A Hopf Bifurcation in a Three-Component Reaction-Diffusion System with a Chemoattraction

Abstract

We consider a three-component reaction-diffusion system with a chemoattraction. The purpose of this work is to analyze the chemotactic effects due to the gradient of the chemotactic sensitivity and the shape of the interface. Conditions for existence of stationary solutions and the Hopf bifurcation in the interfacial problem as the bifurcation parameters vary are obtained analytically.

1. Introduction

We are interested in the effects of diffusivity and chemotaxis on the competition of several species for limited resources. Chemotaxis is an oriented movement of cells in response to a concentration gradient of chemical substances in their environment. It was observed that diffusivity and chemotaxis of cells play a dominant role in cell growth; when several species of cells compete for limited resources, the species with a smaller diffusion rate and larger chemotaxis rate grow better, even when other species have superior growth kinetics.

Mathematical modeling on chemotaxis was initiated in 1970 by Keller and Segel (see [1]) with the use of the following system of PDEs: 𝑢_𝑡,𝑣=∇⋅∇𝑢−𝜒𝑢∇𝑣_𝑡=𝐷∇²𝑣+𝑎𝑢−𝑏𝑣,𝑡>0,𝐱∈ℝ^𝑛, (1) where 𝐷 is a diffusion coefficient, 𝑎 and 𝑏 are positive constants, and 𝜒 is the chemotaxis coefficient. In many biological processes, cells often interact with combinations of repulsive and attractive signalling chemicals to produce various interesting biological patterns. In this paper, we consider the attraction chemotaxis system [2–4]: 𝜀𝜎𝑢_𝑡=𝜀²∇²(𝑣((𝑤𝑣𝑢−𝜀𝜅∇𝑢⋅∇𝜒)−𝑢+𝐻𝑢−𝑎))−𝑣,_𝑡=∇²𝑣+𝜇𝑢−𝑣,𝑡>0,𝐱∈ℝ^𝑛,𝑤_𝑡=∇²𝑤+𝑢+𝑣−𝑤−𝑠₀,𝑡>0,𝐱∈ℝ^𝑛, (2) where 𝜀, 𝜎, 𝜇, and 𝑠₀ are positive constants, 𝐻 is a Heaviside step function, and 𝑎^′(𝑤)>0 for all 𝑤. Here, ∇ is the gradient operator, 𝜒 is the chemical sensitivity function of the chemical repulsion satisfying 𝜒^′(𝑣)≥0 for 𝑣>0, and 𝜅 is a positive constant.

Chemotaxis describes the direct migration of cells along the concentration gradient of a specific chemical produced by the cells. The prototype of the population-based chemotaxis model was described in the above mentioned work of Keller and Segal [1].

Schaaf [5] discussed the existence of nonconstant equilibrium solutions which exhibit aggregating patterns in a bounded domain. In [4, 6], equations describing the dynamics of the interfaces near equilibrium and the stability of the planar standing pulse solutions in the channel domain are obtained for sufficiently small𝜀. Results for several versions of the Keller-Segel system and its related models are discussed in Horstmann [7, 8] and Ward [9]. The effect of chemotaxis or that of lateral inhibition on an activator in reaction-diffusion systems has been studied by several authors (see [10–13]).

In the present work, chemotaxis growth under the influence of lateral inhibition in a three-component reaction-diffusion system is considered. We derive a free boundary problem of this system when 𝜀=0 and then find conditions which are necessary for occurrence of the Hopf bifurcation of chemotaxis and the lateral inhibition on an activator. We derive an evolutional equation of interfaces that is controlled by the two inhibitors 𝑣 and 𝑤.

Suppose that there is only one interfacial curve 𝑥=𝜂(𝑡) in [0,∞) in such a way that [0,∞)=Ω₁∪𝜂(𝑡)∪Ω₀, where Ω₁={𝑥∈[0,∞)∶𝑢(𝑥,𝑡)>𝑎(𝑤(𝑥,𝑡))} and Ω₀={𝑥∈[0,∞)∶𝑢(𝑥,𝑡)<𝑎(𝑤(𝑥,𝑡))}. Let (𝑥₀,𝑡₀) lie on this curve; that is, 𝑥₀=𝜂(𝑡₀). Using a stretching transformation at (𝑥₀,𝑡₀) we make the following substitutions: 𝜉=𝑥−𝑥₀𝜀,𝜌=𝑡−𝑡₀𝜀. (3) Then, the system (2) at (𝑥₀,𝑡₀) becomes 𝜎𝑢_𝜌=𝑢_𝜉𝜉−𝜅𝜒^′𝑣₀𝑣_𝑥𝑢_𝜉+𝐹𝑢,𝑣₀,𝑤₀,𝐹(𝑢,𝑣,𝑤)=−𝑢+𝐻(𝑢−𝑎(𝑤))−𝑣 (4) and the boundary conditions are 𝑢(±∞)=ℎ_±𝑣₀ (5) when 𝜀 tends to zero, where 𝑣₀=𝑣(𝑥₀,𝑡₀) and 𝑤₀=𝑤(𝑥₀,𝑡₀). We put the equation into a traveling coordinate system by setting 𝑧=𝜉−𝜃𝜌 with velocity 𝜃. Thus, 𝑈(𝑧)=𝑢(𝜉,𝜌) satisfies the following conditions: 𝑈_𝑧𝑧+𝜎𝜃−𝜅𝜒^′𝑣₀𝑣_𝑥𝑈_𝑧+𝐹𝑈,𝑣₀,𝑤₀𝑈()=0,±∞=ℎ_±𝑣₀. (6)

The existence of a solution 𝑈(𝑧) is given in [12, 14] and 𝜃 satisfies 𝜎𝜃=𝐶(𝑣₀)+𝜅𝜒^′(𝑣₀)𝑣_𝑥(𝑥₀,𝑡₀). Hence, the velocity of the one-dimensional interface 𝜂(𝑡) is given by 𝑑𝜂(𝑡)=1𝑑𝑡𝜎𝐶𝑣_𝑖+𝜅𝜒^′𝑣_𝑖𝑣_𝑥𝜂(𝑡),𝑡,𝑥∈𝜂(𝑡), (7) where 𝑣_𝑖 is the value of 𝑣 on the interface 𝜂(𝑡) and 𝐶 is a continuously differentiable function defined on an interval 𝐼∶=(−𝑎(𝑤),1−𝑎(𝑤)), which is given by [14–16] 𝐶𝑣𝜂𝑤𝜂𝑤𝜂𝜂;𝑎=−1−2𝑎−2𝑣𝑣𝜂(𝑤)𝑤𝜂𝜂.+𝑎1−𝑎−𝑣 (8)

Hence, a free boundary problem of (2) when 𝜀 is equal to zero is given by 𝑣_𝑡=∇²𝑣−𝜇+1𝑣+𝜇,𝑡>0,𝑥∈Ω₁,𝑣_𝑡=∇²𝑣−𝜇+1𝑣,𝑡>0,𝑥∈Ω₀,𝑣𝜂(𝑡)𝜂(𝑡),𝑣−0,𝑡=𝑣+0,𝑡_𝑥𝜂(𝑡)−0,𝑡=𝑣_𝑥𝜂(𝑡)+0,𝑡,lim_𝑥→∞𝑣()𝑤𝑥,𝑡=0,_𝑡=∇²𝑤−𝑤+1−𝑠₀,𝑡>0,𝑥∈Ω₁,𝑤_𝑡=∇²𝑤−𝑤−𝑠₀,𝑡>0,𝑥∈Ω₀,𝑤𝜂(𝑡)𝜂(𝑡),𝑤−0,𝑡=𝑤+0,𝑡_𝑥𝜂(𝑡)−0,𝑡=𝑤_𝑥𝜂(𝑡)+0,𝑡,lim_𝑥→∞𝑤(𝑥,𝑡)=−𝑠₀. (9)

In this paper, we establish the existence of the Hopf bifurcation described above by an application of the implicit function theorem along the lines of the results in [17]. In order to apply the implicit function theorem, we require more regularity of the solution than that obtained in the papers [4, 6, 13]. Our approach to the problem of well-posedness and to the Hopf bifurcation is to write (9) in the form of an abstract evolution equation on a Banach space, which is the product of a function space and an interval of real numbers. Once we have done this, we are able to apply standard results from the theory of nonlinear evolution equations (see for instance, [18]) to show the well-posedness of the problem and, more importantly, to give an analysis of the Hopf bifurcation.

The organization of the paper is as follows. In Section 2, a change of variables is given which regularizes problem (9) in such a way that results from the theory of nonlinear evolution equations can be applied. In this way, we obtain a regularity of the solution which is sufficient for an analysis of the bifurcation. In Section 3, we show the existence of equilibrium solutions for (9) and obtain the linearization of problem (9). In the last section, we investigate the conditions to obtain the periodic solutions and the bifurcation of the interface problem as the parameter 𝜎 varies.

2. Regularization of the Interface Equation

Now, we consider the existence problem of (9): 𝑣_𝑡=𝜕²𝑣𝜕𝑥²−(𝑡),𝑤𝜇+1𝑣+𝜇𝐻𝑥−𝜂_𝑡=𝜕²𝑤𝜕𝑥²−𝑤+𝐻𝑥−𝜂(𝑡)−𝑠₀,𝑣_𝑥(0,𝑡)=0,lim_𝑥→∞𝑣_𝑥()𝑤𝑥,𝑡=0,𝑡>0,_𝑥(0,𝑡)=0,lim_𝑥→∞𝑤_𝑥(𝑥,𝑡)=−𝑠₀,𝑡>0,𝜎𝜂^′(𝑡)=𝐶𝑣𝜂;𝑎𝑤𝜂+𝜅𝜒^′𝑣𝜂(𝑡),𝑡𝑣_𝑥𝜂(𝑡),(0),𝑡𝑡>0;𝜂=𝜂₀. (10)

Let 𝑤(𝑥,𝑡)=𝑤(𝑥,𝑡)+𝑠₀. Let 𝐴 be an operator defined by 𝐴∶=−(𝑑²/𝑑𝑥²)+𝜇+1 with domain 𝐷(𝐴)={𝑣∈𝐻^2,2(ℝ)∶𝑣_𝑥(0,𝑡)=0,lim_𝑥→∞𝑣_𝑥(𝑥,𝑡)=0}. Let 𝐴₀∶=−(𝜕²/𝜕𝑥²)+1 with domain 𝐷(𝐴₀)={𝑤∈𝐻^2,2((0,∞))∶𝑤_𝑥(0,𝑡)=0,lim_𝑥→∞𝑤_𝑥(𝑥,𝑡)=0}. In order to apply semigroup theory to (10), we choose the space 𝑋∶=𝐿₂(0,∞) with norm ‖⋅‖₂.

To get differential dependence on initial conditions, we decompose 𝑣 in (10) into two parts: 𝑢 which is a solution to a more regular problem and 𝑔 which is less regular but explicitly known in terms of the Green function 𝐺 of the operator 𝐴. Namely, we define 𝑔∶[0,∞)×[0,∞)→ℝ by 𝑔𝑥,𝜂∶=𝐴⁻¹𝜇𝐻⋅−𝜂(𝑥)=𝜇∫^∞₀𝐺𝑥,𝑦𝐻𝑦−𝜂𝑑𝑦, (11) where 𝐺∶[0,∞)×[0,∞)→ℝ is a Green’s function of 𝐴 satisfying the Neumann boundary conditions, and 𝛾∶[0,∞)→ℝ is given by 𝛾𝜂∶=𝑔𝜂,𝜂. (12) If we take a transformation 𝑢(𝑡)(𝑥)=𝑣(𝑥,𝑡)−𝑔(𝑥,𝜂(𝑡)), we have (𝑢_𝑥)(𝑡)(𝑥)=𝑣_𝑥(𝑥,𝑡)−𝑔_𝑥(𝑥,𝜂(𝑡)). Since 𝐺_𝑥(𝑥,𝜂) is discontinuous at 𝑥=𝜂, we cannot obtain one step more regular than that of (10).

To overcome this difficulty, let 𝑝(𝑥,𝑡)=𝑣_𝑥(𝑥,𝑡). Then 𝑝 satisfies 𝑝_𝑡+𝐴𝑝=𝜇𝛿(𝑥−𝜂), where 𝐴=−(𝑑²/𝑑𝑥²)+𝜇+1with domain 𝐷(𝐴)={𝑝∈𝐻^1,2(ℝ)∶𝑝(0,𝑡)=0,lim_𝑥→∞𝑝(𝑥,𝑡)=0}. Define 𝑔∶[0,∞)×[0,∞)→ℝ𝑔𝑥,𝜂∶=𝐴⁻¹𝜇𝛿⋅−𝜂(𝑥)=𝜇^∞₀𝐺𝑥,𝑦𝛿𝑦−𝜂𝑑𝑦, (13) where 𝐺∶[0,∞)×[0,∞)→ℝ is a Green’s function of 𝐴 satisfying the Dirichlet boundary conditions, and 𝛾∶[0,∞)→ℝ is given by 𝛾𝜂∶=𝑔𝜂,𝜂. (14) We define 𝑗∶[0,∞)×[0,∞)→ℝ, 𝑗𝑟,𝜂∶=𝐴₀⁻¹𝐻⋅−𝜂(𝑥)=^∞₀𝐽𝑥,𝑦𝐻𝑦−𝜂𝑑𝑦 (15) and 𝛼∶[0,∞)×ℂ→ℂ𝛼𝜂∶=𝑗𝜂,𝜂, (16) where 𝐽∶[0,∞)²→ℝ is a Green’s function of 𝐴₀ satisfying the boundary conditions.

Applying the transformations 𝑢(𝑡)(𝑥)=𝑣(𝑥,𝑡)−𝑔(𝑥,𝜂(𝑡)), 𝑧(𝑡)(𝑥)=𝑝(𝑥,𝑡)−𝑔(𝑥,𝜂(𝑡)), and 𝑞(𝑡)(𝑥)=𝑤(𝑥,𝑡)−𝑗(𝑥,𝜂(𝑡)) to (10), we get 𝑢_𝑡𝜇+𝐴𝑢=𝜎𝐺×𝐶𝑢𝜂𝜂𝑞𝜂𝜂𝑥,𝜂+𝛾;𝑎+𝛼−𝑠₀+𝜅𝜒^′𝑢𝜂𝜂𝑧𝜂+𝛾𝜂,𝑧+𝛾_𝑡1+𝐴𝑧=−𝜎𝜇𝜂𝐺×𝐶𝑢𝜂𝜂𝑞𝜂𝜂𝑥,𝜂+𝛾;𝑎+𝛼−𝑠₀+𝜅𝜒^′𝑢𝜂𝜂𝑧𝜂+𝛾𝜂,𝑞+𝛾_𝑡+𝐴₀1𝑞=𝜎𝐽×𝐶𝑢𝜂𝜂𝑞𝜂𝜂𝑥,𝜂+𝛾;𝑎+𝛼−𝑠₀+𝜅𝜒^′𝑢𝜂𝜂𝑧𝜂+𝛾𝜂,𝜂+𝛾_′(𝑡)=1𝜎𝐶𝑢𝜂+𝛾𝜂;𝑎𝑞𝜂+𝛼𝜂−𝑠₀+𝜅𝜒^′𝑢𝜂+𝛾𝜂⋅𝑧𝜂+𝛾𝜂,𝑡>0. (17) Thus, we obtain an abstract evolution equation equivalent to (10): 𝑑+𝐴=1𝑑𝑡𝑢,𝑧,𝑞,𝜂𝑢,𝑧,𝑞,𝜂𝜎𝑓,(0)=𝑢𝑢,𝑧,𝑞,𝜂𝑢,𝑧,𝑞,𝜂₀(𝑥),𝑧₀(𝑥),𝑞₀(𝑥),𝜂₀, (18) where 𝐴 is a 4×4 matrix with the main diagonal entries being the operators 𝐴, 𝐴, 𝐴₀, and 𝑂 (the zero operator), and all the other terms are zero. The nonlinear forcing term 𝑓 is 𝑓=𝑓𝑢,𝑧,𝑞,𝜂₁𝜂⋅𝑓₂₁𝑢,𝑧,𝑞,𝜂+𝑓₂₂𝑓𝑢,𝑧,𝑞,𝜂₂𝜂⋅𝑓₂₁𝑢,𝑧,𝑞,𝜂+𝑓₂₂𝑓𝑢,𝑧,𝑞,𝜂₃𝜂⋅𝑓₂₁𝑢,𝑧,𝑞,𝜂+𝑓₂₂𝑓𝑢,𝑧,𝑞,𝜂₂₁𝑢,𝑧,𝑞,𝜂+𝑓₂₂,𝑢,𝑧,𝑞,𝜂 (19) where 𝑓₁∶(0,∞)→𝑋, 𝑓₁(𝜂)(𝑥)∶=𝜇𝐺(𝑥,𝜂), 𝑓₂∶(0,∞)→𝑋, 𝑓₂(𝜂)(𝑥)∶=−(𝜇/𝜂)𝐺(𝑥,𝜂), 𝑓₃∶(0,∞)→𝑋, 𝑓₃(𝜂)(𝑥)∶=𝐽(𝑥,𝜂), 𝑓₂₁∶𝑊→ℂ, 𝑓₂₁(𝑢,𝑧,𝑞,𝜂)∶= 𝐶(𝑢(𝜂)+𝛾(𝜂);𝑎(𝑞(𝜂)+𝛼(𝜂)−𝑠₀)), and 𝑓₂₂∶𝑊→ℂ, 𝑓₂₂(𝑢,𝑧,𝑞,𝜂)∶=𝜅𝜒^′(𝑢(𝜂)+𝛾(𝜂))⋅(𝑧(𝜂)+𝛾(𝜂)) and 𝑊∶={(𝑢,𝑧,𝑞,𝜂)∈𝐶¹(0,∞)×𝐶¹(0,∞)×𝐶¹(0,∞)×(0,∞)∶𝑢(𝜂)+𝛾(𝜂)∈𝐼, 𝑧(𝜂)+𝛾(𝜂)∈𝐼, 𝑞(𝜂)+𝛼(𝜂)−𝑠₀∈𝐼}⊂_open𝐶¹(ℝ)×𝐶¹(ℝ)×𝐶¹(ℝ)×ℝ.

The well-posedness of solutions of (18) is shown in [4, 10, 11], using the fractional powers of degree 𝜃∈(3/4,1] of 𝐴, 𝐴₀, and 𝐴 and the methods of the theory of semigroups of operators. Moreover, the nonlinear term 𝑓 is a continuously differentiable function from 𝑊∩𝑋^𝜃 to 𝑋, where 𝑋∶=𝐷(𝐴)=𝐷(𝐴)×𝐷(𝐴)×𝐷(𝐴₀)×ℝ, 𝑋^𝜃_𝐴∶=𝐷(𝐴^𝜃), 𝑋^𝜃₀∶=𝐷(𝐴^𝜃₀), and 𝑋^𝜃∶=𝐷(𝐴^𝜃)=𝑋^𝜃_𝐴×𝑋^𝜃_𝐴×𝑋^𝜃₀×ℝ.

The velocity of𝜂is denoted by 𝐶𝑢𝜂+𝛾𝜂;𝑎𝑞𝜂+𝛼𝜂−𝑠₀=𝐶𝑆𝑢,𝑞,𝜂=−1−2𝑆𝑢,𝑞,𝜂𝑆,𝑢,𝑞,𝜂1−𝑆𝑢,𝑞,𝜂 (20) where 𝑆(𝑢,𝑞,𝜂)=𝑢(𝜂)+𝛾(𝜂)+𝑎(𝑞(𝜂)+𝛼(𝜂)−𝑠₀).

The derivative of𝑓can be obtained following [19].

Lemma 1. The functions 𝐺(⋅,𝜂)∶(0,∞)→𝑋, 𝐺(⋅,𝜂)∶(0,∞)→𝑋, 𝐽(⋅,𝜂)∶(0,∞)→𝑋, 𝐶(⋅)∶𝑊→ℂ, and 𝑓∶𝑊→𝑋×ℝ are continuously differentiable with derivatives given by 𝐷𝑓₂₁𝑢,𝑧,𝑞,𝜂𝑢,𝑧,𝑞,𝜂=𝐶_𝑆𝑆𝑢,𝑞,𝜂⋅𝑢^′𝜂𝜂+𝑢𝜂+𝛾^′𝜂𝜂+𝑎^′𝑞𝜂+𝛼𝜂−𝑠₀⋅𝑞^′𝜂𝜂+𝑞𝜂+𝛼^′𝜂𝜂,𝐷𝑓₂₂𝑢,𝑧,𝑞,𝜂𝑢,𝑧,𝑞,𝜂=𝜅𝜒^′𝑢𝜂+𝛾𝜂𝑧^′𝜂𝜂+𝑧𝜂+𝛾^′𝜂𝜂,+𝜅𝜒^′′𝑢𝜂𝜂⋅𝑢𝜂+𝛾+𝑢^′𝜂𝜂𝜂𝑧𝜂+𝛾𝜂,𝜂=𝑓𝜂+𝛾𝐷𝑓𝑢,𝑧,𝑞,𝜂𝑢,𝑧,𝑞,₂₁𝑢,𝑧,𝑞,𝜂+𝑓₂₂⋅𝑓𝑢,𝑧,𝑞,𝜂_′1𝜂,𝑓^′₂𝜂,𝑓^′₃𝜂𝜂+,0𝐷𝑓₂₁𝑢,𝑧,𝑞,𝜂+𝐷𝑓₂₂𝜂⋅𝑓𝑢,𝑧,𝑞,𝜂𝑢,𝑧,𝑞,₁𝜂,𝑓₂𝜂,𝑓₃𝜂,1. (21)

3. Equilibrium Solutions and Linearization of the Interface Equation

In this section, we will examine the existence of equilibrium solutions of (18). We look for (𝑢^∗,𝑧^∗,𝑞^∗,𝜂^∗)∈𝐷(𝐴)∩𝑊 satisfying the following equations: 1𝐴𝑢=𝜎×𝐶𝑢𝜂𝜂𝑞𝜂𝜂𝜇𝐺𝑥,𝜂+𝛾;𝑎+𝛼−𝑠₀+𝜅𝜒^′𝑢𝜂𝜂⋅𝑧𝜂+𝛾𝜂,1+𝛾𝐴𝑧=−𝜎𝜇𝜂𝐺×𝐶𝑢𝜂𝜂𝑞𝜂𝜂𝑥,𝜂+𝛾;𝑎+𝛼−𝑠₀+𝜅𝜒^′𝑢𝜂𝜂⋅𝑧𝜂+𝛾𝜂,𝐴+𝛾₀1𝑞=𝜎𝐽⋅,𝜂^∗𝐶𝑢𝜂+𝛾𝜂;𝑎𝑞𝜂+𝛼𝜂−𝑠₀+𝜅𝜒^′𝑢𝜂𝜂⋅𝑧𝜂+𝛾𝜂,𝑢𝜂𝜂𝑞𝜂𝜂+𝛾0=𝐶+𝛾;𝑎+𝛼−𝑠₀+𝜅𝜒^′𝑢𝜂𝜂⋅𝑧𝜂+𝛾𝜂,𝑢+𝛾_′(0)=0=𝑢^′(∞)(0)(∞),𝑞,𝑧=0=𝑧_′(0)=0=𝑞^′(∞). (22)

Theorem 2. Suppose that (1/2)−𝑎(1−𝑠₀)<𝜇/(1+𝜇) and 𝐶^′(𝛾(𝜂); 𝑎(𝛼(𝜂)−𝑠₀)) + 𝜅𝜒^′′(𝛾(𝜂))𝛾(𝜂)−𝜅1+𝜇𝜒^′(𝛾(𝜂))>0 for all 𝜂>0. Then (18) has at least one equilibrium solution (0,0,0,𝜂^∗) for 𝜅<𝜅_𝑐, where 𝜅_𝑐 is a solution of 𝐶𝛾(∞);𝑎𝛼(∞)−𝑠₀+𝜅_𝑐𝜒^′𝛾(∞)𝛾^′(∞)+𝜇𝐺(∞,∞)=0. (23)
The linearization of 𝑓 at the stationary solution (0,0,0,𝜂^∗) is 𝐷𝑓0,0,0,𝜂^∗𝜂=𝑢,𝑧,𝑞,𝜇𝐺⋅,𝜂^∗𝑄𝜂−𝜇𝑢,𝑧,𝑞,𝜂^∗𝐺⋅,𝜂^∗𝑄𝜂𝑗𝑢,𝑧,𝑞,⋅,𝜂^∗𝑄𝜂𝑄𝜂,𝑢,𝑧,𝑞,𝑢,𝑧,𝑞, (24) where 𝑄(𝑢,𝑧,𝑞,𝜂)=(4+𝜅𝜒^′′(𝛾(𝜂^∗)𝛾(𝜂^∗))(𝑢(𝜂^∗)+𝛾^′(𝜂^∗)𝜂)+𝜅𝜒^′(𝛾(𝜂^∗))(𝑧(𝜂^∗)+𝛾^′(𝜂^∗)𝜂)+4𝑎^′(𝛼(𝜂^∗) − 𝑠₀)(𝑞(𝜂^∗)+𝛼^′(𝜂^∗)𝜂). The pair (0,0,0,𝜂^∗) corresponds to a unique steady state (𝑣^∗,𝑝^∗,𝑤^∗,𝜂^∗) of (10) for 𝜎≠0 with 𝑣^∗(𝑥)=𝑔(𝑥,𝜂^∗), 𝑝^∗(𝑥)=𝑔(𝑥,𝜂^∗), and 𝑤^∗(𝑥)=𝑗(𝑥,𝜂^∗)−𝑠₀.

Proof. From the system of (22), we have 𝑢^∗=0, 𝑧^∗=0, and 𝑞^∗=0. In order to show existence of 𝜂^∗, we define Γ𝜂,𝜅∶=𝐶𝛾𝜂;𝑎𝛼𝜂−𝑠₀+𝜅𝜒^′𝛾𝜂⋅𝛾𝜂. (25) Then 𝜕Γ=𝐶𝜕𝜂𝜂,𝜅^′𝛾𝜂;𝑎𝛼𝜂−𝑠₀+𝜅𝜒^′′𝛾𝜂𝛾𝜂⋅𝛾^′𝜂+𝜅𝜒^′𝛾𝜂𝛾^′𝜂+𝐶^′𝛾𝜂;𝑎𝛼𝜂−𝑠₀⋅𝑎^′𝛼𝜂−𝑠₀⋅𝛼^′𝜂. (26) Since 𝛾^′(𝜂)<0 and 𝛼^′(𝜂)<0 for all 𝜂>0, Γ(𝜂,𝜅)=0 is solvable with 𝜂^∗ if Γ(0,𝜅)>0, Γ(∞,𝜅)<0, and (𝜕/𝜕𝜂)Γ(𝜂,𝜅)<0, which means that 𝐶(𝛾(0); 𝑎(𝛼(0)−𝑠₀))>0, 𝐶(𝛾(∞);𝑎(𝛼(∞)−𝑠₀)) + 𝜅𝜒^′(𝛾(∞))𝛾(∞)<0, and 𝐶^′(𝛾(𝜂);𝑎(𝛼(𝜂)−𝑠₀))+𝜅𝜒^′′(𝛾(𝜂))𝛾(𝜂)−𝜅1+𝜇𝜒^′(𝛾(𝜂))>0.
Let𝜅_𝑐be a solution of 𝐶𝛾(∞);𝑎𝛼(∞)−𝑠₀+𝜅_𝑐𝜒^′𝛾(∞)𝛾^′(∞)+𝜇𝐺(∞,∞)=0. (27) Then Γ(∞,𝜅)<Γ(∞,𝜅_𝑐)<Γ(0,𝜅_𝑐) with Γ(∞,𝜅_𝑐)=0. Hence, 𝜂^∗ exists for 𝜅<𝜅_𝑐.
The formula for 𝐷𝑓(0,0,0,𝜂^∗) follows from the relation 𝐶^′(1/2)=4, and the corresponding steady state (𝑣^∗,𝑝^∗,𝑤^∗,𝜂^∗) for (10) is obtained by using Theorem  2.1 in [19].

4. A Hopf Bifurcation

In this section, we show that there exists a Hopf bifurcation from the curve 𝜎↦(0,0,0,𝜂^∗) of the equilibrium solution. First, let us introduce the following relevant definition.

Definition 3. Under the assumptions of Theorem 2, define (for 1≥𝜃>3/4) the linear operator 𝐵 from 𝑋^𝜃 to 𝑋 by 𝐵∶=𝐷𝑓0,0,0,𝜂^∗. (28) We then define (0,0,0,𝜂^∗) to be a Hopf point for (18) if and only if there exist an 𝜖₀>0 and a 𝐶¹-curve −𝜖₀+𝜏^∗,𝜏^∗+𝜖₀↦𝜆(𝜏),𝜙(𝜏)∈ℂ×𝑋_𝐂 (29) (𝑌_𝐂 denotes the complexification of the real space 𝑌) of eigendata for −𝐴+𝜏𝐵 with (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=1/𝜎.

Next, we check (18) for the Hopf points. For this, we solve the eigenvalue problem: −𝐴𝑢,𝑧,𝑞,𝜂+𝜏𝐵𝑢,𝑧,𝑞,𝜂=𝜆𝐼₄𝑢,𝑧,𝑞,𝜂, (30) where 𝐼₄ is a 4×4identity matrix. This is equivalent to (𝐴+𝜆)𝑢=𝜏𝜇𝐺⋅,𝜂^∗×𝑑₂𝑢𝜂^∗+𝛾^′𝜂^∗𝜂+𝜅𝑑₁𝑧𝜂^∗+𝛾^′𝜂^∗𝜂+𝑎₁𝑞𝜂^∗+𝛼^′𝜂^∗,(𝐴+𝜆)𝑧=−𝜏𝜇𝜂^∗𝐺⋅,𝜂^∗×𝑑₂𝑢𝜂^∗+𝛾^′𝜂^∗𝜂+𝜅𝑑₁𝑧𝜂^∗+𝛾^′𝜂^∗𝜂+𝑎₁𝑞𝜂^∗+𝛼^′𝜂^∗,𝐴₀+𝜆𝑞=𝜏𝐽⋅,𝜂^∗×𝑑₂𝑢𝜂^∗+𝛾^′𝜂^∗𝜂+𝜅𝑑₁𝑧𝜂^∗+𝛾^′𝜂^∗𝜂+𝑎₁𝑞𝜂^∗+𝛼^′𝜂^∗,𝜆𝜂=𝜏𝑑₂𝑢𝜂^∗+𝛾^′𝜂^∗𝜂+𝜅𝑑₁𝑧𝜂^∗+𝛾^′𝜂^∗𝜂+𝑎₁𝑞𝜂^∗+𝛼^′𝜂^∗, (31) where 𝑑₁=𝜒^′(𝛾(𝜂^∗)), 𝑑₂=4+𝜅𝜒^′′(𝛾(𝜂^∗))𝛾(𝜂^∗), and 𝑎₁=4𝑎^′(𝛼(𝜂^∗)−𝑠₀).

In the following theorem, we show that an equilibrium solution is a Hopf point.

Theorem 4. Suppose that (1/2)−𝑎(1−𝑠₀)<(𝜇/(1+𝜇))and 𝐶^′(𝛾(𝜂);𝑎(𝛼(𝜂)−𝑠₀)) + 𝜅𝜒^′′(𝛾(𝜂))𝛾(𝜂)>𝜅1+𝜇𝜒^′(𝛾(𝜂)) for all 𝜂>0. Assume that 𝐶^′(𝛾(𝜂^∗);𝑎(𝛼(𝜂^∗)−𝑠₀))+𝜅𝜒^′′(𝛾(𝜂^∗))𝛾(𝜂^∗)>(𝜅/𝜂^∗)𝜒^′(𝛾(𝜂^∗)). Additionally, suppose that the operator −𝐴+𝜏^∗𝐵 has a unique pair {±𝑖𝛽}, 𝛽>0 of purely imaginary eigenvalues for some 𝜏^∗>0. Then, (0,0,0,𝜂^∗,𝜏^∗) is a Hopf point for (18).

Proof. 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,𝑧₀,𝑞₀,𝜂₀)∈𝐷(𝐴)×𝐷(𝐴)×𝐷(𝐴₀)×ℝ. First, we note that 𝜂₀≠0. Otherwise, by (31), (𝐴+𝑖𝛽)𝜓₀=𝜇𝑖𝛽𝜂₀𝐺(⋅,𝜂^∗)=0 and (𝐴+𝑖𝛽)𝑧₀=−(𝜇/𝜂^∗)𝑖𝛽𝜂₀𝐺(⋅,𝜂^∗)=0, which is not possible because 𝐴 is symmetric. So, without loss of generality, let 𝜂₀=1. Then 𝐸(𝜓₀,𝑧₀,𝑞₀,𝑖𝛽,𝜏^∗)=0 by (31), where𝐸∶𝐷(𝐴)_𝐂×𝐷(𝐴)_𝐂×𝐷𝐴_0𝐂×ℂ×ℝ⟶𝑋_𝐂×𝑋_𝐂×𝑋_𝐂𝐸×ℂ,𝑢,𝑧,𝑞,𝜆,𝜏∶=^{()𝐴+𝜆𝑢−𝜏𝜇𝐺⋅,𝜂∗𝑑2𝑢𝜂∗+𝛾′𝜂∗+𝜅𝑑1𝑧𝜂∗+𝛾′𝜂∗+𝑎1⋅𝑞𝜂∗𝜂+𝛼∗()𝜇𝐴+𝜆𝑧+𝜏𝜂∗𝐺⋅,𝜂∗𝑑2𝑢𝜂∗+𝛾′𝜂∗+𝜅𝑑1𝑧𝜂∗+𝛾′𝜂∗+𝑎1⋅𝑞𝜂∗𝜂+𝛼∗}_{𝐴0+𝜆𝑞−𝜏𝐽⋅,𝜂∗𝑑2𝑢𝜂∗+𝛾′𝜂∗+𝜅𝑑1𝑧𝜂∗+𝛾′𝜂∗+𝑎1⋅𝑞𝜂∗𝜂+𝛼∗𝑑𝜆−𝜏2𝑢𝜂∗+𝛾′𝜂∗+𝜅𝑑1𝑧𝜂∗+𝛾′𝜂∗+𝑎1⋅𝑞𝜂∗𝜂+𝛼∗}. (32)
The equation 𝐸(𝑢,𝑧,𝑞,𝜆,𝜏)=0 is equivalent to 𝜆 being an eigenvalue of −𝐴+𝜏𝐵 with eigenfunction (𝑢,𝑧,𝑞,1). We will apply the implicit function theorem to 𝐸. For this, we check that 𝐸 is of 𝐶¹-class and that 𝐷_{(𝑢,𝑧,𝑞,𝜆)}𝐸𝜓₀,𝑧₀,𝑞₀,𝑖𝛽,𝜏^∗∈𝐿𝐷(𝐴)_𝐂×𝐷(𝐴)_𝐂×𝐷𝐴_0𝐂𝑋×ℂ×ℝ,_𝐂×𝑋_𝐂×𝑋_𝐂×ℂ (33) is an isomorphism. In addition, the mapping𝐷_{(𝑢,𝑧,𝑞,𝜆)}𝐸𝜓₀,𝑧₀,𝑞₀,𝑖𝛽,𝜏^∗(^𝑢,_𝑧,_𝑞,_𝜆)=^{𝐴+𝑖𝛽𝑢+𝜆𝜓0−𝜏∗𝜇𝐺⋅,𝜂∗𝑑2𝑢𝜂∗+𝜅𝑑1𝑧𝜂∗+𝑎1𝑞𝜂∗𝐴+𝑖𝛽𝑧+𝜆𝑧0+𝜏∗𝜇𝜂∗𝐺⋅,𝜂∗𝑑2𝑢𝜂∗+𝜅𝑑1𝑧𝜂∗+𝑎1𝑞𝜂∗}_{𝐴0+𝑖𝛽𝑞+𝜆𝑞0−𝜏∗𝐽⋅,𝜂∗𝑑2𝑢𝜂∗+𝜅𝑑1𝑧𝜂∗+𝑎1𝑞𝜂∗𝜆−𝜏∗𝑑2𝑢𝜂∗+𝜅𝑑1𝑧𝜂∗+𝑎1𝑞𝜂∗} (34)is a compact perturbation of the mapping 𝑢,𝑧,𝑞,𝜆↦𝐴+𝑖𝛽𝑢,𝐴+𝑖𝛽𝑧,𝐴₀+𝑖𝛽𝑞,𝜆 (35) which is invertible. Thus, 𝐷_{(𝑢,𝑧,𝑞,𝜆)}𝐸(𝜓₀,𝑧₀,𝑞₀,𝑖𝛽,𝜏^∗) is a Fredholm operator of index0. Therefore, in order to verify (33), it suffices to show that the system of equations 𝐷_{(𝑢,𝑧,𝑞,𝜆)}𝐸𝜓₀,𝑧₀,𝑞₀,𝑖𝛽,𝜏^∗𝑢,𝑧,𝑞,𝜆=0 (36)
which is equivalent to 𝐴+𝑖𝛽𝑢+𝜆𝜓₀=𝜏^∗𝜇𝐺⋅,𝜂^∗×𝑑₂𝑢𝜂^∗+𝜅𝑑₁𝑧𝜂^∗+𝑎₁𝑞𝜂^∗,𝐴+𝑖𝛽𝑧+𝜆𝜉₀=−𝜏^∗𝜇𝜂^∗𝐺⋅,𝜂^∗×𝑑₂𝑢𝜂^∗+𝜅𝑑₁𝑧𝜂^∗+𝑎₁𝑞𝜂^∗,𝐴₀+𝑖𝛽𝑞+𝜆𝑞₀=𝜏^∗𝐽⋅,𝜂^∗×𝑑₂𝑢𝜂^∗+𝜅𝑑₁𝑧𝜂^∗+𝑎₁𝑞𝜂^∗,𝜆=𝜏^∗𝑑₂𝑢𝜂^∗+𝜅𝑑₁𝑧𝜂^∗+𝑎₁𝑞𝜂^∗ (37) necessarily implies that 𝑢=0, 𝑧=0,   𝑞=0, and 𝜆=0. If we define 𝜙∶=𝜓₀−𝜇𝐺(⋅,𝜂^∗), 𝜉∶=𝑧₀+(𝜇/𝜂^∗)𝐺(⋅,𝜂^∗), and 𝜌=𝑞₀−𝐽(⋅,𝜂^∗),  then (37) becomes 𝐴𝐴+𝑖𝛽𝑢+𝜆𝜙=0,𝐴+𝑖𝛽𝑧+𝜆𝜉=0,₀𝜆+𝑖𝛽𝑞+𝜆𝜌=0,𝜏^∗=𝑑₂𝑢𝜂^∗+𝜅𝑑₁𝑧𝜂^∗+𝑎₁𝑞𝜂^∗. (38)(39)(40)(41) On the other hand, since 𝐸(𝜓₀,𝑧₀,𝑞₀,𝑖𝛽,𝜏^∗)=0, 𝜙,  𝜉 and 𝜌 are solutions to the equations, we have 𝐴+𝑖𝛽𝜙=−𝜇𝛿_𝜂∗,𝜇𝐴+𝑖𝛽𝜉=𝜂^∗𝛿_𝜂∗,𝐴₀+𝑖𝛽𝜌=−𝛿_𝜂∗,𝑖𝛽𝜏^∗=𝑑₂𝜙𝜂^∗+𝜇𝐺𝜂^∗,𝜂^∗+𝛾^′𝜂^∗+𝜅𝑑₁𝜉𝜂^∗−𝜇𝜂^∗𝐺𝜂^∗,𝜂^∗+𝛾^′𝜂^∗+𝑎₁𝜌𝜂^∗+𝐽𝜂^∗,𝜂^∗+𝛼^′𝜂^∗. (42)(43)(44)(45)
Multiplying (39) and (43) by 𝜙 and (38) and (42) by 𝜉 and subtracting one from the other, we obtain 𝑢𝜂^∗=−𝜂^∗𝑧𝜂^∗,𝜙𝜂^∗=−𝜂^∗𝜉𝜂^∗. (46) Multiplying (38) by 𝑑₂𝜙, (39) by −𝜂^∗𝜅𝑑₁𝜌, and (40) by 𝑎₁𝜌 and adding the resultants to each, we have −𝑑₂𝜇𝑢𝜂^∗−𝜅𝑑₁𝜇𝑧𝜂^∗−𝑎₁𝑞𝜂^∗+𝜆𝑑₂𝜙²−𝜂^∗𝜅𝑑₁𝜉²+𝑎₁𝜌²+2𝑖𝛽𝑑₂𝑢𝜙−𝜂^∗𝜅𝑑₁𝑧𝜉+𝑎₁𝑞𝜌=0. (47) Multiplying (42) by 𝑑₂𝜙, (43) by −𝜂^∗𝜅𝑑₁𝜌, and (44) by 𝑎₁𝜌 and adding the resultants to each, we obtain 𝑑₂𝐴^1/2𝜙²−𝜂^∗𝜅𝑑₁𝐴^1/2𝜉²+𝑎₁𝐴₀^1/2𝜌²+𝑖𝛽𝑑₂𝜙²−𝜂^∗𝜅𝑑₁𝜉²+𝑎₁𝜌²=−𝑑₂𝜇𝜙𝜂^∗−𝜇𝜅𝑑₁𝜉𝜂^∗−𝑎₁𝜌𝜂^∗. (48) From (45), we get 𝜇𝜏^∗=𝑑₂𝜙²−𝜂^∗𝜅𝑑₁𝜉²+𝑎₁𝜌², (49) and thus (47) implies that 𝑑₂𝑢𝜙−𝜂^∗𝜅𝑑₁𝑧𝜉+𝑎₁𝑞𝜌=0. (50)
Now, multiplying (38) by 𝑑₂𝑢, (42) by −𝜂^∗𝜅𝑑₁𝑧, and (40) by 𝑎₁𝑞 and adding the resultants to each, we have 𝑑₂𝐴^1/2𝑢²−𝜂^∗𝜅𝑑₁𝐴^1/2𝑧²+𝑎₁𝐴₀^1/2𝑞²+𝑖𝛽𝑑₂‖𝑢‖²−𝜂^∗𝜅𝑑₁‖𝑧‖²+𝑎₁𝑞²+𝜆𝑑₂𝜙𝑢−𝜂^∗𝜅𝑑₁𝜉𝑧+𝑎₁𝜌𝑞=0. (51) From (50), we get 𝑑₂𝐴^1/2𝑢²−𝜂^∗𝜅𝑑₁𝐴^1/2𝑧²+𝑎₁𝐴₀^1/2𝑞²𝑑=0₂‖𝑢‖²−𝜂^∗𝜅𝑑₁‖𝑧‖²+𝑎₁𝑞²=0. (52) Multiplying (42) by 𝜙 and (43) by 𝜉, we get 𝐴^1/2𝜙²+𝑖𝛽𝜙²=−𝜇𝜙𝜂^∗,𝐴^1/2𝜉²+𝑖𝛽𝜉²=𝜇𝜂^∗𝜉𝜂^∗, (53) and applying (46) to the above equation, we have 𝐴^1/2𝜙²=𝜂^∗2𝐴^1/2𝜉²,𝜙²=𝜂^∗2𝜉². (54) Now, multiplying (38) by 2𝑖𝛽𝑢 and (42) by 𝜆𝑢 and subtracting the resultants to each other, we obtain 2𝑖𝛽𝐴^1/2𝑢²−𝜂^∗2𝐴^1/2𝑧²−2𝛽²‖𝑢‖²−𝜂^∗2‖𝑧‖²+𝜆𝜙²−𝜂^∗2𝜉². (55) Applying (54) to the above equation, we have 𝐴^1/2𝑢²−𝜂^∗2𝐴^1/2𝑧²=0,‖𝑢‖²−𝜂^∗2‖𝑧‖²=0, (56) and thus (52) implies that 𝑑₂−𝜅𝑑₁𝜂^∗‖𝑢‖²+𝑎₁𝑞²=0. (57) Since 𝑑₂−(𝜅𝑑₁/𝜂^∗)>0 and 𝑎₁>0, we have 𝑢=0 and 𝑞=0, and so, 𝑧=0 and 𝜆=0.

Theorem 5. Under the same condition as in Theorem 4, (0,0,0,𝜂^∗,𝜏^∗) satisfies the transversality condition. Hence, this is a Hopf point for (18).

Proof. By the implicit differentiation of 𝐸(𝜓₀(𝜏),𝑧₀(𝜏),𝑞₀(𝜏),𝜆(𝜏),𝜏)=0, we find𝐷_{(𝑢,𝑧,𝑞,𝜆)}𝐸𝜓₀,𝑧₀,𝑞₀,𝑖𝛽,𝜏^∗𝜓^′₀𝜏^∗,𝑧^′₀𝜏^∗,𝑞^′₀𝜏^∗,𝜆^′𝜏^∗=𝜇𝐺⋅,𝜂^∗𝑑₂𝜓₀𝜂^∗+𝛾^′𝜂^∗+𝜅𝑑₁𝑧₀𝜂^∗+𝛾^′𝜂^∗+𝑎₁𝑞₀𝜂^∗+𝛼𝜂^∗−𝜇𝜂^∗𝐺⋅,𝜂^∗𝑑₂𝜓₀𝜂^∗+𝛾^′𝜂^∗+𝜅𝑑₁𝑧₀𝜂^∗+𝛾^′𝜂^∗+𝑎₁𝑞₀𝜂^∗+𝛼𝜂^∗𝐽⋅,𝜂^∗𝑑₂𝜓₀𝜂^∗+𝛾^′𝜂^∗+𝜅𝑑₁𝑧₀𝜂^∗+𝛾^′𝜂^∗+𝑎₁𝑞₀𝜂^∗+𝛼𝜂^∗𝑑₂𝜓₀𝜂^∗+𝛾^′𝜂^∗+𝜅𝑑₁𝑧₀𝜂^∗+𝛾^′𝜂^∗+𝑎₁𝑞₀𝜂^∗+𝛼𝜂^∗. (58)
This means that the functions 𝑢∶=𝜓^′₀(𝜏^∗),  𝑧∶=𝑧^′₀(𝜏^∗),   𝑞∶=𝑞^′₀(𝜏^∗), and 𝜆∶=𝜆^′(𝜏^∗) satisfy the equations 𝐴+𝑖𝛽𝑢+𝜆𝜓₀−𝜏^∗𝜇𝐺⋅,𝜂^∗×𝑑₂𝑢𝜂^∗+𝜅𝑑₁𝑧𝜂^∗+𝑎₁𝑞𝜂^∗=𝜇𝐺⋅,𝜂^∗𝑑₂𝜓₀𝜂^∗+𝛾^′𝜂^∗+𝜅𝑑₁𝑧₀𝜂^∗+𝛾^′𝜂^∗+𝑎₁𝑞₀𝜂^∗+𝛼𝜂^∗,𝐴+𝑖𝛽𝑧+𝜆𝜉₀+𝜏^∗𝜇𝜂^∗𝐺⋅,𝜂^∗×𝑑₂𝑢𝜂^∗+𝜅𝑑₁𝑧𝜂^∗+𝑎₁𝑞𝜂^∗𝜇=−𝜂^∗𝐺⋅,𝜂^∗𝑑₂𝜓₀𝜂^∗+𝛾^′𝜂^∗+𝜅𝑑₁𝑧₀𝜂^∗+𝛾^′𝜂^∗+𝑎₁𝑞₀𝜂^∗+𝛼𝜂^∗,𝐴₀+𝑖𝛽𝑞+𝜆𝜌₀−𝜏^∗𝐽⋅,𝜂^∗×𝑑₂𝑢𝜂^∗+𝜅𝑑₁𝑧𝜂^∗+𝑎₁𝑞𝜂^∗=𝐽⋅,𝜂^∗𝑑₂𝜓₀𝜂^∗+𝛾^′𝜂^∗+𝜅𝑑₁𝑧₀𝜂^∗+𝛾^′𝜂^∗+𝑎₁𝑞₀𝜂^∗+𝛼𝜂^∗,𝜆−𝜏^∗𝑑₂𝑢𝜂^∗+𝜅𝑑₁𝑧𝜂^∗+𝑎₁𝑞𝜂^∗=𝑑₂𝜓₀𝜂^∗+𝛾^′𝜂^∗+𝜅𝑑₁𝑧₀𝜂^∗+𝛾^′𝜂^∗+𝑎₁𝑞₀𝜂^∗+𝛼𝜂^∗. (59) By letting 𝜙∶=𝜓₀−𝜇𝐺(⋅,𝜂^∗),   𝜉=𝑧₀+(𝜇/𝜂^∗)𝐺(⋅,𝜂^∗), and 𝜌=𝑞₀−𝐽(⋅,𝜂^∗) as before, we obtain 𝐴𝐴+𝑖𝛽𝑢+𝜆𝜙=0,𝐴+𝑖𝛽𝑧+𝜆𝜉=0,₀+𝑖𝛽𝑞+𝜆𝜌=0,𝜆−𝜏^∗𝑑₂𝑢𝜂^∗+𝜅𝑑₁𝑧𝜂^∗+𝑎₁𝑞𝜂^∗=𝑖𝛽𝜏^∗. (60)(61)(62)(63) Multiplying (60) by 𝑑₂𝜙, (61) by−𝜂^∗𝜅𝑑₁𝜉, and (62) by 𝑎₁𝜌 and adding the resultants to each, we obtain −𝑑₂𝜇𝑢𝜂^∗−𝜅𝑑₁𝜇𝑧𝜂^∗−𝑎₁𝑞𝜂^∗+𝜆𝑑₂𝜙²−𝜂^∗𝜅𝑑₁𝜉²+𝑎₁𝜌²+2𝑖𝛽𝑑₂𝑢𝜙−𝜂^∗𝜅𝑑₁𝑧𝜉+𝑎₁𝑞𝜌=0. (64) From (49) and (63), the above equation implies that 𝜇𝑖𝛽(𝜏^∗)²+2𝑖𝛽𝑑₂𝑢𝜙−𝜂^∗𝜅𝑑₁𝑧𝜉+𝑎₁𝑞𝜌=0. (65)
Multiplying (60) by𝑑₂𝑢, (61) by −𝜂^∗𝜅𝑑₁𝑧 and (62) by 𝑎₁𝑞 and adding the resultants to each, we have 𝑑₂𝐴^1/2𝑢²−𝜂^∗𝜅𝑑₁𝐴^1/2𝑧²+𝑎₁𝐴₀^1/2𝑞²+𝑖𝛽𝑑₂‖𝑢‖²−𝜂^∗𝜅𝑑₁‖𝑧‖²+𝑎₁𝑞²+𝜆𝑑₂𝑢𝜙−𝜂^∗𝜅𝑑₁𝑧𝜉+𝑎₁𝑞𝜌=0. (66) From (65), we have 𝑑₂𝐴^1/2𝑢²−𝜂^∗𝜅𝑑₁𝐴^1/2𝑧²+𝑎₁𝐴₀^1/2𝑞²+𝑖𝛽𝑑₂‖𝑢‖²−𝜂^∗𝜅𝑑₁‖𝑧‖²+𝑎₁𝑞²=𝜆𝜇2(𝜏^∗)², (67) and the real part is 𝑑₂𝐴^1/2𝑢²−𝜂^∗𝜅𝑑₁𝐴^1/2𝑧²+𝑎₁𝐴₀^1/2𝑞²=𝜇2(𝜏^∗)²Re𝜆. (68) Now, multiplying (60) by 2𝑖𝛽𝑢 and (61) by 𝜆𝑢 and applying (54) to resultants, we obtain 𝐴^1/2𝑢²−𝜂^∗2𝐴^1/2𝑧²=0,‖𝑢‖²−𝜂^∗2‖𝑧‖²=0, (69) and thus (68) implies that 𝜇2(𝜏^∗)²Re𝜆=𝑑₂−𝜅𝑑₁𝜂^∗𝐴^1/2𝑢²+𝑎₁𝐴₀^1/2𝑞² (70) which is positive since 𝑑₂−(𝜅𝑑₁/𝜂^∗)>0 and 𝑎₁>0. We have Re𝜆^′(𝜏^∗)>0 for 𝛽>0, and thus, by the Hopf-bifurcation theorem in [19], there exists a family of periodic solutions which bifurcates from the stationary solution as 𝜏 passes 𝜏^∗.

Now, we show that there exists a unique 𝜏^∗>0 such that (0,0,𝜂^∗,𝜏^∗) is a Hopf point; thus 𝜏^∗ is the origin of a branch of nontrivial periodic orbits.

Lemma 6. Suppose that 𝑑₂−(𝜅𝑑₁/𝜂^∗)>0. Let 𝐺_𝛽, 𝐺_𝛽, and 𝐽_𝛽 be Green functions of the differential operators 𝐴+𝑖𝛽, 𝐴+𝑖/𝛽 and 𝐴₀+𝑖𝛽 satisfying (42), (43), and (44), respectively. Then, 𝑑₂Re(𝐺_𝛽(𝜂^∗,𝜂^∗))−(𝜅𝑑₁/𝜂^∗)Re(𝐺_𝛽(𝜂^∗,𝜂^∗)) and Re(𝐽_𝛽(𝜂^∗,𝜂^∗)) are strictly decreasing in 𝛽∈ℝ⁺ with Re𝐺₀𝜂^∗,𝜂^∗=𝐺𝜂^∗,𝜂^∗,lim_𝛽→∞Re𝐺_𝛽𝜂^∗,𝜂^∗=0. (71) Moreover, 𝑑₂Im(𝐺_𝛽(𝜂^∗,𝜂^∗))−(𝜅𝑑₁/𝜂^∗)Im(𝐺_𝛽(𝜂^∗,𝜂^∗))>0 and Im(𝐽_𝛽(𝜂^∗,𝜂^∗))<0 for 𝛽>0.

Proof. First, we have (𝐴+𝑖𝛽)⁻¹=(𝐴−𝑖𝛽)(𝐴²+𝛽²)⁻¹. So, if 𝐿(𝛽)∶=Re(𝐴+𝑖𝛽)⁻¹, then 𝐿(𝛽)=𝐴(𝐴²+𝛽²)⁻¹. Moreover, 𝐿(𝛽)→𝐴⁻¹ as 𝛽→0 and 𝐿(𝛽)→0 as 𝛽→∞, which results in the corresponding limiting behavior for Re(𝐺_𝛽(𝜂^∗,𝜂^∗)).
To show that 𝛽↦(𝑑₂Re(𝐺_𝛽(𝜂^∗,𝜂^∗))−(𝜅𝑑₁/𝜂^∗)Re(𝐺_𝛽(𝜂^∗,𝜂^∗))) is strictly increasing, we define ℎ(𝛽)(𝑥)∶=𝑑₂𝐺_𝛽(𝑥,𝜂^∗)−(𝜅𝑑₁/𝜂^∗)𝐺_𝛽(𝑥,𝜂^∗)−𝑑₂𝐺(𝑥,𝜂^∗)+(𝜅𝑑₁/𝜂^∗)𝐺(𝑥,𝜂^∗). Then (in the weak sense initially) 𝐴+𝑖𝛽ℎ𝛽=−𝑖𝛽𝑑₂𝐺⋅,𝜂^∗−𝜅𝑑₁𝜂^∗𝐺⋅,𝜂^∗. (72) As a result, ℎ(𝛽)∈𝐷(𝐴)_𝐂 and ℎ∶ℝ⁺→𝐷(𝐴)_𝐂 is differentiable with 𝑖ℎ(𝛽)+(𝐴+𝑖𝛽)ℎ^′(𝛽) = −𝑖(𝑑₂𝐺(⋅,𝜂^∗)−(𝜅𝑑₁/𝜂^∗)𝐺(⋅,𝜂^∗)), and therefore 𝐴+𝑖𝛽ℎ^′𝛽=−𝑖𝑑₂𝐺_𝛽⋅,𝜂^∗−𝜅𝑑₁𝜂^∗𝐺_𝛽⋅,𝜂^∗. (73) Thus, we get −𝑖𝑑₂−𝜅𝑑₁𝜂^∗ℎ^′𝛽𝜂^∗=𝐴+𝑖𝛽²ℎ^′𝛽ℎ^′𝛽(𝑥)=ℎ𝑑𝑥𝐴+𝑖𝛽^′𝛽⋅𝐴+𝑖𝛽ℎ^′𝛽=𝑑𝑥𝐴ℎ^′𝛽²−𝛽²ℎ^′𝛽²𝑑𝑥+2𝑖𝛽𝐴^1/2ℎ^′𝛽². (74) It follows that −𝑑₂−𝜅𝑑₁𝜂^∗Reℎ^′𝛽𝜂^∗=2𝛽𝐴^1/2ℎ^′𝛽²>0. (75) Since (𝑑₂−(𝜅𝑑₁/𝜂^∗))>0, we have (𝜕/𝜕𝛽)(𝑑₂Re(𝐺_𝛽(𝜂^∗,𝜂^∗))−(𝜅𝑑₁/𝜂^∗)Re(𝐺_𝛽(𝜂^∗,𝜂^∗)))<0 for 𝛽>0.
In order to show (𝑑₂Im(𝐺_𝛽(𝜂^∗,𝜂^∗))−(𝜅𝑑₁/𝜂^∗)Im(𝐺_𝛽(𝜂^∗,𝜂^∗)))>0, from (72), we have −𝑖𝛽𝑑₂−𝜅𝑑₁𝜂^∗ℎ𝛽𝜂^∗=𝐴𝐴+𝑖𝛽ℎ𝛽(𝑥)ℎ𝛽(𝑥)=𝛽𝑑𝑥𝐴ℎ²+𝑖𝛽𝐴^1/2ℎ𝛽² (76) which implies that −𝛽(𝑑₂−(𝜅𝑑₁/𝜂^∗))Imℎ(𝛽)(𝜂^∗)=‖𝐴ℎ(𝛽)‖²>0. Since (𝑑₂−(𝜅𝑑₁/𝜂^∗))>0, we have Imℎ(𝛽)(𝜂^∗)<0 for𝛽>0.
Let 𝑘(𝛽)(𝑥)∶=𝐽_𝛽(𝑥,𝜂^∗)−𝐽(𝑥,𝜂^∗). Then we have (𝜕/𝜕𝛽)(Re(𝐽_𝛽(𝜂^∗,𝜂^∗)))<0 and Im𝐽_𝛽(𝜂^∗,𝜂^∗)<0 for 𝛽>0.

Theorem 7. Under the same condition as in Theorem 4, for a unique critical point 𝜏^∗>0, there exists a unique, purely imaginary eigenvalue 𝜆=𝑖𝛽 of (31) with 𝛽>0.

Proof. We only need to show that the function (𝑢,𝑧,𝑞,𝛽,𝜏)↦𝐸(𝑢,𝑧,𝑞,𝑖𝛽,𝜏) has a unique zero with 𝛽>0 and 𝜏>0. This means solving the system of (31) with 𝜆=𝑖𝛽,  𝑢=𝑣−𝜇𝐺(⋅,𝜂^∗),  𝑧=𝑝+(𝜇/𝜂^∗)𝐺(⋅,𝜂^∗), and 𝑞=𝑤−𝐽(⋅,𝜂^∗), 𝐴+𝑖𝛽𝑣=−𝜇𝛿_𝜂∗,𝜇𝐴+𝑖𝛽𝑧=𝜂^∗𝛿_𝜂∗,𝐴₀+𝑖𝛽𝑞=−𝛿_𝜂∗,𝑖𝛽𝜏^∗=𝑑₂𝑣𝜂^∗+𝜇𝐺𝜂^∗,𝜂^∗+𝛾^′𝜂^∗+𝜅𝑑₁𝑧𝜂^∗−𝜇𝜂^∗𝐺𝜂^∗,𝜂^∗+𝛾^′𝜂^∗+𝑎₁𝑞𝜂^∗+𝐽𝜂^∗,𝜂^∗+𝛼^′𝜂^∗. (77) The real and imaginary parts of the above equation are given by 𝛽𝜏^∗=𝑑₂Im−𝜇𝐺_𝛽𝜂^∗,𝜂^∗+𝜅𝑑₁𝜇Im𝜂^∗𝐺_𝛽𝜂^∗,𝜂^∗−𝑎₁Im𝐽_𝛽𝜂^∗,𝜂^∗,0=𝑑₂Re−𝜇𝐺_𝛽𝜂^∗,𝜂^∗+𝜇𝐺𝜂^∗,𝜂^∗+𝛾^′𝜂^∗+𝜅𝑑₁𝜇Re𝜂^∗𝐺_𝛽𝜂^∗,𝜂^∗−𝜇𝜂^∗𝐺𝜂^∗,𝜂^∗+𝛼^′𝜂^∗+𝑎₁Re−𝐽_𝛽𝜂^∗,𝜂^∗+𝐽𝜂^∗,𝜂^∗+𝛼^′𝜂^∗. (78) Since 𝑑₂Im(−𝜇𝐺_𝛽(𝜂^∗,𝜂^∗))+𝜅𝑑₁Im((𝜇/𝜂^∗)𝐺_𝛽(𝜂^∗,𝜂^∗))−𝑎₁Im(𝐽_𝛽(𝜂^∗,𝜂^∗))>0 by Lemma 6, there is a critical point 𝜏^∗, provided the existence of 𝛽. We now define 𝑇𝛽=𝑑₂Re−𝜇𝐺_𝛽𝜂^∗,𝜂^∗+𝜇𝐺𝜂^∗,𝜂^∗+𝛾^′𝜂^∗+𝜅𝑑₁𝜇Re𝜂^∗𝐺_𝛽𝜂^∗,𝜂^∗−𝜇𝜂^∗𝐺𝜂^∗,𝜂^∗+𝛾^′𝜂^∗+𝑎₁Re−𝐽_𝛽𝜂^∗,𝜂^∗+𝐽𝜂^∗,𝜂^∗+𝛼^′𝜂^∗. (79) Using Lemma 6, we have 𝑇^′(𝛽)>0 for 𝛽>0 and 𝑇(0)=𝑑₂𝛾^′(𝜂^∗)+𝜅𝑑₁𝛾^′(𝜂^∗) + 𝑎₁𝛼^′(𝜂^∗)=(𝑑₂−𝜅𝑑₁1+𝜇)𝛾^′(𝜂^∗)+𝑎₁𝛼^′(𝜂^∗)<0 if 𝑑₂>𝜅𝑑₁1+𝜇. Moreover, lim_𝛽→∞𝑇𝛽=𝑑₂𝜇𝐺𝜂^∗,𝜂^∗+𝛾^′𝜂^∗+𝜅𝑑₁−𝜇𝜂^∗𝐺𝜂^∗,𝜂^∗+𝛾^′𝜂^∗+𝑎₁𝐽𝜂^∗,𝜂^∗+𝛼^′𝜂^∗=𝑑₂−𝜅𝑑₁𝜂^∗𝜇𝐺𝜂^∗,𝜂^∗+𝛾^′𝜂^∗+𝜅𝑑₁𝛾^′𝜂^∗+𝑎₁𝐽𝜂^∗,𝜂^∗+𝛼^′𝜂^∗>0 (80) for 𝑑₂>(𝜅𝑑₁/𝜂^∗) and 𝑎₁>0. Hence, there exists a unique 𝛽>0.

The following theorem summarizes the results above.

Theorem 8. Suppose that (1/2)−𝑎(1−𝑠₀)<(𝜇/(1+𝜇)) and 𝐶^′(𝛾(𝜂);𝑎(𝛼(𝜂)−𝑠₀)) + 𝜅𝜒^′′(𝛾(𝜂))𝛾(𝜂)>𝜅1+𝜇𝜒^′(𝛾(𝜂)) for all 𝜂>0. Then (18) and (10) have at least one stationary solution (𝑢^∗,𝑧^∗,𝑞^∗,𝜂^∗), where 𝑢^∗=𝑧^∗=𝑞^∗=0, and (𝑣^∗,𝑝^∗,𝑤^∗,𝜂^∗) where 𝑣^∗(𝑥)=𝑔(𝑥,𝜂^∗), 𝑝^∗(𝑥)=𝑔(𝑥,𝜂^∗) and 𝑤^∗(𝑥)=𝑗(𝑥,𝜂^∗)−𝑠₀, for all 𝜏 and for 𝜅<𝜅_𝑐, respectively, where 𝜅_𝑐 is a solution of 𝐶𝛾(∞);𝑎𝛼(∞)−𝑠₀+𝜅_𝑐𝜒^′𝛾(∞)𝛾^′(∞)+𝜇𝐺(∞,∞)=0. (81) Assume that 𝐶^′(𝛾(𝜂^∗);𝑎(𝛼(𝜂^∗)−𝑠₀))+𝜅𝜒^′′(𝛾(𝜂^∗))𝛾(𝜂^∗)>(𝜅/𝜂^∗)𝜒^′(𝛾(𝜂^∗)). Then there exists a unique 𝜏^∗ such that the linearization −𝐴+𝜏^∗𝐵 has a purely imaginary pair of eigenvalues. The point (0,0,0,𝜂^∗,𝜏^∗) is then a Hopf point for (18), and there exists a 𝐶⁰-curve of nontrivial periodic orbits for (18) and (10), bifurcating from (0,0,0,𝜂^∗,𝜏^∗) and (𝑣^∗,𝑧^∗,𝑤^∗,𝜂^∗,𝜏^∗), respectively.

Acknowledgment

This paper was supported by 63 Research Fund, Sungkyunkwan University, 2012.