Hopf Bifurcation with the Spatial Average of an Activator in a Radially Symmetric Free Boundary Problem

Abstract

An interface problem derived by a bistable reaction-diffusion system with the spatial average of an activator is studied on an 𝑛-dimensional ball. We analyze the existence of the radially symmetric solutions and the occurrence of Hopf bifurcation as a parameter varies in two and three-dimensional spaces.

1. Introduction

The study of interfacial patterns is important in several areas of biology, chemistry, physics, and other fields [1–4]. Internal layers (or free boundary), which separate two stable bulk states by a sharp transition near interfaces, are often observed in bistable reaction-diffusion equations when the reaction rate is faster than the diffusion effect. We consider a reaction-diffusion system with a sufficiently small positive constant 𝜀 [5, 6]𝜎𝜀𝑢_𝑡=𝜀²∇²(),𝑣𝑢+𝑓𝑢,𝑣_𝑡=𝐷∇²𝑣+𝑔(𝑢,𝑣)𝑡>0,𝐱∈Ω,0=𝜕𝑢=𝜕𝜈𝜕𝑣,𝜕𝜈 (1.1)

where 𝜀, 𝜎, and 𝐷 are positive constants, Ω={𝐱∈ℝ^𝑛∶|𝐱|<𝑅} is the ball in 𝑛-dimensional space, and 𝜈 stands for the unit outward normal on the boundary 𝜕Ω. The nonlinear functions are𝑓(𝑢,𝑣)=−𝑢+𝐻𝑢−𝑎₀−𝑣+1−𝜇⟨𝑢⟩,𝑔(𝑢,𝑣)=𝜇𝑢−𝑣, (1.2)

where 0<𝜇<1 and ⟨𝑢⟩ denotes the spatial average, describing a global feedback effect, namely, ⟨𝑢⟩=1|Ω|_Ω1𝑢𝑑Ω=|Ω|_Ω𝑢(𝐱,𝑡)𝑑𝐱. (1.3)

The system (1.1) with (1.2) is a model for flow discharges proposed by et al. [7], in which 𝑢 is interpreted by the current density and 𝑣 by the voltage drop across the gas gap [7, 8] in a gas-discharge system. This system also exhibits a codimension-two Turing Hopf bifurcation [9], where the conditions of a spatial Turing instability [10] with a certain wavelength 𝜀 and a temporal Hopf bifurcation with a certain frequency 1/√𝜀𝜎 are met simultaneously. Equation (1.1) determines the dynamics of an internal layer, and equation (1.1) together with (1.2) represents a basic model of globally coupled bistable medium which is relevant for current density dynamics in large area bistable semiconductor systems [11–14]. The internal layer has a physical reason as the current filament has a sharp profile with a narrow transition layer connecting flat on- and off-states.

When 𝜀 in (1.1) is sufficiently small for the case of without the spatial average, the singular limit analysis is applied to show the existence and the stability of localized radially symmetric equilibrium solutions [15, 16]. In one-dimensional space for the case of without the spatial average, such equilibrium solutions should undergo certain instabilities, and the loss of stability resulting from a Hopf bifurcation produces a kind of periodic oscillation in the location of the internal layers [2, 17–19]. As the parameter 𝐷 varies, the stability of the spherically symmetric solutions and their symmetry-breaking bifurcations into layer solutions for the case of without the spatial average have been examined in [5, 6].

In this paper, the free boundary problem of (1.1) with (1.2) for the case when 𝜀=0 in two- and three-dimensional space will be studied. Suppose that there is only one (𝑛−1)-dimensional hypersurface 𝜂(𝑡) which is a single closed curve given in the domain Ω in such a way that Ω×(0,∞)=Ω⁺(𝑡)∪𝜂(𝑡)∪Ω⁻(𝑡), where Ω⁻(𝑡)={(𝐱,𝑡)∈Ω×(0,∞)∶𝑢(𝐱,𝑡)>𝑎₀} and Ω⁺(𝑡)={(𝐱,𝑡)∈Ω×(0,∞)∶𝑢(𝐱,𝑡)<𝑎₀}. When 𝐷=1 and 𝜀=0 in (1.1), the spatial average ⟨𝑢⟩ satisfies𝜇⟨𝑢𝐻−⟨𝑣⟩,𝐻=1⟩𝑥−𝜂𝑥−𝜂Ω_𝑛Ω𝑛𝐻𝑛𝑥−𝜂𝑑𝑥=1−𝑅^𝑛, (1.4)

where |Ω₂|=𝜋𝑅² and |Ω₃|=(4/3)𝜋𝑅³. The spatial average of 𝑣 is a solution of ⟨𝑣⟩′(𝑡)=𝜇⟨𝑢⟩−⟨𝑣⟩=1−(𝜂/𝑅)^𝑛−2⟨𝑣⟩. Equation of 𝜂(𝑡) is given by (see [20, 21])𝑑𝜂(𝑡)𝑣𝑑𝑡⋅𝜈=𝐶_𝑖,𝑥∈𝜂(𝑡), (1.5)

where 𝜈 is the outward normal vector on 𝜂(𝑡), 𝑣_𝑖 is the value of 𝑣 on the interface 𝜂(𝑡), and 𝐶(𝑣) is the velocity of the interface. The reaction terms (1.2) satisfy the bistable condition, that is, the nullclines of 𝑓(𝑢,𝑣)=0 and 𝑔(𝑢,𝑣)=0 must have three intersection points and the nullcline 𝑓(𝑢,𝑣)=0 is the triple-valued function of 𝑢 which is called ℎ⁺, ℎ⁻, and ℎ⁰. From [2, 20, 22], the trajectory with a unique value of 𝐶=𝐶(𝑣) exists which is given by 𝐶(𝑣)=ℎ⁺−2ℎ⁰+ℎ⁻. Furthermore, the velocity of the interface 𝐶(𝑣) is a continuously differentiable function defined on an interval 𝐼∶=(−𝑎₀,1−𝑎₀), and thus the velocity of the interface can be normalized by𝐶𝑣𝜂,𝜂,⟨𝑣⟩,𝑡=−1−2𝑎₀−2𝑉𝜎𝑉+𝑎₀1−𝑎₀,−𝑉 (1.6)

where 𝑉=𝑣(𝜂)−((1−𝜇)/𝜇)(1−(𝜂/𝑅)^𝑛−⟨𝑣⟩).

An analysis of the dynamics of this process has been shown (see, e.g., [2, 5, 6, 15]) to lead a free boundary problem consisting of the initial-boundary value problem𝑣_𝑡=∇²𝑣+𝑔ℎ^±,𝑣,(𝐱,𝑡)∈Ω^±(𝑡),𝑣(𝐱,0)=𝑣₀(𝐱),𝑣𝜂(𝑡)𝜂(𝑡),𝑑−0,𝑡=𝑣+0,𝑡𝑣𝜂(𝑡)=𝑑𝑑𝜈−0,𝑡𝑣𝜂(𝑡),𝜂𝑑𝜈+0,𝑡_′(𝑡)𝑣𝜂,𝑤=𝐶,𝜂,𝑤,𝑡_′(𝑡)𝜂=1−𝑅^𝑛−2𝑤,𝑤(0)=𝑤₀, (1.7)

where 𝑤=⟨𝑣⟩.

The organization of the paper is as follows. In Section 2, a change of variables is given which regularizes problem (1.7) in such a way that results from the theory of nonlinear evolution equations can be applied. In this way, we obtain enough regularity of the solution for an analysis of the bifurcation. In Section 3, we show the existence of radially symmetric localized equilibrium solutions for (1.7) and obtain the linearization of problem (1.7). In the last section we show the existence of the periodic solutions and the bifurcation of the interface problem as a parameter 𝜎 varies in two and three dimensions.

2. Regularized System

We look for an existence problem of radially symmetric equilibrium solutions of (1.7) with |𝐱|=𝑟, where the center and the interface are located at the origin and 𝑟=𝜂, respectively. The problem is given by𝑣_𝑡=𝜕²𝑣𝜕𝑟²+𝑛−1𝑟𝜕𝑣−(𝑡)+𝜂(𝑡)𝜕𝑟𝜇+1𝑣+𝜇𝐻𝑟−𝜂1−𝜇1−𝑅ⁿ(𝑡),𝜂−𝑤_′(𝑡)=𝐶𝑣𝜂,𝜂,𝑤,𝑡,𝑡>0,𝜂(0)=𝜂₀,𝑤^′(𝑡)(𝑡)𝜂(𝑡)=−2𝑤+1−𝑅^𝑛,𝑤(0)=𝑤₀,𝜕𝑣𝑣()𝜕𝑟0,𝑡=0=𝜕𝑣𝑣()𝜕𝑟𝑅,𝑡,0<𝑟<𝑅,𝑡>0. (2.1)

As a first step we obtain more regularity for the solution by semigroup methods, considering 𝐴∶=−(𝜕²/𝜕𝑟²)+((𝑛−1)/𝑟)(𝜕/𝜕𝑟)+𝜇+1 as a densely defined operator 𝐴∶𝐷(𝐴)⊂𝑋→𝑋, where 𝐷(𝐴)={𝑣∈𝐻^2,2((0,𝑅))∶(𝜕𝑣/𝜕𝑟)(0,𝑡)=0=(𝜕𝑣/𝜕𝑟)(𝑅,𝑡)} and 𝑋∶=𝐿₂((0,𝑅)) with norm ‖⋅‖₂.

We define 𝑔∶[0,𝑅]×[0,𝑅]×ℂ→ℂ,𝑔𝑟,𝜂,𝑤∶=𝐴⁻¹𝜇𝐻(𝑟)+𝜂⋅−𝜂1−𝜇1−𝑅^𝑛−𝑤=𝜇^𝑅_𝜂𝐺𝑟,𝑦𝑑𝑦+1−𝜇𝜂1+𝜇1−𝑅^𝑛−𝑤, (2.2)

and 𝛾∶[0,𝑅]×ℂ→ℂ,𝛾𝜂,𝑤∶=𝑔𝜂,𝜂,𝑤. (2.3)

Here 𝐺∶[0,𝑅]²→ℝ is a Green's function of 𝐴 satisfying the Neumann boundary conditions: for 𝑛=2,𝐺(𝑟,𝑧)=𝑧𝐼₀𝑟1+𝜇𝐾₀𝑧+𝐾1+𝜇¹𝑅1+𝜇𝐼₁𝑅𝐼1+𝜇⁰𝑧1+𝜇,0<𝑟<𝑧,𝑧𝐼₀𝑧1+𝜇𝐾₀𝑟+𝐾1+𝜇¹𝑅1+𝜇𝐼₁𝑅𝐼1+𝜇⁰𝑟1+𝜇,𝑧<𝑟<𝑅, (2.4)

and for 𝑛=3,𝐺()=1𝑟,𝑧𝑌_𝑅𝑧sinh𝑟1+𝜇𝑟𝑅1+𝜇1+𝜇cosh(𝑅−𝑧)1+𝜇−sinh(𝑅−𝑧),11+𝜇0<𝑟<𝑧𝑌_𝑅𝑧sinh𝑧1+𝜇𝑟𝑅1+𝜇1+𝜇cosh(𝑅−𝑟)1+𝜇−sinh(𝑅−𝑟)1+𝜇,𝑧<𝑟<𝑅, (2.5)

where 𝐼_𝑖 and 𝐾_𝑖 are modified Bessel functions (𝑖=0,1) and 𝑌_𝑅 is given by𝑌_𝑅=1+𝜇𝑅1+𝜇cosh𝑅1+𝜇−sinh𝑅1+𝜇. (2.6)

Moreover, 𝑔(⋅,𝜂,𝑤)∈𝐷(𝐴) for all 𝜂 and 𝑤, (𝜕𝑔/𝜕𝜂)(𝑟,𝜂,𝑤)∈𝐻^1,∞((0,𝑅)²×ℂ) and 𝛾∈𝐶^∞([0,𝑅]×ℂ).

Applying the transformation 𝑢(𝑡)(𝑟)=𝑣(𝑟,𝑡)−𝑔(𝑟,𝜂(𝑡),𝑤(𝑡)), then we obtain an equivalent abstract evolution equation of(2.1)𝑑+𝐴,(0)=𝑢𝑑𝑡𝑢,𝜂,𝑤𝑢,𝜂,𝑤=𝐹𝑢,𝜂,𝑤𝑢,𝜂,𝑤₀,𝜂₀,𝑤₀, (2.7)

where 𝐴 is a 3×3 matrix defined on 𝐷(𝐴)=𝐷(𝐴)×(0,𝑅)×ℂ and given by2𝐴01−𝜇1+𝜇000002 (2.8)

The nonlinear forcing term 𝐹 defined on the set 𝑊∶={(𝑢,𝜂,𝑤)∈𝐶¹([0,𝑅])×(0,𝑅)×ℂ∶𝑢(𝜂)+𝛾(𝜂,𝑤)−((1−𝜇)/𝜇)(1−(𝜂/𝑅)^𝑛−𝑤)∈𝐈}⊂_open𝐶¹([0,𝑅])×(0,𝑅)×ℂ as𝐹=𝑓𝑢,𝜂,𝑤²𝑢,𝜂,𝑤⋅𝑓₁𝜂+1−𝜇𝑛1+𝜇𝑅^𝑛𝜂^𝑛−1+1−𝜇𝜂1+𝜇1−𝑅^𝑛𝑓₂𝜂𝑢,𝜂,𝑤1−𝑅^𝑛, (2.9)

where 𝑓₁∶(0,𝑅)→𝑋,𝑓₁(𝜂)(𝑟)∶=𝜇𝐺(𝑟,𝜂), and 𝑓₂∶𝑊→ℂ,𝑓₂(𝑢,𝜂,𝑤)∶=𝐶(𝑢(𝜂),𝜂,𝑤). We define 𝜉(𝜂)∶=𝜇∫^𝑅_𝜂𝐺(𝜂,𝑦)𝑑𝑦and then 𝛾(𝜂,𝑤)=𝜉(𝜂)+((1−𝜇)/(1+𝜇))(1−(𝜂/𝑅)^𝑛−𝑤). Let 𝜒(𝑢,𝜂)∶=𝑢(𝜂)+𝜉(𝜂)−((1−𝜇)/(1+𝜇))(𝜂/𝑅)^𝑛+((1−𝜇)/𝜇)(𝜂/𝑅)^𝑛, then the velocity of 𝜂 is written by𝐶𝑢𝜂𝜒1,𝜂,𝑤=𝐶𝑢,𝜂,𝑤=−𝜎1−2𝑎₀−2𝜒𝑢,𝜂−1−𝜇/𝜇1+𝜇(1−𝑤)𝑎₀+𝜒𝑢,𝜂−1−𝜇/𝜇1+𝜇(1−𝑤)1−𝑎₀−𝜒−).𝑢,𝜂1−𝜇/𝜇1+𝜇(1−𝑤 (2.10)

Lemma 2.1. The functions 𝑓₁∶(0,𝑅)→𝑋, 𝑓₂∶𝑊→ℂ and 𝐹∶𝑊→𝑋×ℂ×ℝ are continuously differentiable with derivatives given by 𝑓^′₁𝜂=𝜇𝜕𝐺,𝜕𝑧⋅,𝜂𝐷𝑓₂𝑢,𝜂,𝑤𝑢,𝜂,𝑤=𝐶_𝜒𝜒𝑢,𝜂,𝑤𝑢^′𝜂𝜂+𝑢𝜂+𝜉^′𝜂𝜂+1−𝜇𝜇1+𝜇𝑛𝜂^𝑛−1𝑅^𝑛𝜂+𝐶_𝑤𝜒𝑤𝑢,𝜂,𝑤𝑤,𝐷𝐹𝑢,𝜂,𝑤𝑢,𝜂,=𝑓₂𝑢,𝜂,𝑤⋅𝑓^′₁𝜂+1−𝜇𝑛()1+𝜇𝑛−1𝑅^𝑛𝜂^𝑛−2,0,0𝜂+𝐷𝑓₂𝑢,𝜂,𝑤𝑢,𝜂,𝑤⋅𝑓₁𝜂+1−𝜇𝑛1+𝜇𝑅^𝑛𝜂^𝑛−1+−,1,01−𝜇𝑛1+𝜇𝑅^𝑛𝜂^𝑛−1𝑛,0,−𝑅^𝑛𝜂^𝑛−1𝜂, (2.11) where 𝐶_𝜒=𝜕𝐶/𝜕𝜒 and 𝐶_𝑤=𝜕𝐶/𝜕𝑤.

The well posedness of solutions was shown in [23] applying the semigroup theory using domains of fractional powers 𝛼∈(3/4,1] of 𝐴 and 𝐴[24]. Moreover, they obtained that 𝐹∶𝑊∩𝐷(𝐴^𝛼)→𝑋×ℂ×ℝ is a continuously differentiable function, where 𝐷(𝐴^𝛼)=𝐷(𝐴^𝛼)×(0,𝑅)×ℂ.

3. Radially Symmetric Equilibrium Solutions and Linearization

The steady states are solutions of the following problem:𝐴𝑢^∗=𝜇𝐺⋅,𝜂^∗+1−𝜇𝑛1+𝜇𝑅^𝑛𝜂^∗𝑛−1𝐶𝑢^∗𝜂^∗,𝜂^∗,𝑤^∗+1−𝜇1+𝜇−2𝑤^∗𝜂+1−^∗𝑅^𝑛,0=𝐶𝑢^∗𝜂^∗+𝛾𝜂^∗,𝑤^∗,0=−2𝑤^∗𝜂+1−^∗𝑅^𝑛,𝑢^∗′(0)=0=𝑢^∗′(𝑅) (3.1)

for (𝑢^∗,𝜂^∗,𝑤^∗)∈𝐷(𝐴)∩𝑊.

Lemma 3.1. Define 𝜉(𝜂)∶=𝜇∫^𝑅_𝜂𝐺(𝜂,𝑦)𝑑𝑦<0. Then 𝜉′(𝜂)<0, and 𝜉′(𝜂)+𝜇𝐺(𝜂,𝜂)>0 for 0<𝜂<𝑅.

Proof. For 𝑛=2, the derivative of 𝜉 is given by 𝜉^′𝜂=𝜇𝜂−𝐼₀𝜂1+𝜇𝐾₀𝜂1+𝜇+𝐼₁𝜂1+𝜇𝐾₁𝜂𝐾1+𝜇−𝜇𝜂¹𝑅1+𝜇𝐼₁𝑅𝐼1+𝜇²⁰𝜂1+𝜇+𝐼²₁𝜂1+𝜇. (3.2)Let ℎ(𝜂)=𝐼₀(𝜂1+𝜇)𝐾₀(𝜂1+𝜇)−𝐼₁(𝜂1+𝜇)𝐾₁(𝜂1+𝜇). Then lim_𝜂→0ℎ(𝜂)=∞, and ℎ(𝑅)>0. The derivative of ℎ is ℎ^′𝜂=−1+𝜇𝐼₀𝜂1+𝜇+𝐼₁𝜂1+𝜇𝐾₁𝜂1+𝜇+2𝐼₁𝜂1+𝜇𝐾₀𝜂1+𝜇≤−21+𝜇𝐼₁𝜂1+𝜇𝐾₁𝜂1+𝜇−𝐾₀𝜂1+𝜇≤0. (3.3)Thus ℎ(𝜂)>0, and thus 𝜉′(𝜂)<0 for 0<𝜂<𝑅. Moreover, 𝜉^′𝜂𝐼+𝜇𝐺𝜂,𝜂=𝜂¹𝜂1+𝜇𝐼₁𝑅𝐼1+𝜇¹𝑅1+𝜇𝐾₁𝜂1+𝜇−𝐼₁𝜂1+𝜇𝐾₁𝑅1+𝜇. (3.4) Since 𝐼_𝑖 are increasing functions and 𝐾_𝑖 are decreasing functions (𝑖=0,1), 𝐼₁𝑅1+𝜇𝐾₁𝜂1+𝜇−𝐼₁𝜂1+𝜇𝐾₁𝑅=𝐼1+𝜇₁𝑅1+𝜇−𝐼₁𝜂1+𝜇𝐾₁𝜂+𝐾1+𝜇₁𝜂1+𝜇−𝐾₁𝑅1+𝜇𝐼₁𝜂1+𝜇>0, (3.5) and thus 𝜉′(𝜂)+𝜇𝐺(𝜂,𝜂)>0 for 0<𝜂<𝑅. For 𝑛=3, the derivative of 𝜉 is 𝜉^′𝜂=𝜇𝑀₂𝑅1+𝜇+11−1−2𝜂1+𝜇+2𝜂²1+𝜇𝑒^2𝜂1+𝜇−𝜇𝑀₂𝑅1+𝜇−1𝑒^2𝑅1+𝜇−1+2𝜂1+𝜇+2𝜂²1+𝜇𝑒^{2(𝑅−𝜂)1+𝜇}, (3.6) where 𝑀₂=2𝜂²(1+𝜇)1+𝜇[𝑅1+𝜇+1+(𝑅1+𝜇−1)𝑒^2𝑅1+𝜇]. Since 𝑒^2𝜂1+𝜇≥(1+2𝜂1+𝜇+2𝜂²(1+𝜇)), we have 𝜉′(𝜂)≤−(𝜇/𝑀₂)(1+𝑅1+𝜇)4𝜂⁴(1+𝜇²)<0 for 0<𝜂<𝑅. Moreover, 𝜉′(𝜂)+𝜇𝐺(𝜂,𝜂)>2(𝜇/𝑀₂)(𝑅−𝜂)1+𝜇[(𝜂1+𝜇+1)+(𝜂1+𝜇−1)𝑒^2𝜂1+𝜇]>0 for 0<𝜂<𝑅.

Theorem 3.2. Suppose that (i) 0<(1/2)−𝑎₀<(2𝜇−1)/2𝜇 and 𝜉′(𝜂)+(1−𝜇)/2𝜇(1+𝜇)(𝑛/𝑅^𝑛)𝜂^𝑛−1<0 or (ii) 0<(2𝜇−1)/2𝜇<(1/2)−𝑎₀ and 𝜉′(𝜂)+((1−𝜇)/2𝜇(1+𝜇))(𝑛/𝑅^𝑛)𝜂^𝑛−1>0. Then the stationary problem of (2.7) has the only stationary solution (𝑢^∗,𝜂^∗,𝑤^∗) for all 𝜎≠0 with 𝑢^∗=0 and 2𝑤^∗=1−(𝜂^∗/𝑅)^𝑛,𝜂^∗∈(0,𝑅). The linearization of 𝐹 at (0,𝜂^∗,𝑤^∗) is DF0,𝜂^∗,𝑤^∗𝑤=4𝑢,𝜂,𝜎𝑢𝜂^∗+𝛾_𝜂𝜂^∗,𝑤^∗𝜂+𝛾_𝑤𝜂^∗,𝑤^∗𝑤+1−𝜇𝜇𝑛𝑅^𝑛𝜂^∗𝑛−1𝑤⋅𝜂+𝜇𝐺⋅,𝜂^∗+1−𝜇𝑛1+𝜇𝑅^𝑛𝜂^∗𝑛−1−1−𝜇2𝑛1+𝜇𝑤+𝑅^𝑛𝜂^∗𝑛−1𝜂4𝜎𝑢𝜂^∗+𝛾_𝜂𝜂^∗,𝑤^∗𝜂+𝛾_𝑤𝜂^∗,𝑤^∗𝑤+1−𝜇𝜇𝑛𝑅^𝑛𝜂^∗𝑛−1𝑤𝑛𝜂+−2𝑤−𝑅^𝑛𝜂^∗𝑛−1𝜂. (3.7) The pair (0,𝜂^∗,𝑤^∗) corresponds to a unique steady state (𝑣^∗,𝜂^∗,𝑤^∗) of (2.1) for 𝜎≠0 with 𝑣^∗(𝑟)=𝑔(𝑟,𝜂^∗,𝑤^∗).

Proof. From the system (3.1), 𝜂^∗ and 𝑤^∗ are solutions of the following equations: 𝑢^∗=0,𝐶0,𝜂^∗,𝑤^∗=02𝑤^∗𝜂=1−^∗𝑅^𝑛. (3.8)We only check the existence of 𝜂^∗ of (2.10) and (3.8), and thus we let Γ𝜂1∶=2−𝑎₀−𝜒0,𝜂+1−𝜇𝜇1+𝜇1−1−𝜂/𝑅^𝑛2. (3.9)Then Γ(𝑅)=12−𝑎₀(0)=1,Γ2−𝑎₀−𝜉(0)+1−𝜇2𝜇1+𝜇,Γ^′𝜂=−𝜉^′𝜂−1−𝜇𝑛2𝜇1+𝜇𝑅^𝑛𝜂^𝑛−1. (3.10) Since 𝜉′(𝜂)<0 and 𝜉(0)=𝜇/(1+𝜇) for 𝑛=2,3,there is a unique 𝜂^∗∈(0,𝑅) when Γ′(𝜂)<0, Γ(0)>0, Γ(𝑅)<0, or Γ′(𝜂)>0, Γ(0)<0,Γ(𝑅)>0.
The formula for 𝐷𝐹(0,𝜂^∗,𝑤^∗) follows from Lemma 2.1, the relation 𝐶_𝜒(0,𝜂^∗,𝑤^∗)=4/𝜎, and 𝐶_𝑤(0,𝜂^∗,𝑤^∗)=4(1−𝜇)/𝜎𝜇(1+𝜇). The corresponding steady state (𝑣^∗,𝜂^∗,𝑤^∗) for (2.1) is obtained using the transformation and Theorem 2.1 in [17].

Definition 3.3. Suppose that 𝑎₀ and 𝜇 satisfy 0<(1/2)−𝑎₀<(2𝜇−1)/2𝜇 and 𝜉^′(𝜂)+((1−𝜇)/2𝜇(1+𝜇))(𝑛/𝑅^𝑛)𝜂^𝑛−1<0. One defines (for 1≥𝛼>3/4) the operator 𝐵 that is a linear operator from 𝐷(𝐴^𝛼) to 𝐷(𝐴) as 𝜎𝐵∶=4𝐷𝐹0,𝜂^∗,𝑤^∗. (3.11) One then defines (0,𝜂^∗,𝑤^∗) to be a Hopf point for (2.7) if there exists an 𝜖₀>0 and a 𝐶¹-curve −𝜖₀+𝜏^∗,𝜏^∗+𝜖₀⟼𝜆(𝜏),𝜙(𝜏)∈ℂ×𝐷𝐴_𝐂 (3.12) (𝑌_𝐂 denotes the complexification of the real space 𝑌) of eigendata for −𝐴+𝜏𝐵 such that (i)(−𝐴+𝜏𝐵)(𝜙(𝜏))=𝜆(𝜏)𝜙(𝜏),(−𝐴+𝜏𝐵)(𝜙(𝜏))=𝜆(𝜏)𝜙(𝜏), (ii)𝜆(𝜏^∗)=𝑖𝛽 with 𝛽>0,(iii)Re(𝜆)≠0 for all 𝜆 in the spectrum of (−𝐴+𝜏^∗𝐵)⧵{±𝑖𝛽}, (iv)Re𝜆′(𝜏^∗)≠0 (transversality), where 𝜏=4/𝜎.

4. Hopf Bifurcation Analysis

We will show that there is a Hopf bifurcation from the curve 𝜎↦(0,𝜂^∗,𝑤^∗) of radially symmetric stationary solution. The linearized eigenvalue problem of (2.7) is−𝐴𝑢,𝜂,𝑤+𝜏𝐵𝑢,𝜂,𝑤=𝜆𝐼₃𝑢,𝜂,𝑤, (4.1)

where 𝐼₃ is a 3 by 3 identity matrix. This is equivalent to(𝐴+𝜆)𝑢=𝜏𝑢𝜂^∗+𝛾_𝜂𝜂^∗,𝑤^∗𝜂+𝛾_𝑤𝜂^∗,𝑤^∗𝑤+1−𝜇𝜇𝑛𝑤+𝑅^𝑛𝜂^∗𝑛−1𝜂⋅𝜇𝐺⋅,𝜂^∗+1−𝜇𝑛1+𝜇𝑅^𝑛𝜂^∗𝑛−1+1−𝜇𝑛1+𝜇−2𝑤−𝑅^𝑛𝜂^∗𝑛−1𝜂,𝜆𝜂=𝜏𝑢𝜂^∗+𝛾_𝜂𝜂^∗,𝑤^∗𝜂+𝛾_𝑤𝜂^∗,𝑤^∗𝑤+1−𝜇𝜇𝑛𝑤+𝑅^𝑛𝜂^∗𝑛−1𝜂,𝑛𝜆𝑤=−2𝑤−𝑅^𝑛𝜂^∗𝑛−1𝜂. (4.2)

Our main theorem is stated as follows.

Theorem 4.1. Suppose that 𝑎₀ and 𝜇 satisfy 0<(1/2)−𝑎₀<((2𝜇−1)/2𝜇) and 𝜉′(𝜂)+((1−𝜇)/2𝜇(1+𝜇))(𝑛/𝑅^𝑛)𝜂^𝑛−1<0, the problem (2.7), and (2.1), has a unique stationary solution (𝑢^∗,𝜂^∗,𝑤^∗), where 𝑢^∗=0 and 𝑤^∗=(1/2)(1−(𝑛/𝑅^𝑛)(𝜂^∗)^𝑛−1), and (𝑣^∗,𝜂^∗,𝑤^∗), respectively, for all 𝜏>0. Then there exists a unique 𝜏^∗ such that the linearization −𝐴+𝜏^∗𝐵 has a purely imaginary pair of eigenvalues 𝛽. The point (0,𝜂^∗,𝑤^∗,𝜏^∗) is then a Hopf point for (2.7), and there exists a 𝐶⁰-curve of nontrivial periodic orbits for (2.7) and (2.1), bifurcating from (0,𝜂^∗,𝑤^∗,𝜏^∗) and (𝑣^∗,𝜂^∗,𝑤^∗,𝜏^∗), respectively.

We will show the following three theorems that verify the above theorem. The next theorem shows that the steady state is the only Hopf point.

Theorem 4.2. For 𝜏^∗∈ℝ⧵{0}, the operator −𝐴+𝜏^∗𝐵 has a unique pair of purely imaginary eigenvalues {±𝑖𝛽}. Then the point (0,𝜂^∗,𝑤^∗,𝜏^∗) satisfies the conditions (i), (ii), and (iii) in Definition 3.3.

Proof. In the sequel, we denote 𝑏_𝑛=(𝑛/𝑅^𝑛)(𝜂^∗)^𝑛−1, 𝑛=2,3. We assume without loss of generality that 𝛽>0 and Φ^∗ is the (normalized) eigenfunction of −𝐴+𝜏^∗𝐵 with eigenvalue 𝑖𝛽. We have to show that (Φ^∗,𝑖𝛽) can be extended to a 𝐶¹-curve 𝜏↦(𝜙(𝜏),𝜆(𝜏)) of eigendata for −𝐴+𝜏𝐵 with Re(𝜆′(𝜏^∗))≠0. For this, let Φ^∗∶=(𝜓₀,𝜂₀,𝑤₀)∈𝐷(𝐴). First, we note that if 𝑤₀=0, then 𝜂₀=0 (vice versa) in the last equation of (4.2). We see that 𝜂₀≠0 and 𝑤₀≠0, for otherwise, by (4.2), (𝐴+𝑖𝛽)𝜓₀=𝑖𝛽(𝜇𝐺(⋅,𝜂^∗)𝜂₀+((1−𝜇)/(1+𝜇))𝑏_𝑛+((1−𝜇)/(1+𝜇))𝑤₀)=0, which is not possible because 𝐴 is symmetric. So without loss of generality, let 𝜂₀=1. Define 𝐸∶𝐷(𝐴)_𝐂×ℂ×ℂ×ℝ⟶𝑋_𝐂𝐸()()𝑢𝜂×ℂ×ℂ,𝑢,𝑤,𝜆,𝜏∶=𝐴+𝜆𝑢−𝜏^∗+𝛾_𝜂+𝛾_𝑤𝑤+1−𝜇𝜇𝑏_𝑛+𝑤𝜇𝐺⋅,𝜂^∗+1−𝜇𝑏1+𝜇^𝑛+1−𝜇1+𝜇2𝑤+𝑏_𝑛𝜆−𝜏𝑢𝜂^∗+𝛾_𝜂+𝛾_𝑤𝑤+1−𝜇𝜇𝑏_𝑛+𝑤𝜆𝑤+2𝑤+𝑏_𝑛. (4.3) The equation 𝐸(𝑢,𝑤,𝜆,𝜏)=0 is equivalent to 𝜆 being an eigenvalue of −𝐴+𝜏𝐵 with eigenfunction (𝑢,1,𝑤). By (4.2), we have 𝐸(𝜓₀,𝑤₀,𝑖𝛽,𝜏^∗)=0 which is equivalent to 𝐴+𝑖𝛽𝜓₀=𝑖𝛽𝜇𝐺⋅,𝜂^∗+1−𝜇𝑏1+𝜇^𝑛+𝑤₀,𝑖𝛽=𝜏^∗𝜓₀𝜂^∗+𝛾_𝜂+𝛾_𝑤𝑤₀+1−𝜇𝜇𝑏_𝑛+𝑤₀,𝑖𝛽𝑤₀=−2𝑤₀−𝑏_𝑛. (4.4) To apply the implicit function theorem to 𝐸, we have to check that 𝐸 is in 𝐶¹ and that 𝐷_{(𝑢,𝑤,𝜆)}𝐸𝜓₀,𝑤₀,𝑖𝛽,𝜏^∗∈𝐿𝐷(𝐴)_𝐂×ℂ×ℂ,𝑋_𝐂×ℂ×ℂisanisomorphism. (4.5) In addition, the mapping 𝐷_{(𝑢,𝑤,𝜆)}𝐸𝜓₀,𝑤₀,𝑖𝛽,𝜏^∗𝜆=𝑢,𝑤,𝐴+𝑖𝛽𝑢+𝜆𝜓₀−𝜏^∗𝑢𝜂^∗+𝛾_𝑤𝑤+1−𝜇𝜇𝑤𝜇𝐺⋅,𝜂^∗+1−𝜇𝑏1+𝜇^𝑛+21−𝜇𝑤1+𝜇𝜆−𝜏^∗𝑢𝜂^∗+𝛾_𝑤𝑤+1−𝜇𝜇𝑤𝜆𝑤₀+𝑖𝛽𝑤+2𝑤 (4.6) is a compact perturbation of the mapping 𝑢,𝑤,𝜆⟼𝐴+𝑖𝛽𝑢,𝑤,𝜆 (4.7) which is invertible. In order to verify (4.5), it suffices to show that the system 𝐷_{(𝑢,𝑤,𝜆)}𝐸𝜓₀,𝑤₀,𝑖𝛽,𝜏^∗𝑢,𝑤,𝜆=0, (4.8) which are 𝐴+𝑖𝛽𝑢+𝜆𝜓₀=𝜏^∗𝑢𝜂^∗+𝛾_𝑤𝑤+1−𝜇𝜇𝑤𝜇𝐺⋅,𝜂^∗+1−𝜇𝑏1+𝜇^𝑛−21−𝜇1+𝜇𝑤,𝜆=𝜏^∗𝑢𝜂^∗+𝛾_𝑤𝑤+1−𝜇𝜇𝑤,𝜆𝑤₀=−𝑖𝛽𝑤−2𝑤, (4.9) necessarily implies that 𝑢=0,𝑤=0, and 𝜆=0. We define 𝜙∶=𝜓₀−(𝜇𝐺(⋅,𝜂^∗)+((1−𝜇)/(1+𝜇))𝑏_𝑛+((1−𝜇)/(1+𝜇))𝑤₀), then the first equation of (4.9) is given by 𝐴+𝑖𝛽𝑢+𝜆𝜙=𝑖𝛽1−𝜇1+𝜇𝑤. (4.10) Since (𝑣,𝜂,𝑤,𝜆)=(𝜓₀,1,𝑤,𝑖𝛽) solves (4.2), 𝜙 is a solution to the equation 𝐴+𝑖𝛽𝜙=−𝜇𝛿_𝜂∗−1−𝜇𝑏_𝑛+𝑤₀, (4.11)𝑖𝛽𝜏^∗=𝜙𝜂^∗+1−𝜇𝜇𝑏_𝑛+𝑤₀, (4.12)2+𝑖𝛽𝑤₀=−𝑏_𝑛. (4.13) Multiply (4.11) by 𝑟^𝑛−1(𝑛=2,3) and integrate, then ^𝑅₀−𝜙_𝑟𝑟−𝑛−1𝑟𝜙_𝑟+𝜇+1+𝑖𝛽𝜙𝑟^𝑛−1𝑑𝑟=^𝑅₀−𝜇𝛿_𝜂∗−1−𝜇𝑏_𝑛+𝑤₀𝑟^𝑛−1𝑑𝑟 (4.14) which implies that ^𝑅₀𝜕𝜕𝑟−𝑟^𝑛−1𝜙_𝑟+𝜇+1+𝑖𝛽𝑟^𝑛−1𝜙𝑑𝑟=−𝜇𝜂^∗𝑛−1−1−𝜇𝑏_𝑛+𝑤₀𝑅^𝑛𝑛. (4.15) Therefore we obtain 𝜇+1+𝑖𝛽𝑟^𝑛−1𝜙=−𝜇𝜂^∗𝑛−1−1−𝜇𝑏_𝑛+𝑤₀𝑅^𝑛𝑛. (4.16) Multiply (4.10) by 𝑟^𝑛−1(𝑛=2,3) and (4.11) by 𝑟^𝑛−1𝜙. Now we integrate the resulting equation to obtain 𝜇+1+𝑖𝛽𝑟^𝑛−1𝑢=−𝜆𝑟^𝑛−1𝜙+𝑖𝛽1−𝜇⋅𝑅1+𝜇𝑛^𝑛𝑤. (4.17)
Multiply (4.10) and (4.11) by 𝑟^𝑛−1𝜙, and then eliminate the term 𝜆(𝐴+𝑖𝛽)𝑟𝜙². Integrating the resulting equation, we have0=𝜆𝜇𝜂^∗𝑛−1𝜙𝜂^∗+𝜆1−𝜇𝑏_𝑛+𝑤₀𝑟^𝑛−1𝜙+𝜇𝜂^∗𝑛−1𝑢𝜂^∗+𝑏𝜇+1+𝑖𝛽1−𝜇_𝑛+𝑤₀𝜇+1+𝑖𝛽𝑟^𝑛−1𝑢+𝑖𝛽1−𝜇𝑤𝑟1+𝜇𝜇+1+𝑖𝛽^𝑛−1𝜙=𝜇𝜂^∗𝑛−1𝜆𝜙𝜂^∗−𝑖𝛽1−𝜇𝜆1+𝜇𝑤+𝜇+1+𝑖𝛽𝜏^∗−1−𝜇𝜇𝑤1+𝜇 (4.18) by (4.16) and (4.17). Using (4.9), (4.12), and (4.13) in the above equation, then we obtain 0=𝜆𝜇+1+𝑖2𝛽𝜏^∗−𝜇𝐺𝜂^∗,𝜂^∗+𝜉^′𝜂^∗−1−𝜇𝜇𝑏_𝑛+1−𝜇𝜇𝑏_𝑛2+𝑖𝛽². (4.19) Suppose that 𝜆≠0. Then the real and imaginary parts of (4.19) are given by 0=𝜇+1𝜏^∗−𝜇𝐺𝜂^∗,𝜂^∗+𝜉^′𝜂^∗−1−𝜇𝑏_𝑛𝜇⋅𝛽⁴+9𝛽²+124+𝛽²²,10=𝜏^∗−1−𝜇𝜇⋅2𝑏_𝑛4+𝛽²². (4.20) From these equations, we have 0=𝜇𝐺𝜂^∗,𝜂^∗+𝜉^′𝜂^∗+1−𝜇𝑏_𝑛𝜇⋅𝛽⁴+9𝛽²+10−2𝜇4+𝛽²². (4.21) This leads to a contradiction that the right hand side is positive for 𝛽>0 and 0<𝜇<1. Therefore, we should have 𝜆=0. Thus 𝑤=0 and 𝑢=0.

Theorem 4.3. Under the same condition as in Definition 3.3, (0,𝜂^∗,𝑤^∗,𝜏^∗) satisfies the transversality condition. Hence this is a Hopf point for (2.7).

Proof. By implicit differentiation of 𝐸(𝜓₀(𝜏),𝑤(𝜏),𝜆(𝜏),𝜏)=0, 𝐷_{(𝑢,𝑤,𝜆)}𝐸𝜓₀,𝑤₀,𝑖𝛽,𝜏^∗𝜓^′₀𝜏^∗,𝑤^′𝜏^∗,𝜆^′𝜏^∗=𝜇𝐺𝜂^∗,𝜂^∗+1−𝜇𝑏1+𝜇^𝑛𝜓₀𝜂^∗+𝛾_𝜂+𝛾_𝑤𝑤₀+1−𝜇𝜇𝑏_𝑛+1−𝜇𝜇𝑤^′𝜏^∗𝜓₀𝜂^∗+𝛾_𝜂+𝛾_𝑤𝑤₀+1−𝜇𝜇𝑏_𝑛+1−𝜇𝜇𝑤^′𝜏^∗0. (4.22) This means that the functions 𝑢∶=𝜓^′₀(𝜏^∗),𝑤∶=𝑤^′(𝜏^∗), and 𝜆∶=𝜆′(𝜏^∗) satisfy the equations 𝐴+𝑖𝛽𝑢+𝜆𝜓₀−𝜏^∗𝑢𝜂^∗+𝛾_𝑤𝑤+1−𝜇𝜇𝑤𝜇𝐺𝜂^∗,𝜂^∗+1−𝜇𝑏1+𝜇^𝑛+21−𝜇𝑤=𝜓1+𝜇₀𝜂^∗+𝛾_𝜂+𝛾_𝑤𝑤₀+1−𝜇𝜇𝑏_𝑛+1−𝜇𝜇𝑤₀𝜇𝐺𝜂^∗,𝜂^∗+1−𝜇𝑏1+𝜇^𝑛,𝜆−𝜏^∗𝑢𝜂^∗+𝛾_𝑤𝑤+1−𝜇𝜇𝑤=𝜓₀𝜂^∗+𝛾_𝜂+𝛾_𝑤𝑤₀+1−𝜇𝜇𝑏_𝑛+𝑤₀,𝜆𝑤₀+2+𝑖𝛽𝑤=0. (4.23) By letting 𝜙∶=𝜓₀−(𝜇𝐺(⋅,𝜂)+((1−𝜇)/(1+𝜇))𝑏_𝑛+((1−𝜇)/(1+𝜇))𝑤₀), then 𝐴+𝑖𝛽𝑢+𝜆𝜙=𝑖𝛽1−𝜇1+𝜇𝑤. (4.24) From (4.9) and (4.23), we obtain 𝜆𝜏^∗=𝑢𝜂^∗+𝛾_𝑤+1−𝜇𝜇𝑤+𝑖𝛽𝜏^∗2. (4.25) We now multiply (4.24) by 𝑟^𝑛−1𝜙 and (4.11) by 𝑟^𝑛−1𝑢 and then subtract these two equations, then we get 𝜆𝑟^𝑛−1𝜙²=𝜇𝑟^𝑛−1𝑢(𝑟)𝛿_𝜂∗+1−𝜇𝑏_𝑛+𝑤₀𝑟^𝑛−1𝑢+𝑖𝛽1−𝜇1+𝜇𝑤𝑟^𝑛−1𝜙. (4.26) Comparing to (4.11) and then integrating, we have 0=−𝜇𝜂^∗𝑛−1𝜙𝜂^∗𝜆−𝜆1−𝜇𝑏_𝑛+𝑤₀𝑟^𝑛−1𝜙−𝜇𝜂^∗𝑛−1𝑢𝜂^∗−𝑏𝜇+1+𝑖𝛽1−𝜇_𝑛+𝑤₀𝜇+1+𝑖𝛽𝑟^𝑛−1𝑢−𝑖𝛽1−𝜇𝑤𝑟1+𝜇𝜇+1+𝑖𝛽^𝑛−1𝜙. (4.27) Using (4.16) and (4.17) in the above equation, then 0=𝜇𝜂^∗𝑛−1𝜆𝜙𝜂^∗+𝜇+1+𝑖𝛽𝑢𝜂^∗−𝑖𝛽1−𝜇𝑤.1+𝜇 (4.28) By substituting (4.12), (4.13), and (4.23), we have 𝜇+1+𝑖𝛽𝑖𝛽𝜏^∗2=𝜆𝜇+1+2𝑖𝛽𝜏^∗−𝜇𝐺𝜂^∗,𝜂^∗−𝜉^′𝜂^∗−1−𝜇𝜇𝑏_𝑛+1−𝜇𝜇𝑏_𝑛2+𝑖𝛽². (4.29) The real part of 𝜆 is given by 𝛽𝜏^∗𝜇+1²+𝛽²Re𝜆=𝜆²𝜇+1𝑄−𝛽𝑃, (4.30) where 𝑃=𝜇+1𝜏^∗−𝜇𝐺𝜂^∗,𝜂^∗−𝜉^′𝜂^∗−1−𝜇𝜇𝑏_𝑛+1−𝜇𝑏_𝑛𝜇⋅4−𝛽²4+𝛽²²,𝑄=2𝛽𝜏^∗−1−𝜇𝜇⋅4𝑏_𝑛𝛽4+𝛽²²,𝛽𝜇+1𝑄−𝛽𝑃=𝜇+1𝜏^∗+𝛽𝜇𝐺𝜂^∗,𝜂^∗+𝜉^′𝜂^∗+1−𝜇𝑏_𝑛𝛽𝜇4+𝛽²²𝛽⁴+9𝛽²+8−4𝜇. (4.31) Since (𝜇+1)𝑄−𝛽𝑃>0 for 0<𝜇<1, we have Re𝜆>0. Therefore, Re𝜆′(𝜏^∗)>0 for 𝛽>0 and for 0<𝜇<1, and thus by the Hopf-bifurcation theorem in [17], there exists a family of periodic solutions which bifurcates from the stationary solution as 𝜏 passes 𝜏^∗.

The next theorem shows that a critical Hopf point 𝜏^∗ exists uniquely.

Theorem 4.4. Under the same condition as in Definition 3.3, there exists a unique, purely imaginary eigenvalue 𝜆=𝑖𝛽 of (4.2) with 𝛽>0 for a unique critical point 𝜏^∗>0 in order for (0,𝜂^∗,𝑤^∗,𝜏^∗) to be a Hopf point.

Proof. We only need to show that the function (𝑢,𝛽,𝜏)↦𝐸(𝑢,𝑤,𝑖𝛽,𝜏) has a unique zero with 𝛽>0 and 𝜏>0. This means solving the system (4.2) with 𝜆=𝑖𝛽, 𝜂₀=1, and 𝜓₀=𝜙+𝜇𝐺(⋅,𝜂^∗)+((1−𝜇)/(1+𝜇))𝑏_𝑛+((1−𝜇)/(1+𝜇))𝑤₀, 𝐴+𝑖𝛽𝜙=−𝜇𝛿_𝜂∗−1−𝜇𝑏_𝑛+𝑤₀,𝑖𝛽𝜏^∗=𝜙𝜂^∗+𝜇𝐺𝜂^∗,𝜂^∗+𝜉^′𝜂^∗+1−𝜇𝜇𝑏_𝑛+𝑤₀,2+𝑖𝛽𝑤₀=−𝑏_𝑛. (4.32) The second equation becomes 𝑖𝛽𝜏^∗=−𝜇𝐺_𝛽𝜂^∗,𝜂^∗+𝜇𝐺𝜂^∗,𝜂^∗+𝜉^′𝜂^∗−1𝑏𝜇+1+𝑖𝛽1−𝜇_𝑛+𝑤₀+1−𝜇𝜇𝑏_𝑛+𝑤₀, (4.33) where 𝐺_𝛽 is a Green's function of the differential operator 𝐴+𝑖𝛽. The real and imaginary parts of this above equation are given by 𝛽𝜏^∗=−𝜇Im𝐺_𝛽𝜂^∗,𝜂^∗+1−𝜇𝑏_𝑛𝛽𝜇4+𝛽²+1−𝜇𝑏_𝑛𝛽1−𝜇+𝛽²4+𝛽²1+𝜇²+𝛽²=−𝜇Im𝐺_𝛽𝜂^∗,𝜂^∗+1−𝜇𝑏_𝑛𝛽𝜇4+𝛽²1+𝜇²+𝛽²1+𝜇𝛽²+3𝜇+1,0=−𝜇Re𝐺_𝛽𝜇^∗,𝜇^∗+𝜇𝐺𝜂^∗,𝜂^∗+𝜉^′𝜂^∗+1−𝜇𝑏_𝑛2+𝛽²𝜇4+𝛽²−1−𝜇𝑏_𝑛𝜇+12+𝛽²+𝛽²𝜇+1²+𝛽²4+𝛽²=−𝜇Re𝐺_𝛽𝜇^∗,𝜇^∗+𝜇𝐺𝜂^∗,𝜂^∗+𝜉^′𝜂^∗+1−𝜇𝑏_𝑛𝛽𝜇4+𝛽²1+𝜇²+𝛽²×𝛽⁴+3𝛽²+21+𝜇. (4.34) Since Im𝐺_𝛽(𝜂^∗,𝜂^∗)<0 in [17, Lemma 12] and 𝜇>0, there is a unique 𝜏 in the first equation if it does guarantee the existence of 𝛽. Now, we let 𝑇𝛽∶=−𝜇Re𝐺_𝛽𝜇^∗,𝜇^∗+𝜇𝐺𝜂^∗,𝜂^∗+𝜉^′𝜂^∗+1−𝜇𝑏_𝑛𝛽𝜇4+𝛽²1+𝜇²+𝛽²×𝛽⁴+3𝛽²+21+𝜇, (4.35) then 𝑇(∞)=𝜇𝐺(𝜂^∗,𝜂^∗)+𝜉′(𝜂^∗)>0 and 𝑇(0)=𝜉′(𝜂^∗)+(1−𝜇)𝑏_𝑛/2𝜇(1+𝜇)<0 by assumption. If we show that 𝑇′(𝛽)<0, then the existence of 𝛽 is proved: 𝑇′𝛽=−𝜇Re𝐺_𝛽𝜂^∗,𝜂^∗′+21−𝜇𝑏_𝑛𝛽𝜇4+𝛽²²1+𝜇²+𝛽²²×𝜇²+2𝜇+2𝛽⁴+41+𝜇1+2𝜇𝛽²+21+𝜇1+4𝜇−𝜇². (4.36) Since 1+4𝜇−𝜇²>0 for 0<𝜇<1 and (Re𝐺_𝛽(𝜂^∗,𝜂^∗))′<0 in [17, Lemma 12], we have 𝑇′(𝛽)>0 for all 𝛽>0.

There is a unique pure imaginary eigenvalue 𝛽>0 and the critical point 𝜏^∗ of (2.1) and thus there exists a family of periodic solutions which bifurcates from the stationary solution as 𝜏 passes 𝜏^∗ under the condition of Theorem 4.4. Thus we also found the relationship between 𝜇 and 𝑎₀ for which Hopf bifurcation occurs for the problem (2.1).

Acknowledgment

This paper was supported by Kyonggi University Grant 2008.