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: [email protected])Electronic address: [email protected]
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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)
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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)
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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)
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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)
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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)
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(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)
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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)
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(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)
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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
#:
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