a

� �: ð19Þ

Thus, as b > b� we have

b� < b < min b1;dð3a2 � 5Þ

a

� �:

Based on Lemma 1, we have f ðbÞ > 0, that is, DH > 0. Combining Lemma 2 with ðA5Þ, we obtain that for any n; m 2 N; Dnm > 0. Following from b > b� and Lemma 3, we have that for any n; m 2 N; Tnm < 0. Thus, all eigenvalues ofMn;m have negative real parts. That is, the equilibrium point ð0;0Þ of system (3) is locally asymptotically stable. This completes the proof.


Let X ¼ ð0; lpÞ ðl 2 RþÞ in 1-D Lengyel–Epstein system (1), similar to the proof of Theorem 1, we obtain the following conclusion. h

Corollary 1. As b > b�, the equilibrium point of system (1) is locally asymptot ically stable, if ðA1Þ and ðA2Þ or ðA1Þ and ðA3Þ hold.

Remark 1. In [15], Du and Wang show that the equilibrium point of system (1) is locally asymptotical ly stable if one of the following condition s holds:

(i) b > b�; 53 6 a2

61þ5l2

3l2�1;

(ii) b > b�; a2 P 1þ5l2

3l2�1; 0 < d < l2ab

3l2�11þ5l2

a2�1.

Obviously, 53 6 a2

61þ5l2

3l2�1in (i) implies that 3l2 � 1 > 0, that is, l >

ffiffi3p

3 . On the other hand, if the condition (ii) holds, from

the proof in [15], we notice that l still satisfies 3l2 � 1 > 0. However, the parameter l in Corollary 1 is positive. Thus, weenlarge the parameter range of l.

Theorem 3. As b < b�, the equilibriu m point ð0;0Þ of system (3) is unstable.

Proof. Following from b < b�, we obtain

T00 > 0 and D00 > 0;

which means that at least two of eigenvalues of the operator Mn;m have positive real parts, that is, the equilibrium point ð0;0Þis unstable. This completes the proof.

In the following, we analyze the Hopf bifurcation occurring at ð0;0Þ by choosing b as the bifurcation parameter. h

Theorem 4. Under the conditions of Theorems 1 and 2. As b passes through the critical value b�, there is a Hopf bifurcation ofsystem (3) at its zero equilibrium point.

Proof. Let k ¼ �ixðx > 0Þ be the purely imaginary roots of Eq. (9). Substituti ng k ¼ ixðx > 0Þ into Eq. (9), and separating the real and imaginary parts, we have

Dnm �x2 ¼ 0;xTnm ¼ 0;

(ð20Þ

which yields Tnm ¼ 0 and Dnm ¼ x2 > 0; n; m 2 N. According to Tnm ¼ 0, we get the critical value b ¼3a2�5�ð1þrdÞð1þa2Þn2þm2

l2

ar andwhen n ¼ m ¼ 0, the maximum value of b is 3a2�5

ar ¼ b�. Furthermore, the purely imaginary roots must be simple.It is easy to know that Tnþ1 m < Tnm and Tn mþ1 < Tnm; ðn;m 2 NÞ. Combining with b ¼ b�, we have

T00 ¼3a2 � 5� arb

1þ a2 ¼ 0 ð21Þ

and so

Tpq < 0; p; q 2 N; and ðp; qÞ– ð0;0Þ: ð22Þ

From Theorems 1 and 2, we obtain that for any n; m 2 N, Dnm > 0, which implies that there is no zero root of Eq. (9). It follows from the analysis above that system (3) has a pair of purely simple imaginary eigenvalues k ¼ �ixðx > 0Þ and all the other eigenvalues have negative real parts. Taking the first derivative of Eq. (10) with respect to the parameter b at its critical value b� gives

Reðk0ðbÞÞjb¼b� ¼�ar

2ð1þ a2Þ < 0: ð23Þ

According to the Hopf bifurcation theorem, there is a Hopf bifurcation of system (3) at (0,0) as b passes through the critical value b�. This completes the proof.

Similar to the analysis in [16], we obtain the following conclusion about direction and stability of Hopf bifurcation. h

Theorem 5. Under the condition ðA1Þ. If one of ðA2Þ and ðA3Þ or ðA4Þ and ðA5Þ hold, then system (3) undergoes a Hopf bifurcation atð0;0Þ when b ¼ b�.

(i) If 53 < a2 < 27þ

ffiffiffiffiffiffi769p

4 , then the direction of the Hopf bifurcation is subcritical and the bifurcatio n periodic solutions are orbitally stable.


(ii) If a2 > 27þffiffiffiffiffiffi769p

4 , then the direction of the Hopf bifurcation is supercritical and the bifurcatio n periodic solutions are orbitally unstable.

3. Turing Instability

The corresponding ODE model of system (3) is

ut ¼ 4a� u� 4ðuþaÞðvþ1þa2Þ1þðuþaÞ2

;

v t ¼ rb uþ a� ðuþaÞðvþ1þa2Þ1þðuþaÞ2

� �:

8<: ð24Þ

Theorem 6. If ðA1Þ; ðA4Þ and ðA8Þ hold, then Turing instability happens as b > b�. Here ðA8Þ 9 n; m 2 N and ðn;mÞ – ð0;0Þ,satisfying H1 <

n2þm2

l2< H2.

Fig. 1. Graph of u(t,x), a ¼ 15; b ¼ 5; d ¼ 6; l ¼ 0:5; r ¼ 2, the solution tends to the constant steady state u = 3.

Fig. 2. Graph of v(t,x). a ¼ 15; b ¼ 5; d ¼ 6; l ¼ 0:5; r ¼ 2, the solution tends to the constant steady state v = 10.

2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.58.5

9

9.5

10

10.5

11

11.5

12U−V

U

V

Fig. 3. Graph of (u,v). a ¼ 15; b ¼ 5; d ¼ 0:5; l ¼ 1; r ¼ 2, the solution tends to the constant steady state (u,v) = (3,10).

2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.58.5

9

9.5

10

10.5

11

11.5

12U−V

U

V

Fig. 4. Graph of (u,v). a ¼ 15; b ¼ 5; d ¼ 9; l ¼ 0:7; r ¼ 2, the solution tends to the constant steady state (u,v) = (3,10).


Proof. From [16], system (24) is locally asymptoti cally stable as b > b�. If ðA1Þ and ðA4Þ hold, accordin g to the proof of The-orem 2, we obtain DH > 0. Based on Lemma 2, FðHÞ ¼ 0 has two roots. Furthermore, by ðA8Þ, we have that 9 n; m 2 N andðn;mÞ– ð0;0Þ, s.t. Dnm < 0, that is, there are at least two real eigenvalues of Mn;m, one of which is positive and the other isnegative. Therefore, the equilibriu m point of diffusion system (3) must be unstable. According to the Turing instability theorem, Turing instability happens. h

2.6 2.8 3 3.2 3.4 3.6 3.8 49

9.5

10

10.5

11

11.5

12

12.5

13

13.5U−V

U

V

Fig. 5. Graph of (u,v). a ¼ 16; b ¼ 1:5; d ¼ 1:5; l ¼ 1; r ¼ 5:6, the solution tends to the constant steady state (u,v) = (3.2,11.24).

0 1 2 3 4 5 6 7 85

10

15

20

25

30

35U−V

U

V

Fig. 6. Graph of (u,v). a ¼ 20; b ¼ 3; d ¼ 1; l ¼ 1; r ¼ 2, the solution is unstable.


2.75 2.8 2.85 2.9 2.95 3 3.05 3.1 3.15 3.2 3.259.2

9.4

9.6

9.8

10

10.2

10.4

10.6

10.8U−V

u

v

Fig. 7. Graph of (u,v). a ¼ 15; b ¼ 113 ; d ¼ 1; l ¼ 1; r ¼ 2, the solution tends to the spatially homogenous periodic orbit.

10 20 30 40 50 60

5

10

15

20

25

30

35

40

45

50

55

60

10 20 30 40 50 60

5

10

15

20

25

30

35

40

45

50

55

60

Fig. 8. Patterns of u and v for a ¼ 15; b ¼ 1:2; l ¼ 0:9, r ¼ 8; d ¼ 2.


Remark 2. The methods of this paper may extend to study other reaction–diffusion models such as [18–20].

4. Numerical example s

Example 1. Consider system (1) with l ¼ 1; a ¼ 15; b ¼ 5; d ¼ 6, and r ¼ 2. Obviously, the condition (ii) in [15] holds, and so system (1) is locally asymptotically stable. Reset l ¼ 0:5 <

ffiffi3p

3 instead of l ¼ 1, the values of other parameters are as the same as above. Now it is obvious that the paramete rs satisfy ðA1Þ and ðA3Þ, not satisfying the condition (i) and (ii) in [16].By Corollary 1, system (1) is still locally asymptotically stable as illustrated in Figs. 1 and 2. But, the stability of system can not be obtained by Du and Wang [15].

0 200 400 6001.5

2

2.5

3

3.5

4

t

u(t)

0 200 400 6005

6

7

8

9

10

11

12

13

14

tv(t)

Fig. 9. Graph of u(t) and v(t) for a ¼ 15; b ¼ 1:2; r ¼ 8, the solution tends to the constant steady state (u,v) = (3,10).

10 20 30 40 50 60

5

10

15

20

25

30

35

40

45

50

55

60

10 20 30 40 50 60

5

10

15

20

25

30

35

40

45

50

55

60



Example 2. In system (2), taking a ¼ 15, r ¼ 2; l ¼ 1; b ¼ 5, and d ¼ 0:5. By simple calculating, we obtain b� ¼ 11

3 ð< bÞ; ab3a2�5 ¼

1522. Obviously, ðA1Þ and ðA2Þ hold. Following from Theorem 1, the equilibrium point of system (2) is

locally asymptotical ly stable, as shown by Fig. 3.

Example 3. Consider system (2) with a ¼ 15, r ¼ 2; b ¼ 5; d ¼ 9, and l ¼ 0:7. After simple computation, we obtain ab

13a2þ5�4affiffiffiffiffiffiffiffiffiffiffiffiffiffi10ða2þ1Þp ¼ 7:5, 3a2�5


Þ¼ 5:5. Distinctly, b > b�; ðA1Þ and ðA4Þ hold. When n ¼ m ¼ 0, we obtain that

n2þm2

l2< H1. n; m 2 N and ðn;mÞ – ð0;0Þ can yield n2þm2

l2P 1

l2> H2, that is, ðA5Þ holds. By Theorem 2, the equilibrium point of

system (2) is locally asymptotically stable as illustrated in Fig. 4.

10 20 30 40 50 60

5

10

15

20

25

30

35

40

45

50

55

60

10 20 30 40 50 60

5

10

15

20

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Fig. 11. Patterns of u and v for a ¼ 15; b ¼ 1:2; l ¼ 0:9, r ¼ 8; d ¼ 4:5.

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Fig. 12. Patterns of u and v for a ¼ 15; b ¼ 1:2; l ¼ 0:9, r ¼ 8; d ¼ 6:8.


Example 4. Taking a ¼ 16; r ¼ 5:6; l ¼ 1, b ¼ 1:5, and d ¼ 1:5 in [17]. By simple computati on, we obtain that ab

3a2�5 ¼ 0:1866; 1r ¼ 0:1786, ab

13a2þ5�4affiffiffiffiffiffiffiffiffiffiffiffiffiffi10ða2þ1Þp ¼ 1:9869 and b� ¼ 1:4353. Clearly, b > b�; a2 ¼ 10:24 satisfies ðA1Þ and

d ¼ 1:5 satisfies ðA3Þ. By Theorem 1, the equilibrium point of system (2) is locally asymptotical ly stable as shown byFig. 5. That is, the stability analysis is in accordance with numerical example in [17].

Example 5. Consider system (2) with a ¼ 20, r ¼ 2; b ¼ 3; d ¼ 1; l ¼ 1. By simple calculating, we have b� ¼ 3a2�5ar ¼ 43

8 .Clearly, b < b�. By Theorem 3, system (2) is unstable as shown by Fig. 6.

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Fig. 14. Patterns of u and v for a ¼ 15; r ¼ 8; l ¼ 0:9; b ¼ 2, d ¼ 4:5.


Example 6. Taking a ¼ 15; r ¼ 2; l ¼ 1, b ¼ 113 and d ¼ 1. By simple computation, we obtain b� ¼ 11

3 ;ab

3a2�5 ¼ 0:5,1r ¼ 0:5; ab

13a2þ5�4affiffiffiffiffiffiffiffiffiffiffiffiffiffi10ða2þ1Þp ¼ 5:5. Distinctly, b ¼ b�; a2 ¼ 9 satisfies ðA1Þ and d ¼ 1 satisfies ðA3Þ. By Theorem 4, system (2)

undergoes Hopf bifurcation at b ¼ b� as illustrated in Fig. 7. Since a2 ¼ 9, accordin g to Theorem 5, 53 < a2 < 27þ

ffiffiffiffiffiffi769p

4 is satisfied,then the direction of the Hopf bifurcation is subcritical and the bifurcation periodic solutions are orbitally stable.

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Fig. 15. Patterns of u and v for a ¼ 15; r ¼ 8; l ¼ 1:2; b ¼ 2, d ¼ 4:5.

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Fig. 16. Patterns of u and v for a ¼ 15; r ¼ 8; l ¼ 2; b ¼ 2, d ¼ 4:5.


Example 7. In system (2), taking a ¼ 15, r ¼ 8; d ¼ 2; b ¼ 1:2 and l ¼ 0:9. According to calculating,b� ¼ 11

12 ;ab

13a2þ5�4affiffiffiffiffiffiffiffiffiffiffiffiffiffi10ða2þ1Þp ¼ 1:8, 3a2�5


Þ¼ 1:375; H1 ¼ 0:6634 and H2 ¼ 1:3566. Clearly, b > b�; ðA1Þ and ðA4Þ

hold. 9 n ¼ 1; m ¼ 0, s.t. n; m; l satisfy H1 <n2þm2

l2< H2, which implies that ðA8Þ holds. Following from Theorem 6, the system

(2) is Turing unstable and generates spatial steady state pattern shown as Fig. 8 while the corresponding ODE system is stable as illustrated in Fig. 9. The calculations are performed on zero flux boundary condition and a two dimensional grid 60� 60 with grid spacings Dx ¼ Dy ¼ 0:5 and time step Dt ¼ 0:001. Fig. 8 shows pattern with mixed polygons and stripes.In Figs. 10–13, the stripes in the pattern are more obvious, in contrast, polygons become less and less for the value of diffusion coefficient d is increasing and the values of a; b; r; l are fixed. In Figs. 14–16, varying the value of l, the stripes in the pattern are reducing while blobs and polygons are increasing, it implies that adjusting the boundary of x or y can influencethe shape of the pattern.

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Fig. 17. Patterns of u and v for a ¼ 15; r ¼ 8; l ¼ 0:9, b ¼ 1:2; d ¼ 2; a1 ¼ �4; a2 ¼ 2; s ¼ 0:2.

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Fig. 18. Patterns of u and v for s ¼ 3, all other parameter values are same as in Fig. 17.


In order to show the influence of parameters over the Turing pattern form better and intuitively, we consider the following system with delay

ut ¼ @2u@x2 þ @2u

@y2 þ a� u� 4uv1þu2 þ a1ðuðt � s; xÞ � u�Þ;

v t ¼ rd @2v@x2 þ @2v

@y2

� �þ rb u� uv

1þu2

� �þ a2ðvðt � s; xÞ � v�Þ;

8<: ð25Þ

where ðu�;v�Þ ¼ ða5 ;1þ a2

25Þ; a1 and a2 are parameters. The differenc e between system (2) and system (25) is the time delay.Considering system (25), we take a ¼ 15; r ¼ 8; d ¼ 2; b ¼ 1:2; l ¼ 0:9; a1 ¼ �4; a2 ¼ 2; s ¼ 0:2, polygons pattern obtained in Fig. 17. Because of s, the formation of Fig. 17 is quite different form Fig. 8 with more regular polygons and stripes.Therefore, the appearance of delay can change the pattern formation. We consider different s, all other parameter values are

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Fig. 19. Patterns of u and v for s ¼ 10, all other parameter values are same as in Fig. 17.


same as in Fig. 17, the patterns are shown in Figs. 18 and 19. It is obvious that the formatio n of the three figures is completely different, we observe that the range of s has great effect on the pattern formation.

5. Conclusion

The paper analyzes local stability, existence and direction of Hopf bifurcation and Turing instability of 2-D L–Ereaction–diffusion system with homogeneous Neumann boundary condition in detail. Some sufficient condition s ensuring the stability and bifurcation are provided by using stability and bifurcatio n theory. We extend or improve the previously known results. Numerical examples are given to show the effectiven ess of the obtained results.

Acknowled gments

The work is supported by National Natural Science Foundation of China under Grants 61174155 and 11032009. The work is also sponsored by Qing Lan Project of Jiangsu.

References

[1] P. Grindrod, Patterns and Waves , Clarendon Press, Oxford, 1991 .[2] M.C. Cross, P.C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65 (1993) 851–1112.[3] J.D. Murray, Mathematical Biology , Springer, Berlin, 1993 .[4] A. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. Ser. 237 (641) (1952) 37–72.[5] V. Castets, E. Dulos, J. Boissonade, P. De Kepper, Experimental evidence of a sustained Turing-type equilibrium chemical pattern, Phys. Rev. Lett. 64

(1990) 2953–2956.[6] I. Lengyel, I.R. Epstein, Modeling of Turing structures in the chlorite–iodide–malonic acid-starch reaction system, Science 251 (1991) 650–652.[7] I. Lengyel, I.R. Epstein, A chemical approach to designing Turing patterns in reaction–diffusion system, Proc. Natl. Acad. Sci. USA 89 (1992) 3977–3979.[8] T.K. Callahan, E. Knobloch, Pattern formation in three-dimensional reaction–diffusion system, Physica D 132 (1999) 339–362.[9] O. Jensen, E. Mosekilde, P. orckmans, G. Dewel, Computer simulations of Turing structures in the chlorite–iodide–malonic-acid system, Phys. Scr. 53

(1996) 243–251.[10] S.L. Judd, M. Silble, Simple and superlattice Turing patterns in reaction–diffusion system: bifurcation, bistability and parameter collapse, Physica D 136

(2000) 45–65.[11] D.J. Wellkind, L.E. Stephenson, Chemical Turing pattern formation analysis: comparison of theory with experiment, SIAM J. Appl. Math. 61 (2000) 387–

431.[12] W.M. Ni, M.X. Tang, Turing patterns in the Lengyel–Epstein system for the CIMA reaction, Am. Math. Soc. 357 (2005) 3953–3969.[13] J. Jang, W.M. Ni, M.X. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyal–Epstein model, J. Dyn. Differ. Equat. 16 (2004) 297–

320.[14] F.Q. Yi, J.J. Wei, J.P. Shi, Global asymptotical behavior of the Lengyel–Epstein reaction–diffusion system, Appl. Math. Lett. 22 (2009) 52–55.[15] L.L. Du, M.X. Wang, Hopf bifurcation analysis in the 1-D Lengyel–Epstein reaction–diffusion model, J. Math. Anal. Appl. 366 (2010) 473–485.[16] F.Q. Yi, J.J. Wei, J.P. Shi, Diffusion-driven instability and bifurcation in the Lengyel–Epstein system, Nonlinear Anal.: Real World Appl. 9 (2008) 1038–

1051.[17] S. Kar, J.K. Bhattacharjee, D.D. Ray, A model reaction–diffusion system under spatial perturbation: theoretical and numerical investigation, Eur. Phys. J.

B 43 (2005) 109–114.

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[18] W. Han, Y.G. Kao, L.S. Wang, Global exponential robust stability of static interval neural networks with S-type distributed delays, J. Franklin Inst. 348 (2011) 2072–2081.

[19] Q.K. Song, J.D. Cao, Global exponential robust stability of Cohen–Grossberg neural network with time-varying delays and reaction–diffusion terms, J.Franklin Inst. 343 (2006) 705–719.

[20] J.L. Qiu, J.D. Cao, Delay-dependent exponential stability for a class of neural networks with time delays and reaction–diffusion terms, J. Franklin Inst.346 (2009) 301–314.







hopf bifurcation and turing instability of 2-d lengyel–epstein system with reaction–diffusion...

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