Nonlinear DynDOI 10.1007/s11071-014-1438-6

ORIGINAL PAPER

Spatio-temporal dynamics of a reaction-diffusion system fora predator–prey model with hyperbolic mortality

Tonghua Zhang · Yepeng Xing · Hong Zang ·Maoan Han

Received: 30 December 2013 / Accepted: 2 May 2014© Springer Science+Business Media Dordrecht 2014

Abstract We investigate the effects of diffusion onthe spatial dynamics of a predator–prey model withhyperbolic mortality in predator population. More pre-cisely, we aim to study the formation of some elemen-tary two-dimensional patterns such as hexagonal spotsand stripe patterns. Based on the linear stability analy-sis, we first identify the region of parameters in whichTuring instability occurs. When control parameter isin the Turing space, we analyse the existence of stablepatterns for the excited model by the amplitude equa-tions. Then, for control parameter away from the Tur-ing space, we numerically investigate the initial value-controlled patterns. Our results will enrich the patterndynamics in predator–prey models and provide a deepinsight into the dynamics of predator–prey interactions.

Keywords Turing instability · Pattern formation ·Predator–prey model · Control of patterns

T. Zhang (B)Department of Mathematics, Swinbunre Universityof Technology, Hawthorn, VIC 3122, Australiae-mail:tonghuazhang@swin.edu.au

Y. Xing · M. Han · T. ZhangDepartment of Mathematics, Shanghai Normal University,Shanghai 200234, Chinae-mail:mahan@shnu.edu.cn

H. ZangHubei Key Lab of Intelligent Robot,Wuhan Institute of Technology, Wuhan, China

1 Introduction

Since Lotka Volterra’s pioneering work on popula-tion dynamics, numerous mathematical models havebeen proposed to study the relation between preda-tor population and prey population [23]. This type ofmodels is generally known as predator–prey models.Due to the importance of such models in ecology,predator–prey models have been intensively studiedand will continue to be one of the dominant themesin the area [2,4,10,21,32]. Another reason of attract-ing so much attention is that such models can alsobe adopted to study the Turing instability driven bythe diffusion, which comes naturally in ecosystemsdue to the spatial migrations of the predator and prey[17,21,23,24,32,33,37–39].

In the development of mathematical models for pop-ulation dynamics, the functional response and mor-tality rate of the predator are essential. In practice,Holling types of functional responses are popular [13,14,24,39], but others are also used to better describecertain cases, for example authors of [1,5,37] useda response function in term of the square root ofthe prey population to reflect the fact of the exis-tence of the herb behaviour. Although linear mortal-ity rate is intensively used by researchers [11,23,24],other mortality rates, such as quadratic mortality, havealso been used. For example, Edward and Yool usedhyperbolic mortality rate in modelling the planktonpopulation dynamics [6]. Furthermore, in the mod-elling of a system involving two or more than two

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T. Zhang et al.

species in biology and reactants in chemistry, diffu-sion plays an important role as it explains the diver-sity of species in nature [8,23,27,34]. Different dif-fusion rates may lead to non-uniform steady state,namely formation of patterns. For that reason, thestudy of reaction-diffusion equations has attractedgreat attention in the past couple of decades in manydifferent areas, [3,8,9,12,15,17,19,22,26,27,29,31,33–36,38] . The original idea of generating patternsfrom diffusion was proposed by Turing [34] more than60 years ago, and since then it has been an activearea [18,20,25,28,30], and numerous works have beendone numerically [24,28] and theoretically [8,26,27].Recent years, the control and design of patterns arebecoming a growing area, such as authors of [12,19]discussed the global feedback control of patterns andauthor of [35] listed 5 ways of controlling the formationof patterns.

In this paper, we investigate the spatial dynamics ofa reaction-diffusion predator–prey system with hyper-bolic mortality rate. We first perform a general linearstability analysis and find the Turing space in whichTuring instability occurs; then, near the critical valueof the control parameter, we investigate the hexagonaland stripe patterns of the system using the amplitudeequations; we also discover some new patterns whencontrol parameter is far away from its critical value. Atthe end of this paper, we carry out numerical simula-tions to illustrate the patterns we found and concludeour paper with discussions.

2 The mathematical model and bifurcationanalysis

2.1 The spatial model

We consider the following generalized Rosenzweig andMacArthur model:⎧⎨

⎩

dudt = au

(1 − u

k

) − buvc+u ,

dvdt = muv

c+u − h(v),(2.1)

where as uasual u and v are the population densities ofprey and predator, respectively; a is the birth rate, k is acarrying capacity and b is the maximum uptake rate ofthe prey; c is the prey density at which predator has themaximum kill rate; and m is the birth rate of predatorand function h(v) reflects predator death rate. Pleasenotice that when h(v) = nv, model (2.1) is exactly

Fig. 1 Linear, hyperbolic and quadratic mortality rates

the Rosenzweig and MacArthur model. Then, after thenondimensionalization and reduction of parameters

u → kU, v → ac

bV,

k

c→ β,

a

m→ α, t → T

mas what it was performed by Nagano and Maeda [24],model (2.1) becomes⎧⎨

⎩

dUdt = αU

(1 − U − V

1+βU

)≡ F(U, V ),

dVdt = V

(βU

1+βU − h(V )V

)≡ G(U, V ),

(2.2)

where h(V ) is given by

h(V ) = γ V 2

e + ηV

for hyperbolic mortality with γ is the death rate of thepredator, e and η are coefficients of light attenuation bywater and self-shading in the context of plankton mor-tality. Please notice that when η = 0 and e �= 0, it isquadratic mortality; when η �= 0 and e = 0, it gives thelinear mortality; and when both η and e are not zero, itis a mortality of the hyperbolic type, see Fig. 1 wherewe compare three different mortality rates. As seen,when population density is low, the linear rate domi-nates the mortality which is used in many biologicalmodels [23,24]; when population density is relativelyhigh, the quadratic rate dominates the mortality [37];while hyperbolic rate dominates the mortality whenpopulation density is large [6]. Please also note that

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Spatio-temporal dynamics of a reaction-diffusion system

all parameters are nonnegative. For simplicity of dis-cussion, in this paper, we shall concentrate the case ofη = γ and e = 1.

For considering the spatial effect on the populationdynamics, we have the spatial version of the model(2.2) as the following initial boundary value problem:⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

dUdt = d1∇2U + F(U, V ),

dVdt = d2∇2V + G(U, V ),

∂U∂ν

= ∂V∂ν

= 0,

U (∂Ω, 0) > 0, V (∂Ω, 0) > 0, X ∈ Ω,

(2.3)

where d j are the diffusion coefficients of U and V ,respectively; and we denote by δ the ratio δ = d2/d1

of them; ∇2 = ∑nj=1

∂2

∂x2j

is the Laplacian opera-

tor and ν is the outward unit normal vector of theboundary ∂Ω . The third equation of (2.3) is the zero-flux boundary condition, which is made to match thefact that there are no individual species leaving thedomain.

2.2 Linear stability analysis and the turing space

Notice that β > 0, the corresponding non-diffusivemodel has always three equilibria, which consist of twoboundary equilibria (0, 0) and (1, 0), and a positiveequilibrium (U∗, V ∗), where

U∗ = βγ − γ − β + √(βγ − γ − β)2 + 4βγ 2

2βγ,

V ∗ = β

γU∗.

Remark 1 Please notice that in the case of hyperbolicmortality, the positive equilibrium always exists, whichis different from the cases of linear and quadratic mor-talities. For example, when h(V ) = γ V , the posi-tive equilibrium exists if and only if 0 < γ < 1 andβ >

γ1−γ

[24].

We next investigate the linear stability and find theTuring space, in which the uniform steady state loses itsstability due to the diffusive effect and Turing patternoccurs [27]. If we denote the Jacobian matrix associatedwith this equilibrium by

A =(

a10 a01

b10 b01

)

, (2.4)

where at U = U∗

a10 = αU (β − 1 − 2βU )

1 + βU, a01 = − αU

1 + βU,

b10 = β2U

γ (1 + βU )2 , b01 = − βU

(1 + βU )2 ,

then for the non-diffusive model when

tr0 = a10 + b01 < 0 and 0 = a10b01 − a01b10 > 0,

the equilibrium is stable. Please notice that at (U, V ) =(U∗, V ∗), the following holds:

0 = αβU 2

γ (1 + βU )3

√

(βγ − γ − β)2 + 4βγ > 0

and

a10 + b01 = U

(1 + βU )

(

α(β−1−2βU )− β

1 + βU

)

.

It implies that tr0 < 0 if and only if either (1) whenβ − 1 − 2βU ≤ 0, or (2) when β − 1 − 2βU > 0and α < αH , which is defined below. However, we caneasily see that case (1) can not occur as a10 can not benonpositive. Let

β0 = 1 + γ + √1 + 4γ

2 − γ.

Then from case (2), we have

α < αH = β

(β − 1 − 2βU∗)(1 + βU∗),

β > β0, and 0 < γ < 2,

particularly, when α = αH Hopf bifurcation occurssince dtr0

dα> 0.

With diffusion, the linearisation of the model isgiven by{

dUdt = d1∇2U + a10U + a01V,

dVdt = d2∇2V + b10U + b01V,

(2.5)

for which we consider a particular solution of the fol-lowing form [8,37](

UV

)

=(

U∗V ∗

)

+ε

(Uk

Vk

)

eσk t eik·r + c.c. + O(ε2

),

(2.6)

where σk is the growth rate, k = |k| is the wave num-ber, r is the directional vector and c.c. stands for thecomplex conjugate. Then for each k, we have the char-acteristic equation

σ 2k − trkσk + k = 0, (2.7)

where

trk = tr0 − k2(d1 + d2), (2.8a)

k = 0 − k2(d2a10 + d1b01) + d1d2k4. (2.8b)

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T. Zhang et al.

Fig. 2 Turing space whend1 = 0.001, d2 = 0.1,

γ = 0.5

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

β

αHopf curve

Turing Curve

Turing Space

It is easy to verify that (2.7) has two roots given by

σk = 1

2

(

trk ±√

tr2k − 4k

)

(2.9)

which, together with the fact from (2.8a) thattrk ≤ tr0 < 0, implies that Turing instability occursonly when k < 0 at the critical value of the wavenumber k = kc. It follows from (2.8b) that

k2c = d2a10 + d1b01

2d1d2(2.10)

and at which

kc = 0 − (d2a10 + d1b01)2

4d1d2. (2.11)

Please notice that by (2.11), the critical wavenumbersatisfies k2

c = √0/δ. Hence, Turing instability occurs

when

d2a10 + d1b01 > 2√

d1d20 (2.12)

or in the form of

a10δ + b01 > 2√

δ0 with δ = d2

d1.

Let

αTi =β

(γχ1 + 2β + (−1)i 2

√γχ1β + β2

)

γχ21 (1 + βu)δ

with

χ1 = (1 − β + 2βU∗) (2.13)

and Ω = {0 < α < αT 1 or α > αT 2} ⋂{0 < α

< αH }. Then, we obtain the Turing space specified by

{(α, β, γ )}={(α, β, γ ) : α∈Ω,β >β0, 0<γ <2}.(2.14)

The red region in Fig. 2 illustrates the Turing spacewhen diffusion rate δ = 10, γ = 0.5 and 0 < β < 20.

When parameters are chosen from the red region, theassociated system can have hexagonal, stripe patternsor the coexistence of them, which will be seen fromnext section and Figs. 5 and 6. We also show the dis-persion relations in Fig. 3, where we use dashdottedred lines to represent Im(σk), the imaginary part of thegrowth rate, and solid blue lines to represent Re(σk),

the maximum real part of the growth rate. Furthermore,the first sub-figure of Fig. 3 shows the case of α < αc,

in which Re(σk) < 0; the second for α = αc in whichRe(σk) = 0 at the critical wavenumber and Re(σk) < 0otherwise; the third for α > αc in which Re(σk) > 0for wavenumber k falling in certain interval. For thechoice of parameters made in the simulation part, weidentify that the critical wave number is 3.64 at whichthe maximum real part of the growth rate first becomeszero, see the second sub-figure.

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Spatio-temporal dynamics of a reaction-diffusion system

Fig. 3 Dispersion relationswhen d1 = 0.001,

d2 = 0.1, γ = 0.5

0 5 10−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Re(

σ k)/Im

(σk)

α<αc

Re(σk)

Im(σk)

0 5 10−0.15

−0.1

−0.05

0

0.05

0.1

wave number, k

α=αc

Re(σk)

Im(σk)

0 5 10−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

α>αc

Re(σk)

Im(σk)

3 Spatial dynamics and pattern selection

3.1 Amplitude equations

As pointed out in references [9,15,33,37,38], the evo-lution of a dynamical system is critically slowing downwhen control parameter is close to its critical value, andthe amplitude equations can be employed to describethe dynamics of the slow model near the onset. Next, weuse a method of multiple time scales to derive the asso-ciated amplitude equations, which allows us to studysome elementary patterns such as the hexagonal andstripe patterns.

First for model (2.3), we introduce small perturba-tion U = u +U∗, V = v+ V ∗ at the equilibrium point(U∗, V ∗), and then do the Taylor series expansion andtruncate it up to the third order⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

∂u∂t = a10u + a01v + a20u2 + a11uv + a02v

2

+ a30u3 + a21u2v + a03v2 + d1∇2u,

∂v∂t = b10u + b01v + b20u2 + b11uv + b02v

2

+ b30u3 + b21u2v + b03v3 + d2∇2v,

(3.1)

where the coefficients are given by

a20 = 2α

(β(1 − U∗)(1 + βU∗)2 − 1

)

,

a11 = − α

(1 + βU∗)2 , a02 = 0,

a30 = − 6αβ2V ∗

(1 + βU∗)4 , a21 = 2αβ

(1 + βU∗)3 ,

a12 = a03 = 0,

b20 = − 2β2V ∗

(1 + βU∗)3 , b11 = β

(1 + βU∗)2 ,

b02 = − 2γ

(1 + γ V ∗)2 ,

b30 = 6β3V ∗

(1 + βU∗)4 , b21 = − 2β2

(1 + βU∗)3 ,

b12 = 0, b03 = − 2γ

(1 + γ V ∗)3 .

Then, let the linear operator L be

L =(

a10 + d1∇2 a01

b10 b01 + d2∇2

)

(3.2)

and H be given by

H =⎛

⎜⎝

∑

i+ j=2,3ai j uiv j

∑

i+ j=2,3bi j uiv j

⎞

⎟⎠ . (3.3)

The model equations can be rewritten into the followingform:∂X∂t

= LX + H. (3.4)

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T. Zhang et al.

Next, near the Turing bifurcation threshold, weexpand the control parameter α as

αT − α = εα1 + ε2α2 + ε3α3 + o(ε3), (3.5)

where |ε| 1. Similarly, expand the solution X, linearoperator L and the nonlinear term H into Taylor seriesat ε = 0

X = ε

(u1

v1

)

+ ε2(

u2

v2

)

+ ε3(

u3

v3

)

+ o(ε3), (3.6)

H = ε2h2 + ε3h3 + o(ε3), (3.7)

L = LT + (αT − α)M, (3.8)

where

h2 =(

h12

h22

)

=(

aT20u2

1 + aT11u1v1 + aT

02v2

bT20u2

1 + bT11u1v1 + b02v

21

)

(3.9)

and

h3 =(

h13

h23

)

=

⎛

⎜⎜⎜⎜⎜⎜⎝

∑

i+ j=3aT

i j uiv j + 2(aT

20u1u2

+ aT02v1v2) + aT

11(u1v2 + u2v1)∑

i+ j=3bT

i j uiv j + 2(bT

20u1u2

+ bT02v1v2) + bT

11(u1v2 + u2v1)

⎞

⎟⎟⎟⎟⎟⎟⎠

−(

α1(a′20u2

1 + a′11u1v1 + a′

02v21)

α1(b′20u2

1 + b′11u1v1 + b′

02v21)

)

(3.10)

are terms corresponding to the second and third ordersof ε in the expansion of the nonlinear term H, and a′

i j =dai jdα

. For the linear operator

L = LT + (αT − α)M,

we have

LT =(

a10 + d1∇2 a01

b10 b01 + d2∇2

)

α=αT

,

M =(

m11 m12

m21 m22

)

, (3.11)

with

m11 = −u(β − 1 − 2βu)

1 + βu, m12 = u

1 + βu,

m21 = 0, m22 = 0

at u = U∗.Finally, we introduce multiple time scales

∂

∂t= ε

∂

∂T1+ ε2 ∂

∂T2+ o(ε2). (3.12)

Then, substituting Eqs. (3.6)–(3.12) and (3.2)–(3.11) into Eq. (3.4) and collecting like terms of εi

yield

ε : LT

(u1

v1

)

= 0,

ε2 : LT

(u2

v2

)

= ∂

∂T1

(u1

v1

)

− α1M(

u1

v1

)

− h2,

ε3 : LT

(u3

v3

)

= ∂

∂T1

(u2

v2

)

+ ∂

∂T2

(u1

v1

)

−α1M(

u2

v2

)

− α2M(

u1

v1

)

− h3. (3.13)

In what follows, we seek the amplitude equationsby solving system (3.13). Since LT has an eigenvectorassociated with the zero eigenvalue

( f, 1)T , f = a10δ − b01

2b10,

the general solution of the first system of (3.13) can bewritten as

(u1

v1

)

=(

f1

)⎛

⎝3∑

j=1

W j eik j ·r + c.c.

⎞

⎠ , (3.14)

where W j is the amplitude of the mode eik j ·r. Noticethat the second system of (3.13) is nonhomogeneous,and L∗

T , the adjoint operator of LT , has zero eigenvec-tors in the form of(

1g

)

e−ik j ·r + c.c., j = 1, 2, 3

with g = b01−a10δ2δb10

. Let

(Fu

Fv

)

= ∂

∂T1

(u1

v1

)

−β1

(m11u1 + m12v1

m21u1 + m22v1

)

−(

h12

h22

)

.

Then, applying the Fredholm solvability conditiongives

(1, g)

(F j

u

F jv

)

= 0,

where F ju and F j

v are the coefficients of eik j ·r in Fu

and Fv , respectively. It follows after some routine cal-culation that for jl = 1, 2, 3 and jl �= lm if l �= m

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Spatio-temporal dynamics of a reaction-diffusion system

( f + g)∂W j1

∂T1= α1h3W j1 − 2(h1 + gh2)W j2 W j3 ,

(3.15)

where h1 = −( f 2aT20 + f aT

11 +aT02), h2 = −( f 2bT

20 +f bT

11 + bT02), h3 = f m11 + m12 + g( f m21 + m22).

Notice the forms of u1 and v1 given by Eq. (3.14).We have a particular solution for the second system of(3.13) as follows:

(u2

v2

)

=(

U0

V0

)

+3∑

j=1

(U j

Vj

)

eik j ·r+3∑

j=1

(U j j

Vj j

)

ei2k j ·r

+(

U12

V12

)

ei(k1−k2)·r +(

U23

V23

)

ei(k2−k3)·r

+(

U31

V31

)

ei(k3−k1)·r + c.c. (3.16)

with the coefficients being given below at α = αT

(U0

V0

)

=(

2(b01h1−a01h2)0

2(a10h2−b10h1)0

) ⎛

⎝3∑

j=1

|W j |2⎞

⎠

≡(

zu0

zv0

)⎛

⎝3∑

j=1

|W j |2⎞

⎠ , U j = f Vj ,

(X j j

Y j j

)

≡(

zu1

zv1

)

W 2j

= 1

(a10 − 4d1k2c )(b01 − 4d2k2

c ) − a01b10

×(

(b01 − 4d2k2c )h1 − a01h2

(a10 − 4d1k2c )h2 − b10h1

)

W 2j

and(

X jk

Y jk

)

≡(

zu2

zv2

)

W j W k

= 1

(a10 − 3d1k2c )(b01 − 3d2k2

c ) − a01b10

×(

2((b01−3d2k2c )h1−a01h2)

2((a10−3d1k2c )h2−b10h1)

)

W j W k .

Again, apply the Fredholm solvability condition tothe third system of (3.13). We have for j = 1

( f + g)

(∂Vj

∂T1+ ∂W j

∂T2

)

= h3(α1Vj + α2W j )

+ h4W l W m + H(V l W m + V m W l)

− (G1|W1|2 + G2(|W2|2 + |W3|2))W j , (3.17)

with

h4 = −2α1(a′20 f 2 + a′

11 f + a′02

+ g(b′20 f 2 + b′

11 f + b′02)),

H = −2(h1 + gh2),

G1 = −(3a30 f 3 + 2a11 f zv0 + a11 f zv1

+ 4a20 f zu0 + 2a20 f zu1

+ 3a21 f 2 + 4a02zv0 + 2a02zv1 + 2a11zu0

+ a11zu1 + 3a12 f + 3a03)

− g(3b30 f 3 + 2b11 f zv0 + b11 f zv1

+ 4b20 f zu0 + 2b20 f zu1

+ 3b21 f 2 + 4b02zv0 + 2b02zv1 + 2b11zu0

+ b11zu1 + 3b12 f + 3b03),

G2 = −(6a30 f 3 + 2a11 f zv0 + a11 f zv2

+ 4a20 f zu0 + 2a20 f zu2

+ 6a21 f 2 + 4a02zv0 + 2a02zv2

+ 2a11zu0 + a11zu2 + 6a12 f + 6a03)

− g(6b30 f 3 + 2b11 f zv0 + b11 f zv2

+ 4b20 f zu0 + 2b20 f zu2

− 6b21 f 2 + 4b02zv0 + 2b02zv2 + 2b11zu0

+ b11zu2 + 6b12 f + 6b03).

The combination of equations (3.15) and (3.17)gives the amplitude equations (3.18) for the amplitudeA j ,

τ0∂ A j

∂t= μA j + h Al Am

−(g1|A1|2+g2(|A2|2+|A3|2)

)A j , (3.18)

where

τ0 = f + g

αT, μ = αT − α

αTh3, h = H

αT f, gi = Gi

αT f 2 .

Please notice that system (3.18) is in complex form.Following to reference [27], for the purpose of conve-nience of discussion, we convert it into the real formby A j = ρ j exp(iϕ j ) with ρ j are the real amplitudesand ϕ j the phase angles.⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

τ0∂ϕ∂t =−h

ρ21ρ2

2+ρ21ρ2

3+ρ22ρ2

3ρ1ρ2ρ3

sin ϕ,

τ0∂ρ1∂t =μρ1+hρ2ρ3 cos ϕ−g1ρ

31 −g2(ρ

22 +ρ2

3 )ρ1,

τ0∂ρ2∂t =μρ2+hρ1ρ3 cos ϕ−g1ρ

32 −g2(ρ

21 +ρ2

3 )ρ2,

τ0∂ρ3∂t =μρ3+hρ1ρ2 cos ϕ−g1ρ

33 −g2(ρ

21 +ρ2

2 )ρ3,

(3.19)

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T. Zhang et al.

Fig. 4 Stability of patterns

−0.0055 0 0.0198 0.04 0.08 0.124 0.160

0.02

0.04

0.06

0.08

0.1

0.12

μ

ρ

ρS

ρH

0

ρH

πρ

Sq

where ϕ = ϕ1+ϕ2+ϕ3. Since we are only interested inthe stable steady states and notice the fact that hρi �= 0,from the first equation of (3.19), we have ϕ = 0 or π.

Also, noticing the fact that τ0 > 0, it implies that whenh > 0, the state corresponding to ϕ = 0 is stable, butthe one corresponding to ϕ = π when h < 0. Then,system of amplitude equations (3.19) becomes⎧⎪⎪⎨

⎪⎪⎩

τ0∂ρ1∂t = μρ1 + |h|ρ2ρ3 − g1ρ

31 − g2(ρ

22 + ρ2

3 )ρ1,

τ0∂ρ2∂t = μρ2 + |h|ρ1ρ3 − g1ρ

32 − g2(ρ

21 + ρ2

3 )ρ2,

τ0∂ρ3∂t = μρ3 + |h|ρ1ρ2 − g1ρ

33 − g2(ρ

21 + ρ2

2 )ρ3.

(3.20)

Please notice that generally the amplitude equations arevalid only when the control parameter is in the Turingspace. It is easy to see that the above system of ordi-nary differential equations (3.20) has five equilibria,which corresponds five kinds of steady states [7,27,37].Notice the symmetry of the system, we have

(1) System (3.20) always has an equilibrium O =(0, 0, 0), which is the original of the system andand is stable for μ < μ2 = 0 and unstable forμ > μ2;

(2) When μg1 > 0, system (3.20) has an equilibrium

(√

μg1

, 0, 0) corresponding to stripe type of patterns

of system (2.3), furthermore when 0 < g1 < g2

and μ > μ3 = h2g1(g2−g1)2 , it is stable;

(3) System (3.20) has an equilibrium (ρ−, ρ−, ρ−),

where

ρ− = |h| − √h2 + 4(g1 + 2g2)μ

2(g1 + 2g2)

when either (g1 + 2g2) > 0 and 0 > μ > μ1 =−h2

4(g1+2g2); or (g1+2g2) < 0 and 0 < μ < μ1, and

when it exits, it corresponds to pattern of hexagonalspots of (2.3). Since in this case, one of eigenvalues

λ3 = ρ−√

h2 + 4(g1 + 2g2)μ > 0,

the equilibrium is always unstable;(4) System (3.20) has an equilibrium (ρ+, ρ+, ρ+),

where

ρ+ = |h| + √h2 + 4(g1 + 2g2)μ

2(g1 + 2g2)

when μ > μ1 and g1 + 2g2 > 0, and it is asso-ciated with hexagonal patterns of (2.3). Routinecalculation gives the three eigenvalues as follows:

λ1,2 = 2

(g1 + 2g2)

((g2−g1)μ−(2g1+g2)|h|ρ+)

λ3 = −√

h2 + 4(g1 + 2g2)μρ+ < 0

So, this equilibrium is stable if one of the following

(a) 2g1 + g2 > 0, g2 > g1 and μ < μ4;(b) 2g1 + g2 > 0, g2 ≤ g1;

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Spatio-temporal dynamics of a reaction-diffusion system

Fig. 5 The time evolution of the three types of patterns in theprey with control parameter μ is a = 0.019, b = 0.4, c = 3,respectively. Most left column corresponds time instant t = 500,

middle column corresponds to time instant t = 1500 and mostright one corresponds to time instant t = 5000

(c) 2g1 + g2 = 0 and μ < μ4;(d) 2g1 + g2 < 0 and μ4 < μ < μ5;

hold, and where μ4 = 2g1+g2(g2−g1)2 h2 and μ5 =

(2g1+g2)h2

2(g1+2g2)(g2−g1);

(5) When g2 > g1 and μ > g1ρ21 system have equi-

librium (ρ1, ρ2, ρ2) associated with square type ofpatterns of (2.3), where

ρ1 = |h|g2 − g1

, ρ2 =√

μ − g1ρ21

g1 + g2,

and they are always unstable in the Turing space.

From the above analysis, we see when μ < μ1,only the stationary state is stable; when control para-meter α crosses the first critical value so that μ1 <

μ < μ2 = 0, hexagonal steady states appear, and wehave the first bistable region in which both the station-ary and hexagonal states are stable; as control parame-ter increases further so that μ2 < μ < μ3, then thestationary state loses its stability and the hexagonal isstill stable; further, increase control parameter so thatμ3 < μ < μ4, stripe patterns are observed and thesecond bistable region appears in which we have stablehexagonal and stripe patterns; finally, when μ > μ4,

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Fig. 6 The time evolution of the the prey when μ = 0.019 and t a = 20, b = 100, c = 500, d = 1500 and e = 5000, respectively

hexagons lose their stability and only the stripes arestable, please see Fig. 4. In this figure, parameters areset as follows: β = 6, d1 = 0.001, d2 = 0.1, γ =0.5. Under this setting, we have αc = 0.1156, h =1.6203, g1 = 12.5128, g2 = 53.2076 and μ1 =−0.005519, μ2 = 0, μ3 = 0.01983, μ4 = 0.1240. Ifrestrict the control parameter to the Turing space, then0 < μ < 0.4578. Also in this figure, we use the dottedlines for unstable states and solid lines for stable states;ρS stands for the stripe patterns, ρH for hexagon andρSq for squares.

3.2 Pattern selection

Our simulations are performed for model (2.3) in two-dimensional space. Details about the simulation set-tings and qualitative analysis follow.

All simulations employ the non-zero initial andNeumann boundary conditions with a system size of6π × 6π space units with the spatial mesh consistingof 200 × 200 grid point. The system is solved numer-ically using the finite-difference method with the timestep of 0.05. In these simulations, parameters are set as

in previous section, and we vary μ, the control para-meter to generate different patterns. Initial values arethe uniform steady state (U∗, V ∗) with ±(0.1–1 %)perturbation to break the symmetry. In Fig. 5, we onlyillustrate the density distributions of the prey as, in oursimulations, the predator shows the very similar dis-tributions. As seen, we observed three types of pat-terns: spots (5a.3), stripes (5b.3) and squares (5c.3). Inall three figures, we showed the patterns when time ist = 5000. The time evolution of these patterns are alsoillustrated in Fig. 5. When μ = 0.019 ∈ (μ2, μ3),

from previous analysis or Fig. 4, we know that only thespot pattern is stable, which is also confirmed by thesimulation. Please see Fig. 5a.1 corresponding to timet = 500 and (5a.2) corresponding to time t = 1500. Ifμ = 0.4 ∈ (μ4, μc), the Fig. 4 suggests that the stripetype of pattern dominates the region, see Fig. (5b.3)which agrees well with the theoretical analysis in previ-ous section. As seen in Fig. 5b.1 corresponding to timet = 100 and (5b.2) corresponding to time t = 500,

all patterns except the stripe quickly lose stability toform stripes. When μ grows further beyond the Tur-ing space, namely μ > 0.4578, all elementary pat-terns such as spots and stripes lose stability and stable

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Spatio-temporal dynamics of a reaction-diffusion system

Fig. 7 Initial value-controlled pattern formation: μ = 3 andt (1) =600, (2) =1040, (3) =2500, respectively. For a (U0, V0) =(U∗, V ∗) − 0.01 × (1, 1) if (x − 100)2 + (y − 100)2 <

800, otherwise (U0, V0) = (U∗, V ∗). For b (U0, V0) =

(U∗, V ∗) − 0.01 × (1, 1) if (x − 100)2 + (y − 100)2 < 100,(x − 50)2 + (y − 50)2 < 100, (x − 50)2 + (y − 150)2 < 100,(x −150)2 +(y−50)2 < 100 or (x −150)2 +(y−150)2 < 100,otherwise (U0, V0) = (U∗, V ∗)

squares, which is unstable in the Turing space, appear,please see Fig. 5c.3 where μ = 3 is way out of theTuring space.

Finally, we test the robustness of the stability to theinitial perturbation. For example, we only apply the±1 % perturbation to the 20 × 20 mesh point arealocated at the centre of the grid. We still see the abovepatterns, which implies that their stabilities are robustalthough they are in a different way of evolution, seeFig. 6 for example where the control parameter is in theTuring space. As seen pattern formation begins with theperturbed area, and then extends to area surrounding it,then the whole area. It starts with stripes, which thenbreaks down into spots. In Fig. 6, we showed patternsat three different time instances: t = 20, 100 and 500time units, where few spots grow to form more andmore stripes; then stripes break down into more spots,see sub-figure (e). In our simulations, we also tried dif-ferent values of the control parameter, and we saw thesimilar manner of patterns growing.

As seen above, the external perturbation can triggerthe formation of patterns, in which the perturbation is

treated as a ‘seed.’ If we understand the dynamics ofthe system well, we can take advantage of such seedto control the formation of patterns [12,19,35]. Whenthe control parameter μ is out of the Turing space, weuse initial values to control the formation of patterns,and we found that when 0.4578 < μ < 8, stable pat-terns can still exist. For example, when μ = 3 and weperturbed one or five circular areas of the grid, someinteresting patterns have been observed in Fig. 7.

4 Discussions and conclusions

In the present study, for a spatial predator–prey modelwith hyperbolic mortality, we investigated the patternformation driven by the diffusions. We first classi-fied the Turing space (the red area in Fig. 2), whichis bounded by the Hopf bifurcation curve and Turinginstability curve. Then, by the amplitude equations, weanalysed the stability of some elementary bifurcations:spots, stripes and squares. At the end, we numericallyinvestigated the initial value-controlled patterns simu-

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lations when control parameter is beyond the Turingspace.

Furthermore, we noticed the dynamical differencebetween the different mortality rates. For example,the existence of the positive equilibrium, which existsfor the linear [24] and quadratic mortality rates [37]only when certain conditions are satisfied; however,it always exists for hyperbolic mortality, which is usedwhen modelling the plankton population dynamics [6].From an ecological point of view, in general homo-geneous cases with linear and quadratic mortalities,both predator and prey populations will vanish even-tually because of the non-existence of positive equi-librium. Even when it exists under certain condition,linear mortality in the spatial model can not induceTuring instability though interesting lattice, and spiralpatterns have been discovered [24]. In this case, the pos-itive equilibrium is either uniformly stable or unstable,but neither of the forms can induce Turing instability.In ecology, we have two corresponding phenomena: (1)predator and prey population coexist in the concerning2-dimension region, or (2) densities of predator andprey have oscillation behaviour, which corresponds toirregular patterns. Quadratic mortality can not producesquares [37], though Turing instability can be inducedby it. With hyperbolic mortality, positive equilibriumexists unconditionally, and when it is stable, predatorand prey coexist always. Furthermore, stable Turingpatterns form, which not only implies both predator andprey persist in space, but also some ecological impli-cations: spots (see Figs. 5, 6) are assumed as a predatordefence function, and stripes (see Figs. 6, 7) are relatedto the social communication and predator defence [16].The ecological implications of squares and spirals arenot clear at the moment.

Acknowledgments The authors thank the referees for theirvaluable suggestions and comments concerning improving thework. The authors would also like to acknowledge the sup-port from the National Natural Science Foundation of China(11101318 and 11001212).

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