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

Authors: YoonMee Ham; Sang-Gu Lee; Quoc Phong Vu
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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:
(1)ut=∇·(∇u-χu∇v),vt=D∇2v+au-bv,t>0,x∈ℝn,
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]:
(2)ɛσut=ɛ2∇2u-ɛκ∇(u·∇χ(v))-u+H(u-a(w))-v,vt=∇2v+μu-v,t>0,x∈ℝn,wt=∇2w+u+v-w-s0,t>0,x∈ℝn,
where ɛ, 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 smallɛ. 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 whenɛ=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 curvex=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:
(3)ξ=x-x0ε,ρ=t-t0ε.
Then, the system (2) at (x0,t0) becomes
(4)σuρ=uξξ-κχ′(v0)vxuξ+F(u,v0,w0),F(u,v,w)=-u+H(u-a(w))-v
and the boundary conditions are
(5)u(±∞)=h±(v0)
when ɛ 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:
(6)Uzz+(σθ-κχ′(v0)vx)Uz+F(U,v0,w0)=0,U(±∞)=h±(v0).The existence of a solutionU(z) is given in [12, 14] and θ satisfies σθ=C(v0)+κχ′(v0)vx(x0,t0). Hence, the velocity of the one-dimensional interface η(t) is given by
(7)dη(t)dt=1σ(C(vi)+κχ′(vi)vx(η(t),t)),x∈η(t),
where vi 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]
(8)C(v(η);a(w(η)))=-1-2a(w(η))-2v(η)(v(η)+a(w))(1-a(w(η))-v(η)).Hence, a free boundary problem of (2) when ɛ is equal to zero is given by
(9)vt=∇2v-(μ+1)v+μ,t>0,x∈Ω1,vt=∇2v-(μ+1)v,t>0,x∈Ω0,v(η(t)-0,t)=v(η(t)+0,t),vx(η(t)-0,t)=vx(η(t)+0,t),limx→∞v(x,t)=0,wt=∇2w-w+1-s0,t>0,x∈Ω1,wt=∇2w-w-s0,t>0,x∈Ω0,w(η(t)-0,t)=w(η(t)+0,t),wx(η(t)-0,t)=wx(η(t)+0,t),limx→∞w(x,t)=-s0.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 Section2, 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):
(10)vt=∂2v∂x2-(μ+1)v+μH(x-η(t)),wt=∂2w∂x2-w+H(x-η(t))-s0,vx(0,t)=0,limx→∞vx(x,t)=0,t>0,wx(0,t)=0,limx→∞wx(x,t)=-s0,t>0,ση′(t)=C(v(η);a(w(η)))+κχ′(v(η(t),t))vx(η(t),t),(v(η(t),t)f)t>0;η(0)=η0.Letw^(x,t)=w(x,t)+s0. Let A be an operator defined by A:=-(d2/dx2)+μ+1 with domain D(A)={v∈H2,2(ℝ):vx(0,t)=0,limx→∞vx(x,t)=0}. Let A0:=-(∂2/∂x2)+1 with domain D(A0)={w∈H2,2((0,∞)):w^x(0,t)=0,limx→∞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 decomposev 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
(11)g(x,η)∶=A-1(μH(·-η)(x))=μ∫0∞G(x,y)H(y-η)dy,‍
where G:[0,∞)×[0,∞)→ℝ is a Green’s function of A satisfying the Neumann boundary conditions, and γ:[0,∞)→ℝ is given by
(12)γ(η)∶=g(η,η).
If we take a transformation u(t)(x)=v(x,t)-g(x,η(t)), we have (ux)(t)(x)=vx(x,t)-gx(x,η(t)). Since Gx(x,η) is discontinuous at x=η, we cannot obtain one step more regular than that of (10).To overcome this difficulty, letp(x,t)=vx(x,t). Then p satisfies pt+Ap=μδ(x-η), where A=-(d2/dx2)+μ+1with domain D(A)={p∈H1,2(ℝ):p(0,t)=0,limx→∞p(x,t)=0}. Define g^:[0,∞)×[0,∞)→ℝ(13)g^(x,η)∶=A-1(μδ(·-η)(x))=μ∫0∞‍G^(x,y)δ(y-η)dy,
where G^:[0,∞)×[0,∞)→ℝ is a Green’s function of A satisfying the Dirichlet boundary conditions, and γ^:[0,∞)→ℝ is given by
(14)γ^(η)∶=g^(η,η).
We define j:[0,∞)×[0,∞)→ℝ,
(15)j(r,η)∶=A0-1(H(·-η)(x))=∫0∞‍J(x,y)H(y-η)dy
and α:[0,∞)×ℂ→ℂ(16)α(η)∶=j(η,η),
where J:[0,∞)2→ℝ is a Green’s function of A0 satisfying the boundary conditions.Applying the transformationsu(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
(17)ut+Au=μσG(x,η)×(κχ′(u(η)+γ(η))C(u(η)+γ(η);a(q(η)+α(η)-s0))+κχ′(u(η)+γ(η))(z(η)+γ^(η))),zt+Az=-1σμηG^(x,η)×(κχ′(u(η)+γ(η))C(u(η)+γ(η);a(q(η)+α(η)-s0))+κχ′(u(η)+γ(η))(z(η)+γ^(η))),qt+A0q=1σJ(x,η)×(κχ′(u(η)+γ(η))C(u(η)+γ(η);a(q(η)+α(η)-s0))+κχ′(u(η)+γ(η))(z(η)+γ^(η))),η′(t)=1σ(κχ′(u(η)+γ(η))C(u(η)+γ(η);a(q(η)+α(η)-s0))+κχ′(u(η)+γ(η))·(z(η)+γ^(η))),hhhhhhhhhhhhhhhhhhhhhhhhhhhihht>0.
Thus, we obtain an abstract evolution equation equivalent to (10):
(18)ddt(u,z,q,η)+A~(u,z,q,η)=1σf(u,z,q,η),(u,z,q,η)(0)=(u0(x),z0(x),q0(x),η0),
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
(19)f(u,z,q,η)=(f1(η)·(f21(u,z,q,η)+f22(u,z,q,η))f2(η)·(f21(u,z,q,η)+f22(u,z,q,η))f3(η)·(f21(u,z,q,η)+f22(u,z,q,η))f21(u,z,q,η)+f22(u,z,q,η)),
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, XAθ:=D(Aθ), X0θ:=D(A0θ), and X~θ:=D(A~θ)=XAθ×XAθ×X0θ×ℝ.The velocity ofηis denoted by
(20)C(u(η)+γ(η);a(q(η)+α(η)-s0))=C(S(u,q,η))=-1-2S(u,q,η)S(u,q,η)(1-S(u,q,η)),
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 functionsG(·,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
(21)Df21(u,z,q,η)(u~,z~,q~,η~)=CS(S(u,q,η))·(u′(η)η~+u~(η)+γ′(η)η~)+a′(q(η)+α(η)-s0)·(q′(η)η^+q~(η)+α′(η)η~),Df22(u,z,q,η)(u^,z^,q^,η^)=κχ′(u(η)+γ(η))(z′(η)η^+z^(η)+γ^′(η)η^),+κχ′′(u(η)+γ(η))·(u^(η)+u′(η)η^+γ(η)η^))(z(η)+γ^(η)),Df(u,z,q,η)(u^,z^,q^,η^)=(f21(u,z,q,η)+f22(u,z,q,η))·(f1′(η),f2′(η),f3′(η),0)η^+(Df21(u,z,q,η)+Df22(u,z,q,η))(u^,z^,q^,η^)·(f1(η),f2(η),f3(η),1).


## 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:
(22)Au=1σμG(x,η)×((z(η)+γ^(η))C(u(η)+γ(η);a(q(η)+α(η)-s0))+κχ′(u(η)+γ(η))·(z(η)+γ^(η))),Az=-1σμηG^(x,η)×(κχ′(u(η)+γ(η))C(z(η)+γ^(η))(u(η)+γ(η);a(q(η)+α(η)-s0))+κχ′(u(η)+γ(η))·(z(η)+γ^(η))),A0q=1σJ(·,η*)(κχ′(u(η)+γ(η))C(u(η)+γ(η);a(q(η)+α(η)-s0))hhhhhhhh+κχ′(u(η)+γ(η))·(z(η)+γ^(η))),0=C(u(η)+γ(η);a(q(η)+α(η)-s0))+κχ′(u(η)+γ(η))·(z(η)+γ^(η)),u′(0)=0=u′(∞),z(0)=0=z(∞),q′(0)=0=q′(∞).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<κc, where κc is a solution of
(23)C(γ(∞);a(α(∞)-s0))+κcχ′(γ(∞))(γ′(∞)+μG(∞,∞))=0.
The linearization off at the stationary solution (0,0,0,η*) is
(24)Df(0,0,0,η*)(u^,z^,q^,η^)=(μG(·,η*)Q(u^,z^,q^,η^)-μη*G^(·,η*)Q(u^,z^,q^,η^)j(·,η*)Q(u^,z^,q^,η^)Q(u^,z^,q^,η^)),
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
(25)Γ(η,κ)∶=C(γ(η);a(α(η)-s0))+κχ′(γ(η))·γ^(η).
Then
(26)∂∂ηΓ(η,κ)=(C′(γ(η);a(α(η)-s0))+κχ′′(γ(η))γ^(η))·γ′(η)+κχ′(γ(η))γ^′(η)+C′(γ(η);a(α(η)-s0))·a′(α(η)-s0)·α′(η).
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.
Letκcbe a solution of
(27)C(γ(∞);a(α(∞)-s0))+κcχ′(γ(∞))(γ′(∞)+μG(∞,∞))=0.
Then Γ(∞,κ)<Γ(∞,κc)<Γ(0,κc) with Γ(∞,κc)=0. Hence, η* exists for κ<κc.
The formula forDf(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 curveσ↦(0,0,0,η*) of the equilibrium solution. First, let us introduce the following relevant definition.Definition 3.
Under the assumptions of Theorem2, define (for 1≥θ>3/4) the linear operator B from X~θ to X~ by
(28)B∶=Df(0,0,0,η*).
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
(29)(-ϵ0+τ*,τ*+ϵ0)↦(λ(τ),ϕ(τ))∈ℂ×X~C
(YC 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:
(30)-A~(u,z,q,η)+τB(u,z,q,η)=λI4(u,z,q,η),
where I4 is a 4×4identity matrix. This is equivalent to
(31)(A+λ)u=τμG(·,η*)×(d2(u(η*)+γ′(η*)η)+κd1(z(η*)+γ^′(η*)η)+a1(q(η*)+α′(η*))),(A+λ)z=-τμη*G^(·,η*)×((q(η*)+α′(η*))d2(u(η*)+γ′(η*)η)+κd1(z(η*)+γ^′(η*)η)+a1(q(η*)+α′(η*))),(A0+λ)q=τJ(·,η*)×((q(η*)+α′(η*))d2(u(η*)+γ′(η*)η)+κd1(z(η*)+γ^′(η*)η)+a1(q(η*)+α′(η*))),λη=τ((q(η*)+α′(η*))d2(u(η*)+γ′(η*)η)+κd1(z(η*)+γ^′(η*)η)+a1(q(η*)+α′(η*))),
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, thatβ>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, letΦ*=(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), where(32)E:D(A)C×D(A)C×D(A0)C×ℂ×ℝ⟶XC×XC×XC×ℂ,E(u,z,q,λ,τ):=((A+λ)u-τμG(·,η*)(d2(u(η*)+γ′(η*))+κd1(z(η*)+γ^′(η*)+a1·(q(η*)+α(η*)))(A+λ)z+τμη*G^(·,η*)(d2(u(η*)+γ′(η*))+κd1(z(η*)+γ^′(η*)+a1·(q(η*)+α(η*)))(A0+λ)q-τJ(·,η*)(d2(u(η*)+γ′(η*))+κd1(z(η*)+γ^′(η*)+a1·(q(η*)+α(η*)))λ-τ(d2(u(η*)+γ′(η*))+κd1(z(η*)+γ^′(η*)+a1·(q(η*)+α(η*)))).
The equationE(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
(33)D(u,z,q,λ)E(ψ0,z0,q0,iβ,τ*)∈L(D(A)C×D(A)C×D(A0)C×ℂ×ℝ,XC×XC×XC×ℂ)
is an isomorphism. In addition, the mapping(34)D(u,z,q,λ)E(ψ0,z0,q0,iβ,τ*)(u^,z^,q^,λ^)=((A+iβ)u^+λ^ψ0-τ*μG(·,η*)(d2u^(η*)+κd1z^(η*)+a1q^(η*))(A+iβ)z^+λ^z0+τ*μη*G^(·,η*)(d2u^(η*)+κd1z^(η*)+a1q^(η*))(A0+iβ)q^+λ^q0-τ*J(·,η*)(d2u^(η*)+κd1z^(η*)+a1q^(η*))λ^-τ*(d2u^(η*)+κd1z^(η*)+a1q^(η*)))is a compact perturbation of the mapping
(35)(u^,z^,q^,λ^)↦((A+iβ)u^,(A+iβ)z^,(A0+iβ)q^,λ^)
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
(36)D(u,z,q,λ)E(ψ0,z0,q0,iβ,τ*)(u^,z^,q^,λ^)=0
which is equivalent to(37)(A+iβ)u^+λ^ψ0=τ*μG(·,η*)×(d2u^(η*)+κd1z^(η*)+a1q^(η*)),(A+iβ)z^+λ^ξ0=-τ*μη*G^(·,η*)×(d2u^(η*)+κd1z^(η*)+a1q^(η*)),(A0+iβ)q^+λ^q0=τ*J(·,η*)×(d2u^(η*)+κd1z^(η*)+a1q^(η*)),λ^=τ*(d2u^(η*)+κd1z^(η*)+a1q^(η*))
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
(38)(A+iβ)u^+λ^ϕ=0,(39)(A+iβ)z^+λ^ξ=0,(40)(A0+iβ)q^+λ^ρ=0,(41)λ^τ*=d2u^(η*)+κd1z^(η*)+a1q^(η*).
On the other hand, since E(ψ0,z0,q0,iβ,τ*)=0, ϕ,  ξ and ρ are solutions to the equations, we have
(42)(A+iβ)ϕ=-μδη*,(43)(A+iβ)ξ=μη*δη*,(44)(A0+iβ)ρ=-δη*,(45)iβτ*=d2(ϕ(η*)+μG(η*,η*)+γ′(η*))+κd1(ξ(η*)-μη*G^(η*,η*)+γ^′(η*))+a1(ρ(η*)+J(η*,η*)+α′(η*)).
Multiplying (39) and (43) by ϕ and (38) and (42) by ξ and subtracting one from the other, we obtain
(46)u^(η*)=-η*z^(η*),ϕ(η*)=-η*ξ(η*).
Multiplying (38) by d2ϕ¯, (39) by -η*κd1ρ¯, and (40) by a1ρ¯ and adding the resultants to each, we have
(47)-d2μu^(η*)-κd1μz^(η*)-a1q^(η*)+λ^(d2∥ϕ∥2-η*κd1∥ξ∥2+a1∥ρ∥2)+2iβ∫‍(d2u^ϕ¯-η*κd1z^ξ¯+a1q^ρ¯)=0.
Multiplying (42) by d2ϕ¯, (43) by -η*κd1ρ¯, and (44) by a1ρ¯ and adding the resultants to each, we obtain
(48)d2∥A1/2ϕ∥2-η*κd1∥A1/2ξ∥2+a1∥A01/2ρ∥2+iβ(d2∥ϕ∥2-η*κd1∥ξ∥2+a1∥ρ∥2)=-d2μϕ(η*)¯-μκd1ξ(η*)¯-a1ρ(η*)¯.
From (45), we get
(49)μτ*=d2∥ϕ∥2-η*κd1∥ξ∥2+a1∥ρ∥2,
and thus (47) implies that
(50)∫‍(d2u^ϕ¯-η*κd1z^ξ¯+a1q^ρ¯)=0.
Now, multiplying (38) by d2u^¯, (42) by -η*κd1z^¯, and (40) by a1q^¯ and adding the resultants to each, we have
(51)(d2∥A1/2u^∥2-η*κd1∥A1/2z^∥2+a1∥A01/2q^∥2)+iβ(d2∥u^∥2-η*κd1∥z^∥2+a1∥q^∥2)+λ^∫‍(d2ϕu^¯-η*κd1ξz^¯+a1ρq^¯)=0.
From (50), we get
(52)d2∥A1/2u^∥2-η*κd1∥A1/2z^∥2+a1∥A01/2q^∥2=0d2∥u^∥2-η*κd1∥z^∥2+a1∥q^∥2=0.
Multiplying (42) by ϕ¯ and (43) by ξ¯, we get
(53)∥A1/2ϕ∥2+iβ∥ϕ∥2=-μϕ¯(η*),∥A1/2ξ∥2+iβ∥ξ∥2=μη*ξ¯(η*),
and applying (46) to the above equation, we have
(54)∥A1/2ϕ∥2=(η*)2∥A1/2ξ∥2,∥ϕ∥2=(η*)2∥ξ∥2.
Now, multiplying (38) by 2iβu^¯ and (42) by λ^u^¯ and subtracting the resultants to each other, we obtain
(55)2iβ(∥A1/2u^∥2-(η*)2∥A1/2z^∥2)-2β2(∥u^∥2-(η*)2∥z^∥2)+λ^(∥ϕ∥2-(η*)2∥ξ∥2).
Applying (54) to the above equation, we have
(56)∥A1/2u^∥2-(η*)2∥A1/2z^∥2=0,∥u^∥2-(η*)2∥z^∥2=0,
and thus (52) implies that
(57)(d2-κd1η*)∥u^∥2+a1∥q^∥2=0.
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 Theorem4, (0,0,0,η*,τ*) satisfies the transversality condition. Hence, this is a Hopf point for (18).Proof.
By the implicit differentiation ofE(ψ0(τ),z0(τ),q0(τ),λ(τ),τ)=0, we find(58)D(u,z,q,λ)E(ψ0,z0,q0,iβ,τ*)(ψ0′(τ*),z0′(τ*),q0′(τ*),λ′(τ*))=(μG(·,η*)(d2(ψ0(η*)+γ′(η*))+κd1(z0(η*)+γ^′(η*))+a1(q0(η*)+α(η*))-μη*G^(·,η*)(d2(ψ0(η*)+γ′(η*))+κd1(z0(η*)+γ^′(η*))+a1(q0(η*)+α(η*))J(·,η*)(d2(ψ0(η*)+γ′(η*))+κd1(z0(η*)+γ^′(η*))+a1(q0(η*)+α(η*))d2(ψ0(η*)+γ′(η*))+κd1(z0(η*)+γ^′(η*))+a1(q0(η*)+α(η*))).
This means that the functionsu~:=ψ0′(τ*),  z~:=z0′(τ*),   q~:=q0′(τ*), and λ~:=λ′(τ*) satisfy the equations
(59)(A+iβ)u~+λ~ψ0-τ*μG(·,η*)×(d2u~(η*)+κd1z~(η*)+a1q~(η*))=μG(·,η*)(d2(ψ0(η*)+γ′(η*))+κd1(z0(η*)+γ^′(η*))+a1(q0(η*)+α(η*)),(A+iβ)z~+λ~ξ0+τ*μη*G^(·,η*)×(d2u~(η*)+κd1z~(η*)+a1q~(η*))=-μη*G^(·,η*)(d2(ψ0(η*)+γ′(η*))+κd1(z0(η*)+γ^′(η*))+a1(q0(η*)+α(η*)),(A0+iβ)q~+λ~ρ0-τ*J(·,η*)×(d2u~(η*)+κd1z~(η*)+a1q~(η*))=J(·,η*)(d2(ψ0(η*)+γ′(η*))+κd1(z0(η*)+γ^′(η*))+a1(q0(η*)+α(η*)),λ~-τ*(d2u~(η*)+κd1z(η*)+a1q~(η*))=d2(ψ0(η*)+γ′(η*))+κd1(z0(η*)+γ^′(η*))+a1(q0(η*)+α(η*)).
By letting ϕ:=ψ0-μG(·,η*),   ξ=z0+(μ/η*)G^(·,η*), and ρ=q0-J(·,η*) as before, we obtain
(60)(A+iβ)u~+λ~ϕ=0,(61)(A+iβ)z~+λ~ξ=0,(62)(A0+iβ)q~+λ~ρ=0,(63)λ~-τ*(d2u~(η*)+κd1z~(η*)+a1q~(η*))=iβτ*.
Multiplying (60) by d2ϕ¯, (61) by-η*κd1ξ¯, and (62) by a1ρ¯ and adding the resultants to each, we obtain
(64)-d2μu~(η*)-κd1μz~(η*)-a1q~(η*)+λ~(d2∥ϕ∥2-η*κd1∥ξ∥2+a1∥ρ∥2)+2iβ∫‍(d2u~ϕ¯-η*κd1z~ξ¯+a1q~ρ¯)=0.
From (49) and (63), the above equation implies that
(65)iβμ(τ*)2+2iβ∫‍(d2u~ϕ¯-η*κd1z~ξ¯+a1q~ρ¯)=0.
Multiplying (60) byd2u~¯, (61) by -η*κd1z~¯ and (62) by a1q~¯ and adding the resultants to each, we have
(66)d2∥A1/2u~∥2-η*κd1∥A1/2z~∥2+a1∥A01/2q~∥2+iβ(d2∥u~∥2-η*κd1∥z~∥2+a1∥q~∥2)+λ~∫‍(d2u~ϕ¯-η*κd1z~ξ¯+a1q~ρ¯)=0.
From (65), we have
(67)d2∥A1/2u~∥2-η*κd1∥A1/2z~∥2+a1∥A01/2q~∥2+iβ(d2∥u~∥2-η*κd1∥z~∥2+a1∥q~∥2)=λ~μ2(τ*)2,
and the real part is
(68)d2∥A1/2u~∥2-η*κd1∥A1/2z~∥2+a1∥A01/2q~∥2=μ2(τ*)2Reλ~.
Now, multiplying (60) by 2iβu~¯ and (61) by λ~u~¯ and applying (54) to resultants, we obtain
(69)∥A1/2u~∥2-(η*)2∥A1/2z~∥2=0,∥u~∥2-(η*)2∥z~∥2=0,
and thus (68) implies that
(70)μ2(τ*)2Reλ~=(d2-κd1η*)∥A1/2u~∥2+a1∥A01/2q~∥2
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 uniqueτ*>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 thatd2-(κd1/η*)>0. Let Gβ, G^β, and Jβ be Green functions of the differential operators A+iβ, A+i/β and A0+iβ satisfying (42), (43), and (44), respectively. Then, d2Re(Gβ(η*,η*))-(κd1/η*)Re(G^β(η*,η*)) and Re(Jβ(η*,η*)) are strictly decreasing in β∈ℝ+ with
(71)ReG0(η*,η*)=G(η*,η*),limβ→∞ReGβ(η*,η*)=0.
Moreover, d2Im(Gβ(η*,η*))-(κd1/η*)Im(G^β(η*,η*))>0 and Im(Jβ(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(Gβ(η*,η*)).
To show thatβ↦(d2Re(Gβ(η*,η*))-(κd1/η*)Re(G^β(η*,η*))) is strictly increasing, we define h(β)(x):=d2Gβ(x,η*)-(κd1/η*)G^β(x,η*)-d2G(x,η*)+(κd1/η*)G^(x,η*). Then (in the weak sense initially)
(72)(A+iβ)h(β)=-iβ(d2G(·,η*)-κd1η*G^(·,η*)).
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
(73)(A+iβ)h′(β)=-i(d2Gβ(·,η*)-κd1η*G^β(·,η*)).
Thus, we get
(74)-i(d2-κd1η*)h′(β)(η*)¯=∫‍(A+iβ)2h′(β)h′(β)(x)¯dx=∫‍(A+iβ)h′(β)·(A+iβ)h′(β)¯dx=∥Ah′(β)∥2-β2∥h′(β)∥2dx+2iβ∥A1/2h′(β)∥2.
It follows that
(75)-(d2-κd1η*)Re(h′(β)(η*))=2β∥A1/2h′(β)∥2>0.
Since (d2-(κd1/η*))>0, we have (∂/∂β)(d2Re(Gβ(η*,η*))-(κd1/η*)Re(G^β(η*,η*)))<0 for β>0.
In order to show(d2Im(Gβ(η*,η*))-(κd1/η*)Im(G^β(η*,η*)))>0, from (72), we have
(76)-iβ(d2-κd1η*)h(β)(η*)¯=∫‍A(A+iβ)h(β)(x)h(β)(x)¯dx=∥Ah(β)∥2+iβ∥A1/2h(β)∥2
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.
Letk(β)(x):=Jβ(x,η*)-J(x,η*). Then we have (∂/∂β)(Re(Jβ(η*,η*)))<0 and ImJβ(η*,η*)<0 for β>0.Theorem 7.
Under the same condition as in Theorem4, 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*),
(77)(A+iβ)v=-μδη*,(A+iβ)z=μη*δη*,(A0+iβ)q=-δη*,iβτ*=d2(v(η*)+μG(η*,η*)+γ′(η*))+κd1(z(η*)-μη*G^(η*,η*)+γ^′(η*))+a1(q(η*)+J(η*,η*)+α′(η*)).
The real and imaginary parts of the above equation are given by
(78)βτ*=d2Im(-μGβ(η*,η*))+κd1Im(μη*G^β(η*,η*))-a1Im(Jβ(η*,η*)),0=d2(Re(-μGβ(η*,η*))+μG(η*,η*)+γ′(η*))+κd1(Re(μη*G^β(η*,η*))-μη*G^(η*,η*)hhhhhhh+α′(η*)(μη*G^β(η*,η*)))+a1(Re(-Jβ(η*,η*))+J(η*,η*)+α′(η*)).
Since d2Im(-μGβ(η*,η*))+κd1Im((μ/η*)G^β(η*,η*))-a1Im(Jβ(η*,η*))>0 by Lemma 6, there is a critical point τ*, provided the existence of β. We now define
(79)T(β)=d2(Re(-μGβ(η*,η*))+μG(η*,η*)+γ′(η*))+κd1(Re(μη*G^β(η*,η*))hhhhhhhh-μη*G^(η*,η*)+γ^′(η*)(μη*G^β(η*,η*)))+a1(Re(-Jβ(η*,η*))+J(η*,η*)+α′(η*)).
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,
(80)limβ→∞T(β)=d2(μG(η*,η*)+γ′(η*))+κd1(-μη*G^(η*,η*)+γ^′(η*))+a1(J(η*,η*)+α′(η*))=(d2-κd1η*)(μG(η*,η*)+γ′(η*))+κd1γ^′(η*)+a1(J(η*,η*)+α′(η*))>0
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<κc, respectively, where κc is a solution of
(81)C(γ(∞);a(α(∞)-s0))+κcχ′(γ(∞))(γ′(∞)+μG(∞,∞))=0.
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.



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*Source: 830386-2013-12-15.xml*