bifurcation analysis in a delayed diffusive nicholson’s ...thesequenceofeigenvaluesofd4isfdk2g1...

Nonlinear Analysis: Real World Applications 11 (2010) 1692–1703

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Nonlinear Analysis: Real World Applications

journal homepage: www.elsevier.com/locate/nonrwa

Bifurcation analysis in a delayed diffusive Nicholson’sblowflies equationI

Ying Su a, Junjie Wei a,∗, Junping Shi b,ca Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, PR Chinab Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USAc School of Mathematics, Harbin Normal University, Harbin, Heilongjiang, 150080, PR China

a r t i c l e i n f o

Article history:Received 14 November 2007Accepted 23 March 2009

Keywords:Nicholson’s blowflies equationDiffusionDelaySteady state bifurcationHopf bifurcationDirichlet boundary condition

a b s t r a c t

The dynamics of a diffusive Nicholson’s blowflies equation with a finite delay and Dirichletboundary condition have been investigated in this paper. The occurrence of steady statebifurcation with the changes of parameter is proved by applying phase plane ideas. Theexistence of Hopf bifurcation at the positive equilibrium with the changes of specifyparameters is obtained, and the phenomenon that the unstable positive equilibrium statewithout dispersion may become stable with dispersion under certain conditions is foundby analyzing the distribution of the eigenvalues. By the theory of normal form and centermanifold, an explicit algorithm for determining the direction of the Hopf bifurcation andstability of the bifurcating periodic solutions are derived.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In order to describe the population dynamics of Nicholson’s blowflies, Gurney et al. [1] have proposed the following delayequation

dudt= −dmu(t)+ εu(t − τ)e−au(t−τ), (1.1)

where ε is themaximumper capita daily egg production rate, 1/a is the size atwhich the blowfly population reproduces at itsmaximum rate, dm is the per capita daily adult death rate and τ is the generation time. Eq. (1.1) has been extensively studiedin the literature, where its results mainly concern the global attractivity of positive equilibrium and oscillatory behaviors ofsolutions (see [2–7,30,32]). Several studies have also been carried out on Eq. (1.1) with time periodic coefficients (see [8,9])and on discrete Nicholson’s blowflies equation (see [10–15]). After rescaling Eq. (1.1), it takes the form

u = au, t =tτ, τ = dmτ , β =

ε

dm,

and by dropping the tildes, then it may be written as

dudt(t) = −τu(t)+ βτu(t − 1)e−u(t−1). (1.2)

I This research is supported by the National Natural Science Foundation of China, and Specialized Research Fund for the Doctoral Program of HigherEducation, National Science Foundation of US, and Longjiang professorship of Department of Education of Heilongjiang Province.∗ Corresponding author.E-mail address: [email protected] (J. Wei).

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Y. Su et al. / Nonlinear Analysis: Real World Applications 11 (2010) 1692–1703 1693

To explain interactions among organisms, Yang and So [16] extended Eq. (1.2) to the flowing diffusive form

∂u∂t(t, x) = d4u(t, x)− τu(t, x)+ βτu(t − 1, x)e−u(t−1,x). (1.3)

Furthermore, some researchers have studied the phenomenon by using an equation in the following form

dudt(t, x) = Dm∆u(t, x)− dmu(t, x)+ εu(t − τ , x)e−au(t−τ ,x). (1.4)

So and Yang [17] investigated global attractivity of the equilibrium of Eq. (1.3) with the Dirichlet boundary condition

u(t, 0) = u(t, π) = 0, for t ≥ 0 (1.5)

and they [16] studied the stability and existence of Hopf bifurcation of Eq. (1.3) with Neuman boundary condition. Somenumerical and Hopf bifurcation analysis on Eq. (1.3) has been carried out by So, Wu and Yang [18]. Generalized Nicholson’sblowflies models with distributed delay in Eqs. (1.3) and (1.4) have been studied extensively (see [19–22]). For the Dirichletboundary value problem of the diffusive Nicholson’s blowflies equation, So and Yang [17] have proved that there is a uniquepositive steady state solution if and only if (β − 1)τ > λ1, where λ1 is the principal eigenvalue of −4 with Dirichletboundary condition.The purpose of the present paper is to study the bifurcation of Eq. (1.3) with Dirichlet boundary condition (1.5).We prove

the existence of positive steady state bifurcation by a direct calculation presented in Robinson [23]. The conclusions are thatthe problem (1.3) and (1.5) has a unique positive steady state if and only if β > 1 + d/τ , and the Eq. (1.3) with Dirichletboundary condition

u(t, 0) = u(t, π) = lnβ, for t ≥ 0 (1.6)

has no other positive steady state solution except u = lnβ . On the other hand, we provide a detailed analysis of Hopfbifurcation for the problems (1.3) and (1.6) by applying the local Hopf bifurcation theory (see [24]). More specifically, weprove that, as β increases, the positive equilibrium u∗ = lnβ loses its stability and a sequence of Hopf bifurcations occurat u∗. Furthermore, by using the center manifold theory introduced by Lin, So and Wu [25] and normal form method dueto Faria [26], we derive an explicit algorithm for determining the stability and direction of the Hopf bifurcations occurringat u∗.The rest of this paper is organized as follows. In Section 2, existence of positive steady state bifurcation is established. In

Section 3, the occurrence of Hopf bifurcation and the phenomenon that the unstable positive equilibrium state withoutdispersion may become stable with dispersion under certain conditions are found by analyzing the distribution of theeigenvalues. In Section 4, an algorithm for determining the direction and stability of the Hopf bifurcation is derived byusing the center manifold due to Lin, So and Wu [25] and normal form method due to Faria [26]. Finally, some numericalanalysis is given in order to illustrate the theoretic results found.

2. Positive steady state bifurcation

In the present section, we consider equation

∂u∂t(t, x) = d

∂2u∂x2

(t, x)− τu(t, x)+ βτu(t − 1, x)e−u(t−1,x), (2.1)

where (t, x) ∈ D = (0,∞)× [0, π], β, d, τ > 0, with Dirichlet boundary condition

u(t, 0) = u(t, π) = 0, for t ≥ 0. (2.2)

The steady state u(x) of (2.1) and (2.2) satisfies

duxx = τu− τβue−u,u(0) = u(π) = 0. (2.3)

Taking v = ux, we can rewrite the equation in (2.3) into a pair of differential equations:{ux = v,vx =

τ

d(u− βue−u). (2.4)

We can now apply phase plane ideas, treating x as the time variable. It follows from Eq. (2.4) that, on any trajectory,

d2v2 −

τ

2u2 − τβ(ue−u + e−u) = C,

where C is a constant. It is obvious that Eq. (2.4) has two fixed points given by

(u, v) = (0, 0), (lnβ, 0).

1694 Y. Su et al. / Nonlinear Analysis: Real World Applications 11 (2010) 1692–1703

Fig. 2.1. The phase portrait of (2.4).

Clearly, lnβ > 0 if and only if β > 1. This relation is assumed throughout this section. The matrix associated with thelinearized vector field of Eq. (2.4) is given by(

0 1τ

d[1− β(e−u − ue−u)] 0

).

The eigenvalues associated with the fixed point (0, 0) are given by λ1,2 = ±i√τ(β−1)d , and the eigenvalues associated with

the fixed point (lnβ, 0) are given by λ1,2 = ±√τ lnβd . Hence, (0, 0) is a center and (lnβ, 0) is a saddle.

Lemma 2.1. The trajectory starts at u = lnβ, v = 0which moves around clockwise, can strike the v axis at a value v = v < 0.

Proof. Obviously, this lemma holds true if and only if C∗ > −τβ , where C∗ is associated with the trajectory which passesthrough the point (lnβ, 0), and is given by

C∗ = −τ

2(lnβ)2 − τ(1+ lnβ).

Denote s = lnβ with s > 0. Then C∗ > −τβ is equivalent to the inequality

s2 + 2(s+ 1)− 2es < 0 (2.5)

holds. It is easy to verify that (2.5) holds when s > 0. The proof is completed. �

Lemma 2.2. The trajectory starts at u = 0, v = v which moves around clockwise, can strike the u axis at a value u = u.

Proof. Obviously, along this trajectory ux = v < 0, hence, u decreases strictly as x increases. There exists δ > 0 such thatln(β − δ) > 0, then

vx =τ

d(u− βue−au) >

τ

du(1−

β

β − δ

)> 0

holds when u < 0. This implies that v is strictly increasing as a function of xwhen u < 0. Therefore the trajectory can strikethe u axis. �

The phase portrait for (2.4) is given in Fig. 2.1. To satisfy the boundary conditions we need the trajectory that starts onthe v axis at x = 0 andmoves back onto the v axis when x = π . If the trajectory in the right-half plane (resp. left-half plane)moving from v axis strike u axis at a point, denote the point by u0 (resp. u0), the ‘‘time’’ by t1(u0) (resp. t2(u0)).

Proposition 2.3. limu0→0 t1(u0) = limu0→0 t2(u0) =π2 [

dτ(β−1) ]

12 .

Proof.

t1(u0) =

√d2

∫ u0

0

[τ2u2 + τβ(ue−u + e−u)+ C0

]− 12du,

where

C0 = −τ

2u20 − τβ(u0e

−u0 + e−u0).


Let u = u0 sin x, x ∈ (0, π2 ), we obtain

t1(u0) =

√d2

∫ π2

0

[−τ

2u20 cos

2 x+ τβ(u0 sin xe−u0 sin x − u0e−u0)+ τβ(e−u0 sin x − e−u0)]− 12· u0 cos xdx

=

√d2

∫ π2

0

[−τ

2+τβ(u0 sin xe−u0 sin x − u0e−u0)+ τβ(e−u0 sin x − e−u0)

u20 cos2 x

]− 12dx. (2.6)

Then by

limu0→0

u0 sin xe−u0 sin x − u0e−u0 + e−u0 sin x − e−u0

u20 cos2 x= limu0→0

e−u0 − sin2 xe−u0 sin x

2 cos2 x=12,

it follows that

limu0→0

t1(u0) =π

2

[d

τ(β − 1)

] 12

.

For the case t2(u0), the proof is similar, and then we omit it. �

Proposition 2.4. t1(u0) and t2(u0) is strictly increasing as a function of u0 and u0, respectively.

Proof. Denote

∆(u0) =u0 sin xe−u0 sin x − u0e−u0 + e−u0 sin x − e−u0

u20 cos2 x. (2.7)

By (2.6), it suffices to prove that∆(u0) is strictly decreasing as a function of u0. We have

∆′(u0) =e−u0 − sin2 xe−u0 sin x

u0 cos2 x− 2u0 sin xe−u0 sin x − u0e−u0 + e−u0 sin x − e−u0

u30 cos2 x.

Let

A = sin x,

and

F(A) = −A2e−Au0 + e−u0 − 2(Ae−Au0

u0−e−u0

u0

)−2u20(e−Au0 − e−u0)

with A ∈ (0, 1), u0 > 0. Then∆′(u0) can be expressed as

∆′(u0) =F(A)u0 cos2 x

.

Thus, we only need to verify that F(A) < 0 for any u0 > 0. For a fixed u0, clearly,

F(1) = 0, F ′(A) = u0A2e−Au0 > 0.

Therefore, for any u0 > 0, F(A) < 0 where A ∈ (0, 1), which completes the proof. Similarly, we can conclude t2(u0) isstrictly increasing as a function of u0. �

Proposition 2.5. limu0→lnβ t1(u0) = ∞.

Proof. First, we verify that ∆(u0) 6= 12β for any u0 ∈ [0, lnβ), where ∆(u0) is defined by (2.7). We know that ∆(u0) is

strictly decreasing as a function of u0 from the proof of Proposition 2.4. By the methods that we have used in the proof ofabove proposition, we can obtain that∆(u0) is strictly decreasing as a function of x for fixed u0. Since for u0 ∈ [0, lnβ],

limx→ π

2

∆(u0) = limx→ π

2

e−u0

2,

then we obtain

limx→ π

2

∆(lnβ) =12β.

This implies that∆(u0) > 12β for u0 ∈ [0, lnβ). Thus, t1(u0) <∞ for any u0 ∈ [0, lnβ).


Now we compute t1(lnβ). Let µ = sin x, then from (2.6) we derive that

t1(lnβ) =

√2d

∫ 1

0

(lnβ)2

−τ2 (1− µ

2)(lnβ)2 + τβ lnβµ(1β

)µ− τ lnβ + τβ

(1β

)µ− τ

12

dµ.

Hence, from

limµ→1

−τ2 (1− µ

2)(lnβ)2 + τβ lnβµ(1β

)µ− τ lnβ + τβ

(1β

)µ− τ

(1− µ)2(lnβ)2=τ

2lnβ

and∫ 10

11−µdµ = ∞we have t1(lnβ) = ∞. This completes the proof. �

Based on all above analysis, we have the following steady state bifurcation theorem for (2.1) and (2.2).

Theorem 2.6. If β ∈ (1, d/τ + 1], then the problem (2.1) and (2.2) has no positive solution; If β ∈ (d/τ + 1,+∞), then theproblem (2.1) and (2.2) possesses a unique positive steady state.

Remark 2.7. 1. Suppose that β > 1 is satisfied. If we treat d or τ as a parameter, we can also obtain the existence of positivesteady state bifurcation.

2. The conclusion of Theorem 2.6 is a direct corollary of Proposition 2.3 of So and Yang [17]. But we here provide a newmethod to prove the conclusions.

Now, we consider Eq. (2.1) with the following boundary value condition

u(t, 0) = u(t, π) = lnβ, for t ≥ 0. (2.8)

Theorem 2.8. The problem (2.1) and (2.8) has only one positive steady state solution given by u = lnβ .

Proof. In order to satisfy the boundary conditions (2.8) we require that the trajectory starting on the line u = lnβ at x = 0should move back onto the u = lnβ when x = π . In fact, from the first integral of (2.4) we have that the trajectory passingthrough the point (lnβ, 0) is given by

d2v2 −

τ

2u2 − βτ(ue−u + e−u) = −

τ

2(lnβ)2 − τ(lnβ + 1). (2.9)

It follows that

v2 =2d

[τ2u2 + βτ(ue−u + e−u)−

τ

2(lnβ)2 − τ(lnβ + 1)

], (2.10)

and hence, |v| → +∞ as u→+∞. For any u∗ ∈ R, the trajectory given by (2.10) intersects with the line u = u∗ two timesat most. Let (u(x), v(x)) be the trajectory which starts on the line u = lnβ when x = 0. There are two cases: v(0) > 0 andv(0) < 0. In the case of v(0) > 0, u(x) is increasing as long as the trajectory stays in the first quadrant. From the uniquenessand the properties of the trajectory given by (2.10), we have that the trajectory of (u(x), v(x)) shall always stay in the firstquadrant. This shows that the trajectory starting from the line u = lnβ with v(0) > 0 cannot move back onto the lineu = lnβ . When v(0) < 0, it follows that the orbit shall be outside the homoclinic orbit given by (2.10) (see Fig. 2.1), andcome back to u = ln β , but one part of u(x)will be negative. So there is still no positive steady state solution with v(0) < 0.The proof is complete. �

3. Analysis of stability and bifurcation

In this section, we shall carry out the analysis of stability and the existence of Hopf bifurcation of Eq. (2.1). Clearly,u∗ = lnβ is the unique nontrivial equilibrium of Eq. (2.1) when β 6= 1, u∗ is positive when β > 1, and u∗ is negative when0 < β < 1. We first transform the fixed point u = u∗ of Eq. (2.1) to the origin via the translation u = u− u∗ and drop thehats for simplicity of notation, then Eq. (2.1) is transformed into

∂u∂t(t, x) = d4u(t, x)− τu(t, x)− τ lnβ + τu(t − 1, x)e−u(t−1,x) + τ lnβe−u(t−1,x). (3.1)

Then we consider Eq. (3.1) with the Dirichlet boundary condition

u(t, 0) = u(t, π) = 0.


Denote X = L2[0, π] as the Hilbert space with inner product 〈·, ·〉, and C = C([−1, 0], X) with the sup norm a Banachspace. Then in the abstract space C this equation is

ddtu(t) = d4u(t)− τu(t)− τ lnβ + τu(t − 1)e−φ(t−1) + τ lnβe−u(t−1). (3.2)

In X , the sequence of eigenvalues of d4 is {−dk2}∞k=1, with normalized eigenfunctions βk(x) =√2πsin kx. The set {βk} is an

orthonormal basis for X . The linearized equation about the equilibrium point zero is

ddtφ(t) = d4φ(t)− τφ(t)− τ lnβφ(t − 1)+ τφ(t − 1), φ ∈ C, (3.3)

with characteristic equations

λ+ dk2 + τ + (τ lnβ − τ)e−λ = 0, (k = 1, 2, . . .). (1k)

Let λ1,2 = ±iω be solutions of Eq. (1k), then we have{dk2 + τ + (τ lnβ − τ) cosω = 0, k = 1, 2, . . . ,ω − (τ lnβ − τ) sinω = 0, (3.4)

which leads to

tanω = −ω

dk2 + τ, k = 1, 2, . . . , (2k)

β = exp(1−

dk2

τ cosω−

1cosω

).

Let

ω(k)j ∈

((2j+

12

)π, (2j+ 1)π

)and ω

(k)−j ∈

((2j+

32

)π, 2(j+ 1)π

)be the root of Eq. (2k), k = 1, 2, . . . ; j = 0, 1, 2, . . . , and define

β(k)j = exp

(1−

dk2

τ cosω(k)j−

1

cosω(k)j

), k = 1, 2, . . . ; j = 0, 1, 2, . . . ,

and

β(k)−j = exp

(1−

dk2

τ cosω(k)−j

−1

cosω(k)−j

), k = 1, 2, . . . ; j = 0, 1, 2, . . . .

Then we know that ±iω(k)j (resp. ± iω(k)−j ) are the purely imaginary roots of Eq. (1k) with β = β

(k)j (resp. β = β

(k)−j ), and

Eq. (1k) has no other purely imaginary root. It is not difficult to verify that β(k+1)j > β(k)j , β

(k)j+1 > β

(k)j and β

(k)j > e2 for

k = 1, 2, . . . ; j = 0, 1, 2, . . . .We reorder⋃∞

k=1{β(k)j }∞

j=0 as {β0, β1, β2, . . .}, so that βm ≤ βm+1,m ≥ 0. Clearly, β0 = β(1)0 .

Similarly, it is not difficult to verify that β(k+1)−j < β

(k)−j , β

(k)−(j+1) < β

(k)−j and β

(k)−j < 1 for k = 1, 2, . . . ; j = 0, 1, 2, . . . .We

reorder⋃∞

k=1{β(k)−j }∞

j=0 as {β−0, β−1, β−2, . . .}, so that β−m ≥ β−(m+1),m ≥ 0. Clearly, β−0 = β(1)−0 . Let λ(β) = γ (β)+ iω(β)

be the root of Eq. (1k) satisfying γ (β(k)j ) = 0 (resp. γ (β(k)−j ) = 0) and ω(β

(k)j ) = ω

(k)j (resp. ω(β(k)

−j ) = ω(k)−j )when β is close

to β(k)j (resp. β(k)−j ). Then we have the following transversality result:

Lemma 3.1. γ ′(β(k)j ) > 0 and γ′(β

(k)−j ) < 0.

Proof. Substituting λ(β) into Eq. (1k) and taking the derivative associated with β , then replacing β by β(k)j yields

γ ′(β(k)j ) = Re

−τe−λ 1β

1− (τ lnβ − τ)e−λ

∣∣∣∣∣β=β

(k)j

=τ

βRe

[− cosω(k)j + i sinω

(k)j

1− (τ lnβ(k)j − τ) cosω(k)j + (τ lnβ

(k)j − τ) sinω

(k)j

]


=τ

β

τ lnβ(k)j − τ − cosω(k)j

[1− (τ lnβ(k)j − τ) cosω(k)j ]

2 + (τ lnβ(k)j − τ)2 sin2 ω

(k)j

=τ

β(k)j

ω(k)j

sinω(k)j− cosω(k)j

[1− (τ lnβ(k)j − τ) cosω(k)j ]

2 + (τ lnβ(k)j − τ)2 sin2 ω

(k)j

.

Obviously, γ ′(β(k)j ) > 0, as sinω(k)j > 0 and cosω(k)j < 0.

Similarly, one can obtain the second inequality. �

It is easy to verify that λ = 0 is a root of Eq. (1k) when

β =: βk = e−dk2τ .

Let λ = λ(β) be one of the roots of Eq. (1k) satisfying λ(βk) = 0. By substituting λ(β) into Eq. (1k) and taking the derivativewith respect to β yields

dλ(β)dβ

∣∣∣∣β=βk

= −τ

βk[1+ (dk2 + τ)]< 0.

Obviously, βk+1 < βk < 1 for k ≥ 1. From cosω(1)−j > 0 it follows that

dτ<

d

τ cosω(1)−j

+1

cosω(1)−j

− 1.

This implies that

β1 = e−dτ > exp

(1−

d

τ cosω(1)−j

−1

cosω(1)−j

)= β

(1)−j .

Denote

ω∗ = ω(1)0 , β

∗= β0, and β∗ = β1.

From Lemma 3.1 and above analysis, we have the following conclusions on the distribution of the roots of Eq. (1k).

Lemma 3.2. (i) If β ∈ (β∗, β∗), then all roots of Eq. (1k)(k ≥ 1) have negative real parts.(ii) Eq. (1k) (k ≥ 1) have purely imaginary roots if and only if β = βm or β = β−m, m = 0, 1, 2, . . . .When β = β∗, all theroots of Eq. (1k), (k ≥ 1), except ±iω∗, have negative real parts.

(iii) If β > β∗, then Eq. (1k) (k ≥ 1) have at least a pair of roots with positive real parts; If β < β∗, then Eq. (1k) (k ≥ 1) haveat least a positive root.

Proof. First, it is easy to verify that all roots of all the equations (1k) have negative real parts when β = e, for any k ≥ 1.From the definition of βm and β−m, Eq. (1k) (k ≥ 1) have purely imaginary roots if and only if β = βm or β = β−m. Inaddition, by the definitions of β∗ = β0 and β∗ = β1 > β−0, we know that β∗ is the smallest value of β > e such thatEq. (1k) have roots appearing on the imaginary axis, and β∗ is also the largest value of β < e such that Eq. (1k) have a rootappearing on the imaginary axis. Hence, by the a result of Ruan andWei [31, Corollary 2.4], we have that all roots of Eq. (1k)have negative real parts for β ∈ (β∗, β∗), and all the roots of Eq. (1k), (k ≥ 1), except±iω∗, have negative real parts whenβ = β∗. Since γ ′(β(k)j ) > 0, Eq. (1k) (k ≥ 1) have at least a pair of roots with positive real parts for β ∈ (β

∗,∞). Similarly,since λ′(βk) < 0 and γ ′(β

(k)−j ) < 0, Eq. (1k) (k ≥ 1) have at least one positive root for β ∈ (0, β∗). �

Applying Lemmas 3.1 and 3.2, we have the following result on the dynamics of Eq. (3.1).

Theorem 3.3. If β ∈ (β∗, β∗), then the zero solution of Eq. (3.1) is asymptotically stable, and unstable when β > β∗ or β < β∗,as well as Eq. (3.1) undergoes a Hopf bifurcation at the origin when β = β∗.

Theorem 3.3 shows that the stability of the nontrivial equilibrium of Eq. (2.1) and the existence of Hopf bifurcation asthe parameter β varies. Next we discuss the effect of dispersion on the stability.When the diffusion coefficient d = 0, then Eq. (2.1) is reduced to Eq. (1.2), and its characteristic equation is (1k) with

k = 0 given by

λ+ τ + (τ lnβ − τ)e−λ = 0. (3.5)


Let λ1,2 = ±iω be solutions of Eq. (3.5), then we have{τ + (τ lnβ − τ) cosω = 0,ω − (τ lnβ − τ) sinω = 0, (3.6)

which leads to

tanω = −ω

τ, (3.7)

β = exp(1−

1cosω

).

Then, we can obtain the following conclusions.

Lemma 3.4. There exists a sequence of values of β denoted by

. . . , β(0)−1, β

(0)−0, β

(0)0 , β

(0)1 , . . . ,

such that all roots of Eq. (3.5) have negative real parts when β ∈ (1, β(0)0 ), and Eq. (3.5) has at least a root with positive real partwhen

β ∈ (0, 1) ∪ (β(0)0 ,∞),

where

β(0)j = exp

(1−

1cosωj

), β

(0)−j = exp

(1−

1cosω−j

),

and

ωj ∈

((2j+

12

)π, (2j+ 1)π

)and ω−j ∈

((2j+

32

)π, 2(j+ 1)π

), (j = 0, 1, 2, . . .)

are the root of Eq. (3.7).

Clearly, β∗ < 1 < β(0)0 < β∗. By combining Theorem 3.3 and Lemma 3.4, we can obtain the following results on the

effect of dispersion on the stability.

Theorem 3.5. When β ∈ (β∗, 1) ∪ (β(0)0 , β

∗), u = lnβ is an asymptotically stable nontrivial equilibrium for Eq. (1.2), and anunstable one for the problem (2.1) and (2.8).

4. Direction and stability of the Hopf bifurcation

In this section, we shall study the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions byemploying the center manifold theorem due to Lin, So and Wu [25] and normal form method due to Faria [26] for partialdifferential equations with delay. To study the qualitative behavior near the critical point

β∗ = exp(1−

dτ cosω∗

−1

cosω∗w

),

we change the parameter β by taking β = β∗+α. Then Eq. (1k) has a pair of eigenvalues λ(α), λ(α), λ(α) = γ (α)+ iσ(α)of class C1, with λ(0) = iω∗ and γ ′(0) > 0 by Lemma 3.1.We use the same notations as in Faria et al. [26]. LetΛ = {iω∗,−iω∗}, for u ∈ C, define

L(u) = −τu(0)− (τ lnβ∗ − τ)u(−1)

and

F(u, α) = τ ln(β∗ + α)e−u(t−1) + τu(t − 1)e−u(t−1) − τ ln(β∗ + α)+ (τ lnβ∗ − τ)u(−1)

= −τ

β∗αu(−1)+

τ lnβ∗

2u2(−1)− τu2(−1)−

τ lnβ∗

3!u3(−1)+

τ

2u3(−1)+ O(|u|3 + |(α, u)|3).

The Eq. (3.2) is written as

ddtu(t) = d∆u(t)+ L(ut)+ F(ut , α). (4.1)


From the definition of the associated FDE, (see [26] Definition 4.1), the FDE associated with (3.2) by Λ at the equilibriumpoint u = 0 is given by

x(t) = −(d+ τ)x(t)− (τ lnβ∗ − τ)x(t − 1)+ 〈F(xtβ1, α), β1〉, x ∈ C([−1, 0];R). (4.2)

Defining

R(xt) = −(d+ τ)x(t)− (τ lnβ∗ − τ)x(t − 1),

Eq. (4.2) is written in the following form

x = R(xt)+ 〈F(xtβ1, α), β1〉. (4.3)

From [27–29] and [26], we obtain dim P = dim P0 = 2, and P0 = spanΦ , where

Φ(θ) = (φ1(θ), φ2(θ)), with φ1(θ) = eiω∗θ , φ2(θ) = e−iω

∗θ ,

Ψ (0) =(ψ1(0)ψ2(0)

), with ψ1(0) = ψ2(0) = [1− τ(lnβ∗ − 1)e−iω

∗

]−1, where the bar denotes the complex conjugation.

In BC = P ⊕ Kerπ which is decomposed byΛ, then Eq. (4.1) becomes{z = Bz + Ψ (0)〈F(Φzβ1 + y, α), β1〉 ,ddty = A1y+ (I − π)X0F(Φzβ1 + y, α), z ∈ C2, y ∈ Q1

(4.4)

and in BC = P ⊕ Kerπ0, Eq. (4.2) becomes{z = Bz + Ψ (0)〈F(Φzβ1 + y, α), β1〉,ddty = A0,1y+ (I − π0)X0〈F((Φz + y)β1, α), β1〉, z ∈ C2, y ∈ Q 1

(4.5)

where B = diag (iω∗,−iω∗).The existence of a two-dimensional local center manifold for Eq. (4.1) tangent to P at u = 0, α = 0 follows from [25].Let

z = Bz +12g12 (z, 0, α)+

13!g13 (z, 0, α)+ · · · , z ∈ C2, (4.6)

be the normal form of Eq. (4.1) on the center manifold [26], and

z = Bz +12g10,2(z, 0, α)+

13!g10,3(z, 0, α)+ · · · , z ∈ C2, (4.7)

be the normal form of Eq. (4.2) on the center manifold [27].

Theorem 4.1.

g13 (z, 0, α) = g10,3(z, 0, 0)+

(cz21z2cz1z22

)+ O(|z|α2),

where

c = 3ψ1(0)(τ lnβ∗ − 2τ)∑k>1

c2k

[e−iω

∗ 2(τ lnβ∗ − 2τ)dk2 + τ lnβ∗

+ e−3iω∗ τ lnβ∗ − 2τ2iω∗ + dk2 + τ + (τ lnβ∗ − τ)e−2iω∗

], (4.8)

the ‘‘bar’’ denotes the complex conjugation, and the ck’s are given by the following expression:

ck = 〈β1βk, β1〉 =

0, if k even,

−

(2π

) 32 4k(k2 − 4)

, if k odd.

Proof. From [26]

f13(z, 0, α) = f

10,3(z, 0, α)+

32Ψ (0)〈D1F2(Φzβ1, α)

∑k>1

hk(z, α)βk, β1〉, (4.9)

where h(z, α) =∑k≥1 hk(z, α)βk, h(z, α) is the unique solution of

(M22h)(z, α) = f22 (z, 0, α),


and

F2(u, α) = τ lnβ∗u2(−1)−2ταβ∗u(−1)− 2τu2(−1).

From the definition of F2 and f 22 (z, 0, 0), we have

D1F2(v, α)(u) = −2ταβ∗u(−1)+ 2(τ lnβ∗ − 2τ)u(−1)v(−1),

〈D1F2(Φzβ1, 0)(ψβk), β1〉 = 〈2(τ lnβ∗ − 2τ)(e−iω∗

z1 + eiω∗

z2)β1ψ(−1)βk, β1〉= 2(τ lnβ∗ − 2τ)(e−iω

∗

z1 + eiω∗

z2)ψ(−1) · ck, (4.10)

f 22 (z, 0, 0) = −ΦΨ (0)〈F2(Φzβ1, 0), β1〉β1 + X0F2(Φzβ1, 0),

〈f 22 (z, 0, 0), βk〉 = 〈X0(τ lnβ∗− 2τ)(e−iω

∗

z1 + eiω∗

z2)2β12, βk〉= X0(τ lnβ∗ − 2τ)(e−iω

∗

z1 + eiω∗

z2)2 · ck, k > 1.

Nowwe need to compute hk(z, 0), by solving the equation (M22h)(z, 0) = f22 (z, 0, 0). The definition ofM

22 (see [26]) leads to

Dzhk(z, 0)Bz − hk(z, 0) = 0,hk(z, 0)(0)+ (dk2 + τ)hk(z, 0)(0)+ (τ lnβ∗ − τ)hk(z, 0)(−1)= (τ lnβ∗ − 2τ)(e−iω

∗

z1 + eiω∗

z2)2,(4.11k)

where k > 1 and hk(z, 0)(0) = ddθ hk(z, 0)(θ)|θ=0. For each k > 1, it is easy to solve (4.11k) by setting hk(z, 0)(θ) =∑

|q|=2 hq,k(θ)zq,

hk(z, 0)(θ) = ck

[τ lnβ∗ − 2τ

2iω∗ + dk2 + τ + (τ lnβ∗ − τ)e−2iω∗z21 +

τ lnβ∗ − 2τ−2iω∗ + dk2 + τ + (τ lnβ∗ − τ)e2iω∗

z22

+2(τ lnβ∗ − 2τ)dk2 + τ lnβ∗

z1z2

].

By using (4.9) and (4.10), we obtain

c = 3ψ1(0)(τ lnβ∗ − 2τ)∑k>1

c2k

[e−iω

∗ 2(τ lnβ∗ − 2τ)dk2 + τ lnβ∗

+ e−3iω∗ τ lnβ∗ − 2τ2iω∗ + dk2 + τ + (τ lnβ∗ − τ)e−2iω∗

]. �

For Eq. (3.2), the normal form on the center manifold is written in polar coordinates (ρ, ξ) as{ρ = γ ′(0)αρ + Kρ3 + O(α2ρ + |(ρ, α)|4),

ξ = −iω∗ + O(|(ρ, α)|).(4.12)

Denote K ∗ = 13!Re g

10,3(z, 0, 0), from [26], we have K = K

∗+

13!Re c , where c is defined by (4.8). Now we compute K

∗,using [27]. Since, for v ∈ C([0, 1], R),

R(v) = −dv(0)− τv(0)− (τ lnβ∗ − τ)v(−1).Thus

R(1) = −d− τ lnβ∗,

R(e2iω∗θ ) = −d− τ − (τ lnβ∗ − τ)e−2iω

∗

.

For v ∈ C([0, 1], R), α ∈ R denoteF(v, α) = 〈F(vβ1, α), β1〉.

Then

F(v, α) = −τα

β∗v(−1)+

43

(2π

) 32(τ lnβ∗

2− τ

)v2(−1)+

32π

(τ

2−τ lnβ∗

6

)v3(−1)+ O(|α|2 + |(v, α)|3).

And henceF(x1eiω

∗θ+ x2e−iω

∗θ+ x3 · 1+ x4e2iω

∗θ , 0)

=43

(2π

) 32(τ lnβ∗

2− τ

)(x1e−iω

∗

+ x2eiω∗

+ x3 · 1+ x4e−2iω∗

)2

+32π

(τ

2−τ lnβ∗

6

)(x1e−iω

∗

+ x2eiω∗

+ x3 · 1+ x4e−2iω∗

)3 + · · ·

:= B(2,0,0,0)x21 + B(1,1,0,0)x1x2 + B(1,0,1,0)x1x3 + B(0,1,0,1)x2x4 + B(2,1,0,0)x21x2 + · · · .


By comparing the corresponding coefficients, we obtain

B(2,1,0,0) =32π

(τ

2−τ lnβ∗

6

)e−iω

∗

,

B(1,1,0,0) =43

(2π

) 32(τ lnβ∗

2− τ

),

B(0,1,0,1) = B(1,0,1,0) =43

(2π

) 32(τ lnβ∗

2− τ

)e−iω

∗

,

B(2,0,0,0) =43

(2π

) 32(τ lnβ∗

2− τ

)e−2iω

∗

.

From [27],

K ∗ = Re[

11− R(θeiω∗θ )

(B(2,1,0,0) −B(1,1,0,0)B(1,0,1,0)

R(1)+B(2,0,0,0)B(0,1,0,1)2iω∗ − R(e2iω∗θ )

)

]= Re

[ψ1(0)(B(2,1,0,0) −

B(1,1,0,0)B(1,0,1,0)R(1)

+B(2,0,0,0)B(0,1,0,1)2iω∗ − R(e2iω∗θ )

)

]

= Re

ψ1(0) 32π

(τ

2−τ lnβ∗

6

)e−iω

∗

+

169

( 2π

)3 ( τ lnβ∗2 − τ

)2e−iω

∗

d+ τ lnβ∗+

169

( 2π

)3 ( τ lnβ∗2 − τ

)2e−3iω

∗

2iω∗ + d+ τ + (τ lnβ∗ − τ)e−2iω∗

.

Set

m = [d+ τ + (τ lnβ∗ − τ) cos 2ω∗]2 + [2ω∗ − (τ lnβ∗ − τ) sin 2ω∗]2,mk = [dk2 + τ + (τ lnβ∗ − τ) cos 2ω∗]2 + [2ω∗ − (τ lnβ∗ − τ) sin 2ω∗]2,n = [1− (τ lnβ∗ − τ) cosω∗]2 + (τ lnβ∗ − τ)2 sin2 ω∗,

then

K ∗ =1− (τ lnβ∗ − τ) cosω∗

n(d+ τ lnβ∗)

[32π(d+ τ lnβ∗)

(τ

2−τ lnβ∗

6

)+169

(2π

)3 (τ lnβ∗

2− τ

)2]cosω∗

−(τ lnβ∗ − τ)2

n(d+ τ lnβ∗)

[32π(d+ τ lnβ∗)

(τ

2−τ lnβ∗

6

)+169

(2π

)3 (τ lnβ∗

2− τ

)2]sin2 ω∗

+1mn169

(2π

)3 (τ lnβ∗

2− τ

)2 [{cos 3ω∗[d+ τ + (τ lnβ∗ − τ) cos 2ω∗]

− [2ω∗ − (τ lnβ∗ − τ) sin 2ω∗] sin 3ω∗}[1− (τ lnβ∗ − τ) cosω∗]− (τ lnβ∗ − τ) sinω∗{cos 3ω∗[2ω∗ − (τ lnβ∗ − τ) sin 2ω∗]+ [d+ τ + (τ lnβ∗τ − τ) cos 2ω∗] sin 3ω∗}

].

Thus, from this and (4.8) it follows that

K = K ∗ +13!Re c = K ∗ +

τ lnβ∗ − 2τ2

×

∑k>1

c2k

[2(τ lnβ∗ − 2τ) [1− (τ lnβ∗ − τ) cosω∗] cosω∗

n(dk2 + τ lnβ∗)−2(τ lnβ∗ − 2τ)(τ lnβ∗ − τ) sin2 ω∗

n(dk2 + τ lnβ∗)

+τ lnβ∗ − 2τmkn

{cos 3ω∗[dk2 + τ + (τ lnβ∗ − τ) cos 2ω∗][1− (τ lnβ∗ − τ) cosω∗]

− sin 3ω∗[2ω∗ − (τ lnβ∗ − τ) sin 2ω∗][1− (τ lnβ∗ − τ) cosω∗]− sinω∗ cos 3ω∗(τ lnβ∗ − τ)[2ω∗ − (τ lnβ∗ − τ) sin 2ω∗]

− sinω∗ sin 3ω∗(τ lnβ∗ − τ)[dk2 + τ + (τ lnβ∗ − τ) cos 2ω∗]}]. (4.13)

We have shown that all roots of the characteristic equations (1k), except±iω∗, have negative real partswhenβ = β∗. Hencewe have the following result on the properties of the Hopf bifurcation.


Theorem 4.2. For Eq. (3.2) a generic Hopf bifurcation occurs from u = 0, β = β∗. The direction of the bifurcation is β > β∗

(resp. β < β∗) and the bifurcating periodic solutions are stable (resp. unstable) if K < 0 (resp. > 0).

So far, we have derived the formula (4.13) for determining the properties of the Hopf bifurcation occurring at the positiveequilibriumwhen β = β∗ to Eq. (2.1). To illustrate the analytical results found, wewill consider some particular cases of Eq.(3.2). We choose the coefficients as follows: τ = 2, d = 0.7, then ω∗ = 2.4124, β∗ = 16.6163. Furthermore, we obtainK = −0.0611 < 0. If we choose τ = 2, d = 0.35, then ω∗ = 2.3551, β∗ = 14.3462. We obtain K = −0.0716 < 0.Therefore, in both those cases, for Eq. (3.2) a generic supercritical Hopf bifurcation occurs from u = 0, β = β∗.

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