bifurcation and optimal harvesting of a diffusive predator–prey system with delays and interval...

Bifurcation and optimal harvesting of a diffusive predator–prey systemwith delays and interval biological parameters$

Xuebing Zhang a,b, Hongyong Zhao a,n

a Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People's Republic of Chinab Department of Basic Course, Huaian College of Information Technology, Huaian 223003, People's Republic of China

H I G H L I G H T S

� Consider the model with imprecise data as form of an interval in nature.� Apply PDE theory to discuss the stability and bifurcation of the model.� A Hopf bifurcation occurs as the delays increase through a certain threshold.� Pay attention to the exploitation or harvesting of biological resources.

a r t i c l e i n f o

Article history:Received 21 February 2014Received in revised form17 August 2014Accepted 18 August 2014Available online 27 August 2014

Keywords:Three speciesReaction–diffusionHopf bifurcationStability

a b s t r a c t

This paper deals with a delayed reaction–diffusion three-species Lotka–Volterra model with intervalbiological parameters and harvesting. Sufficient conditions for the local stability of the positiveequilibrium and the existence of Hopf bifurcation are obtained by analyzing the associated characteristicequation. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of thebifurcating periodic solutions are derived by applying the normal form method and center manifoldtheorem. Then an optimal control problem has been considered. Finally, numerical simulation results arepresented to validate the theoretical analysis. Numerical evidence shows that the presence of harvestingcan impact the existence of species and over harvesting can result in the extinction of the prey or thepredator which is in line with reality.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Since Lotka (1925) and Volterra (1926) proposed the firstpredator–prey model, the predator–prey model, especially thetwo-species predator–prey systems have been investigated exten-sively (see, for example, Hu et al., 2011; Zuo and Wei, 2011; He,1996; Song and Wei, 2005; Baurmann et al., 2007). However, thereis often interaction among multiple populations in nature. There-fore, it is more realistic to consider a multiple-species predator–prey system. For the three-species prey–predator system, therelationship among three groups may be competitive, one pre-dator and two preys, two predators competing for the same prey,or a food chain (see, for example, Zhang et al., 2012; Ma et al.,2012; Kumar et al., 2002; Liu and Yuan, 2006; Azar et al., 1995; Xuand Liao, 2013; So, 1979).

Recently, Zaman and Saker (2011) proposed a three-speciesprey–predator system in the following form:

dy1ðtÞdt

¼ y1ðtÞðr1�a11y1ðtÞ�a13y3ðtÞÞ;dy2ðtÞdt

¼ y2ðtÞðr2�a22y2ðtÞ�a23y3ðtÞÞ;dy3ðtÞdt

¼ y3ðtÞð�r3þa31y1ðtÞþa32y2ðtÞÞ;

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

ð1:1Þ

where ri are the natural growth rate of yi ði¼ 1;2Þ, a13; a23 are thepredation coefficients for yi, where y1 is density-independent, buty2 is density-dependent with intra-specific coefficients a22 andgrows logistically with growth rate r2 having a carrying capacityr2=a22; r3 is the natural death rate for the predator in the absenceof prey; a31=a13 and a32=a23 are the conversion factors.

In Zaman and Saker (2011), the authors discussed stability andperiodicity of system (1.1). However, they did not investigate thedelay and space in population dynamics.

It is well known that delays which occur in the interactionbetween predator–prey play a complicated role on a predator–preysystem. Many researchers have incorporated it into biological

Contents lists available at ScienceDirect

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Journal of Theoretical Biology

http://dx.doi.org/10.1016/j.jtbi.2014.08.0310022-5193/& 2014 Elsevier Ltd. All rights reserved.

☆The work is supported by National Natural Science Foundation of China underGrants 61174155 and 11032009. The work is also sponsored by Qing Lan Project ofJiangsu.

n Corresponding author: Tel.: +08613851617295.E-mail address: [email protected] (H. Zhao).

Journal of Theoretical Biology 363 (2014) 390–403

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models (see, for example, Teng and Chen, 2001; Chen et al., 1995;Meng and Chen, 2006; Zhang et al., 2009, 2013; Xu et al., 2004; Zuo,2013; Beretta and Kuang, 1996; Yan and Zhang, 2008; Wei andWang, 2006; Liu et al., in press) as delays could affect the stability ofa predator–prey system by creating instability, oscillation, and chaosphenomena. Therefore, more realistic models of population inter-actions should take into account the seasonality of the changingenvironment and the effects of delays.

On the other hand, in real life the species is spatially hetero-geneous and hence individuals will tend to migrate towardsregions of lower population density to add the possibility ofsurvival (Wu, 1996). For this reason, diffusion cannot be ignoredin studying the predator–prey system. There have been someexcellent papers with diffusion in a predator–prey system (see, forexample, Ma et al., 2012; Chang and Wei, 2012; Sambath et al.,2013; Tian and Weng, 2011; Tian and Zhang, 2013).

In addition, as we all know that the harvesting has a strongimpact on the dynamics of a model. In a certain extent, harvestingstrategy can control the long-term stationary density of populationefficiently. However, it can also lead to the incorporation of apositive extinction probability and, therefore, to potential extinc-tion in finite time. Moreover harvesting can drive the populationdensity to a dangerously low level at which extinction becomessure, no matter how the harvester affects the population after-wards. Therefore, determining socially acceptable harvesting pol-icy is undoubtedly one of the most challenging and mostcontroversial problems in the management of renewableresources, which attract ecologist's and economists' interests.There have been many investigations regarding optimal harvestingpolicy for a prey–predator system (see, for example, Chang andWei, 2012; Bai and Wang, 2007; Leung, 1995; Fister and Lenhart,2006; Song and Chen, 2001).

Motivated by the above discussions, we shall consider system(1.1) with delays and spatial diffusion. Let u1ðtÞ, u2ðtÞ and u3ðtÞrepresent the populations of preys and predator at time t,respectively. Assume that the predator species need time τ topossess the ability of predation after it was born. Then in this casethe corresponding model with homogeneous Neumann boundaryconditions resulting from considering one spatial variable andincorporating diffusion terms Δui ði¼ 1;2;3Þ and harvesting termsqiEiui ði¼ 1;2;3Þ to system (1.1)

∂u1

∂t¼ d1Δu1þu1ðr1�a11u1�a13u3ðt�τÞÞ�q1E1u1;

∂u2

∂t¼ d2Δu2þu2ðr2�a22u2�a23u3ðt�τÞÞ�q2E2u2;

∂u1

∂t¼ d3Δu3þu3ð�r3þa31u1þa32u2Þ�q3E3u3;

0oxoπ; t40;∂ui

∂xðx; tÞ ¼ 0 ði¼ 1;2;3Þ; x¼ 0; π; tZ0;

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

ð1:2Þ

with initial conditions

u1ðs; xÞ ¼ ψ1ðs; xÞZ0;u2ðs; xÞ ¼ ψ2ðs; xÞZ0;u3ðs; xÞ ¼ ψ3ðs; xÞZ0;ðs; xÞA ½�τ;0� � ½0; π�;

8>>>><>>>>:

ð1:3Þ

where d1; d2; d340 denote the diffusion coefficients of the first prey,second prey and the predator, respectively, d ri; aij; Ei ði; j¼ 1;2;3Þare all positive parameters.

The system (1.2) has certain parameters like most systemconsidered so far, it has been assumed that all biological andenvironmental parameters are constants precisely known. How-ever, in real life any biological or environmental parameters aredifficult to be precise. Since the lack of precise numerical informa-tion of the biological parameters such as prey population growth

rate, predator population decay rate and predation coefficient, weconsider the model with imprecise data as form of an interval innature.

Assume rjA ½rjl; rju� ðj¼ 1;2;3Þ, aijA ½aijl; aiju� ði; j¼ 1;2;3Þ. Alsorjl40 ðj¼ 1;2;3Þ and aijl40 ði; j¼ 1;2;3Þ. Based on Pal et al.(2013), system (1.2) can be written in the following form:

∂u1

∂t¼ d1Δu1þu1ððr1lÞ1�p1 ðr1uÞp1 �ða11lÞ1�p4 ða11uÞp4u1

�ða13lÞ1�p5 ða13uÞp5u3ðt�τÞÞ�q1E1u1;

∂u2

∂t¼ d2Δu2þu2ððr2lÞ1�p2 ðr2uÞp2 �ða22lÞ1�p6 ða22uÞp6u2

�ða23lÞ1�p7 ða23uÞp7u3ðt�τÞÞ�q2E2u2;

∂u1

∂t¼ d3Δu3þu3ð�ðr3lÞ1�p3 ðr3uÞp3 �ða31lÞ1�p8 ða311uÞp8u1

�ða32lÞ1�p9 ða32uÞp9u2Þ�q3E3u3;

0oxoπ; t40;∂ui

∂xðx; tÞ ¼ 0 ði¼ 1;2;3Þ; x¼ 0; π; tZ0;

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

ð1:4Þ

where piA ½0;1�. In the initial conditions (1.3), we assume that

ψ jðs; xÞAC¼ Cð½�τ;0�;XÞand X is defined by

X ¼ fuAW2;2ð0; πÞ : uxð0Þ ¼ uxðπÞ ¼ 0gwith the inner product ⟨�; �⟩.

To simplify the notations, in the following, we assume that

r1 ¼ ðr1lÞ1�p1 ðr1uÞp1 ; r2 ¼ ðr2lÞ1�p2 ðr2uÞp2 ;r3 ¼ ðr3lÞ1�p3 ðr3uÞp3 ; a11 ¼ ða11lÞ1�p4 ða11uÞp4 ;a13 ¼ ða13lÞ1�p5 ða13uÞp5 ; a22 ¼ ða22lÞ1�p6 ða21uÞp6 ;a23 ¼ ða23lÞ1�p7 ða23uÞp7 ; a31 ¼ ða31lÞ1�p8 ða31uÞp8 ;a32 ¼ ða32lÞ1�p9 ða32uÞp9 ;

for piA ½0;1� ði¼ 1;2;…;9Þ.Then system (1.4) become as

∂u1

∂t¼ d1Δu1þu1ðr1� a11u1� a13u3ðt�τÞÞ�q1E1u1;

∂u2

∂t¼ d2Δu2þu2ðr2� a22u2� a23u3ðt�τÞÞ�q2E2u2;

∂u1

∂t¼ d3Δu3þu3ð� r3þ a31u1þ a32u2Þ�q3E3u3;

0oxoπ; t40;∂ui

∂xðx; tÞ ¼ 0 ði¼ 1;2;3Þ; x¼ 0; π; tZ0:

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

ð1:5Þ

In this paper, we will study the stability and the local Hopfbifurcation for system (1.5) with delay τ as the bifurcation para-meter. To the best of our knowledge, it is the first time to deal withthe research of Hopf bifurcation for system (1.5).

The structure of this paper is arranged as follows. In Section 2,we consider the existence, uniqueness of positive solutions ofsystem (1.5). In Section 3, we study the local stability and theexistence of Hopf bifurcation. In Section 4, we give formuladetermining the direction of Hopf bifurcation and the stability ofspatially homogeneous and inhomogeneous bifurcating periodicsolutions. In Section 5, an optimal harvesting policy is consideredby using variational calculus. Finally, to support our theoreticalpredictions, some numerical simulations are given which supportthe analysis of Sections 3–5.

2. The existence, uniqueness of nonnegative solutions

In this section, we will discuss the existence, uniqueness ofnonnegative solutions. To do so, we need the following conceptsand results.

X. Zhang, H. Zhao / Journal of Theoretical Biology 363 (2014) 390–403 391

Definition 1. A pair of smooth functions U ¼ ðu1;u2;u3ÞT andU ¼ ðu1;u2;u3ÞT are called upper-lower solutions of (1.5), ifui4ui ði¼ 1;2;3Þ in ½�τ;1Þ � ½0; π�, and the following differentialinequalities hold:

∂u1

∂t�d1Δu1Zu1ðr1� a11u1� a13u3ðt�τÞÞ�q1E1u1;

∂u2

∂t�d2Δu2Zu2ðr2� a22u2� a23u3ðt�τÞÞ�q2E2u2;

∂u3

∂t�d3Δu3Zu3ð� r3þ a31u1þ a32u2Þ�q3E3u3;

∂u1∂t

�d1Δu1ru1ðr1� a11u1� a13u3ðt�τÞÞ�q1E1u1;

∂u2

∂t�d2Δu2ru2ðr2� a22u2� a23u3ðt�τÞÞ�q2E2u2;

∂u3∂t

�d3Δu3ru3ð� r3þ a31u1þ a32u2Þ�q3E3u3;

uðs; xÞrψ iðs; xÞruðs; xÞ; ðs; xÞA ½�τ;0� � ½0; π�:

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

ð2:1Þ

Furthermore, we make the following assumption:

ðH1Þa23ðr1�q1E1Þ� a13ðr2�q2E2Þ40;a11ðr3þq3E3Þ� a31ðr1�q1E1Þ40;a32ðr2�q2E2Þ� a22ðr3þq3E3Þ40:

8><>:

Lemma 1. Let U and U be a pair of coupled upper and lowersolutions for the system (1.5) and suppose that the initial functionsψ i ði¼ 1;2;3Þ are Holder continuous in ½�τ;0� � ½0; π�. Then system(1.5) has exactly one regular solution Uðx; tÞ ¼ ðu1ðx; tÞ;u2ðx; tÞ;u3ðx; tÞÞT satisfying UrUrU in ½0; π� � ½�τ;1Þ as (H1) holds.

Proof. Let U ¼ ð0;0;0ÞT and U ¼ ðu1;u2;u3ÞT , where ui ði¼ 1;2;3Þsatisfies

u14max Jψ J1;r1�q1E1

a11

� �;

u24max Jψ J2;r2�q2E2

a22

� �u34max

½0;π�fψ3ð0; xÞg

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

ð2:2Þ

where Jψ J i ¼max½� τ;0��½0;π�fψ iðs; xÞg, i¼1,2. Then U ¼ ð0;0;0ÞT andU ¼ ðu1;u2;u3ÞT are clearly a pair of lower-upper solution of (1.5),hence, 0ruiðx; tÞrui ði¼ 1;2;3Þ for ðt; xÞA ½�τ;1� � ½0; π�. Andalso, by the maximal principle, we have uiðx; tÞ40 ði¼ 1;2;3Þ forall t40; xA ½0; π� if ψ ið0; xÞ40 ði¼ 1;2;3Þ. This completes theproof. □

3. Local stability and Hopf bifurcation

In this section, we will discuss the local stability and Hopfbifurcation of system (1.5) by analyzing the corresponding char-acteristic equations.

Lemma 2. If (H1) holds, then system (1.5) has a unique positiveequilibrium point Enðun

1;un

2;un

3ÞT .

Proof. From system (1.5), we can obtain a unique equilibriumpoint Enðun

1;un

2;un

3ÞT , where

un

1 ¼a32a23ðr1�q1E1Þ� a13a32ðr2�q2E2Þþ a13a22ðr3þq3E3Þ

a11a23a32þ a13a22a31;

un

2 ¼a11a23ðr3þq3E3Þ� a31a23ðr1�q1E1Þþ a13a31ðr2�q2E2Þ

a11a23a32þ a13a22a31;

un

3 ¼a31a22ðr1�q1E1Þþ a11a32ðr2�q2E2Þ� a11a22ðr3þq3E3Þ

a11a23a32þ a13a22a31:

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

ð3:1Þ

It is easy to obtain that the equilibrium point En is positive if (H1)holds. This completes the proof. □

Let ~uj ¼ uj�un

j ðj¼ 1;2;3Þ and drop bars for the simplicity ofnotations. Then system (1.5) can be rewritten as the followingform:

∂u1

∂t¼ d1Δu1� a11un

1� a11u21� a13u3ðt�τÞu1

� a13un

1u3ðt�τÞ;∂u2

∂t¼ d2Δu2� a22un

2u2� a22u22� a23u3ðt�τÞu2

� a23un

2u3ðt�τÞ;∂u3

∂t¼ d3Δu3þ a31u1u3þ a32u2u3þ a31u1un

3þ a32u2un

3;

0oxoπ; t40;∂ui

∂xðx; tÞ ¼ 0 ði¼ 1;2;3Þ; x¼ 0; π; tZ0:

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

ð3:2Þ

Thus, the constant steady state En of system (1.5) is transformedinto the zero equilibrium ð0;0;0ÞT of system (3.2).

In the following, we will analysis stability and bifurcation of thezero equilibrium point of system (3.2).

Denote

UðtÞ ¼ ðu1ðtÞ;u2ðtÞ;u3ðtÞÞT ¼ ðu1ðt; �Þ;u2ðt; �Þ;u3ðt; �ÞÞT ;then (3.2) can be rewritten as an abstract differential equation inthe phase space C¼ Cð½�τ;0�;XÞ of the form

_U ¼DΔUðtÞþLðUtÞþ f ðUtÞ; ð3:3Þwhere

D¼ diagfd1; d2; d3g;

Δ¼ diagf∂2=∂x2; ∂2=∂x2; ∂2=∂x2g;

UtðθÞ ¼UðtþθÞ; �τrθr0;

L : C-X

and f : C-X are given, respectively, by

LðφÞ ¼� a11un

1φ1ð0Þ� a13un

1φ3ð�τÞ� a22un

2φ2ð0Þ� a23un

2φ3ð�τÞa31un

3φ1ð0Þþ a32un

3φ2ð0Þ

0B@

1CA ð3:4Þ

and

f ðφÞ ¼� a11φ

21ð0Þ� a13φ1ð0Þφ3ð�τÞ

� a22φ22ð0Þ� a23φ2ð0Þφ3ð�τÞ

a31φ1ð0Þφ3ð0Þþ a32φ2ð0Þφ3ð0Þ

0B@

1CA: ð3:5Þ

For φðθÞ ¼ UtðθÞ, φ¼ ðφ1;φ2;φ3ÞT AC, the linearized system of(3.3) at the zero equilibrium is

_U ¼DΔUðtÞþLðUtÞ; ð3:6Þand its characteristic equation is

λy�DΔy�Lðeλ�yÞ ¼ 0; ð3:7Þwhere yAdomðΔÞ, and ya0; domðΔÞ � X.

From the properties of the Laplacian operator defined on thebounded domain, the operator Δ on X has the eigenvalues�k2; kAN09f0;1;2…g with the relative eigenfunctions, where

β1k ¼γk00

0B@

1CA; β2k ¼

0γk0

0B@

1CA; β3k ¼

00γk

0B@

1CA; γk ¼ cos ðkxÞ: ð3:8Þ

Clearly, ðβ1k ; β2k ; β3k Þ10 constructs a basis of the phase space X.Therefore, any element y in X can be expanded as Fourier series in

X. Zhang, H. Zhao / Journal of Theoretical Biology 363 (2014) 390–403392

the following form:

y¼ ∑1

k ¼ 0YTk

β1kβ2k

β3k

0BB@

1CCA; Yk ¼

⟨y; β1k ⟩

⟨y; β2k ⟩

⟨y; β3k ⟩

0BB@

1CCA: ð3:9Þ

By calculation

LðφT ðβ1k ; β2k ; β3k ÞT Þ ¼ LðφÞT ðβ1k ; β2k ; β3k ÞT ; kAN0: ð3:10ÞAccording to (3.8) and (3.9), (3.7) is equivalent to

∑1

k ¼ 0YTk λI3þDk2�

� a11 ~u1 0 � a13 ~u1e� λτ

0 � a22 ~u2 � a23 ~u2e� λτ

a31 ~u3 a32 ~u3 0

0B@

1CA

264

375

β1kβ2k

β3k

0BB@

1CCA¼ 0

ð3:11ÞHence, we conclude that (3.7) is equivalent to the following

form:

λ3þAkλ2þBkλþCkþðGλþDkÞe� λτ ¼ 0; ð3:12Þ

where

Ak ¼ ðd1þd2þd3Þk2þ a11un

1þ a22un

2;

Bk ¼ ðd1k2þ a11un

1Þðd2k2þ a22un

2Þþðd1k2þ a11un

2Þd3k2þðd2k2þ a22 ~u2Þd3k2;

Ck ¼ ðd1k2þ a11un

1Þðd2k2þ a22un

2Þd3k2;

Dk ¼ ðd1k2þ a11un

1Þa23a32un

2un

3þðd2k2þ a22un

2Þa13a31 ~u1 ~u3;

G¼ a13a31un

1un

3þ a23a32un

2un

3: ð3:13ÞObviously, for 8kAN0, λ¼ 0 is not a root of (3.12).

As τ¼ 0, Eq. (3.12) is equivalent to the following cubic equation:

λ3þAkλ2þðBkþGÞλþCkþDk ¼ 0: ð3:14Þ

Clearly, Ak40, BkþG40, CkþDk40 for any kAN0. In addition, itis easy to show that

AkðBkþGÞ�ðCkþDkÞ ¼ ðd1k2þd2k2þ a11un

1þ a22un

2Þ�ðd1k2þ a11un

1Þðd2k2þ a22un

2Þþðd1k2þ a11un

1þd2k2þ a22un

2þd3k2Þðd1k2

þd2k2þ a11un

1þ a22un

2Þd3k2

þðd1k2þ a11un

1Þa13a31un

1un

3

þðd2k2þ a22un

2Þa23a32un

2un

3

þd3k2ða13a31un

1un

3þ a23a32un

2un

3Þ40

By the Routh–Hurwitz criteria, all the roots of Eq. (3.14) havenegative real parts. Therefore, we have the following result.

Lemma 3. Assume that (H1) holds, then the zero equilibrium point ofsystem (3.2) with τ¼ 0 is locally asymptotically stable.

Now we discuss the effect of the delay τ on the stability of thetrivial solution of (3.2). Assume that iω is a root of Eq. (3.12). Thenω should satisfy the following equation for some kAN0:

� iω3�Akω2þ iBkωþCkþðGiωþDkÞð cos ðωτÞ� i sin ðωτÞÞ ¼ 0;

ð3:15Þwhich implies that

�Akω2þCk ¼ �Dk cos ðωτÞ�Gω sin ðωτÞ;

�ω3þBkω¼ �Gω cos ðωτÞþDk sin ðωτÞ:

(ð3:16Þ

From (3.16), adding up the squares of both we have

ω6þðA2k�2BkÞω4þðB2

k�2AkCk�G2Þω2þC2k�D2

k ¼ 0: ð3:17ÞLet z¼ ω2 and denote

Pk ¼ A2k�2Bk; Qk ¼ B2

k�2CkAk�G2; Vk ¼ C2k�D2

k : ð3:18ÞThen (3.17) can be rewritten into the following form:

z3þPkz2þQkzþVk ¼ 0: ð3:19Þ

When k¼0, Eq. (3.19) becomes

z3þP0z2þQ0zþV0 ¼ 0: ð3:20ÞDenote

hðzÞ ¼ z3þP0z2þQ0zþV0: ð3:21ÞMake the following assumption:

ðH2Þ a11a22un

1un

2� a23a32un

2un

3� a13a31un

1un

340;

ðH3Þ d3a11a22un

1un

2�d1a23a32un

2un

3�d2a13a31un

1un

3

�un

1un

2un

3ða11a23a32þ a13a31a22Þ40:

Lemma 4. Eq. (3.21) has a unique positive root.

Proof. Clearly, from (3.13)

P0 ¼ ða11un

1Þ2þða22un

2Þ240:

Obviously,

V0 ¼ C20�D2

0 ¼ ðC0þD0ÞðC0�D0Þ ¼ �ðun

1un

2un

3ða11a23a32þ a13a31a22ÞÞ2o0:

According to Descartes's rule of signs (Rene, 1968), Eq. (3.21)has a unique positive root. This completes the proof. □

According to Lemma 4, Eq. (3.21) has a unique positive root,denoted by z0, and thus Eq. (3.17) has a unique positive rootω0 ¼

ffiffiffiffiffiz0

p. By (3.16), we have

cos ðω0τn

0Þ ¼A0D0ω

20þGω4

0�C0D0�B0Gω20

D20þG2ω2

0

� bn;

sin ðω0τn

0Þ ¼D0B0ω0þGA0ω

20�D0ω

30�GC0

GðD20þG2ω2

0Þ� an;

8>>>><>>>>:

ð3:22Þ

thus

τj0 ¼

arccosðbnÞþ2jπω0

; anZ0;

2π�arccosðbnÞþ2jπω0

; ano0;

8>>><>>>:

ð3:23Þ

where jAN0, then 7 iω0 is a pair of purely imaginary roots of

(3.17) with τ¼ τj0. Clearly, sequence fτj0g1j ¼ 0 is increasing and

limj-þ1

τj0 ¼ þ1: ð3:24Þ

Thus, we can define

τ0 ¼ τ00 ¼minfτj0g: ð3:25Þ

Lemma 5. If (H1)–(H3) hold, then for all kZ1, (3.19) has no positivereal root.

Proof. From (3.13), we can obtain

Pk ¼ A2k�2Bk ¼ ðd1k2þ a11un

1Þ2þðd2k2þ a22un

2Þ2þðd3k2Þ240:

Qk ¼ B2k�2CkAk�G2 ¼ ½ðd1k2þ a11un

1Þðd2k2þ a22un

2Þ�2

þ½ðd1k2þ a11un

1Þd3k2�2þ½ðd2k2þ a22un

2Þd3k2�2

�ða13a31un

1un

3þ a23a32un

2un

3Þ2:


By (H2), we know that Qk40,

Vk ¼ C2k�D2

K ¼ ðCk�DK ÞðCkþDK Þ:

Obviously, CkþDK 40,

Ck�DK ¼ d1d2d3k6þd3ðd2a11un

1þd1a12un

2Þk4

þðd3a11a22un

1un

2�d1a23a32un

2un

3�d2a13a31un

1un

3Þk2�un

1un

2un

3ða11a23a32þ a13a31a22Þ:

By (H3), we have Ck�DK 40.According to Descartes's rule of signs (Rene, 1968), Eq. (3.19)

has no positive real root for 8kZ1. This completes the proof. □

Lemma 6. Let λðτÞ ¼ αðτÞ7 iωðτÞ be the root of (3.12) near τ¼ τj0satisfying αðτj0Þ ¼ 0, ωðτj0Þ ¼ω0. Suppose that h0ðz0Þa0, where h(z) isdefined by (3.21). Then, the following transversality condition holds:

dðRe λðτÞÞdτ τ ¼ τj0

a0�� ð3:26Þ

and the sign of dðRe λðτÞÞ=dτjτ ¼ τj0

is consistent with that of h0ðz0Þ.

Proof. Differentiating the two sides of (3.19) with respect to τyields

dλdτ

� ��1

¼ ð3λ2þ2A0λþB0ÞeλτþGλðGλþD0Þ

�τ

λ:

From (3.16) and (3.18), we can easily obtain

dðRe λðτÞÞdτ

� �1

τ ¼ τj0

¼ Reð3λ2þ2A0λþB0Þeλτ

λðGλþD0Þ

" #τ ¼ τj0 ;λ ¼ iω0

þReG

λðGλþD0Þ

� τ ¼ τj0 ;λ ¼ iω0

¼ �Gω20½ðB0�3ω2

0Þ cos ðω0τj0Þ�2A0ω0 sin ðω0τ

j0Þ�

G2ω40þD2

0ω20

þD0ω0½2A0ω0 cos ðω0τj0ÞþðB0�3ω2

0Þ sin ðω0τj0Þ��G2ω2

0

G2ω40þD2

0ω20

¼ 3ω60þ2ðA2

0�2B0Þω40þðB2

0�2A0C0�G2Þω20

G2ω40þD2

0ω20

¼ z0ð3z20þ2P0z0þQ0ÞG2ω4

0þD20ω

20

¼ z0h0ðz0Þ

G2ω40þD2

0ω20

:

Thus, we have

signdðRe λðτÞÞ

dτ

� �τ ¼ τj0

¼ signdðRe λðτÞÞ

dτ

� ��1

τ ¼ τj0

¼ signz0h

0ðz0ÞG2ω4

0þD20ω

20

( ):

Noting that G2ω40þD2

0ω20, z040, we conclude that the sign of

½dðRe λðτÞÞ=dτ�τ ¼ τj0

is determined by that of h0ðz0Þ. This completesthe proof. □

On the basis of Lemmas 3–6, we have the following result.

Theorem 1. If the conditions (H1)–(H3) hold and h0ðz0Þa0, then thefollowing statements are true:

(i) when τA ½0; τ0Þ, the trivial equilibrium point of (3.2) is asympto-tically stable;

(ii) the Hopf bifurcation occurs at τ¼ τ0, that is, system (3.2) has abranch of periodic solutions bifurcating from the zero solutionnear τ¼ τ0.

Next, we discuss the effect of diffusion on spatially inhomoge-neous Hopf bifurcation. Consider Eq. (3.19) again. Noting thatPk40, so, if there exist k0AN9f1;2;3…g such that Ck�Dko0,

then according to Descartes's rule of signs, Eq. (3.19) has a uniquepositive root, denoted by zk0 , and hence Eq. (3.17) has a uniquepositive root ωk0 ¼

ffiffiffiffiffiffizk0

p . By (3.16), we have

cos ðωk0 τÞ ¼Ak0Dk0ω

2k0þGω4

k0�Ck0Dk0 �B0Gω2

k0

D20þG2ω2

k0

; ð3:27Þ

thus

τjk0 ¼arccosðbnÞþ2jπ

ωk0; ð3:28Þ

where j¼ 0;1;…, then 7 iωk0 is a pair of purely imaginary roots of(3.12) with τ¼ τjk0 . We assume that h0ðzk0 Þa0. Using the similarargument as that in the proof of Lemma 6, the following transver-sally condition holds:

dðRe λðτÞÞdτ τ ¼ τj

k0

a0:�� ð3:29Þ

Thus, we have the following result.

Theorem 2. If (H1) holds, assume furthermore that h0ðzk0 Þa0. Thensystem (3.2) undergoes a spatially inhomogeneous Hopf bifurcationat the zero equilibrium point when τ¼ τjk0 ðjAN0Þ, i.e., a family ofspatially inhomogeneous periodic solutions bifurcating from the zeroequilibrium point when τ crosses through the critical valuesτ¼ τjk0 ðjAN0Þ.

4. Direction and stability of Hopf bifurcation

In the previous section, we have shown that system (3.2)admits a series of periodic solutions bifurcating from the zeroequilibrium point at the critical value τk

j ðk; jAN0Þ. In this section,we derive explicit formulae to determine the properties of theHopf bifurcation at critical value τk

j ðk; jAN0Þ. By using the normalform theory and center manifold reduction for PFDEs developedby Wu (1996). Throughout this section, we also assume that thecondition (H1) holds.

For fixed jAN0, denote τkj by τn and introduce the new

parameter μ¼ τ�τn. Normalizing the delay τ by the time-scalingt-t=τ. Then (3.2) can be rewritten as

dUðtÞdt

¼ τnDΔUðtÞþLðτnÞðUtÞþFðUt ; μÞ; ð4:1Þ

where LðμÞðφÞ : C-X and Fð�; μÞ : C-X are given by

LðμÞðφÞ ¼ μ

� a11un

1φ1ð0Þ� a13un

1φ3ð�1Þ� a22un

2φ2ð0Þ� a13un

2φ3ð�1Þa31un

3φ1ð0Þþ a32un

3φ2ð0Þ

0B@

1CA; ð4:2Þ

Fðφ; μÞ ¼ μDΔφð0ÞþLðμÞφþ f ðφ; μÞ ð4:3Þ

and

f ðφ; μÞ ¼ ðτnþμÞ� a11φ

21ð0Þ� a13φ1ð0Þφ3ð�1Þ

� a22φ22ð0Þ� a23φ2ð0Þφ3ð�1Þ

a31φ1ð0Þφ3ð0Þþ a32φ2ð0Þφ3ð0Þ

0B@

1CA ð4:4Þ

for φ¼ ðφ1;φ2;φ3ÞT AC.Then linearized system (4.1) at ð0;0;0Þ is

dUðtÞdt

¼ τnDΔUðtÞþLðτnÞðUtÞ: ð4:5Þ

Based on the discussion in Section 3, we can easily know that forτ¼ τn, the characteristic equation of (3.12) has a pair of simplepurely imaginary eigenvalues Λk ¼ fiωkτ

n; � iωkτng.


Let C≔Cð½�1;0�;R3Þ, considering the following FDE on C:

_z ¼ �τnDk2zþLðτnÞðztÞ: ð4:6Þ

Obviously, LðτnÞ is a continuous linear function mappingCð½�1;0�;R3Þ into R3. By the Riesz representation theorem, thereexists a 3�3 matrix function ηðθ; τÞ (�1rθr0), whose elementsare of bounded variation such that

�τnDk2φð0ÞþLðτnÞðφÞ ¼Z 0

�1½dηkðθ; τnÞ�φðθÞ for φAC: ð4:7Þ

In fact, we can choose

ηðθ; τnÞ ¼ τn

�d1k2� a11un

1 0 0

0 �d2k2� a22un

2 0

a31un

3 a32un

3 �d3k2

0BBB@

1CCCAδðθÞ

�τn0 0 � a13un

1

0 0 � a23un

2

0 0 0

0B@

1CAδðθþ1Þ; ð4:8Þ

where δ is the Dirac delta function.Let AðτnÞ denote the infinitesimal generator of the semigroup

induced by the solutions of (4.6) and An be the formal adjoint ofAðτnÞ under the bilinear pairing

ðψ ;ϕÞ ¼ ðψð0Þ;ϕð0ÞÞ�Z 0

�1

Z θ

ξ ¼ 0ψðξ�θÞ dηkðθÞϕðξÞ dξ

¼ ðψð0Þ;ϕð0ÞÞþτnZ 0

�1ψðθþ1Þ

0 0 � a13un

1

0 0 � a23un

2

0 0 0

0B@

1CAϕðθÞ dθ;

ð4:9Þ

for ϕAC;ψACn ¼ Cð½0;1�;R3Þ. Then AðτnÞ and An are a pair ofadjoint operators. From the discussion in Section 3, we know thatAðτnÞ has a pair of simple purely imaginary eigenvalues 7 iωkτ

n

and they are also eigenvalues of An since AðτnÞ and An are a pair ofadjoint operators. Let P and Pn be the center spaces, that is, thegeneralized eigenspaces of AðτnÞ and An associated with Λ0,respectively. Then Pn is the adjoint space of P and dim P ¼dim Pn ¼ 2. Direct computations give the following results.

Lemma 7. Let

α¼ a23un

2ðd1k2þ a11un

1þ iωkÞa13un

1ðd2k2þ a22un

2þ iωkÞ; β¼ �ðd1k2þ a11un

1þ iωkÞeiωkτn

a13un

1;

αn ¼ a32ðiωkþd1k2þ a11un

1Þa31ðiωkþd2k

2þ a22un

2Þ; βn ¼ iωkþd1k

2þ a11un

1

a31un

3:

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

ð4:10Þ

Then

p1ðθÞ ¼ eiωkτnθð1;α; βÞT ; p2ðθÞ ¼ p1ðθÞ; �1rθr0; ð4:11Þ

is a basis of P associated with Λk and

q1ðsÞ ¼ ð1; αn; βnÞe� iωkτns; q2ðsÞ ¼ q1ðsÞ; 0rsr1; ð4:12Þ

is a basis of Q associated with Λk.

Let Φ¼ ðΦ1;Φ2Þ and Ψn ¼ ðΨn

1;Ψn

2ÞT with

Φ1ðθÞ ¼p1ðθÞþp2ðθÞ

2¼

Refeiωkτnθg

Refαeiωkτnθg

Refβeiωkτnθg

0B@

1CA

¼

cos ωkτnθ

1a13un

1ððd2k2 þ a22un

2Þ2 þω2

kÞða23un

2ðððd1k2þ a11un

1Þðd2k2

þ a22un

2Þþω2k Þ cos ðωkτ

nθÞþωkððd1k2þ a11un


2ÞÞ sin ðωkτnθÞÞÞ

1a13un

1ððωk cos ðωkτ

nÞþðd1k2þ a11un

1Þ sin ðωkτnÞÞ sin ðωkτ

nθÞ

�ððd1k2þ a11un

1Þ cos ðωkτnÞ�ωk sin ðωkτ

nÞÞ cos ðωkτnθÞ

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA;

Φ2ðθÞ ¼p1ðθÞ�p2ðθÞ

2i¼ Imfeiωkτ

nθgImfαeiωkτ

nθg

!

¼

sin ωkτnθ

1a13un

1ððd2k2 þ a22un

2Þ2 þω2

kÞða23un

2ðððd1k2þ a11un

1Þðd2k2

þ a22un

2Þþω2k Þ sin ðωkτ

nθÞþωkððd2k2þ a22un


1ÞÞ cos ðωkτnθÞÞÞ

1a13un

1ð�ðωk cos ðωkτ

nÞþðd1k2þ a11un

1Þ sin ðωkτnÞÞ cos ðωkτ

nθÞ

þðððd1k2þ a11un

1Þ cos ðωkτnÞ�ωk sin ðωkτ

nÞÞ sin ðωkτnθÞÞ

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA;

for θA ½�1;0�, and

Ψn

1ðsÞ ¼q1ðsÞþq2ðsÞ

2¼

Refe� iωkτnsg

Refαne� iωkτnsg

Refβne� iωkτnsg

0B@

1CA

T

¼

cos ωkτns

1a31ððd2k2 þ a22un

2Þ2 þω2

kÞða32ðððd1k2þ a11un

1Þðd2k2

þ a22un

2Þþω2k Þ cos ðωkτ

nsÞþωkððd2k2þ a22un

2ÞÞ�ðd1k2þ a11un

1ÞÞ sin ðωkτnsÞÞÞ

ðd1k2 þ a11un

1Þ cos ðωkτnsÞþωk sin ðωkτ

nsÞa31un

3

0BBBBBBBBBB@

1CCCCCCCCCCA

T

;

Ψn

2ðsÞ ¼q1ðsÞ�q2ðsÞ

2i¼

Imfe� iωkτnsg

Imfαe� iωkτnsg

Imfβeiωkτnsg

0B@

1CA

T

¼

� sin ωkτns

1a31ððd2k2 þ a22un

2Þ2 þω2

kÞða32ð�ððd1k2þ a11un

1Þðd2k2

þ a22un

2Þþω2k Þ sin ðωkτ

nsÞþωkððd2k2

þ a22un

2d�ðd1k2þ a11un

1ÞÞ cos ðωkτnsÞÞÞ

� ðd1k2 þ a11un

1Þ sin ðωkτnsÞþωk cos ðωkτ

nsÞa31un

3

0BBBBBBBBBB@

1CCCCCCCCCCA

T

;

for sA ½0;1�. From (4.9), we can obtain ðΨ n

1;Φ1Þ and ðΨ n

1;Φ2Þ. Notingthat

ðq1;p1Þ ¼ ðΨn

1;Φ1Þ�ðΨn

2;Φ2Þþ i½ðΨn

1;Φ2ÞþðΨn

2;Φ1Þ�

and

ðq1;p1Þ ¼ 1þααnþββn�τne� iωkτn

βða13un

1þαna23un

2Þ≔Dn:

Therefore, we have

ðΨ n

1;Φ1Þ�ðΨn

2;Φ2Þ ¼ RefDng;

ðΨ n

1;Φ2ÞþðΨ n

2;Φ1Þ ¼ ImfDng:Now, we define ðΨ n;ΦÞ ¼ ðΨ n

j ;ΦkÞ ðj; k¼ 1;2Þ and construct anew basis Ψ for Q by

Ψ ¼ ðΨ1;Ψ 2ÞT ¼ ðΨn;ΦÞ�1Ψn:


Obviously, ðΨ ;ΦÞ is a second order identity matrix. In addition,define f k ¼ ðξ1k ; ξ2k ; ξ3k Þ, where

ξ1k ¼cos ðkxÞ

00

0B@

1CA; ξ2k ¼

0cos ðkxÞ

0

0B@

1CA; ξ2k ¼

00

cos ðkxÞ

0B@

1CA:

Let c � f k be defined by

c � f k ¼ c1ξ1kþc2ξ2kþc3ξ3k

for c¼ ðc1; c2; c3ÞT ; cjAR ðj¼ 1;2;3Þ.Then the center space of linear equation (4.5) is given by PCNC,

where

PCNφ¼ΦðΨ ; ⟨φ; f k⟩Þ � f k; φAC; ð4:13Þand C¼ PCNC PSC, here PSC denotes the complementary sub-space of PCNC.

Let Aτn be defined by

AτnφðθÞ ¼ _φðθÞþX0ðθÞ½τnDΔφð0ÞþLðτnÞðφðθÞÞ� _φð0Þ�; φABC;

where X0 : ½�1;0�-BðX;XÞ is given by

X0ðθÞ ¼0; �1rθo0;I; θ¼ 0:

(ð4:14Þ

Then Aτn is the infinitesimal generator induced by the solution of(4.5) and (4.1) can be rewritten as the following operator differ-ential equation:

_Ut ¼ AτnUtþX0FðUt ; μÞ: ð4:15ÞUsing the decomposition C¼ PCNC PSC and (4.13), the solu-

tion of (4.1) can be rewritten as

Ut ¼Φx1ðtÞx2ðtÞ

!� f kþhðx1; x2; μÞ; ð4:16Þ

where

x1ðtÞx2ðtÞ

!¼ ðΨ ; ⟨Ut ; f k⟩Þ; ð4:17Þ

and hðx1; x2; μÞAPsc with hð0;0;0Þ ¼Dhð0;0;0Þ ¼ 0. In particular,the solution of (4.1) on the center manifold is given by

Un

t ¼Φx1ðtÞx2ðtÞ

!� f kþhðx1; x2;0Þ: ð4:18Þ

Setting z¼ x1� ix2 and noticing that p1 ¼Φ1þ iΦ2, then (4.18)can be rewritten as

Un

t ¼12Φ

zþz

iðz�zÞ

!� f kþwðz; zÞ ¼ 1

2ðp1zþp1zÞ � f kþWðz; zÞ; ð4:19Þ

where Wðz; zÞ ¼ hððzþzÞ=2; �ðz�zÞ=2i;0Þ. Moreover, by Wu(1996), z satisfies

_z ¼ iωkτnzþgðz; zÞ; ð4:20Þ

where

gðz; zÞ ¼ ðΨ1ð0Þ� iΨ2ð0ÞÞ⟨FðUn

t ;0Þ; f k⟩: ð4:21ÞLet

Wðz; zÞ ¼W20z2

2þW11zzþW02

z2

2þ⋯ ð4:22Þ

and

gðz; zÞ ¼ g20z2

2þg11zzþg02

z2

2þ⋯: ð4:23Þ

From (4.19), we have

⟨FðUn

t ;0Þ; f k⟩¼τnz2

4

�ða11þ a13βe� iωkτn Þ

�ða22α2þ a23αβe� iωkτ

n Þβa31þαβa32

0B@

1CA1π

Z π

0cos 3ðkxÞ dx

þτnzz4

�ð2a11þ a13βeiωkτn þ a13βe� iωkτ

n Þ�ð2a22ααþ a23αβeiωkτ

n þ a23αβe� iωkτn Þ

a31ðβþβÞþ a32ðαβþαβÞ

0B@

1CA1π

Z π

0cos 3ðkxÞ dx

þτnz2

4

�ða11þ a13βeiωkτn Þ

�ða22α2þ a23αβeiωkτ

n Þβ a31þαβ a32

0B@

1CA1π

Z π

0cos 3ðkxÞ dx

þτn

4

⟨ð� a11ð2W ð1Þ20 ð0Þþ4W ð1Þ

11 ð0ÞÞ� a13ð2W ð3Þ11 ð�1Þ

þW ð3Þ20 ð�1ÞÞþ2W ð1Þ

11 ð0Þβe� iωkτn

þβW ð1Þ20 ð0Þeiωkτ

n Þ cos kx; cos kx⟩⟨ð� a22ð4W ð2Þ

11 ð0Þαþ2W ð2Þ20 ð0ÞαÞ

� a23ð2W ð3Þ11 ð�1ÞαþW ð3Þ

20 ð�1ÞαÞþW ð2Þ

20 ð0Þβþ2W ð2Þ11 ð0Þβe� iωkτ

n Þ cos kx; cos kx⟩⟨ða31ð2W ð3Þ

11 ð0ÞþW ð3Þ20 ð0Þþ2W ð1Þ

11 ð0ÞβþW ð1Þ

20 ð0ÞβÞþ a32ð2W ð3Þ11 ð0ÞαÞ

þW ð3Þ20 ð0Þαþ2W ð2Þ

11 ð0ÞβþW ð2Þ20 ð0ÞÞβÞ cos kx;

cos kx⟩

0BBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCA

z2zþ⋯;

where

⟨WnijðθÞ; cos kx⟩¼ 1

π

Z π

0Wn

ijðθÞðxÞ cos kx dx; iþ j¼ 2; nAN:

Notice thatR π0 cos 3ðkxÞ dx¼ 0, for 8kAN.

Let ðψ1;ψ2;ψ3Þ ¼ Ψ1ð0Þ� iΨ 2ð0Þ. Then by (4.21)–(4.23), we haveg20 ¼ g11 ¼ g02 ¼ 0, when kAN. When k¼0, we can obtain thefollowing quantities:

g20 ¼τn

2½�ða11þ a13αβe� iωkτ

n Þψ1�ða22α2

þ a23αβe� iωkτn Þψ2þðβa31þαβa32Þψ3�;

g11 ¼τn

4½�ð2a11þ a13βeiωkτ

n þ a13βe� iωkτn Þψ1

�ð2a22ααþ a23αβeiωkτn þ a23αβe� iωkτ

n Þψ2

þða31ðβþβÞþ a32αβþ a32αβÞψ3�;

g02 ¼τn

2½�ða11þ a13βe� iωτn Þψ1 �ða22α

2

þ a23αβe� iωτn Þψ2 þða31βþ a32αβÞψ3 �;

g21 ¼τn

2½⟨ð� a11ð2W ð1Þ

20 ð0Þþ4W ð1Þ11 ð0ÞÞ� a13ð2W ð3Þ

11 ð�1ÞþW ð3Þ

20 ð�1ÞÞþ2W ð1Þ11 ð0Þβe� iω0τ

n þβW ð1Þ20 ð0Þeiω0τ

n Þ cos kx;

� cos kx⟩ψ1þ ⟨ð� a22ð4W ð2Þ11 ð0Þαþ2W ð2Þ

20 ð0ÞαÞ� a23ð2W ð3Þ11 ð�1Þα

þW ð3Þ20 ð�1ÞαÞþW ð2Þ

20 ð0Þβþ2W ð2Þ11 ð0Þβe� iω0τ

n Þ cos kx; cos kx⟩ψ2

þ⟨ða31ð2W ð3Þ11 ð0ÞþW ð3Þ

20 ð0Þþ2W ð1Þ11 ð0Þβ

þW ð1Þ20 ð0ÞβÞþ a32ð2W ð3Þ

11 ð0ÞαþW ð3Þ

20 ð0Þαþ2W ð2Þ11 ð0ÞβþW ð2Þ

20 ð0ÞÞβÞ cos kx; cos kx⟩ψ3�:Since W20ðθÞ;W11ðθÞ for θA ½�1;0� appear in g21, we still need

to compute them. It follows easily from (4.22) that

_W ðz; zÞ ¼W20z _zþW11ð_zzþz _z ÞþW02z _zþ⋯ ð4:24Þand

AτnW ¼ AτnW20z2

2þAτnW11zzþAτnW02

z2

2þ⋯: ð4:25Þ


In addition, by Wu (1996), WðzðtÞ; zðtÞ satisfy_W ¼ AτnWþHðz; zÞ; ð4:26Þwhere

Hðz; zÞ ¼H20z2

2þH11zzþH02

z2

2þ⋯

¼ X0FðUn

t ;0Þ�ΦðΨ ; ⟨X0FðUn

t ;0Þ; f k⟩Þ � f k; ð4:27Þwith HijAPSC; iþ j¼ 2.

Thus, from (4.19) and (4.24)–(4.26), we can obtain that

ð2iωkτn�Aτn ÞW20 ¼H20;

�AτnW11 ¼H11:

(ð4:28Þ

Noticing that Aτn has only two eigenvalues 7 iωkτn with zero

real parts, therefore, (4.26) has unique solution Wijðiþ j¼ 2Þ in PSCgiven by

W20 ¼ ð2iωkτn�Aτn Þ�1H20;

W11 ¼ �A�1τn H11:

(ð4:29Þ

From (4.27), we know that for �1rθo0,

Hðz; zÞ ¼ �ΦðθÞΨ ð0Þ⟨FðUn

t ;0Þ; f k⟩ � f k;

¼ � p1ðθÞþp2ðθÞ2

;p1ðθÞ�p2ðθÞ

2i

� �ðΨ1ð0ÞΨ2ð0ÞÞ

�⟨FðUn

t ;0Þ; f k⟩ � f k;

¼ �12½p1ðθÞðΨ1ð0Þ� iΨ 2ð0ÞÞþp2ðθÞðΨ 1ð0Þþ iΨ2ð0ÞÞ�

�⟨FðUn

t ;0Þ; f k⟩ � f k;

¼ �14½g20p1ðθÞþg02p2ðθÞ�z2 � f k�

12½g11p1ðθÞ

þg11p2ðθÞ�zz � f kþ⋯:

Therefore, for �1rθo0,

H20ðθÞ ¼0; kAN

�12 ½g20p1ðθÞþg02p2ðθÞ� � f k; k¼ 0

(ð4:30Þ

H11ðθÞ ¼0; kAN

�12½g11p1ðθÞþg11p2ðθÞ� � f k; k¼ 0

(ð4:31Þ

and

Hðz; zÞð0Þ ¼ FðUn

t ;0Þ�ΦðΨ ; ⟨FðUn

t ;0Þ; f k⟩Þ � f k;

H20ð0Þ ¼

τn

2


� a22α2� a23αβe� iωkτ

n

βa31þαβa32

0B@

1CA cos 2 kx; kAN

τn

2



n

βa31þαβa32

0B@

1CA

�12½g20p1ð0Þþg02p2ð0Þ� � f 0; k¼ 0

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

ð4:32Þ

H11ð0Þ ¼

τn

4

�ð2a11þ a13βeiωkτn þβe� iωkτ

n

a13Þ�ð2a22ααþ a23αβeiωkτ

n

þ a23αβe� iωkτn Þ

a31ðβþβÞþ a32αβþ a32αβ

0BBBB@

1CCCCA cos 2 kx; kAN

τn

4

�ð2a11þ a13βeiωkτn þβe� iωkτ

n


n

þ a23αβe� iωkτn Þ


0BBBB@

1CCCCA

�12½g11p1ð0Þþg11p2ð0Þ� � f 0; k¼ 0

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

ð4:33Þ

By the definition of Aτn , we get from (4.29) that

_W 20ðθÞ ¼ 2iωkτnW20ðθÞþ1

2 ½g20p1ðθÞþg02p2ðθÞ� � f k; �1rθo0:

Noting that p1ðθÞ ¼ p1ð0Þeiωkτn , �1rθr0. Hence

W20ðθÞ ¼i2

g20ωkτn

p1ðθÞþg02

3ωkτnp2ðθÞ

� � f kþEe2iωkτ

nθ ; ð4:34Þ

and

E¼W20ð0Þ; kAN

W20ð0Þ�i2

g20ωkτn

p1ð0Þþg02

3ωkτnp2ð0Þ

� � f 0; k¼ 0

8><>: ð4:35Þ

Using the definition of Aτn , and combining (4.29) and (4.35), weget

2iωkτn ig202ωkτn

p1ð0Þ � f 0þig02

6ωkτnp2ð0Þ � f 0þE

�

�τnDΔig202ωkτn

p1ð0Þ � f 0þig02

6ωkτnp2ð0Þ � f 0þE

�

�LðτnÞ ig202ωkτn

p1ðθÞ � f 0þig02

6ωkτnp2ðθÞ � f 0þEe2iωkτ

nθ

�

¼ τn

2



n

βa31þαβa32

0B@

1CA�1

2½g20p1ð0Þþg02p2ð0Þ� � f 0:

Notice that

τnDΔ½p1ð0Þ � f 0�þLðτnÞ½p1ðθÞ � f 0� ¼ iωkτnp1ð0Þ � f 0

τnDΔ½p2ð0Þ � f 0�þLðτnÞ½p2ðθÞ � f 0� ¼ � iωkτnp2ð0Þ � f 0:

(

Then for kAN0,

2iωkτnE�τnDΔE�LðτnÞðEe2iωkτ

nθÞ

¼ τn

2



n

βa31þαβa32

0B@

1CA cos 2 kx:

From the above expression, we can see easily that

E¼ 12

2iωkþd1k2

þ a11un

1 0 a13un

1e�2iωkτ

n

0 2iωkþd2k2

þ a22un

2 a23un

2e�2iωkτ

n

� a31un

3 � a32un

3 d3k2þ2iωk

0BBBBBBBB@

1CCCCCCCCA

�1

��ða11þ a13βe� iωkτ

n Þ� a22α

2� a23αβe� iωkτn

βa31þαβa32

0B@

1CA cos 2 kx

By the similar way, we have

_W 11ðθÞ ¼ 12 ½g11p1ðθÞþg11p2ðθÞ� � f k; �1rθo0;

and

W11ðθÞ ¼i

2ωkτn½�g11p1ðθÞþg11p2ðθÞ� � f kþF:

Similar to the above, we can obtain that

F ¼ 14

d1k2þ a11un

1 0 a13un

1

0 d2k2þ a22un

2 a23un

2

� a31un

3 a32un

3 d3k2

0BBB@

1CCCA

�1


��ð2a11þ a13βeiωkτ

n þβe� iωkτn


n þ a23αβe� iωkτn Þ


0B@

1CA cos 2 kx:

So far, W20ðθÞ and W11ðθÞ have been expressed by the para-meters of system (3.2). Therefore, g21 can be expressed explicitly.

Theorem 3. System (3.2) has the following Poincare normal form:

_ξ ¼ iω0τnξþc1ð0Þξ∣ξ∣2þoð∣ξ∣5Þ;

where

c1ð0Þ ¼i

2ωkτng20g11�2∣g11∣

2� ∣g02∣2

3

� þg21

2;

so we can compute the following results:

σ2 ¼ �Reðc1ð0ÞÞReðλ0ðτnÞÞ;

β2 ¼ 2 Reðc1ð0ÞÞ;

T2 ¼ � Imðc1ð0ÞÞþσ2 Imðλ0ðτnÞÞωkτn

;

which determine the properties of bifurcating periodic solutions atthe critical values τn, i.e., σ2 determines the directions of the Hopfbifurcation: if σ240 ðσ2o0Þ, then the Hopf bifurcation is super-critical (subcritical) and the bifurcating periodic solutions exist forτ4τn; β2 determines the stability of the bifurcating periodic solu-tions: the bifurcating periodic solutions on the center manifold arestable (unstable), if β2o0 ðβ240Þ; and T2 determines the period ofthe bifurcating periodic solutions: the periodic increase(decrease), ifT240 ðT2o0Þ.

5. Optimal harvesting

In this section, we consider optimal control strategy for system(1.5) with τ¼ 0. From Lemma 1 in Section 2, we know that thecondition (H1) ensures persistence of population under theexploitative condition. In this case, we will discuss the optimalpolicy which makes an optimal sustainable yield possible. Weconsider the following sustainable yield:

h¼maxui ;Ei

Z T

0

ZΩðq1E1ðx; tÞu1ðx; tÞþq2E2ðx; tÞu2ðx; tÞ

þq3E3ðx; tÞu3ðx; tÞÞ dx dt ði¼ 1;2;3Þ ð5:1ÞIn order to find optimal ui and Ei ði¼ 1;2;3Þ, we first construct ourobjective functional

J½uiðx; tÞ; Eiðx; tÞ; λiðx; tÞ� ¼Z T

0

ZΩðq1E1u1þq2E2u2þq3E3u3

þλ1∂u1

∂t�d1Δu1�u1ðr1� a11u1� a13u3Þþq1E1u1

� �

þλ2∂u2

∂t�d2Δu2�u2ðr2� a22u2� a23u3Þþq2E2u2Þ

� �

þλ3∂u3

∂t�d3Δu3�u3ð� r3þ a31u1þ a32u2Þþq3E3u3

� �ði¼ 1;2;3Þ:

Denote

Fðx; t; λi;ui;uit ;ΔuiÞ ¼ λ1∂u1

∂t�d1Δu1�u1ðr1� a11u1� a13u3Þþq1E1u1

� �

þλ2∂u2

∂t�d2Δu2�u2ðr2� a22u2� a23u3Þþq2E2u2Þ

� �

þλ3∂u3

∂t�d3Δu3�u3ð� r3þ a31u1þ a32u2Þþq3E3u3

� �:

Using the variational calculus (Su and Pan, 1993), we can obtainthe conditions of the maximum of J½ui; Ei; λi� are

∂F∂Ei

¼ 0 ði¼ 1;2;3Þ∂F∂λi

¼ 0 ði¼ 1;2;3Þ

∂F∂ui

�∂Fuit∂t

þ ∑3

i ¼ 1

∂F2uxixi

∂x2i¼ 0;

∂ui

∂n ∂Ω ¼ 0:��

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

ð5:2Þ

From the first equation of (5.2), we obtain λi ¼ �1 ði¼ 1;2;3Þ.The third equation of (5.2) follows

q1E1�λ1r1þ2a11λ1u1þλ1q1E1�λ3a31u3�∂λ1∂t

�d1Δλ1 ¼ 0;

q2E2�λ2r2þ2a22λ2u2þλ2q2E2�λ3a32u3�∂λ2∂t

�d2Δλ2 ¼ 0;q3E3þλ3r3� a31λ1u1�λ3a32u2þλ3q3E3

þλ1u1a13þλ2u2a23�∂λ3∂t

�d3Δλ3 ¼ 0:

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

ð5:3Þ

Substituting λi ¼ �1 ði¼ 1;2;3Þ into (5.3), we obtain

u⋆1 ¼ r1a

232þ a31ð2a22 r3þ r2a23� a32 r2Þ� a23a32 r1

2a22a231�2a13a22a31þ2a11a

232�2a11a23a32

;

u⋆2 ¼ r2a

231þ a32ð2a11 r3þ r1a13� a31 r1Þ� a13a31 r2

2a22a231�2a13a22a31þ2a11a

232�2a11a23a32

;

u⋆3 ¼ 2r3a11a22þ a11a23 r2þ r1a13a22� a11a32 r2� a22a31 r1

2a22a231�2a13a22a31þ2a11a

232�2a11a23a32

;

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

ð5:4Þwhich are the optimal population level under exploitation. Usingthe second equation of (5.3), substituting (5.4) into the secondequation of (5.3), we know

E⋆1 ¼ 1q1

ðr1� a11u1� a13u3Þ

E⋆2 ¼ 1q2

ðr2� a22u2� a23u3Þ

E⋆3 ¼ 1q3

ð� r3þ a31u1þ a32u2Þ

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

ð5:5Þ

Therefore, we summarize the above analysis by the followingtheorem.

Theorem 4. If (H1) holds, then there exist optimal harvesting effortsE⋆i 40 ði¼ 1;2;3Þ and corresponding solutions u⋆

i 40 ði¼ 1;2;3Þ,which maximizes (5.1). The optimal harvesting efforts and thecorresponding optimal equilibrium point are given by (5.5) and (5.4).

6. Numerical simulation

In this section, we give numerical simulation of some examplesto illustrate our theoretical results. Let us consider a set of artificialvalues of parameters as follows in appropriate units r1A ½1:2;3�;r2A ½3;5�; r3A ½0:2;6�; a11A ½0:22�; a13A ½0:11�; a22A ½0:15;1�;a23 A ½0:52�; a31A ½0:2;1�; a32A ½0:25;1�.

Example 1. Consider system (1.5) with pi¼0 ði¼ 1;2…;9Þ, d1 ¼ 2,d2 ¼ 3, d3 ¼ 6, q1¼0.2, E1 ¼ 2, q2¼0.5, E2 ¼ 2, q3¼0.4, E3 ¼ 8. By asimple calculation, we can obtain that r1 ¼ 1:2, r2 ¼ 3, r3 ¼ 0:2,a11 ¼ 0:2, a13 ¼ 0:1, a22 ¼ 0:15, a23 ¼ 0:5, a31 ¼ 0:2, a32 ¼ 0:25. It iseasy to obtain that the positive equilibrium point of system (1.5) isEn ¼ ð1:8571;5:7143;0:2857ÞT and τ0 ¼ 5:0055. Obviously, the con-ditions (H1)–(H3) hold. According to Theorem 1, system (1.5) islocally asymptotically stable for τ¼ 4A ½0; τ0Þ and unstable when


τ¼ 5:54τ0. These facts are illustrated by the numerical simulationsin Fig. 1. Fig. 1(b) shows that both the prey and predator populationsreach periodic oscillations around the equilibrium En in finite time.

When τ¼ τ0 ¼ 5:0055, we can compute c1ð0Þ ¼ �18:8668�24:3689i, σ2 ¼ ��18:8668=Reðλ0ðτnÞÞ ¼ 971:103140, β2 ¼ 2 Reðc1ð0ÞÞo0. Therefore, from the discussions in Section 4, we knowthat the bifurcated periodic solutions are orbitally asymptoticallystable on the center manifold. In addition, from Theorem 3, weknow that system (1.5) has a stable center manifold near thepositive equilibrium En for τ near τ0 ¼ 5:0055. Therefore, the centermanifold theory implies that the bifurcated periodic solutions ofsystem (1.5) when τ0 ¼ 5:0055 in the whole phase space areorbitally asymptotically stable, and the Hopf bifurcation is super-critical for σ240.

Example 2. Let p1 ¼ 1, p2 ¼ 1, p3 ¼ 1, p4¼0.5, p5¼0.7, p6¼0.4,p7¼0.1, p8¼0.2, p9¼0.9, d1 ¼ 2, d2 ¼ 3, d3 ¼ 6, q1¼0.2, E1 ¼ 4,q2¼0.5, E2 ¼ 4, q3¼0.4, E3 ¼ 4. Thus, r1 ¼ 3, r2 ¼ 5, r3 ¼ 6,a11 ¼ 0:6325, a13 ¼ 0:5012, a22 ¼ 0:3204, a23 ¼ 0:5743, a31 ¼0:2759, a32 ¼ 0:8706. Clearly, system (1.5) exists a positive equili-brium point En ¼ ð2:8052;7:8409;0:8496ÞT , and τ0 ¼ 0:8093. By thecalculation, the conditions (H1)–(H3) hold. According to Theorem 1,system (1.5) is locally asymptotically stable when τ¼ 0:5A ½0; τ0Þand unstable when τ¼ 0:94τ0. These facts are illustrated by thenumerical simulations in Fig. 2. Numerical simulation shows thatboth the prey and predator populations reach periodic oscillationsaround the equilibrium En in finite time.

In addition, when τ¼ τ0 ¼ 0:8093, we get c1ð0Þ ¼ �0:3910�0:2193i, σ2 ¼ ��0:3910=Reðλ0ðτnÞÞ ¼ 0:677840, β2 ¼ 2 Reðc1ð0ÞÞo0. According to Theorem 3 in Section 4, the bifurcated periodicsolutions of system (1.5) when τ0 ¼ 0:8093 in the whole phasespace are orbitally asymptotically stable, and the Hopf bifurcationis supercritical for σ240.

Example 3. Let p1 ¼ 1, p2 ¼ 1, p3¼0.4, p4¼0.5, p5¼0.7, p6¼0.4,p7¼0.1, p8¼0.2, p9¼0.9, d1 ¼ 2, d2 ¼ 3, d3 ¼ 6, q1¼0.2, E1 ¼ 4,q2¼0.5, E2 ¼ 4, q3¼0.4, E3 ¼ 4. It is easy to see that r1 ¼ 3, r2 ¼ 5,r3 ¼ 0:7796, a11 ¼ 0:6325, a13 ¼ 0:5012, a22 ¼ 0:3204, a23 ¼ 0:5743,a31 ¼ 0:2759, a32 ¼ 0:8706. By calculation, the positive equilibrium

point of system (1.5) is En ¼ ð0:4803;2:5812;3:7835ÞT andτ0 ¼ 0:1591. Obviously, the conditions (H1)–(H3) hold. Accordingto Theorem 1, system (1.5) is locally asymptotically stable whenτ¼ 0:1A ½0; τ0Þ and unstable when τ¼ 0:24τ0, see Fig. 3.

When τ¼ τ0 ¼ 0:1591, according to Theorem 3, clearly,c1ð0Þ ¼ 0:1803�0:5327i, σ2 ¼ �0:1803=Reðλ0ðτnÞÞ ¼ �0:0793o0,β2 ¼ 2 Reðc1ð0ÞÞ40, i.e., the spatially homogeneous bifurcatingperiodic solutions are unstable on the center manifold and theHopf bifurcation is subcritical. We cannot get the bifurcatingperiodic solutions from simulations (Zuo and Wei, 2011).

0 50 100 150 2000

2

4

6

8

10

t

u 1,u2,u

3 u2u1

u3

0 100 200 300 400 500 600 700 8000

2

4

6

8

10

t

u 1,u2,u

3u1 u2 u3

Fig. 2. The numerical simulations of system (1.5) with the parameters given inExample 2. (a) The positive constant steady state En is stable when τ¼ 0:5; (b) thepositive constant steady state En is unstable when τ¼ 0:9.

0 50 100 150 2000

1

2

3

4

5

t

u 1,u2,u

3

u3

u2 u1

0 200 400 600 800 10000

2

4

6

8

t

u 1,u2,u

3

u 1 u 2 u 3

Fig. 3. The numerical simulations of system (1.5) with the parameters given inExample 3. (a) The positive constant steady state En is stable when τ¼ 0:1; (b) thepositive constant steady state En is unstable when τ¼ 0:2.

0 50 100 150 200 250 300 350 4000

2

4

6

8

t

u 1,u2,u

3

u3

u1

u2

0 500 1000 15000

2

4

6

8

t

u 1,u2,u

3

u1 u2 u3

Fig. 1. The numerical simulations of system (1.5) with the parameters given inExample 1. (a) The positive constant steady state En is stable when τ¼ 4; (b) thepositive constant steady state En is unstable when τ¼ 5:5.


Example 4 (The effect of the delay τ). In this example, we discussthe effect of delay τ on the population dynamics of system (1.5).Let the parameters are the same with Example 2. Fig. 4 gives themaximum and minimum species population for different delay forsystem (1.5). From it we can find that when τo1:2 the minimumspecies population of three species is greater than zero, when1:2rτr2 the minimum species population of the predator isequal to zero and when τ42 the minimum species population ofthree species is all equal to zero. These mean that three speciesmay be extinct with τ increasing. As τ¼ 0:9, τ¼ 1:5 and τ¼ 2:5population dynamics and phase portraits for system (1.5) areshown in Fig. 5 which validate the analysis results above.

Example 5 (The effect of the diffusion). Let p1 ¼ 1, p2 ¼ 1, p3¼0.8,p4¼0.5, p5¼0.7, p6¼0.4, p7¼0.1, p8¼0.2, p9¼0.9, d1 ¼ 0:01,d2 ¼ 0:01, d3 ¼ 2, q1¼0.2, E1 ¼ 4, q2¼0.5, E2 ¼ 4, q3¼0.4, E3 ¼ 4. Bycalculation, parameters r1 ¼ 3, r2 ¼ 5, r3 ¼ 3:0390, a11 ¼ 0:6325,a13 ¼ 0:5012, a22 ¼ 0:3204, a23 ¼ 0:5743, a31 ¼ 0:2759, a32 ¼0:8706. Moreover, this set of parameters gives a unique positiveequilibrium En ¼ ð1:4865;4:8576;2:5317ÞT . Obviously, (H1) holds.We can get that τ00 ¼ 0:2397, τ01 ¼ 0:9894. According to Theorem 2in Section 3, a family of spatially inhomogeneous periodic solutionsbifurcating from En when τ crosses through the critical valuesτ1j .

In addition, when τ¼ τ01 ¼ 0:9894, we get c1ð0Þ ¼ �0:0061�0:0281i, σ2 ¼ �ð�0:0061Þ=Reðλ0ðτnÞÞ ¼ 0:020640, β2 ¼ 2 Reðc1ð0ÞÞo0. According to Theorem 3 in Section 4, we know that spatiallyinhomogeneous periodic solutions bifurcating from En whenτ0 ¼ 0:9894 in the whole phase space are orbitally asymptoticallystable and supercritical for σ240. Since τ00oτ10, we cannot get thebifurcating periodic solutions from simulations.

Figs. 6 and 7 compare the population of the prey and predatorbetween the temporal model (i.e., system (1.5) without diffusion)and the spatio-temporal model of system (1.5) when τ¼ 0:22 andτ¼ 0:25. Comparing Fig. 6(a) and (b), we can conclude that theconvergence speed of the temporal model is faster than that ofthe spatio-temporal model. From Fig. 7 we can obtain that theamplitude of the temporal model is smaller than that of thespatio-temporal model. These mean that the diffusion terms havean effect on the system. In fact, we can compute the amplitude ofu1;u2 and u3 are 0.8634, 2.8775, 2.9715 respectively for thetemporal model and the correspond amplitude for the spatio-temporal model are 0.9805, 3.2594, 3.3805 respectively.

Example 6 (The effect of the harvesting effort). In this example, wewant to see how the harvesting effort E1; E2; E3 affect the dynamicsof system (1.5). Let the parameters are the same with Example 2,except E1, E2 and E3.

(1) Choose E2 ¼ 4, E3 ¼ 4, but E1 varies in ½0;15�, the corre-sponding situations of equilibrium points are shown in Fig. 8(a) andstable range of system (1.5) is shown in Fig. 8(b).

Fig. 8(a) shows that with the increase of harvesting effort E1,the populations of the first prey and predator species are decreas-

ingan-dthepo-pu-lat-io-nsofthese-co-ndpr-eyarein-cr-ea-si-ng.Ke-ep-ingon

0 100 200 300 400 5000

2

4

6

8

10

t

u 1,u2,u

3

u1 u2 u3

τ=0.9

0 5 10 15 20 25 30 35 400

2

4

6

8

10

t

u 1,u2,u

3

u1

u2

u3

τ=1.5

0 20 40 60 80 1000

5

10

15

20

25

t

u 1,u2,u

3

u1u2u3

τ=2.5

Fig. 5. Population dynamics and phase portraits for system (1.5): (a) τ¼ 0:9;(b) τ¼ 1:5; (c) τ¼ 2:5.

0 100 200 300 400 5000

1

2

3

4

5

6

7

u,u

,u

τ=0.23

0 100 200 300 400 5000

1

2

3

4

5

6

7

u,u

,u

τ=0.23

Fig. 6. Population dynamics and phase portraits for the temporal model and thespatio-temporal model at τ¼ 0:22.

0 0.5 1 1.5 2 2.5 30

5

10

15

τ

u

uu

of s

peci

esM

axim

um a

nd m

inim

um

Fig. 4. Maximum and minimum species population for different time delay forsystem (1.5).


increasing harvesting effort E1, the predator will be extinct. Fig. 8(b) shows that the stability range of system (1.5) is enlarged withharvesting effort E1 increasing.

(2) Taking E1 ¼ 4, E3 ¼ 4, but E2 varies in ½0;6�, the correspond-ing situations of equilibrium points are shown in Fig. 9(a) andstable range of system (1.5) is shown in Fig. 9(b). Numericalevidence shows that with the increase of E2, the harvest effort E2makes the population of the second prey and predator reduce andthe population of the first prey increases. Keeping on increasingharvest effort E2, the second prey and the predator will be extinct.Fig. 9(b) shows that as the growth of the harvest effort E2 thestability range is increasing at first and then decreases.

0 5 10 15−2

0

2

4

6

8

10

E1

u∗,u∗ ,u∗

u

u

u

0 5 10 150.5

1

1.5

2

2.5

3

3.5

E1

τ

unstable

stable

Fig. 8. The equilibrium point of system (1.5) varies with E1 increasing.

0 1 2 3 4 5 6−2

0

2

4

6

8

10

E2

u∗,u∗

,u∗

u

u

u

0 1 2 3 4 5 60

5

10

15

E2

τ unstable

stable


0 2 4 6 8 10−2

0

2

4

6

8

10

12

E3

u 1∗ ,u 2∗ ,u 3∗

u2∗

u1∗

u3∗

0 2 4 6 8 100

10

20

30

40

50

E3

τ 0

unstable

stable


0 100 200 300 400 5000

0.5

1

1.5

2

2.5

t

u 1

spatio−temporal temporal

0 100 200 300 400 5002

3

4

5

6

7

8

t

u 2


0 100 200 300 400 5000

1

2

3

4

5

6

7

t

u 3


Fig. 7. Population dynamics and phase portraits for the temporal model and thespatio-temporal model at τ¼ 0:25.


(3) Choose E1 ¼ 2, E2 ¼ 2, but E3 varies in ½0;10�, the correspond-ing situations of equilibrium points are shown in Fig. 10. Fig. 10(a) shows that increasing the harvest E3 leads to the decreasing thepopulations of predator species and increasing the populations of thetwo preys. Keeping on increasing E3, the predator will be extinct.Fig. 10(b) shows that as the growth of the harvest effort E3, thestability range is increasing at first and then decreased.

Example 7 (Optimal harvesting). Let p1¼0.5, p2¼0.6, p3¼0.8,p4¼0.5, p5¼0.4, p6¼0.2, p7¼0.3, p9 ¼ 1, d1 ¼ 2, d2 ¼ 3, d3 ¼ 6,q1¼0.2, q2¼0.5, q3¼0.4 and p8 varies in ½0;1Þ. The optimalequilibrium points ðu⋆

1 ;u⋆2 ;u

⋆3 ÞT and optimal harvesting efforts

E⋆1 ; E⋆2 ; E

⋆3 are given in Table 1. From Table 1, we can obtain that

the optimal harvesting efforts E⋆1 and E⋆2 keep going up with p1increasing, at the same time the optimal harvesting effort E⋆3 is

decreasing.Choose p8 ¼ 0:2;0:6, respectively, the phase portraits for system

(1.5) at τ¼ 0 are shown in Fig. 11 and it shows that system (1.5)converges to the optimal equilibrium point with optimal harvest-ing effort quickly.

7. Discussion and conclusion

In this paper, to a three species prey–predator system, weintroduce the delays and diffusion into the system. Through thetheoretical analysis and numerical simulation reveal that thediscrete delays are responsible for the stability switch of themodel system, and a Hopf bifurcation occurs as the delays increasethrough a certain threshold (see Examples 1–3). The increasingdelay τ may lead to the extinct of the prey or the predator(Example 4).

The diffusion we incorporate into the system can effect theconvergence speed and the amplitude of the system. By adjustingthe diffusion coefficient, the spatially inhomogeneous periodicsolutions bifurcate from En. The corresponding facts are obtainedfrom Example 5. All these mean that the diffusion has importantinfluence on survival of the species.

In addition, this work pays attention to the exploitation orharvesting of biological resources. From Example 6, we can obtainthat over harvesting may be lead to the extinct of the species.Consequently, reasonable harvesting policies are indisputably oneof the major and interesting problems from ecological andeconomic point of view. In Section 5, an optimal harvesting policyis studied for system (1.5) at τ¼ 0 (see Example 7).

In this work, we also consider the system with intervalbiological parameters, from Examples 1–3, we know that thedynamic behavior of system (1.5) can be changed throughadjusting the parameter pi ði¼ 1;2…;9Þ. Let p1 ¼ 1, p2 ¼ 1,p4¼0.5, p5¼0.7, p6¼0.4, p7¼0.1, p8¼0.2, p9¼0.9, d1 ¼ 2, d2 ¼ 3,d3 ¼ 6, q1¼0.2, E1 ¼ 4, q2¼0.5, E2 ¼ 4, q3¼0.4, E3 ¼ 4 and p3varies in ½0;1�. The dynamic behavior of system (1.5) is shownin Table 2. Table 2 shows that the positive equilibrium point ofsystem (1.5) varies with p3 increasing and τ0 is increasing as thegrowth of p3. We also can obtain that when p3 ¼ 0;0:2;0:4;0:6the bifurcated periodic solutions of system occurred from theequilibrium at τ¼ τ0 are subcritical and unstable on the centermanifold, as p3 ¼ 0:8;1 the bifurcated periodic solutions becomesupercritical and stable on the center manifold.

Acknowledgment

We would like to express our gratitude to the referees for theirvaluable comments and suggestions that led to a truly significantimprovement of the paper.

0 50 100 150 2000

5

10

15

t

u 1,u

2,u

3

u1 u3

u2

0 50 100 150 2000

2

4

6

8

10

12

t

u 1,u

2,u

3

u2

u1 u3

Fig. 11. Population dynamics and phase portraits for system (1.5) at τ¼ 0 withoptimal harvesting effort: (a) p8¼0.2; (b) p8¼0.6.

Table 2The dynamic behavior of system (1.5) for different p3.

p3 The equilibrium point τ0 c1ð0Þ σ2 β2 T2

0 (0.2222, 1.9972, 4.1092) 0.1499 0.8975–1.6004i �0.4645 1.7951 0.66050.2 (0.3089, 2.1936, 3.9997) 0.1525 0.4577–0.9569i �0.2220 0.9153 0.38940.4 (0.4803, 2.5812, 3.7835) 0.1591 0.1723-0.5158i �0.0758 0.3445 0.20720.6 (0.8186, 3.3466, 3.3566) 0.1777 0.0260–0.2788i �0.0103 0.0520 0.11450.8 (1.4865, 4.8576, 2.5137) 0.2397 �0.0579–0.1721i 0.0240 �0.1159 0.08751 (2.8052, 7.8409, 0.8496) 0.8093 �0.3911–0.2193i 0.6778 �0.7821 0.5228

Table 1Optimal equilibrium points, optimal harvesting efforts for different p8.

p8 Optimal equilibrium point Optimal harvesting effort

0 1.7512, 12.9206, 0.7944 2.9512, 1.2830, 25.57970.2 1.7937, 12.3670, 0.6731 2.9694, 1.7097, 24.55750.4 1.8021, 11.5863, 0.5019 3.1576, 2.3114, 23.08360.6 1.7348, 10.5865, 0.2827 3.6457, 3.0819, 21.14700.8 1.5539, 9.5112, 0.0470 4.5139, 3.9106, 18.9962


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