Delay-driven spatial patterns in a plankton allelopathic systemCanrong Tian Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 22, 013129 (2012); doi: 10.1063/1.3692963 View online: http://dx.doi.org/10.1063/1.3692963 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/22/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spatially dependent parameter estimation and nonlinear data assimilation by autosynchronization of a system ofpartial differential equations Chaos 23, 033101 (2013); 10.1063/1.4812722 Traveling wave governs the stability of spatial pattern in a model of allelopathic competition interactions Chaos 22, 043136 (2012); 10.1063/1.4770064 Control of spatiotemporal patterns in the Gray–Scott model Chaos 19, 043126 (2009); 10.1063/1.3270048 Controlling the onset of traveling pulses in excitable media by nonlocal spatial coupling and time-delayedfeedback Chaos 19, 015110 (2009); 10.1063/1.3096411 Pursuit-evasion predator-prey waves in two spatial dimensions Chaos 14, 988 (2004); 10.1063/1.1793751

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Delay-driven spatial patterns in a plankton allelopathic system

Canrong Tiana)

Department of Basic Sciences, Yancheng Institute of Technology, Yancheng 224003, China

(Received 26 September 2011; accepted 14 February 2012; published online 13 March 2012)

Spatial patterns have received considerable attention in the physical, biological, and social

sciences. Generally speaking, time delay is a prevailing phenomenon in aquatic environments,

since the production of allelopathic substance by competitive species is not instantaneous, but

mediated by some time lag required for maturity of species. A natural question is how delay affects

the spatial patterns. Here, we consider a delayed plankton allelopathic system consisting of two

competitive species and analytically investigate how the time delay affects the stability and spatial

patterns. Based upon a stability analysis, we demonstrate that the delay can induce spatial patterns

under some conditions. Moreover, by use of a series of numerical simulations performed with a

finite difference scheme, we show that the delay plays an important role on pattern selection.VC 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3692963]

In the field of physics, it is shown that the delay may

induce spatial patterns in reaction diffusion system. In ec-

ological system, the effect of time delay is inevitable in

aquatic environments since the production of allelopathic

substance by the competitive species is not instantaneous,

but mediated by some time lag required for maturity of

the species. By the analysis of a real plankton allelopathic

model, we show that the delay induced spatial patterns

can occur in biology.

I. INTRODUCTION

Spatial pattern formation arose from the observation in

chemistry by Turing1 that diffusion can also destabilize equi-

librium solution, a scenario well known as Turing instability.

Although this is counter-intuitive, the interest in the diffusion-

driven instability has long expanded from chemical system to

biological system.2,3 Adding diffusion to a planktonic system,

Levin and Segel4 theoretically demonstrated that diffusion

plays an important role in generating spatial patterns. The

method developed in their paper turned out to be the routine

framework for the study of spatial patterns and has been

applied extensively.5–12 In Refs. 13 and 14, the phenomenon

of spatial patterns has been confirmed to actually occur in

some closed systems. However, there is still no clear experi-

mental evidence to show when or how spatial patterns occur

under natural biological conditions (see Refs. 2 and 3) for a

review).

Time delay is a ubiquitous phenomenon in biological

systems and has an important role in affecting population

dynamics. In a system described by ordinary differential

equations, time delay could change qualitatively the nature

of the equilibrium from a stable equilibrium to an unstable

one and thus induces bifurcation.15,16 In a chemical system

with reaction-diffusion, it shows that short time delay

beyond a critical threshold is also able to trigger spatial-

temporal instability.17 A question arises what is the role of

time delay in the formation of spatial patterns in a biologi-

cal system.

To this end, we consider a reaction-diffusion plankton

allelopathic system,

@u1

@t�d1Du1¼u1ða1�b11u1�b12u2þe1u1u2Þ; ðx;tÞ2XT ;

@u2

@t�d2Du2¼u2ða2�b21u1�b22u2þe2ðu1Þsu2Þ; ðx;tÞ2XT ;

@u1

@g¼@u2

@g¼0; ðx;tÞ2RT ;

u1ðx;tÞ¼w1ðx;tÞ; u2ðx;0Þ¼w2ðxÞ; ðx;tÞ2Xs;

8>>>>>>>>>><>>>>>>>>>>:

(1.1)

where XT :¼ X� ð0; TÞ;RT :¼ ð@XÞ � ð0; TÞ for a fixed

T > 0, Xs ¼ X� ½�s; 0�. Here, u1 and u2 are the population

densities (number of cells per liter) of two competing spe-

cies; a1 and a2 are the rates of cell proliferation per hour,

while b11 and b22 are the rates of intra-specific competition

of the first and the second species, respectively; we denote

b12 and b21 by the rates of inter-specific competition of the

first and the second species, respectively, and ai

biiði ¼ 1; 2Þ

are environmental expressing capacities (representing the

number of cells per liter). Here, e1 and e2 are the rates of

allelopathic stimulation of the first species by the second

and vice versa, respectively. di (i¼ 1, 2) are the diffusion

coefficients. ðu1Þs � u1ðx; t� sÞ; s is a positive constant,

which means that the production of allelopathic substance

by the competitive species will not be instantaneous, but

mediated by some time lag required for maturity of the spe-

cies. When both u1 and u2 have the time lags, the mathe-

matical method and results are similar to that in this paper.

The homogeneous Neumann boundary condition biologi-

cally indicates that there is no population flux across the

boundary.a)Electronic mail: unfoxeses@yahoo.com.cn.

1054-1500/2012/22(1)/013129/7/$30.00 VC 2012 American Institute of Physics22, 013129-1

CHAOS 22, 013129 (2012)

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Motivation of studying the model (1.1) goes as follows.

In aquatic environment, allelopathy may provide a competi-

tive advantage to angiosperms, algae, or cyanobacteria in

their interaction with other primary producers. Field evi-

dence and laboratory studies indicate that allelopathy occurs

in all aquatic habitats (marine and freshwater).18,19 Allelop-

athy includes not only inhibiting interactions, but also stimu-

lating interactions. Based on the two species, Lotka-Volterra

competitive model, Chattopadhyay20 first constructed a set of

differential equations to model plankton allelopathic system,

where each species produces a substance toxic to the other,

but only when the other is present. In Refs 21 and 22, addition

of time delay to the differential equations, a stable limit cycle

induced by time delay was observed. Model (1.1) is built21,22

by introducing diffusion to account for spatial effect.23

The goal of this paper is to explore whether time delay

can drive the emergence of Turing pattern and how different

delay affects the spatially inhomogenous distribution of pop-

ulation density. Our analyses reveal that the delay may

induce spatial patterns and different delay may lead to pat-

tern transition between spots and stripes.

The paper is organized as follows. In Sec. II, we ana-

lyze, from the mathematical point of view, the role of delay

in the generation of spatial patterns, and then derive the con-

ditions for these patterns to generate. In Sec. II, a finite dif-

ference formulation for approximating the governing

equations and some numerical experiments are demonstrated

to confirm our theoretical findings. In addition, we show that

if the delay increases, the selection of spatial patterns con-

verge from the spots to the stripes. The paper ends with

Sec. IV for some discussions.

II. DELAY DRIVEN SPATIAL PATTERNS

In this section, we derive the conditions for spatial pat-

terns to occur. In particular, we show that when the delay is

absent, the problem (1.1) does not generate spatial patterns,

while, in the presence of delay, the formation of spatial pat-

terns is induced.

A. Linear stability analysis

It is easy to know that the uniform equilibrium of the

system (1.1) satisfies the following equations:

u1ða1 � b11u1 � b12u2 þ e1u1u2Þ ¼ 0;

u2ða2 � b21u1 � b22u2 þ e2u1u2Þ ¼ 0:

(

From a routine algebraic computation, sufficient conditions

can be obtained for the existence of the positive uniform

equilibrium of system (1.1).

Lemma 2.1: In system (1.1), suppose that the parameterssatisfy the following hypothesis ðH1Þ:ðH1Þ One of the conditions (2.1) and (2.2) is true.

e1

e2

<b12

b22

<a1

a2

<b11

b21

and

a2e1 þ b12b21 < a1e2 þ b11b22; (2.1)

b12

b22

<a1

a2

<b11

b21

<e1

e2

and

a1e2 þ b12b21 < a2e1 þ b11b22: (2.2)

Then the system (1.1) admits a unique positive uniform equi-librium u� ¼ ðu�1; u�2Þ, where

u�i ¼ �qij �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2

ij � 4pijrij

q=2pij

� �; i; j ¼ 1; 2;

pij ¼ �bijei þ biiej; qij ¼ �aiej þ ajei � biibjj þ bijbji;

rij ¼ aibjj � ajbij: (2.3)

Proof: Assume that u� ¼ ðu�1; u�2Þ is a positive uniform equi-

librium of Eq. (1.1). Then u� ¼ ðu�1; u�2Þ is a solution of the

following system of quadratic equation:

a1 � b11u�1 � b12u�2 þ e1u�1u�2 ¼ 0;

a2 � b21u�1 � b22u�2 þ e2u�1u�2 ¼ 0:

((2.4)

By eliminating the term u�1u�2, we have the following equiva-

lent system of Eq. (2.4), where the two variables are inde-

pendent upon each other.

pijðu�i Þ2 þ qiju

�i þ rij ¼ 0; i; j ¼ 1; 2; i 6¼ j; (2.5)

where pij; qij, and rij are defined in Eq. (2.3).

Since Eqs. (2.1) and (2.2) possess the same conditionb12

b22< a1

a2< b11

b21, we can verify that rij > 0. Also we can check

that p12 > 0 and p21 > 0 cannot hold, because p12 > 0 and

p21 > 0 imply b11

b21< a1

a2< b12

b22, which is a contradiction to

b12

b22< a1

a2< b11

b21. Hence in order to ensure that u� ¼ ðu�1; u�2Þ is

the unique positive solution of Eq. (2.4), it suffices to show

that

p12 < 0; p21 > 0 and q21 < 0 (2.6)

or

p12 > 0; q12 < 0 and q21 < 0: (2.7)

After a direct calculation, Eqs. (2.1) and (2.2), respectively,

ensure that Eqs. (2.6) and (2.7) are valid. Thus the proof is

completed. h

Next, we shall carry out the linear stability analysis of

Eq. (1.1). We set u ¼ u1 � u�1; v ¼ u2 � u�2, and substitute

them in Eq. (1.1). Retaining the linear terms in u and v gives

rise to

@u

@t� d1Du¼ AuþBv; ðx; tÞ 2XT ;

@v

@t� d2Dv¼CuþDvþEuðt� sÞ; ðx; tÞ 2XT ;

@u

@g¼ @v@g¼ 0; ðx; tÞ 2RT ;

uðx; tÞ ¼w1ðx; tÞ� u�1; vðx; tÞ ¼w2ðx; tÞ� u�2; ðx; tÞ 2Xs;

8>>>>>>>>>><>>>>>>>>>>:

(2.8)

013129-2 Canrong Tian Chaos 22, 013129 (2012)

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where

A ¼ �u�1ðb11 þ e1u�2Þ; B ¼ �u�1ðb12 þ e1u�1Þ;C ¼ �u�2b21; D ¼ �u�2ðb22 þ e2u�1Þ; E ¼ �ðu�2Þ

2e2:

(2.9)

Since the boundary condition is homogeneous Neumann on

the domain X, the appropriate eigenfunction of Eq. (2.8) is

ðu; vÞ ¼ ðc1; c2Þekt cos kx; (2.10)

where k is the eigenvalue and k is the wavenumber. Substitu-

tion of this form in Eq. (2.8) yields

ðkc1þ d1k2c1Þekt cos kx¼ ðAc1þBc2Þekt cos kx;

ðkc2þ d2k2c2Þekt cos kx¼ ðCc1þDc2þEe�ksc1Þekt cos kx:

(

(2.11)

Since ekt cos kx 6� 0, Eq. (2.11) is equivalent to the following

set of linear algebraic equations:

k� Aþ d1k2 �B�C� Ee�ks k� Dþ d2k2

� �c1

c2

� �¼ 0

0

� �: (2.12)

Nontrivial solutions to Eq. (2.12) exist if and only if

detk� Aþ d1k2 �B�C� Ee�ks k� Dþ d2k2

� �¼ 0: (2.13)

Denoting

Dðk; sÞ ¼D k2 � ðAþ D� ðd1 þ d2Þk2Þk

þ ð�Aþ d1k2Þð�Dþ d2k2Þ � BC� BEe�ks;

Eq. (2.13) is equivalent to the following equation:

Dðk; sÞ ¼ 0: (2.14)

Equation (2.14) is always named by the characteristic equa-

tion of the delayed reaction diffusion system (1.1). If we

denote k ¼ rþ ix by the complex eigenvalues of Eq. (2.14),

the nonlinear system has asymptotic behavior as follows:

(i) If r � 0, then the system (1.1) is stable.

(ii) If r > 0, then the system (1.1) is unstable.

Thus, spatial patterns generate if the critical point s sat-

isfies r ¼ 0, which is called delay driven spatial patterns.

Moreover the critical value of the delay s is called the Turing

bifurcation. In fact the solution of Eq. (2.8), which is the

time evolution of the perturbation, is the sum of the all

appropriate eigenfunctions in the form,

ðu; vÞ ¼Xðc1; c2Þekt cos kx;

where k is a set of discrete nonnegative values and k!1.

Observing that, near a critical point s, most of the eigenvalues

have negative real part, there might be a set of eigenfunctions

whose eigenvalues have very small real part, called marginal.

This is an extension of Center Manifold Theorem from diffu-

sion driven spatial patterns to delay driven spatial patterns.

In this paper, we assume that the hypothesis ðH2Þ holds

ðH2Þ B < 0; Aþ D < 0; BE < AD� BC < �BE:

Then we have the results on delay driven spatial patterns.

Theorem 2.1. If the system (1.1) satisfies the hypothesisðH1Þ and ðH2Þ, then the delay can induce spatial patterns.

(i) If the delay is absent, that is s ¼ 0, then the positiveequilibrium u� of Eq. (1.1) is asymptotic stable.

(ii) If the delay is present, that is s 6¼ 0, then there existsa critical point s�, when s � s�, the positive equilib-rium u� of Eq. (1.1) is unstable.

Proof: (i) We first show that when s ¼ 0, there is no

spatial pattern. From the above argument, it is sufficient to

show all the roots of Dðk; 0Þ have the negative real parts. We

denote the following functions depending on k:

�AðkÞ ¼ �ðAþ DÞ þ ðd1 þ d2Þk2;

�BðkÞ ¼ ð�Aþ d1k2Þð�Dþ d2k2Þ � BC;

�CðkÞ ¼ �BE:

(2.15)

Then Dðk; sÞ is written in the form,

Dðk; sÞ ¼ k2 þ �AðkÞkþ �BðkÞ þ �CðkÞe�ks: (2.16)

Thus

Dðk; 0Þ ¼ k2 þ �AðkÞkþ �BðkÞ þ �CðkÞ:

In view of the hypothesis ðH2Þ, it is easy to verify that�AðkÞ > 0 and �BðkÞ þ �CðkÞ > 0, which shows that the real

parts of the roots are negative.

(ii) By the use of the instability result for the delayed

differential equations owing to Ref. 24, in order to prove the

instability of the uniform equilibrium, it is sufficient to show

that there exists pure imaginary ix and positive real s such

that Dðix; sÞ ¼ 0.

If ix is a root of (2.16), then we have

x2 � �BðkÞ ¼ �CðkÞ cos xs;

x �AðkÞ ¼ �CðkÞ sin xs;

((2.17)

which leads to

x4 þ ð �AðkÞ2 � 2 �BðkÞÞx2 þ �BðkÞ2 � �CðkÞ2 ¼ 0; (2.18)

where

�AðkÞ2 � 2 �BðkÞ ¼ ð�Aþ d1k2Þ2

þ ð�Dþ d2k2Þ2 þ 2 �BðkÞ �CðkÞ;�BðkÞ2 � �CðkÞ2 ¼ ðAD� BC� ðDd1 þ Ad2Þk2 þ d1d2k4Þ2

� ð�BEÞ2: (2.19)

013129-3 Delay-driven spatial patterns Chaos 22, 013129 (2012)

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In view of the hypothesis ðH2Þ, it is easy to verify that �BðkÞ2 � �CðkÞ2 < 0, which leads to the fact that Eq. (2.18) has a unique

positive real root,

x ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� �AðkÞ2 þ 2 �BðkÞ þ ½ð �AðkÞ2 � 2 �BðkÞÞ2 � 4ð �BðkÞ2 � �CðkÞ2Þ�1=2

2

s; (2.20)

and the equation Dðix; sÞ ¼ 0 has pure imaginary root ixwhen

s ¼ s� þ 2jpk; j ¼ 0; 1; 2 …; s� ¼

arccosx2 � �BðkÞ

�CðkÞx

:

(2.21)

In view of the instability theorem of Ref. 24, we conclude

that if s � s�, then ðu�1; u�1Þ is unstable. Thus the proof is

completed. h

B. Turing parameter space

In view of Theorem 2.1, the fulfillments of the hypothe-

sis ðH1Þ and ðH2Þ are sufficient for the positive uniform equi-

librium ðu�1; u�2Þ being linearly unstable with respect to the

particular case of system (1.1). In order to the numerical sim-

ulations, we take the following values:

a1 ¼ 2; a2 ¼ 1; b11 ¼ 0:07; b12 ¼ 0:05;

b21 ¼ 0:015; b22 ¼ 0:08; e1 ¼ 0:0008; e2 ¼ 0:003;

d1 ¼ 0:01; d2 ¼ 0:01: (2.22)

For this particular choice, the positive uniform equilibrium is

given by

ðu�1; u�2Þ ¼ ð15:97; 23:697Þ: (2.23)

Here we notice that these parameters (the original data given

in Ref. 21) are not the actual values from experimental

observations, but they have the theoretical biological senses.

a1 ¼ 2 and a2 ¼ 1 is that the two species, respectively,

reproduce 2 and 1 cell division per hour. The carrying

capacities a1

b11¼ 28:57 and a2

b22¼ 12:5 are reasonable because

the theoretical maximum densities of the two species are

approximately 14 000 and 6000 cells per liter, respectively.

Other parameters such as intra-species competition coeffi-

cients bii, inter-species competition coefficients bij, and alle-

lopathic coefficients ei are also appropriate to plankton

allelopathy model.

In Figure 1 we depict the parameter spaces where insta-

bilities are expected to appear according to conditions ðH1Þand ðH2Þ above. These graphics are obtained by fixing all pa-

rameters in Eq. (2.22) except for a1 and a2 (Figure 1, top

left), b11 and b21 (Figure 1, top right), b12 and b22 (Figure 1,

bottom left), and e1 and e2 (Figure 1, bottom right).

Similar to Ref. 25, we calculate the wavenumber explic-

itly and determine the pattern selection by the linearization

around the positive uniform equilibrium and taking s as the

Turing bifurcation parameter. From the mathematical view-

point, the Turing bifurcation occurs when ImðkðkÞÞ ¼ 0 and

ReðkðkÞÞ ¼ 0 at k � kc 6¼ 0, where kc is the critical wave-

number and kðkÞ is the root of the characteristic Eq. (2.14).

The characteristic equation is

detk� Aþ d1k2 �B�C� Ee�ks k� Dþ d2k2

� �¼ 0:

If kc is the critical point of the wavenumber and sc is the

Turing bifurcation threshold, the similar argument in Theo-

rem 2.1 leads to

x4 þ ð �AðkÞ2 � 2 �BðkÞÞx2 þ �BðkÞ2 � �CðkÞ2 ¼ 0:

In order to find out the critical wavenumber, we only need to

confirm that

mink2ðd1d2k4 � ðDd1 þ Ad2Þk2 þ AD� BCþ BEÞ ¼ 0;

which is a quadratic polynomial of k2. So we can assume the

critical point k2 in the form,

k2c ¼

Dd1þAd2þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðDd1þAd2Þ2�4d1d2ðAD�BCþBEÞ

q2d1d2

(2.24)

and

xðkcÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� �AðkcÞ2 þ 2 �BðkcÞÞ þ ½ð �AðkcÞ2 � 2 �BðkcÞÞ2 � 4ð �BðkcÞ2 � �CðkcÞ2Þ�1=2

2

s:

In this way, the Turing bifurcation threshold given by sc, sat-

isfies the following explicit function:sc ¼ arccos

xðkcÞ2 � �BðkcÞ�CðkcÞ

�xðkcÞ: (2.25)

013129-4 Canrong Tian Chaos 22, 013129 (2012)

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III. NUMERICAL RESULTS

In this section, using the difference method, we give

some numerical results based on the formulae in Sec. II. The

domain of Eq. (1.1) is confined to a square domain

X ¼ ½0; L� � ½0; L� R2. The wavenumber for this two-

dimensional domain is thereby

k ¼ 2pðm=L; n=LÞ;

jkj ¼ 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðm=LÞ2 þ ðn=LÞ2

q; m; n ¼ 0; 1;…:

We consider system (1.1) in a fixed domain L¼ 60 and

solve it on a grid with 100� 100 sites by a simple Euler

method with a time step of Dt ¼ 0:001, and by discretizing

the Laplacian in the grid with lattice sites denoted by (i, j).The form is

Dujði;jÞ ¼1

s2½alði; jÞuði� 1; jÞ þ arði; jÞuðiþ 1; jÞ

þ adði; jÞuði; j� 1Þ þ auði; jÞuði; jþ 1Þ � 4uði; jÞ�;(3.1)

FIG. 1. Turing parameters’ space for model (1.1). Within the grey zones, Turing instabilities are expected to appear.

FIG. 2. (Color online) Comparison of solution u1 of the system (1.1) between the delay s ¼ 0 (left) and s ¼ 2:5 (right). All the parameters are given in Eq.

(2.23). The step of time iteration is 2 000 000.

013129-5 Delay-driven spatial patterns Chaos 22, 013129 (2012)

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where s is the lattice constant and the matrix elements of

al; ar; ad; au are unity except at the boundary. When (i,j) is at

the left boundary, that is i¼ 0, we define alði; jÞuði� 1; jÞ� uðiþ 1; jÞ, which guarantees zero-flux of reactants in the

left boundary. Similarly we define arði; jÞ, adði; jÞ; auði; jÞsuch that the boundary is no-flux.

According to Eqs. (2.25) and (2.24), we obtain the

Turing bifurcation threshold and the corresponding critical

wavenumber,

sc ¼ 2:4511 and kc ¼ 5:6019:

The initial data is taken as a uniformly distributed random

perturbation around the equilibrium ðu�1; u�2Þ in X, with a var-

iance lower than the amplitude of the final patterns. More

precisely,

u1ðx; tÞ ¼ u�1 þ g1ðx; tÞ; u2ðx; 0Þ ¼ u�2 þ g2ðxÞ; (3.2)

where g1ðx; tÞ 2 ½�2; 2� and g2ðxÞ 2 ½�2; 2�.The numerical results indicate that the positive uniform

equilibrium is stable for s ¼ 0 (Figure 2, left) and unstable

FIG. 3. (Color online) Pattern selection and

transient patterns. From left to right: Spatial

patterns of species u1 (left) and u2 (right).

From top to bottom: Snapshots in the differ-

ent delay s ¼ 2:5 (top), s ¼ 5 and s ¼ 7:5(middle), s ¼ 10 (bottom). All the parame-

ters are given in Eq. (2.22). The step of time

iteration is 2 000 000.

013129-6 Canrong Tian Chaos 22, 013129 (2012)

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for s > s� (Figure 2, right). Figure 2 indicates that the left so-

lution is homogeneous and the right one is inhomogeneous,

that means when the delay is absent, the system (1.1) cannot

induce spatial pattern.

Now by the test of numerical simulations, we show that

the selection of different patterns depends on the value of the

delay s. All the other parameters are given in Eq. (2.22). In

Figure 3 we show the spatial patterns for the different delay sfor system (1.1). It is clear that spatial patterns arise accord-

ing to Figure 3. Of course, the particular shapes and fre-

quency of these patterns are also influenced by the initial

data, but qualitatively speaking, we see that spotted patterns

ðs ¼ 2:5Þ and striped patterns ðs ¼ 5; s ¼ 7:5 and 10) appear.

Therefore, we find that when the delay s increases, the modes

of the patterns convert from the spots to the stripes.

IV. DISCUSSION

In this paper, we have developed a theoretical framework

for studying the phenomenon of pattern formation in a 2D

reaction-diffusion system with time delay. Applying a stabil-

ity analysis and suitable numerical simulations, we investigate

the Turing parameter space, the Turing bifurcation, and the

pattern selection. We have shown that the time delay can lead

to the spatial pattern. The stability of the positive uniform

equilibrium is determined in the Turing parameter space ðH1Þand ðH2Þ. In a biological sense, the existence of stability

switch induced by the delay is found in the region of the

Turing space of the large interactions of the allelopathic

stimulations.

Numerical studies have been employed to support and

extend the obtained theoretical results. For fixed interac-

tions of the allelopathic stimulations, the numerical simula-

tions illustrate the existence of both stable and unstable

equilibrium near the critical point of the delay which is in

good agreement with our theoretical analysis results. For

the time delay far away from the critical point, our theoreti-

cal analysis is not available. However, numerical simula-

tions and Figure 3 show that with increasing the delay from

the Hopf bifurcation, the selection of the spatial pattern

transform from spots to stripes.

We introduce the delay in the plankton allelopathic sys-

tem, and show that the delay induces the Turing patterns.

By the analysis of numerical simulations, we predict that

the delay induces the spatio-temporal pattern, which is sim-

ilar to the temporal-spatial periodic fluctuation observed in

aquatic environment.19 Our study of partial differential

equations (PDE) model has to simplify the real ecological

processes, which may not always be satisfactory. However,

the mathematical model can help to clarify patterns of spe-

cies interactions and guide further experiments. The pro-

posed approach has applicability to other reaction-diffusion

systems including delays, such as predator-prey model and

mutualistic model.

ACKNOWLEDGMENTS

The work is partially supported by PRC Grant NSFC

10801115.

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