Spatial dynamics in a predator-prey model with herd behavior

Sanling Yuan,1,a) Chaoqun Xu,1 and Tonghua Zhang2,b)

1College of Science, University of Shanghai for Science and Technology, Shanghai 200093,People’s Republic China2Mathematics, H38, FEIS, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia

(Received 5 April 2013; accepted 18 June 2013; published online 8 July 2013)

In this paper, a spatial predator-prey model with herd behavior in prey population and quadratic

mortality in predator population is investigated. By the linear stability analysis, we obtain the

condition for stationary pattern. Moreover, using standard multiple-scale analysis, we establish the

amplitude equations for the excited modes, which determine the stability of amplitudes towards

uniform and inhomogeneous perturbations. By numerical simulations, we find that the model

exhibits complex pattern replication: spotted pattern, stripe pattern, and coexistence of the two. The

results may enrich the pattern dynamics in predator-prey models and help us to better understand the

dynamics of predator-prey interactions in a real environment. VC 2013 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4812724]

For ecological systems, one of their main characters is

the relationship between different species and their living

environment and thus modelling predator-prey interac-

tions is one of the important issues in mathematical biol-

ogy. For the classical predator-prey models, the crucial

components are the growth function of prey species in

the absence of predator, the mortality function of preda-

tor species in the absence of prey and the functional

response function of the predator to the prey. Recently,

Braza19 considered a predator-prey system in which the

prey exhibits herd behavior and the predator interacts

with the prey along the outer corridor of the herd of

prey. In this case, the response function is appropriate in

terms of the square root of the number of prey species.

Moreover, for intermediate predators (such as pisci-

vores), the mortality term is suited to quadratic mortal-

ity. In this paper, we investigate a spatial predator-prey

model with herd behavior in prey population and quad-

ratic mortality in predator population and find that this

model exhibits complex pattern replication: spotted

pattern, stripe pattern and coexistence of the two.

Furthermore, we deduce that the Turing pattern is

induced by quadratic mortality. Hopefully, this work will

provide us further understanding the dynamics of

predator-prey interactions in a real environment.

I. INTRODUCTION

Predation is one of the most important types of interac-

tion which has effects on population dynamics of all species.

As a result, predator-prey models have long been and will

continue to be one of the dominant themes in both ecosys-

tems and mathematical ecology.1,2

There are many factors which affect population dynam-

ics in predator-prey models. One crucial component of

predator-prey relationships is the functional response.

Generally, the functional response can be classified into

many different types: Holling I-III types,3,4 Hassell-Varley

type,5 Beddington-DeAngelis type,6,7 Crowley-Martin type8

and the modified forms of these types.9–17 Recently, a

predator-prey model is considered in which the prey exhibits

herd behavior, so that the predator interacts with the prey

along the outer corridor of the herd of prey, which is more

appropriate to model the response functions of prey that

exhibit herd behavior in terms of the square root of the prey

population.18,19

The other crucial component in the predator-prey mod-

els is the formulation of the mortality terms for the predator.

Usually, the predator mortality is described by either the

“linear mortality” version, or the “quadratic mortality”

version.20 There are many authors choose the quadratic mor-

tality which is suited to intermediate predators (such as

piscivores).2,20,21

Recently, many research scholars pointed out that spa-

tial mathematical model is an appropriate tool for investi-

gating fundamental mechanism of complex spatiotemporal

population dynamics.22–36 In their researches, reaction-

diffusion (RD) equations have been widely used to

describe the spatiotemporal dynamics. Since Turing37 first

proposed RD theory to describe the range of spatial

patterns observed in the developing embryo, RD models

have been studied extensively to explain pattern formation

in many areas.38–44 However, predator-prey models with

square root functional response and quadratic mortality

have received surprisingly little attention in the literature.

For such reason, in the present paper, we are intended to

study such model.

The organization of this paper is as follows. In Sec. II,

we introduce a spatial predator-prey model with Neumann

boundary conditions and give a general survey of the linear

stability analysis. Furthermore, we obtain Turing bifurcation

with Neumann boundary conditions. In Sec. III, we carry out

a nonlinear analysis using multiple-scale analysis to derive

the amplitude equations and also present a series of numeri-

cal simulations to reveal that there is a large variety of

a)Electronic address: sanling@usst.edu.cnb)Electronic address: tonghuazhang@groupwise.swin.edu.au

1054-1500/2013/23(3)/033102/10/$30.00 VC 2013 AIP Publishing LLC23, 033102-1

CHAOS 23, 033102 (2013)

Downloaded 14 Jul 2013 to 136.186.72.15. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions

different spatiotemporal dynamics in the spatial model.

Finally, conclusions and discussions are presented in

Sec. IV.

II. THE MODEL AND BIFURCATION ANALYSIS

A. The spatial model

The basic predator-prey model with logistic growth in

the prey and a square root response function is given by19

dX

dt¼ rX 1� X

K

� �� a

ffiffiffiffiXp

Y

1þ thaffiffiffiffiXp ;

dY

dt¼ �sY þ ca

ffiffiffiffiXp

Y

1þ thaffiffiffiffiXp ;

8>>>><>>>>:

(1)

where X(t) and Y(t) stand for the prey and predator den-

sities, respectively, at time t. The parameter r is the growth

rate of the prey, K is its carrying capacity, and s is the death

rate of the predator in the absence of prey. The parameter ais the search efficiency of Y for X, c is the conversion or

consumption rate of prey to predator, and th is average han-

dling time.

Following Refs. 2, 20, and 21, we choose the quadratic

mortality for predator population. Then model (1) will be

converted to the following form:

dX

dt¼ rX 1� X

K

� �� a

ffiffiffiffiXp

Y

1þ thaffiffiffiffiXp ;

dY

dt¼ �sY2 þ ca

ffiffiffiffiXp

Y

1þ thaffiffiffiffiXp ;

8>>>><>>>>:

(2)

where �sY2 represents the quadratic mortality for predator

population.

On the other hand, we assume that the prey and predator

population move randomly, described as Brownian motion.

Then, we propose a spatial model corresponding to Eq. (2)

as follows:

@X

@t¼ rX 1� X

K

� �� a

ffiffiffiffiXp

Y

1þ thaffiffiffiffiXp þ dXr2X;

@Y

@t¼ �sY2 þ ca

ffiffiffiffiXp

Y

1þ thaffiffiffiffiXp þ dYr2Y;

8>>>><>>>>:

(3)

where the nonnegative constants dX and dY are the diffusion

coefficients of X and Y, respectively. r2 ¼ @2

@R21

þ @2

@R22

is the

usual Laplacian operator in two-dimensional space R ¼ðR1;R2Þ which is used to describe the Brownian motion. In

general, to ensure that Turing pattern is determined by

reaction-diffusion mechanism, we choose the following non-

zero initial condition:

XðR; 0Þ > 0; YðR; 0Þ > 0; R 2 X ¼ ½0; L� � ½0; L�;

and Neumann (zero-flux) boundary condition

@X

@�¼ @Y

@�¼ 0;

where L denotes the size of the system in the directions of Xand Y, � is the outward unit normal vector of the boundary

@X.

In order to minimize the number of parameters involved

in model (3), some scaling needs to take place. Following

Ref. 19, the variables are scaled as

x ¼ 1

KX; y ¼ a

rffiffiffiffiKp Y; tnew ¼ rtold;

r1 ¼ffiffiffiffiffir

dX

rR1; r2 ¼

ffiffiffiffiffir

dX

rR2;

and the other parameters are made dimensionless as follows:

snew ¼ffiffiffiffiKp

asold; cnew ¼

affiffiffiffiKp

rcold;

a ¼ thaffiffiffiffiKp

; d ¼ dY

dX:

With these changes, Eq. (3) becomes

@x

@t¼ xð1� xÞ �

ffiffiffixp

y

1þ affiffiffixp þr2x;

@y

@t¼ �sy2 þ c

ffiffiffixp

y

1þ affiffiffixp þ dr2y:

8>>><>>>:

(4)

In Ref. 19, the author used the simplifying assumption

that a¼ 0, which implies that the average handling time is

zero. In line with the work, we also assume that a¼ 0. Then

the working equations are

@x

@t¼ xð1� xÞ �

ffiffiffixp

yþr2x;

@y

@t¼ �sy2 þ c

ffiffiffixp

yþ dr2y:

8>><>>: (5)

B. Linear stability analysis

The corresponding non-diffusive model has at most

three equilibriums, which consist of two boundary equili-

briums (0, 0), (1, 0) and a positive equilibrium ðx�; y�Þ,where

x� ¼ 1� c

s; y� ¼ c

s

ffiffiffiffiffix�p

:

It is clear that the condition for ensuring that x� and y� are

positive is that c < s.

From the biological point of view, we are interested to

studying the stability behavior of the positive equilibrium,

which corresponds to coexistence of prey and predator. The

Jacobian matrix corresponding to this equilibrium is as

follows:

A ¼a11 a12

a21 a22

!;

where

033102-2 Yuan, Xu, and Zhang Chaos 23, 033102 (2013)

Downloaded 14 Jul 2013 to 136.186.72.15. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions

a11 ¼3c

2s� 1; a12 ¼ �

ffiffiffiffiffix�p

; a21 ¼c2

2s; a22 ¼ �c

ffiffiffiffiffix�p

:

The Turing condition is the one in which the positive

steady state is stable for the non-diffusive model, but it is

unstable for the reaction-diffusion model.45 And the condi-

tion for the positive steady state to be stable for the corre-

sponding non-diffusive model is given by

a11 þ a22 < 0

and

a11a22 � a12a21 > 0:

C. Bifurcation analysis

First, we address the temporal stability of the uniform

state to nonuniform perturbations30,46

xy

� �¼ x�

y�

� �þ e

xk

yk

� �expfktþ ikrg þ c:c:þ oðe2Þ; (6)

where k is the growth rate of perturbations in time t, i is the

imaginary unit and i2 ¼ �1; k � k ¼ k2 and k is the wave

number, r ¼ ðr1; r2Þ is the spatial vector in two dimensions,

and c.c. stands for the complex conjugate. After substituting

Eq. (6) into Eq. (5), we can obtain the characteristic equation

of system (5), for the growth rate k, as follows:

ðA� kIÞ xy

� �¼ 0;

where

A ¼ a11 � k2 a12

a21 a22 � dk2

� �:

As a result, we have characteristic polynomial of the original

problem

k2k � trkkk þ Dk ¼ 0; (7)

where

trk ¼ a11 þ a22 � k2ð1þ dÞ ¼ tr0 � k2ð1þ dÞ;Dk ¼ a11a22 � a21a12 � ða11dþ a22Þk2 þ dk4

¼ D0 � ða11dþ a22Þk2 þ dk4:

The roots of Eq. (7) can be obtained by the following form:

kk ¼1

2trk 6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitr2

k � 4Dk

q� �:

We know that the Hopf bifurcation breaks the temporal

symmetry of a system, giving rise to oscillations that are per-

iodic in time and uniform in space. And the Turing bifurca-

tion breaks the spatial symmetry, leading to the formation of

patterns which are oscillatory in space and stationary in

time.29,47

Hopf bifurcation occurs when ImðkkÞ 6¼ 0; ReðkkÞ ¼ 0

at k¼ 0, i.e., a11 þ a22 ¼ 0. Then we can get the critical

value of the Hopf bifurcation parameter s

sH ¼c3 � 3cþ c2

ffiffiffiffiffiffiffiffiffiffiffiffiffic2 þ 3p

2ðc2 � 1Þ :

Turing bifurcation occurs when ImðkkÞ ¼ 0; ReðkkÞ ¼ 0 at

k ¼ kT 6¼ 0, and the wave number kT satisfies k2T ¼

ffiffiffiffiD0

d

q. We

can obtain the critical value of the Turing bifurcation param-

eter sT. The expression of sT is shown in Appendix B. At the

Turing threshold, the spatial symmetry of the system is

broken and gives rise to a form stationary in time and oscilla-

tory in space with the wavelength kT ¼ 2pkT

, see Refs. 29, 30,

and 47.

In Fig. 1, we show the Turing space in c – s plane for

model (5) with d ¼ 10, where

C1 : s ¼ c; C2 : 3c� 2s� 2cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisðs� cÞ

p¼ 0;

C3 :5ð3c� 2sÞ

s� c

ffiffiffiffiffiffiffiffiffiffiffi1� c

s

r� 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10c 1� c

s

� �32

s¼ 0:

C1 is positive equilibrium existence line (the black line), C2

is Hopf bifurcation line (the blue line), C3 is Turing bifurca-

tion line (the red line). For parameters in domain T, above

C2, the positive equilibrium of corresponding non-diffusive

model is stable; below C3, the corresponding solution of the

model (5) is unstable. That is to say, Turing instability

occurs, therefore Turing patterns emerge. This domain is

called as “Turing space.” We show the dispersion relation

corresponding to several values of bifurcation parameter swhile keeping the others fixed as d ¼ 10, c¼ 0.8 in Fig. 2.

In that figure, curve (2) corresponds to the critical value

s2 ¼ 0:9972, curve (4) corresponds to the other critical value

s4 ¼ 0:9263. The positive equilibrium of corresponding non-

diffusive model is unstable when s < s4 (e.g., curve (5) in

Fig. 2), the solution of the model (5) is stable when s > s2

(e.g., curve (1) in Fig. 2). When s4 < s < s2 (e.g., curve (3)

in Fig. 2), the Turing instability occurs.

FIG. 1. The Turing space (marked T) of model (5) with d ¼ 10. Area T

which is located above the positive equilibrium existence line (the black line

C1), and bounded by the Hopf bifurcation line (the blue line C2) and the

Turing bifurcation line (the red line C3).

033102-3 Yuan, Xu, and Zhang Chaos 23, 033102 (2013)

Downloaded 14 Jul 2013 to 136.186.72.15. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions

III. SPATIAL DYNAMICS OF MODEL (5)

A. Amplitude equations

It is pointed out that near a bifurcation, the evolution of

a dynamical system exhibits critical slowing down, which of-

ten admits a simplified description in terms of an amplitude

equation.48 It is also proved that slow modes may be

described in terms of amplitude equations even if the under-

lying bifurcation cannot be realized for a given system.

Notice that the eigenvalues associated with the critical

modes are close to zero when s is close to Turing bifurcation

sT, i.e., the critical modes are slow modes, then the whole dy-

namics can be reduced to the dynamics of the active slow

modes.39,49–52 In the following, we will deduce the ampli-

tude equations by using the standard multiple-scale analysis.

Close to onset s ¼ sT , the solution of model (5) can be

expanded as

X ¼ xy

� �¼ ~X �

X3

j¼1

½Aj expðikj � rÞ þ �Aj expð�ikj � rÞ�; (8)

where ~X ¼ ðf ; 1Þ0 is the eigenvector of the linearized opera-

tor. In other words, ~X defines the direction of the eigenmo-

des in concentration space (i.e., the ratio of x and y). Aj and

the conjugate �Aj are, respectively, the amplitudes associated

with the modes kj and �kj. By the analysis of the symme-

tries, up to the third order in the perturbations, the spatiotem-

poral evolutions of the amplitudes Ajðj ¼ 1; 2; 3Þ are

described through the amplitude equations

s0

@A1

@t¼ lA1 þ h �A2

�A3

�½g1jA1j2 þ g2ðjA2j2 þ jA3j2Þ�A1;

s0

@A2

@t¼ lA2 þ h �A1

�A3

�½g1jA2j2 þ g2ðjA1j2 þ jA3j2Þ�A2;

s0

@A3

@t¼ lA3 þ h �A1

�A2

�½g1jA3j2 þ g2ðjA1j2 þ jA2j2Þ�A3;

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

(9)

where l ¼ ðsT � sÞ=sT is a normalized distance to onset and

s0 is a typical relaxation time. Notably, for model (5), the

stationary state becomes Turing unstable when the bifurca-

tion parameter s decreases, so that l increases when the

bifurcation parameter s decreases.

Amplitude equation (9) allows us to study the existence

and stability of arrays of hexagons and strips. In order to obtain

the amplitude equations, we should first write the linearized

form of model (5) at the equilibrium point ðx�; y�Þ as follows:

@x

@t¼ a11xþa12yþ c

8sx��1

� �x2� 1

2ffiffiffiffiffix�p xy

� c

16sðx�Þ2x3þ 1

8

ffiffiffiffiffiffiffiffiffiffiðx�Þ3

q x2yþoðq3Þþr2x;

@y

@t¼ a21xþa22y� c2

8sx�x2þ c

2ffiffiffiffiffix�p xy� sy2

þ c2

16sðx�Þ2x3� c

8

ffiffiffiffiffiffiffiffiffiffiðx�Þ3

q x2yþoðq3Þþdr2y;

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

(10)

where q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p. Equation (10) can be converted to the

following form:

@

@tX ¼ LXþH; (11)

where L is the linear operator, H is the nonlinear term,

L ¼ a11 þr2 a12

a21 a22 þ dr2

� �;

H ¼

c

8sx�� 1

� �x2 � 1

2ffiffiffiffiffix�p xy� c

16sðx�Þ2x3

þ 1

8

ffiffiffiffiffiffiffiffiffiffiðx�Þ3

q x2yþ oðq3Þ

� c2

8sx�x2 � sy2 þ c

2ffiffiffiffiffix�p xyþ c2

16sðx�Þ2x3

� c

8

ffiffiffiffiffiffiffiffiffiffiðx�Þ3

q x2yþ oðq3Þ

0BBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCA

:

Near the Turing bifurcation threshold, we expand the

bifurcation parameter s in the following term:

sT � s ¼ es1 þ e2s2 þ e3s3 þ oðe3Þ;

where jej � 1. Equally, expanding the variable X and the

nonlinear term H according to this small parameter e, we

have the following results:

X ¼ ex1

y1

� �þ e2 x2

y2

� �þ e3 x3

y3

� �þ oðe3Þ; (12)

H ¼ e2h2 þ e3h3 þ oðe3Þ; (13)

where h2 and h3 are corresponding to the second and the third

orders of e in the expansion of the nonlinear term H. At the

same time, the linear operator L can be expanded as follows:

L ¼ LT þ ðsT � sÞM; (14)

FIG. 2. The relation between ReðkÞ (the real part of the eigenvalue k) and kwith d ¼ 10, c¼ 0.8 and different s. Curve (1): s1 ¼ 1:1000; Curve (2):

s2 ¼ 0:9972; Curve (3): s3 ¼ 0:9500; Curve (4): s4 ¼ 0:9263; Curve (5):

s5 ¼ 0:9000.

033102-4 Yuan, Xu, and Zhang Chaos 23, 033102 (2013)

Downloaded 14 Jul 2013 to 136.186.72.15. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions

where

LT ¼aT

11 þr2 aT12

aT21 aT

22 þ dr2

� �; M ¼ m11 m12

m21 m22

� �:

From the chain rule for differentiation, the derivative with

respect to time should convert to the following term:

@

@t¼ e

@

@T1

þ e2 @

@T2

þ oðe2Þ; (15)

where T1 ¼ et; T2 ¼ e2t.Using Eqs. (12)–(15), we can expand Eq. (11) into a per-

turbation series with respect to e. For the order e, we have

the following linear equation:

LTx1

y1

� �¼ 0: (16)

Similarly, for the order e2 and the order e3, we have the fol-

lowing equations:

LTx2

y2

� �¼ @

@T1

x1

y1

� �� s1M

x1

y1

� �� h2; (17)

LT

x3

y3

!¼ @

@T1

x2

y2

!þ @

@T2

x1

y1

!

�s1Mx2

y2

!� s2M

x1

y1

!� h3: (18)

Solving Eq. (16), we obtain

x1

y1

� �¼ f

1

� � X3

j¼1

Wj expðikj � rÞ þ c:c:

0@

1A; (19)

i.e., ðx1; y1Þ0 is the linear combination of the eigenvectors

that corresponds to the eigenvalue 0 of linear operation LT ,

where Wj is the amplitude of the mode expðikj � rÞ when the

system is under the first-order perturbation and its form is

determined by the perturbational term of the higher order.

According to the Fredholm solvability condition, the

vector function of the right-hand side of Eq. (17) must be or-

thogonal with the zero eigenvectors of operator LþT (the

adjoint operator of LT) to ensure the existence of the nontri-

vial solution of this equation. The zero eigenvectors of oper-

ator LþT are

1

g

� �expð�ikj � rÞ þ c:c:; j ¼ 1; 2; 3:

For Eq. (17), we have

LT

x2

y2

!¼ @

@T1

x1

y1

!� s1

m11x1 þ m12y1

m21x1 þ m22y1

!

�

c

8sx�� 1

� �x2

1 �1

2ffiffiffiffiffix�p x1y1

� c2

8sx�x2

1 þc

2ffiffiffiffiffix�p x1y1 � sy2

1

0BBB@

1CCCA¢

Fx

Fy

!:

The orthogonality condition is

ð1; gÞ Fjx

Fjy

!¼ 0;

where Fjx and Fj

y, separately, represent the coefficients corre-

sponding to expðikj � rÞ in Fx and Fy. Using the orthogonality

condition, we can obtain the following result:

ðf þ gÞ @W1

@T1

¼ s1½fm11 þ m12 þ gðfm21 þ m22Þ�W1

�2ðh1 þ gh2Þ �W2�W3;

ðf þ gÞ @W2

@T1

¼ s1½fm11 þ m12 þ gðfm21 þ m22Þ�W2

�2ðh1 þ gh2Þ �W1�W3;

ðf þ gÞ @W3

@T1

¼ s1½fm11 þ m12 þ gðfm21 þ m22Þ�W3

�2ðh1 þ gh2Þ �W1�W2:

8>>>>>>>>>>>><>>>>>>>>>>>>:Solving Eq. (17), we have

x2

y2

!¼

X0

Y0

!þX3

j¼1

Xj

Yj

!expðikj � rÞ

þX3

j¼1

Xjj

Yjj

!expði2kj � rÞ

þX12

Y12

!expðiðk1 � k2Þ � rÞ

þX23

Y23

!expðiðk2 � k3Þ � rÞ

þX31

Y31

!expðiðk3 � k1Þ � rÞ þ c:c: (20)

The coefficients in Eq. (20) are obtained by solving the sets of

the linear equations about expð0Þ; expðikj � rÞ; expði2kj � rÞ,and expðiðkj � kkÞ � rÞ. With this method, we have

X0

Y0

!¼

zx0

zy0

!ðjW1j2 þ jW2j2 þ jW3j2Þ; Xj ¼ fYj;

Xjj

Yjj

!¼

zx1

zy1

!W2

j ;Xjk

Yjk

!¼

zx2

zy2

!Wj

�Wk:

For Eq. (18), we have

LT

x3

y3

!¼ @

@T1

x2

y2

!þ @

@T2

x1

y1

!�s1

m11x2 þ m12y2

m21x2 þ m22y2

!

�s2

m11x1 þ m12y1

m21x1 þ m22y1

!

�

2c

8sx�� 1

� �x1x2 �

1

2ffiffiffiffiffix�p ðx1y2

þx2y1Þ �c

16sx�x3

1 þ1

8ðx�Þ�3=2x2

1y1

c2

4sx�x1x2 þ

c

2ffiffiffiffiffix�p ðx1y2 þ x2y1Þ

�2sy1y2 þc2

16sx�x3

1 �c

8ðx�Þ�3=2x2

1y1

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA:

033102-5 Yuan, Xu, and Zhang Chaos 23, 033102 (2013)

Downloaded 14 Jul 2013 to 136.186.72.15. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions

Using the Fredholm solvability condition again, we can

obtain

ðf þgÞ @Y1

@T1

þ@W1

@T2

� �¼ ½fm11þm12þgðfm21þm22Þ�ðs1Y1þ s2W1ÞþHð �Y2

�W3þ �Y 3�W2Þ� ½G1jW1j2þG2ðjW2j2jþW3j2Þ�W1:

The other two equations can be obtained through the trans-

formation of the subscript of W.

Notice that the amplitude Ajðj ¼ 1; 2; 3Þ can be

expanded as

Aj ¼ eWj þ e2Yj þ oðe3Þ:

By the expression of Aj and Eq. (15), we can get the ampli-

tude equation corresponding to A1 as follows:

s0

@A1

@t¼ lA1 þ h �A2

�A3 � ½g1jA1j2 þ g2ðjA2j2 þ jA3j2Þ�A1;

where

s0 ¼f þ g

sT ½fm11 þ m12 þ gðfm21 þ m22Þ�; l ¼ sT � s

sT;

h ¼ H

sT ½fm11 þ m12 þ gðfm21 þ m22Þ�;

g1 ¼G1

sT ½fm11 þ m12 þ gðfm21 þ m22Þ�;

g2 ¼G2

sT ½fm11 þ m12 þ gðfm21 þ m22Þ�:

The other two equations of Eqs. (9) can be obtained through

the transformation of the subscript of A (The parameters con-

tained in the above formulae can be computed as specified in

Appendix B).

B. Amplitude stability

Each amplitude in Eqs. (9) can be decomposed to mode

qj ¼ jAjj and a corresponding phase angle uj. Then, substi-

tuting Aj ¼ qj expðiujÞ into Eqs. (9) and separating the real

and imaginary parts, we can get four differential equations of

the real variables as follows:

s0

@u@t¼ �h

q21q

22 þ q2

1q23 þ q2

2q23

q1q2q3

sin u;

s0

@q1

@t¼ lq1 þ hq2q3 cos u� g1q

31 � g2ðq2

2 þ q23Þq1;

s0

@q2

@t¼ lq2 þ hq1q3 cos u� g1q

32 � g2ðq2

1 þ q23Þq2;

s0

@q3

@t¼ lq3 þ hq1q2 cos u� g1q

33 � g2ðq2

1 þ q22Þq3;

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

(21)

where u ¼ u1 þ u2 þ u3.

The dynamical system (21) possesses five kinds of

solutions.45,53

(1) The stationary state (O), given by

q1 ¼ q2 ¼ q3 ¼ 0;

is stable for l < l2 ¼ 0 and unstable for l > l2.

(2) Stripe patterns (S), given by

q1 ¼ffiffiffiffiffilg1

r6¼ 0; q2 ¼ q3 ¼ 0;

is stable for l > l3 ¼ h2g1

ðg2�g1Þ2and unstable for l < l3.

(3) Hexagon patterns ðH0;HpÞ are given by

q1 ¼ q2 ¼ q3 ¼jhj6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih2 þ 4ðg1 þ 2g2Þl

p2ðg1 þ 2g2Þ

;

with u ¼ 0 or p, and exist when l > l1 ¼ �h2

4ðg1þ2g2Þ. The

solution qþ ¼ jhjþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih2þ4ðg1þ2g2Þlp

2ðg1þ2g2Þ is stable only for

l < l4 ¼ 2g1þg2

ðg2�g1Þ2h2, and q� ¼ jhj�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih2þ4ðg1þ2g2Þlp

2ðg1þ2g2Þ is

always unstable.

(4) The mixed states are given by

q1 ¼jhj

g2 � g1

; q2 ¼ q3 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil� g1q2

1

g1 þ g2

s;

with g2 > g1; l > g1q21 and are always unstable.

With fixed parameters d ¼ 10, c¼ 0.8, we can find that

h¼�4.1984, g1 ¼ 163:0647; g2 ¼ 95:9394; l1 ¼ �0:0104;l2 ¼ 0; l3 ¼ 0:6379 and l4 ¼ 1:3886. Summarize the above

analyses, we can show our results in Fig. 3. The system

exists a bistable region ðl1; l2Þ. In other words, when the

control parameter l lies in this region, the Hp patterns and

the stationary state are all stable. The H0 patterns are always

unstable when l > l2. When l lies in region ðl2; l3Þ, the

stripe patterns are unstable, and the Hp patterns are stable. In

region ðl3; l4Þ, the system exists another bistable state

FIG. 3. Bifurcation diagram of model (5) with d ¼ 10, c¼ 0.8. H0: hexagonal

patterns with u ¼ 0; Hp: hexagonal patterns with u ¼ p; S: stripe patterns.

Solid lines: stable states; dashed lines: unstable states. l1 ¼ �0:0104;l2 ¼ 0; l3 ¼ 0:6379; l4 ¼ 1:3886.

033102-6 Yuan, Xu, and Zhang Chaos 23, 033102 (2013)

Downloaded 14 Jul 2013 to 136.186.72.15. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions

(i.e., the bistable state between the hexagon patterns and the

stripe patterns). When the parameter l is more than l4, stripe

patterns emerge in the two-dimensional space.

C. Pattern selection

In this subsection, we perform a series of numerical

simulations of the spatial model (5) in two-dimensional

space, and the qualitative results are shown in the following

part.

All our numerical simulations employ the non-zero ini-

tial and Neumann boundary conditions with a system size of

200� 200 space units. The model (5) is solved numerically

using the Euler method with a time step size of Dt ¼ 0:0005

and space step size Dh ¼ 1:25. We keep d ¼ 10, c¼ 0.8 and

vary s, and the initial density distributions are random spatial

distributions of the species. In the simulations, it was found

that the final distributions of predator and prey are always of

the similar type. As a result, we only show our results of

pattern formation about one species distribution (in this pa-

per, we show the distribution of prey, for instance).

For the different values of s located in the “Turing

space” (the domain T in Fig. 1), we show three categories of

Turing patterns for the distribution of prey x of model (5). In

every pattern, the red (blue) represents the high (low) density

of prey x.

In Fig. 4(a), parameter values (s¼ 0.989) satisfy

l ¼ 0:0082 2 ðl2; l3Þ. In this figure Hp hexagon patterns

prevail over the whole domain eventually. According to the

analysis above, there should be only Hp hexagon patterns

under this circumstance. In other words, the numerical simu-

lation is corresponding to the theoretical analysis. We should

also pay attention to the situation that l is very close to l2

(i.e., s is very close to the critical value s2 ¼ 0:9972). Under

this circumstance, the Hp hexagon patterns come into being

very slowly. This is the universal phenomenon of critical

slowing down. On the other hand, Fig. 4(a) consists of blue

(minimum density of x) hexagons on a red (maximum

FIG. 4. The three categories of Turing patterns of the prey in model (5) with parameters d ¼ 10, c¼ 0.8. (a) s¼ 0.989; (b) s¼ 0.949; (c) s¼ 0.927. Moments:

(a) t¼ 25 000; (b) t¼ 20 000; (c) t¼ 20 000.

FIG. 5. Snapshots of contour pictures of the time evolution of the prey at different instants with d ¼ 10, c¼ 0.8, s¼ 0.989. Moments: (a) t¼ 0; (b) t¼ 5000;

(c) t¼ 6000; (d) t¼ 6500; (e) t¼ 7000; (f) t¼ 25 000.

033102-7 Yuan, Xu, and Zhang Chaos 23, 033102 (2013)

Downloaded 14 Jul 2013 to 136.186.72.15. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions

density of x) background, i.e., the prey is isolated with low

population density. We call this pattern “spots.”

In Fig. 4(b), s¼ 0.949 (l ¼ 0:0484 2 ðl2; l3Þ), a few

stripes emerge, and the remainder of the spots pattern

remains time independent, i.e., spots-stripes pattern. When sis decreased to 0.927 (l ¼ 0:0704 2 ðl2; l3Þ), model dynam-

ics exhibits a transition from spots-stripes growth to stripes

replication, i.e., spots decay and the stripes pattern emerges,

cf. Fig. 4(c). In the two cases, the numerical simulations can-

not correspond to the theoretical analysis. This phenomenon

cannot be explained by the amplitude equations, because the

control parameter s is far away from the critical value

s2 ¼ 0:9972.

From Fig. 4, one can see that for the case c¼ 0.8, on

decreasing the controlled parameter s, the sequence “spots !spots-stripes mixtures ! stripes” is observed. Next, we show

the evolutionary processes of Turing pattern formation of these

three patterns. In Fig. 5, it exhibits a competition between

stripes and spots. The pattern takes a long time to settle down,

starting with a homogeneous state ðx�; y�Þ ¼ ð0:1911; 0:3536Þ

FIG. 6. Snapshots of contour pictures of the time evolution of the prey at different instants with d ¼ 10, c¼ 0.8, s¼ 0.949. Moments: (a) t¼ 0; (b) t¼ 250;

(c) t¼ 1000; (d) t¼ 1500; (e) t¼ 2500; (f) t¼ 20 000.

FIG. 7. Snapshots of contour pictures of the time evolution of the prey at different instants with d ¼ 10, c¼ 0.8, s¼ 0.927. Moments: (a) t¼ 0; (b) t¼ 50;

(c) t¼ 500; (d) t¼ 2500; (e) t¼ 10000; (f) t¼ 20 000.

033102-8 Yuan, Xu, and Zhang Chaos 23, 033102 (2013)

Downloaded 14 Jul 2013 to 136.186.72.15. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions

(cf. Fig. 5(a)), and the random perturbations lead to the forma-

tion of stripes and spots (cf. Figs. 5(b)–5(e)), and ending with

spots only (cf. Fig. 5(f)). In Fig. 6, starting with a homogene-

ous state ðx�; y�Þ ¼ ð0:1570; 0:3340Þ (cf. Fig. 6(a)), the ran-

dom perturbations lead to the formation of spots-stripes

(cf. Figs. 6(c)–6(e)), ending with the time-independent spots-

stripes pattern (cf. Fig. 6(f)). In Fig. 7, we show the spatial pat-

tern of prey with c¼ 0.8, s¼ 0.927 and the initial condition is

ðx�; y�Þ ¼ ð0:1370; 0:3194Þ. The random perturbations lead to

the formation of spots-stripes (cf. Figs. 7(c)–7(e)), and the later

random perturbations make these spots decay, ending with the

time-independent stripes pattern (cf. Fig. 7(f)).

IV. DISCUSSION AND CONCLUSIONS

In summary, this study presents the Turing pattern selec-

tion in a spatial predator-prey model. First, we obtain the

Turing space and establish the amplitude equations for the

excited modes. Second, we illustrate all three categories

(spots, spots-stripes mixtures, and stripes) of Turing patterns

close to the onset of Turing bifurcation via numerical simula-

tions which indicates that the model dynamics exhibits com-

plex pattern replication. It should be noted that, if the

predator mortality described by the linear form, the spatial

predator-prey model cannot give rise to Turing structures

(see Appendix A). In other words, the Turing pattern is

induced by quadratic mortality.

By the above analysis, we can find that the qualitative

dynamics of the model (5) are fundamentally different

when parameter s in model (5) slightly change. The param-

eter s is the (non-dimensional) death rate of the predator.

From the biological point of view, our results show that the

death rate of predator may play a vital role in the spatial

predator-prey model. By varying the value of the predator

mortality s, we obtain three different typical types of pat-

tern: spot pattern (Fig. 4(a)), spot-stripe pattern (Fig. 4(b)),

and stripe pattern (Fig. 4(c)). From the view of population

dynamics, one can see that there exists the spot pattern

replication-the prey x is the isolated zone with low density,

and the remainder region is high density, which means the

prey may break out in the area in Fig. 4(a). In other words,

the prey in this area is safe. The biological significance of

the other cases can be determined in the same way as the

above case.

The methods and results in the present paper may

enrich the research of pattern formation in the predator-

prey models and may well explain the filed observations in

some areas. Further studies are necessary to analyze the

behavior of more complex spatial models such as

predator-prey models with time delay, noise or other

terms.54

ACKNOWLEDGMENTS

This work was supported by the National Natural

Science Foundation of China (11271260,1147015), Shanghai

Leading Academic Discipline Project (No. XTKX2012) and

the Innovation Program of Shanghai Municipal Education

Commission (13ZZ116).

APPENDIX A: ANALYSIS OF THE LINEAR MORTALITYMODEL

If the predator mortality described by the linear form,

i.e., sy, then model (5) is changed to

@x

@t¼ xð1� xÞ �

ffiffiffixp

yþr2x;

@y

@t¼ �syþ c

ffiffiffixp

yþ dr2y:

8>><>>: (A1)

When s < c, the corresponding non-diffusive model has

three equilibriums, which consist of two boundary equili-

briums (0, 0), (1, 0) and a positive equilibrium

ðx�; y�Þ ¼ ðs=c; sðc2�s2Þ=c3Þ. The Jacobian matrix corre-

sponding to the positive equilibrium is

A ¼ a11 a12

a21 a22

� �;

where

a11 ¼c2 � 3s2

2c2; a12 ¼ �

s

c; a21 ¼

c2 � s2

2c; a22 ¼ 0:

A general linear analysis shows that the necessary con-

ditions for yielding Turing patterns are given by

tr0 ¼ a11 þ a22 < 0; (A2a)

D0 ¼ a11a22 � a12a21 > 0; (A2b)

da11 þ a22 � 2ffiffiffiffiffiffiffiffidD0

p> 0: (A2c)

From a22 ¼ 0 and (A2a) we can obtain that a11 need to

be less than zero. Since d is positive, we have that da11 < 0,

which is incompatible with (A2c). That is to say that, when

the predator mortality is linear form, there is no Turing

pattern.

APPENDIX B: COMPUTATIONS OF THE PARAMETERS

Substituting sT for s in a11, a12, a21, a22, we obtain

aT11; aT

12; aT21; aT

22. The expression of some parameters is as

follows:

m ¼ c3 þ dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi27ð27d4 � 9d2c2 þ c4Þ

q; n ¼ m2=3 � 9d2 þ c2

dm1=3;

p ¼ �d1=2n1=2ðm2=3 þ c2Þ þ 9d5=2n1=2 þ 23=2c3=2m1=3

d3=2m1=3n1=2;

q ¼ � 1

3

ffiffiffic

d

rþ

ffiffiffi2p

6ðffiffiffinpþ ffiffiffi

pp Þ;

sT ¼3

2

cd

dþ cq2 þ 2ffiffiffiffifficdp

q3:

m11 ¼3c

2s2T

; m12 ¼c

2s2T

ffiffiffiffiffix�p ; m21 ¼

c2

2s2T

; m22 ¼c2

2s2T

ffiffiffiffiffix�p :

033102-9 Yuan, Xu, and Zhang Chaos 23, 033102 (2013)

Downloaded 14 Jul 2013 to 136.186.72.15. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions

f ¼ daT11 � aT

22

2aT21

; g ¼ aT22 � daT

11

2daT21

;

h1 ¼ f1

2ffiffiffiffiffix�p � f 2 c

8sx�� 1

� �; h2 ¼ f 2 c2

8sx�� f

c

2ffiffiffiffiffix�p þ s:

DT0 ¼ aT

11aT22 � aT

12aT21; k2T

T ¼

ffiffiffiffiffiffiDT

0

d

s;

zx0

zy0

!¼ 2

DT0

aT22h1 � aT

12h2

aT11h2 � aT

21h1

!;

zx1

zy1

� �¼ 1

P

ðaT22 � 4dk2T

T Þh1 � aT12h2

ðaT11 � 4k2T

T Þh2 � aT21h1

!;

P ¼ ðaT11 � 4k2T

T ÞðaT22 � 4dk2T

T Þ � aT12aT

21;

zx2

zy2

� �¼ 1

Q

ðaT22 � 3dk2T

T Þh1 � aT12h2

ðaT11 � 3k2T

T Þh2 � aT21h1

!;

Q ¼ ðaT11 � 3k2T

T ÞðaT22 � 3dk2T

T Þ � aT12aT

21;

H ¼ �2ðh1 þ gh2Þ;

�G1 ¼ 2c

8sx�� 1

� �f � 1

2ffiffiffiffiffix�p

� �ðzx0 þ zx1Þ

� 1

2ffiffiffiffiffix�p f ðzy0 þ zy1Þ þ

3

8ðx�Þ�3=2f 2 � 3c

16sðx�Þ2f 3

þg

"� c2

4sx�f þ c

2ffiffiffiffiffix�p

!ðzx0 þ zx1Þ

þ c

2ffiffiffiffiffix�p f � 2s

� �ðzy0 þ zy1Þ

þ 3c2

16sðx�Þ2f 3 � 3c

8ðx�Þ�3=2f 2

#;

�G2 ¼ 2c

8sx�� 1

� �f � 1

2ffiffiffiffiffix�p

� �zx2

� 1

2ffiffiffiffiffix�p fzy2 þ

3

4ðx�Þ�3=2f 2 � 3c

8sx�f 3

þg

"� c2

4sx�f þ c

2ffiffiffiffiffix�p

!zx2 þ

c

2ffiffiffiffiffix�p f � 2s

� �zy2

þ 3c2

8sx�f 3 � 3c

4ðx�Þ�3=2f 2

#:

1H. I. Freedman, Deterministic Mathematical Models in PopulationEcology (Marcel Dekker, New York, 1980).

2M. Baurmanna, T. Gross, and U. Feudel, J. Theor. Biol. 245, 220 (2007).3C. S. Holling, Can. Entomol. 91, 293 (1959).4C. S. Holling, Can. Entomol. 91, 385 (1959).5M. P. Hassell and G. C. Varley, Nature 223, 1133 (1969).6J. R. Beddington, J. Animal Ecol. 44, 331 (1975).

7D. L. DeAngelis, R. A. Goldsten, and R. V. O’Neill, Ecology 56, 881

(1975).8P. H. Crowley and E. K. Martin, J. North Am. Benthol. Soc. 8, 211 (1989).9A. A. Berryman, Ecology 73, 1530 (1992).

10Y. Kuang and E. Beretta, J. Math. Biol. 36, 389 (1998).11X.-A. Zhang, L.-S. Chen, and A. U. Neumann, Math. Biosci. 168, 201

(2000).12M. A. Aziz-Alaoui and M. D. Okiye, Appl. Math. Lett. 16, 1069 (2003).13A. F. Nindjin, M. A. Aziz-Alaoui, and M. Cadivel, Nonlinear Anal. RWA

7, 1104 (2006).14L.-T. Han, Z.-E. Ma, and H. W. Hethcote, Math. Comput. Model. 34, 849

(2001).15R. Xu and Z.-E. Ma, Nonlinear Anal. RWA 9, 1444 (2008).16M. Liu and K. Wang, Commun. Nonlinear Sci. Numer. Simul. 16, 3792

(2011).17C.-Y. Ji, D.-Q. Jiang, and X.-Y. Li, J. Comput. Appl. Math. 235, 1326

(2011).18V. Ajraldi, M. Pittavino, and E. Venturino, Nonlinear Anal. RWA 12,

2319 (2011).19P. A. Braza, Nonlinear Anal. RWA 13, 1837 (2012).20S. J. Brentnall, K. J. Richards, J. Brindley, and E. Murphy, J. Plankton

Res. 25, 121 (2003).21E. A. Fulton, A. D. M. Smith, and C. R. Johnson, Ecol. Modell. 169, 157

(2003).22L. A. Segel and J. L. Jackson, J. Theor. Biol. 37, 545 (1972).23M. Mimura and J. D. Murray, J. Theor. Biol. 75, 249 (1978).24A. Okubo and L. A. Levin, Diffusion and Ecological Problems: Modern

Perspective, Interdisciplinary Applied Mathematics, 2nd ed. (Springer,

New York, 2001).25J. D. Murray, J. Theor. Biol. 88, 161 (1981).26J. Bascompte, R. V. Sol�e, and N. Mart�ınez, J. Theor. Biol. 187, 213

(1997).27J. Bascompte and R. V. Sol�e, Trends Ecol. Evol. 13, 173 (1998).28G.-Q. Sun, G. Zhang, Z. Jin, and L. Li, Nonlinear Dynam. 58, 75 (2009).29W.-M. Wang, Q.-X. Liu, and Z. Jin, Phys. Rev. E 75, 051913 (2007).30X.-C. Zhang, G.-Q. Sun, and Z. Jin, Phys. Rev. E 85, 021924 (2012).31X.-N. Guan, W.-M. Wang and Y.-L. Cai, Nonlinear Anal. RWA 12, 2385

(2011).32�A. L. Rodrigues and T. Tom�e, Braz. J. Phys. 38, 87 (2008).33C. Argolo, H. Otaviano, I. Gleria, E. Arashiro, and T. Tom�e, Int. J.

Bifurcation Chaos 20, 309 (2010).34R. A. Kraenkel and D. J. Pamplona da Silva, Physica A 389, 60

(2010).35D. J. Pamplona da Silva and R. A. Kraenkel, Physica A 391, 142 (2012).36F. Azevedo, R. A. Kraenkel, and D. J. Pamplona da Silva, Ecol. Complex.

11, 154 (2012).37A. M. Turing, B. Math. Biol. 52, 153 (1990).38Q. Ouyang and H. L. Swinney, Nature 352, 610 (1991).39V. Dufiet and J. Boissonade, Phys. Rev. E 53, 4883 (1996).40C. Varea, J. L. Arag�on, and R. A. Barrio, Phys. Rev. E 56, 1250 (1997).41S. Kondo and R. Asai, Nature 376, 765 (1995).42L. Saunoriene and M. Ragulskis, Phys. Rev. E 84, 056213 (2011).43V. N. Biktashev and M. A. Tsyganov, Phys. Rev. Lett. 107, 134101

(2011).44B. Pe~na and C. P�erez-Garc�ıa, Phys. Rev. E 64, 056213 (2001).45Q. Ouyang, Nonlinear Science and the Pattern Dynamics Introduction

(Peking University Press, Beijing, 2010).46M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).47G.-Q. Sun, Z. Jin, Q.-X. Liu, and L. Li, J. Stat. Mech. 2007, P11011

(2007).48M. Ipsen, F. Hynne, and P. G. Sorensen, Physica D 136, 66 (2000).49W.-M. Wang, W.-J. Wang, Y.-Z. Li, and Y.-J. Tan, Chin. Phys. B 20,

034702 (2011).50W.-M. Wang, Y.-Z. Li, F. Rao, L. Zhang, and Y.-J. Tan, J. Stat. Mech.

2010, P11036 (2010).51G.-Q. Sun, Z. Jin, Q.-X. Liu, and L. Li, Chin. Phys. B 17, 3936 (2008).52J.-F. Zhang, W.-T. Li, and X.-P. Yan, Appl. Math. Comput. 218, 1883

(2011).53S. Ciliberto, P. Coullet, J. Lega, E. Pampaloni, and C. Perez-Garcia, Phys.

Rev. Lett. 65, 2370 (1990).54G.-Q. Sun, Z. Jin, Q.-X. Liu, and B.-L. Li, BioSystem 100, 14 (2010).

033102-10 Yuan, Xu, and Zhang Chaos 23, 033102 (2013)

Downloaded 14 Jul 2013 to 136.186.72.15. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions