Hopf bifurcation in a reaction-diffusive-advection two-species competition model with one delay

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Hopf bifurcation in a reaction-diffusive-advection two-species competition model with one delay

Qiong Meng

^B¹

, Guirong Liu

¹

and Zhen Jin

²

1School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, China

2Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, China

Received 26 October 2020, appeared 19 September 2021 Communicated by Péter L. Simon

Abstract. In this paper, we investigate a reaction-diffusive-advection two-species competition model with one delay and Dirichlet boundary conditions. The existence and multiplicity of spatially non-homogeneous steady-state solutions are obtained. The stability of spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcation with the changes of the time delay are obtained by analyzing the distribu- tion of eigenvalues of the infinitesimal generator associated with the linearized system.

By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic orbits are derived. Finally, numerical simulations are given to illustrate the theoretical results.

Keywords: reaction-diffusive, advection, delay, Hopf bifurcation, spatial heterogeneity.

2020 Mathematics Subject Classification: 34K18, 35K57, 92D25.

1 Introduction

In this paper, we consider a two-species competition model in a reaction-diffusive-advection with one delay

(_∂u₍_x,t₎

∂t =∇ ·[d₁∇u(x,t)−a₁u(x,t)∇m] +u(x,t)[m(x)−b₁u(x,t−r)−c₁v(x,t−r)],

∂v(x,t)

∂t =∇ ·[d₂∇v(x,t)−a₂v(x,t)∇m] +v(x,t)[m(x)−b₂u(x,t−r)−c₂v(x,t−r)], (1.1) where u(x,t),v(x,t) represents the population density at location x ∈ _Ω and time t, time delayr >0 represents the maturation time, and Ωis a bounded domain inR^k (1≤ k≤3)in (1.1) with a smooth boundary ∂Ω. a_i,b_i,c_i,d_i >0(i=1, 2).

In (1.1), we assume that both species have the same per-capita growth rates at placex∈ _Ω, denoted by m(x). This scenario can occur if the two species are competing for the same resources. To reflect the heterogeneity of environment, we assume thatm(x)is a nonconstant function. In some sense,m(x)can reflect the quality and quantity of resources available at the locationx, where the favorable region{x ∈_Ω:m(x)>0}acts as a source and the unfavorable part{x ∈_Ω: m(x)<0}is a sink region, see [26]. Whenm(x)≡1, see [15,18].

BCorresponding author. Email: mengqiong@qq.com

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Under our assumptions in (1.1), the dispersal of the two competitors can be described in terms of their fluxes

Ju= −d₁∇u+a₁u∇m, Jv =−d2∇v+a2v∇m,

respectively, where d1∇u and d2∇v account for random diffusion, and a1u∇m and a2v∇m represent movement upward along the environmental gradient. The two non-negative con- stantsa₁ anda₂ measure the tendency of the two populations to move up along the gradient ofm(x), and d₁ andd2 represent the random diffusion rates of two species, respectively. See [1,2,4–8,10,11,13,17,20,22–29].

Whenb₁ =b₂= c₁ = c₂ =1, r =0 in (1.1), Chen, Hambrock and Lou [6] investigated the following model

(_∂u₍_x,t₎

∂t =∇ ·[d₁∇u(x,t)−a₁u(x,t)∇m] +u(x,t)[m(x)−u(x,t)−v(x,t)],

∂v(x,t)

∂t =∇ ·[d₂∇v(x,t)−a₂v(x,t)∇m] +v(x,t)[m(x)−u(x,t)−v(x,t)]. (1.2) They showed that at least two scenarios can occur: if only one species has a strong tendency to move upward the environmental gradients, the two species can coexist since one species mainly pursues resources at places of locally most favorable environments while the other relies on resources from other parts of the habitat; if both species have such strong biased movements, it can lead to overcrowding of the whole population at places of locally most favorable environments, which causes the extinction of the species with stronger biased movement. These results provided a new mechanism for the coexistence of competing species, and they also implied that selection is against excessive advection along environmental gradients, and an intermediate biased movement rate may evolve.

Whenv=0 in (1.1), Chen, Lou and Wei [8] investigated the following model, (_∂u₍_x,t₎

∂t =∇ ·[d∇u−a₁u∇m] +u(x,t)[m(x)−u(x,t−r)],

u(x,t) =0. (1.3)

They investigated a reaction-diffusion-advection model with time delay effect. The stability and instability of the spatially nonhomogeneous positive steady state were investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator.

Their results implied that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction-diffusion-advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values was also considered, and their results suggested that Hopf bifurcation is more likely to occur when the advection rate increases. See [3,9,12,14–16,18,19,21,30–34].

When d₁ = d₂ = d, a₂ = a₁ in (1.1), we study the following model with homogeneous Dirichlet boundary and initial value conditions











∂u(x,t)

∂t = ∇ ·[d∇u−a₁u∇m] +u(x,t)[m(x)−b₁u(x,t−r)−c₁v(x,t−r)],

∂v(x,t)

∂t =∇ ·[d∇v−a₁v∇m] +v(x,t)[m(x)−b₂u(x,t−r)−c₂v(x,t−r)], x ∈_Ω, t >0,

u(x,t) =v(x,t) =0, x∈_∂Ω, t≥0,

u(x,t) = ϕ₁(x,t)≥0, v(x,t) =ϕ₂(x,t)≥0, (x,t)∈ _Ω×[−r, 0],

(1.4)

with the initial value functions

ϕ_i(x,·)∈ C([−r, 0]_,_R⁺₀) (x∈_Ω)_, ϕ_i(·_,t)∈ H₀¹(_Ω) (t∈[−r, 0])_, i=_{1, 2.}

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In this paper, we mainly investigate whether time delayr can induce Hopf bifurcation for reaction-diffusion-advection model (1.4).

As in [2,8], Let ue = e⁽⁻^a¹^/d⁾^m⁽^x⁾u,ve= e⁽⁻^a¹^/d⁾^m⁽^x⁾v,et = td, dropping the tilde sign, and denotingλ=1/d,a=a₁/d,τ=dr, system (1.4) can be transformed as follows:











∂u

∂t =e⁻âm⁽^x⁾∇ ·[eâm⁽^x⁾∇u] +λu[m(x)−b₁eâm⁽^x⁾u(x,t−τ)−c₁eâm⁽^x⁾v(x,t−τ)]_,

∂v

∂t =e⁻âm⁽^x⁾∇ ·[eâm⁽^x⁾∇v] +λv[m(x)−b₂eâm⁽^x⁾u(x,t−τ)−c₂eâm⁽^x⁾v(x,t−τ)], x∈ _Ω, t>0,

u(x,t) =v(x,t) =0, x ∈∂Ω, t≥0,

u(x,t) = ϕ₁(x,t)≥0, v(x,t) = ϕ₂(x,t)≥0, (x,t)∈_Ω×[−τ, 0].

(1.5)

Throughout the paper, unless otherwise specified,m(x)satisfies the following assumption (H) m∈C²(_Ω), and max_x_∈_Ωm(x)>0.

The following eigenvalue problem

(−e⁻^am⁽^x⁾∇ ·[e^am⁽^x⁾∇u] =−_∆u−a∇m· ∇u =λm(x)u, x ∈_Ω,

u(x) =_0, x∈∂Ω, (1.6)

is crucial to derive our main results. It follows from [2,8,26] that, under assumption (H), (1.6) has a unique positive principal eigenvalue λ∗ admitting a strictly positive eigenfunction ϕ∈C¹₀⁺^δ(_Ω)for someδ ∈(0, 1)andR

Ωϕ²dx =1.

The rest of the paper is organized as follows. In Section 2, we study the existence of positive steady state solutions of (1.5). In Section 3, we focus on the eigenvalue problem of the linearized system of the steady-state solution of (1.5). In Section 4, we study the stability and Hopf bifurcation of the spatially nonhomogeneous positive steady state of (1.5). In Section 5, we derive an explicit formula, which can be used to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic orbits. In Section 6, we give some numerical simulations are illustrated to support our analytical results.

Throughout the paper, we also denote the spaces X = H²(_Ω)∩ H₀¹(_Ω), Y = L²(_Ω). Moreover, we denote the complexification of a linear space Z to be Z_C = Z⊕iZ = {x₁+ ix2 | x₁,x2 ∈ Z}, the domain of a linear operator L by D(L), the kernel of L by N(L), and the range of L by R(L). For Hilbert space Y_C, we use the standard inner product hu,vi = R

Ωu(x)^Tv(x)dx,u,v ∈Y_C².

2 Existence of positive steady state solutions

Denote

L:=∇ ·[_e^am⁽^x⁾∇] +λ∗e^am⁽^x⁾m(_x)_,

whereλ∗is a unique positive principal eigenvalue of problem (1.6) admitting a strictly positive eigenfunction ϕ∈ C₀¹⁺^δ(_Ω)for someδ∈ (0, 1)andR

Ωϕ²dx=1.

Now, we have the following decompositions:

X=N(L)⊕X₁, Y=N(L)⊕Y₁, N(L) =span{ϕ}, X₁ =

y∈X: Z

Ωϕ(x)y(x)dx =0

, Y₁= R(L) =

y∈Y: Z

Ωϕ(x)y(x)dx=0

.

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Clearly, the operator L:X→Y is Fredholm with index zero. L|_X₁ :X₁ →Y₁ is invertible and has a bounded inverse.

In this section, we consider the existence of positive spatially nonhomogeneous steady states solutions of system (1.5), which satisfy

(∇ ·[eâm⁽^x⁾∇u] +λeâm⁽^x⁾u[m(x)−b₁eâm⁽^x⁾u(x,t)−c₁eâm⁽^x⁾v(x,t)] =0,

∇ ·[eâm⁽^x⁾∇v] +λeâm⁽^x⁾v[m(x)−b₂eâm⁽^x⁾u(x,t)−c₂eâm⁽^x⁾v(x,t)] =0, (2.1) Suppose that the solution of (2.1) has the following expressions:

(u_λ =α(λ−λ∗)[ϕ+ (λ−λ∗)ξ(x)],

v_λ = β(λ−λ∗)[ϕ+ (λ−λ∗)η(x)], (2.2) whereα,β∈_R,ξ,η∈X₁. Substitute (2.2) into (2.1) we have











Lξ+m(x)e^am⁽^x⁾[ϕ+ (λ−λ∗)ξ]−λαb₁e^2am⁽^x⁾[ϕ+ (λ−λ∗)ξ]²

−λβc₁e^2am⁽^x⁾[ϕ+ (λ−λ∗)ξ][ϕ+ (λ−λ∗)η] =0, Lη+m(x)e^am⁽^x⁾[ϕ+ (λ−λ∗)η]−λαc2e^2am⁽^x⁾[ϕ+ (λ−λ∗)η]²

−λβb2e^2am⁽^x⁾[ϕ+ (λ−λ∗)ξ][ϕ+ (λ−λ∗)η] =0.

(2.3)

Whenλ=λ∗, (2.3) becomes following equations

(Lξ+m(x)e^am⁽^x⁾ϕ−λ∗αb₁e^2am⁽^x⁾ϕ²−λ∗βc₁e^2am⁽^x⁾ϕ² =0,

Lη+m(x)e^am⁽^x⁾ϕ−λ∗βb₂e^2am⁽^x⁾ϕ²−λ∗αc₂e^2am⁽^x⁾ϕ² =0. (2.4) Multiplying both sides of each equation in (2.4) by ϕand integrating onΩ, we have

α_λ_∗ = ^c²−c₁

b₁c₂−b₂c₁d₁, β_λ_∗ = ^b¹−b₂ b₁c₂−b₂c₁d₁, where d₁ =

R

Ωm(x)e^am⁽^x⁾ϕ²dx λ∗R

Ωe^2am⁽^x⁾ϕ³dx > 0, see [8]. And ξ_λ_∗,η_λ_∗ ∈ X₁ is the unique solution of the following equations

(Lξ+m(x)e^am⁽^x⁾ϕ−λ∗α_λ_∗b₁e^2am⁽^x⁾ϕ²−λ∗β_λ_∗c₁e^2am⁽^x⁾ϕ² =0,

Lη+m(x)e^am⁽^x⁾ϕ−λ∗β_λ_∗b₂e^2am⁽^x⁾ϕ²−λ∗α_λ_∗c₂e^2am⁽^x⁾ϕ² =0. (2.5) To guarantee positive steady states solutions of system (2.1), we need following conditions:

(H1) (λ−λ∗)_b^c²⁻^c¹

1c2−b2c1 >0, (λ−λ∗)_b^b¹⁻^b²

1c2−b2c1 >0.

Theorem 2.1. Assume that (H1) holds. Then there exist a constant δ >0 and a continuously differentiable mapping which defined byλ → (ξ_λ,η_λ,α_λ,β_λ), from(λ∗−δ,λ∗+δ)to X₁²×(_R⁺)² such that system(1.5)has a positive spatially nonhomogeneous steady-state solution:

(u_λ =α_λ(λ−λ∗)[ϕ+ (λ−λ∗)ξ_λ(x)]_,

v_λ =β_λ(λ−λ∗)[ϕ+ (λ−λ∗)η_λ(x)]. (2.6)

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Proof. LetF= (F₁,F₂,F₃,F₄)be defined as the following











F₁(ξ,η,α,β,λ) =Lξ+m(x)e^am⁽^x⁾[ϕ+ (λ−λ∗)ξ]−λαb₁e^2am⁽^x⁾[ϕ+ (λ−λ∗)ξ]²

−λβc₁e^2am⁽^x⁾[ϕ+ (λ−λ∗)ξ][ϕ+ (λ−λ∗)η] =0,

F2(ξ,η,α,β,λ) =Lη+m(x)e^am⁽^x⁾[ϕ+ (λ−λ∗)η]−λαc2e^2am⁽^x⁾[ϕ+ (λ−λ∗)η]²

−λβb₂e^2am⁽^x⁾[ϕ+ (λ−λ∗)ξ][ϕ+ (λ−λ∗)η] =0, F₃(ξ,η,α,β,λ) =hϕ,ξi=0,

F₄(ξ,η,α,β,λ) =hϕ,ηi=0.

It is easy to obtain that from (2.5)

F_i(ξ_λ_∗,η_λ_∗,α_λ_∗,β_λ_∗,λ∗) =0, (i=1, 2, 3, 4). The Fr ´echet derivative ofF at(ξ_λ_∗,η_λ_∗,α_λ_∗,β_λ_∗,λ∗)is

∂F

∂(ξ,η,α,β)

(ξ_λ_∗,η_λ_∗,α_λ_∗,β_λ_∗,λ∗)





 ξb bη bα βb







=







Lξb−λ∗(_bαb₁+βcb ₁)e^am⁽^x⁾ϕ² Lηb−λ∗(_bαc₂+βbb ₂)e^am⁽^x⁾ϕ² hϕ,ξbi

hϕ,ηbi





 .

It is clear that the derivative operator _∂₍_ξ,η,α,β^∂F ₎ ₍

ξ_λ_∗,η_λ_∗,α_λ_∗,β_λ_∗,λ∗) is bijective. By using the implicit function theorem we know that there exist a constantδ >0 and a continuously differentiable mapping which defined by λ → (ξ_λ,η_λ,α_λ,β_λ)_from (λ∗−δ,λ∗+δ)_to X₁²×(_R⁺)² such that system (1.5) has a positive spatially nonhomogeneous steady-state solution (2.6).

3 Eigenvalue problems of the linearized system

For the convenience of discussion, we always suppose thatΛ= (λ∗−δ,λ∗)∪(λ∗,λ∗+δ). Let (u_λ,v_λ)^T is a spatially nonhomogeneous steady-state solution of (1.5) which is deter- mined by (2.6). Let

˜

u=u−u_λ, v˜ =v−v_λ,

dropping the tilde sign, system (1.5) can be transformed as follows:











∂u

∂t =e⁻âm⁽^x⁾∇ ·[eâm⁽^x⁾∇u] +λu(x,t)[m(x)−b₁eâm⁽^x⁾u_λ−c₁eâm⁽^x⁾v_λ]

−λe^am⁽^x⁾u_λ[b₁u(x,t−τ) +c₁v(x,t−τ)],

∂v

∂t =e⁻âm⁽^x⁾∇ ·[eâm⁽^x⁾∇v] +λv(x,t)[m(x)−b₂eâm⁽^x⁾u_λ−c₂eâm⁽^x⁾v_λ]

−λe^am⁽^x⁾v_λ[b₂u(x,t−τ) +c₂v(x,t−τ)].

(3.1)

Denote A_λ,B_λ : A_λ=

A1 0 0 A₂

, B_λψ=

λe^am⁽^x⁾uλ[_b₁ψ₁(−τ) +_c₁ψ₂(−τ)]

λe^am⁽^x⁾v_λ[b₂ψ₁(−τ) +c₂ψ₂(−τ)]

where

A1 =_e⁻âm⁽^x⁾∇ ·[_eâm⁽^x⁾∇] +λ[_m(_x)−b1eâm⁽^x⁾uλ−c1eâm⁽^x⁾vλ]_, A₂ =e⁻âm⁽^x⁾∇ ·[eâm⁽^x⁾∇] +λ[m(x)−b₂eâm⁽^x⁾u_λ−c₂eâm⁽^x⁾v_λ]_,

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andψ= (ψ₁,ψ₂)^T ∈ X_C².

It follows from [14,33] that the semigroup induced by the solutions of the linearized system (3.1) has the infinitesimal generatorT_τ,λ satisfying

T_τ,λψ=ψ,^˙ (3.2)

where

D(T_τ,λ) ={ψ∈C_C∩C_C¹ |ψ(0)∈ X_C, ˙ψ(0) =A_λψ(0)−B_λψ(−τ)}, where

C_C=C([−τ, 0],Y_C²), C_C¹ =C¹([−τ, 0],Y_C²).

Moreover,µ∈ _Can eigenvalue ofT_τ,λ if and only if there existsψ= (ψ₁,ψ₂)^T ∈ X²_C\ {(0, 0)^T} such that

∆(λ,µ,τ)ψ= A_λψ−B_λψe⁻^µτ−µψ=0. (3.3) Lemma 3.1. 0is not an eigenvalue of T_τ,λ.

Proof. If 0 is an eigenvalue of T_τ,λ, that is, there exists some ψ = (ψ₁,ψ₂)^T ∈ X_C² \ {(0, 0)^T} such that

∆(λ, 0,τ)ψ=0. (3.4)

Note that∆(λ∗, 0,τ) = ^L₀⁰_L_andN(_L) =_span{ϕ}. We let thatψtakes the form (

ψ₁= p₁ϕ+ (λ−λ∗)q₁(x),

ψ₂= p₂ϕ+ (λ−λ∗)q₂(x), (3.5) where p₁,p₂ ∈ _R, q₁(x),q₂(x) ∈ X₁. Then substituting (3.5) into (3.4) and let λ = λ∗, by calculation, we have

(Lq₁+[m(x)e^am⁽^x⁾ϕ−λ∗e^2am⁽^x⁾(b₁α_λ_∗+c₁β_λ_∗)ϕ²]p₁−λ∗e^2am⁽^x⁾α_λ_∗ϕ²(b₁p₁+c₁p₂) =0,

Lq₂+[m(x)e^am⁽^x⁾ϕ−λ∗e^2am⁽^x⁾(b₂α_λ_∗+c₂β_λ_∗)ϕ²]p₂−λ∗e^2am⁽^x⁾β_λ_∗ϕ²(b₂p₁+c₂p₂) =_0. ^(3.6) By (2.5), (3.6) becomes

(L(q₁−ξ_λ_∗p₁)−λ∗e^2am⁽^x⁾α_λ_∗ϕ²(b₁p₁+c₁p₂) =_0,

L(q₂−η_λ_∗p₂)−λ∗e^2am⁽^x⁾β_λ_∗ϕ²(b₂p₁+c₂p₂) =0. (3.7) Multiplying both sides of each equation in (3.7) byϕand integrating onΩ, we have

(b₁p₁+c₁p₂=0,

b₂p₁+c₂p₂=0. (3.8)

By the condition (H1), we haveb₁c₂−b₂c₁6=0. So we getp₁ = p₂ =0 from (3.8). By (3.6), we getq₁=q₂ =0. Thenψ₁ =0,ψ₂ =0. The Lemma3.1is now proved.

We will show that the eigenvalues of T_τ,λ could pass through the imaginary axis when time delayτincreases. It is obvious thatT_τ,λ has an imaginary eigenvalueµ=iω (ω6=0)for someτ≥0 if and only if

m(λ,ω,θ)ψ=_∆(λ,ω,θ)ψ= A_λψ−B_λψe⁻^iθ−iωψ=0 (3.9) is solvable for someω >0,θ ∈ [0, 2π),τ= ^θ⁺^2nπ

ω ,n ∈_N₀ ={0, 1, 2, . . .}andψ= (ψ₁,ψ₂)^T ∈ X²_C\ {(_{0, 0})^T}_.

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Lemma 3.2. If (ω,θ,ψ) ∈ (0,∞)×[0, 2π)×(X²_C\ {(0, 0)^T}) solves(3.9), then _λ₋^ω_λ_∗ is bounded forλ∈_Λ.

Proof. Assume that(ω,θ,ψ)∈(0,∞)×[0, 2π)×(X_C² \ {(0, 0)^T})satisfy the following equation hA_λψ−B_λψe⁻^iθ−iωψ,ψi=0. (3.10) Separating the real and imaginary parts of system (3.10), we obtain

ωhψ,ψi=sinθhBλψ,ψi.

|ω|

|λ−λ∗| =λe^am⁽^x⁾|sinθ|

Dα_λ[ϕ+(λ−λ∗)ξ_λ(x)](b1ψ₁+c1ψ₂) β_λ[ϕ+(λ−λ∗)η_λ(x)](b2ψ₁+c2ψ₂)

,ψ

E hψ,ψi

≤(λ∗+δ)e^a^max^x^∈^Ω^m⁽^x⁾max{M,N}max{|b₁|,|c₁|,|b₂|,|c₂|}. where M =max_λ∈_Λ{|α_λ|[kϕk_∞+ (λ+λ∗)kξ_λ(x)k_∞]},

N = max_λ∈_Λ{|β_λ|[kϕk_∞+ (λ+λ∗)kη_λ(x)k_∞]}. The boundedness of _λ₋^ω_λ_∗ follows from the continuity ofλ7→ (α_λ,β_λ,kξ_λ(x)k_∞,kη_λ(x)k_∞). The Lemma3.2is now proved.

Note thatX=N(L)⊕X₁. If(ω,θ,ψ)satisfies (3.9), let ψ= (ψ₁,ψ₂)^T ∈X_C² \ {(0, 0)^T}can be represented as

(

ψ₁= p₁ϕ+ (λ−λ∗)q₁(x),

ψ₂= p₂ϕ+ (λ−λ∗)q₂(x), (3.11) where p₁,p2∈_R, q₁(x),q2(x)∈ X₁. Let

G(q₁,q₂,p₁,p₂,h,θ,λ)ψ= ^m(λ,(λ−λ∗)h,θ)

λ−λ∗ ψ=0, (3.12)

wherem(λ,ω,θ)is defined as in (3.9).

Obviously, we have

G(_q₁_,_q₂_,_p₁_,_p₂_,_h,_θ,λ∗)ψ=_0, that is

(L(q₁−ξ_λ_∗p₁)−λ∗e^2am⁽^x⁾α_λ_∗ϕ²(b₁p₁+c₁p₂)e⁻^iθ−ihϕe^am⁽^x⁾p₁ =0,

L(q₂−η_λ_∗p₂)−λ∗e^2am⁽^x⁾β_λ_∗ϕ²(b₂p₁+c₂p₂)e⁻^iθ−ihϕe^am⁽^x⁾p₂=0. (3.13) Multiplying both sides of each equation in (3.13) by ϕand integrating onΩ, we have

−λ∗d₂e⁻^iθMp=ihp, (3.14)

where p = (p₁,p₂)^T,d₂ =

R

Ωe^2am⁽^x⁾ϕ³dx R

Ωe^am⁽^x⁾ϕ²dx >0, M= ^α^λ^∗^b¹ ^α^λ^∗^c¹

β_λ_∗b2 β_λ_∗c2

. Separating the real and imaginary parts of (3.14), we get

(

λ∗d2sinθMp=hp,

λ∗d₂cosθMp=0. (3.15)

It is easy to obtain the following lemma.

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Lemma 3.3.

(1) Whenθ = ^π₂ _in_(3.15),λ∗d2M has two real eigenvalues h1 =λ∗d1d2, h2=λ∗d1d2(_c₂−c1)(_b₁−b2) b₁c₂−b₂c₁ , and(c₂−c₁,b₁−b₂)^T and(−c₁,b₂)^T are two eigenvectors associated with eigenvalues h₁and h₂, respectively.

(2) Whenθ=^3π₂ in(3.15),−λ∗d₂M has two real eigenvalues h₁=−λ∗d₁d₂, h₂=−λ∗d₁d₂⁽^c²_b⁻^c¹⁾⁽^b¹⁻^b²⁾

1c2−b2c₁ , and(c₂−c₁,b₁−b₂)^T and(−c₁,b₂)^T are two eigenvectors associated with eigenvalues h₁and h₂, respectively.

For each j=1, 2, set

h^j_λ_∗=

(|h_j|, ifλ>λ∗,

−|h_j|, ifλ<λ∗, (3.16)

which satisfiesω_λ^j_∗ = (λ−λ∗)h^j_λ_∗ >0, and their corresponding eigenvectors ((p¹_1λ_∗,p¹_2λ_∗)^T = (c₂−c₁,b₁−b₂)^T, if h¹_λ_∗ =|h₁|,

(p²_1λ_∗,p²_2λ_∗)^T = (−c₁,b2)^T, if h²_λ_∗ =|h2|, (3.17) ((p¹_1λ_∗,p¹_2λ_∗)^T = (c₂−c₁,b₁−b₂)^T, ifh¹_λ_∗ =−|h₁|,

(p²_1λ_∗,p²_2λ_∗)^T = (−c₁,b₂)^T, ifh²_λ_∗ =−|h₂|. (3.18) And set

θ_λ^j_∗= (π

2, ifλ>λ∗,

3π

2 , ifλ>λ∗, (3.19)

which satisfies−e⁻^iθ^λ^j^∗h_λ^j_∗ =ih_λ^j_∗.

Thus,q_1λ^j _∗,q^j_2λ_∗ ∈ X1is the unique solution of the following equations







L(q₁^j −ξ_λ_∗p^j_1λ_∗)−λ∗e^2am⁽^x⁾α_λ_∗φ²(b₁p^j_1λ_∗+c₁p^j_2λ_∗)e⁻^iθ^j^λ^∗ −ih^j_λ_∗φe^am⁽^x⁾p_1λ^j _∗= 0,

L(q₂^j −η_λ_∗p_2λ^j _∗)−λ∗e^2am⁽^x⁾β_λ_∗φ²(b₂p_1λ^j _∗+c₂p_2λ^j _∗)e⁻^iθ^λ^j^∗−ih_λ^j_∗φe^am⁽^x⁾p_2λ^j _∗ =0, (3.20) Remark 3.4.

(1) Whenv=0,b₁=1 in (1.4),h₁in Lemma3.3(1) is the same ash_λ_∗ in (2.20) in [8].

(2) When a₁ = 0 in (1.4), h₁,h₂ in Lemma3.3 (1) are the same as that in Lemma 3.4 (i) in [15].

Then we get the following lemma.

Lemma 3.5. Assume that (H1) holds. For j=1, 2, the following equation (G(q^j₁,q₂^j,p₁^j,p₂^j,θ^j,h^j,λ∗) =_0,

q₁^j, q₂^j ∈X₁, p^j₁, p₂^j, h^j ∈_R, θ^j ∈[0, 2π] ^(3.21) has a unique solution(q_1λ^j _∗,q_2λ^j _∗,p_1λ^j _∗,p_2λ^j _∗,θ_λ^j_∗,h_λ^j_∗), see(3.16)–(3.20).

Theorem 3.6. Assume that (H1) holds. Then for j = 1, 2, there exist a constant δ > ₀ and a continuously differentiable mapping which defined by λ → (q_1λ^j ,q_2λ^j ,p^j_1λ,p_2λ^j ,θ_λ^j,h_λ^j) from Λ to X²₁×_R²×_R⁺×_Rsuch that G(q_1λ^j ,q_2λ^j ,p_1λ^j ,p^j_2λ,θ_λ^j,h_λ^j,λ) =0.

(9)

Proof. LetG= (g₁,g₂,g₃,g₄,g₅,g₆)be defined as the following:











g₁= m(_λ,(λ−λ∗)h_λ^j,θ_λ^j)q_1λ^j +m(x)e^am⁽^x⁾ϕp_1λ^j

−λeâm⁽^x⁾ϕ[α_λb₁(ϕ+ (λ−λ∗)ξ) +β_λc₁(ϕ+ (λ−λ∗)ξ)]p^j_1λ +λϕα_λeâm⁽^x⁾(ϕ+ (λ−λ∗)ξ)(b₁p_1λ^j +c₁p^j_2λ)e⁻îθ^λ^j −ih^j_λϕp_1λ^j =_0, g₂= m(λ,(λ−λ∗)h_λ^j,θ_λ^j)q_2λ^j +m(x)eâm⁽^x⁾ϕp_2λ^j

−λeâm⁽^x⁾ϕ[α_λb₂(ϕ+ (λ−λ∗)ξ) +β_λc₂(ϕ+ (λ−λ∗)ξ)]p^j_2λ +λϕα_λeâm⁽^x⁾(ϕ+ (λ−λ∗)ξ)(b₂p_1λ^j +c₂p^j_2λ)e⁻îθ^λ^j −ih^j_λϕp_2λ^j =0, g₃=Rehϕ,q^j_1λi=0, g₄ =Imhϕ,q_1λ^j i=0,

g₅=Rehϕ,q^j_2λi=0, g₆ =Imhϕ,q_2λ^j i=0.

The Fréchet derivative ofGat(q_1λ^j _∗,q_2λ^j _∗,p_1λ^j _∗,p^j_2λ_∗,θ_λ^j_∗,h^j_λ_∗,λ∗)is

∂G(q_1λ^j _∗,q^j_2λ_∗,p_1λ^j _∗,p_2λ^j _∗,θ_λ^j_∗,h_λ^j_∗,λ∗)

∂(q^j_1λ,q_2λ^j ,p^j_1λ,p_2λ^j ,θ_λ^j,h^j_λ)





 ˆ q^j_1λ qˆ^j_2λ

ˆ p_1λ^j

ˆ p_2λ^j θˆ_λ^j hˆ_λ^j







=







e⁻^am⁽^x⁾Lqˆ_1λ^j +g_e₁pˆ^j_1λ+g_e₂pˆ^j_2λ+_eg₃θˆ_λ^j +g_e₄hˆ^j_λ e⁻^am⁽^x⁾Lqˆ_2λ^j +g_e5pˆ^j_1λ+g_e6pˆ^j_2λ+_eg7θˆ_λ^j +g_e8hˆ^j_λ Reh_{ϕ, ˆ}q_1λ^j i

Imhϕ, ˆq_1λ^j i Rehϕ, ˆq_2λ^j i Imhϕ, ˆq_2λ^j i





 ,

where











ge₁=m(x)eâm⁽^x⁾ϕ−λ∗eâm⁽^x⁾(α_λ_∗b₁+β_λ_∗c₁)ϕ²−λ∗α_λ_∗eâm⁽^x⁾b₁e⁻îθ^λ^j^∗−ih^j_λ_∗ϕ,

ge₂=−λ∗α_λ_∗eâm⁽^x⁾c₁e⁻îθ^λ^j^∗, eg₃= −iθ_λ^j_∗λ∗ϕ²α_λ_∗(b₁p^j_1λ_∗+b₂p^j_2λ_∗)e⁻îθ^j^λ^∗, ge₄ =−iϕp^j_1λ_∗, ge₅=m(x)eâm⁽^x⁾ϕ−λ∗eâm⁽^x⁾(α_λ_∗b₂+β_λ_∗c₂)ϕ²−λ∗α_λ_∗eâm⁽^x⁾b₂e⁻îθ^λ^j^∗−ih^j_λ_∗ϕ,

ge₆=−λ∗α_λ_∗eâm⁽^x⁾c₂e⁻îθ^λ^j^∗, eg₇= −iθ_λ^j_∗λ∗ϕ²α_λ_∗(b₁p_1λ_∗+b₂p_2λ_∗)e⁻îθ^j^λ^∗, ge₈ =−iϕp^j_2λ_∗. It is clear that the derivative operator

∂G(q_1λ^j _∗,q_2λ^j _∗,p_1λ^j _∗,p_2λ^j _∗,θ_λ^j_∗,h_λ^j_∗,λ∗)

∂(q_1λ^j ,q_2λ^j ,p^j_1λ,p_2λ^j ,θ_λ^j,h^j_λ)

is bijective. By using the implicit function theorem we know that there exist a constantδ >0 and a continuously differentiable mapping which defined by λ → (q^j_1λ,q^j_2λ,p_1λ^j ,p^j_2λ,θ_λ^j,h_λ^j), from from Λ to X₁²×_R²×_R⁺×_R such that G(q_1λ^j ,q_2λ^j ,p^j_1λ,p_2λ^j ,θ_λ^j,h^j_λ,λ) = 0. The proof of Theorem3.6is complete.

From Theorem3.6, we derive the following result.

Theorem 3.7. Assume that (H1) holds. For j =1, 2,λ∈_Λand n∈_N₀, let

τ_n^j = ^θ

j

1λ+2nπ ω_λ^j

, ω_λ^j = (λ−λ∗)h^j_λ,