a Modified Holling-Tanner Predator-Prey System with Multiple Delays

Volume 2012, Article ID 236484,19pages doi:10.1155/2012/236484

Research Article

Stability and Hopf Bifurcation in

a Modified Holling-Tanner Predator-Prey System with Multiple Delays

Zizhen Zhang,

Huizhong Yang,

and Juan Liu

1Key Laboratory of Advanced Process Control for Light Industry of Ministry of Education, Jiangnan University, Wuxi 214122, China

2School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 233030, China

3Department of Science, Bengbu College, Bengbu 233030, China

Correspondence should be addressed to Huizhong Yang,yanghzjiangnan@163.com Received 9 August 2012; Revised 17 September 2012; Accepted 4 October 2012 Academic Editor: Kunquan Lan

Copyrightq2012 Zizhen Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A modified Holling-Tanner predator-prey system with multiple delays is investigated. By analyzing the associated characteristic equation, the local stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays are established. Direction and stability of the periodic solutions are obtained by using normal form and center manifold theory. Finally, numerical simulations are carried out to substantiate the analytical results.

1. Introduction

Predator-prey dynamics has long been and will continue to be of interest to both applied mathematicians and ecologists due to its universal existence and importance 1,2. Many population models investigating the dynamic relationship between predators and their preys have been proposed and studied. For example, Lotka-Volterra model 3–5, Leslie- Gower model6–10, and Holling-Tanner model11–16. Among these widely used models, Holling-Tanner model plays a special role in view of the interesting dynamics it possesses.

Holling-Tanner model for predator-prey interaction is governed by the following nonlinear coupled ordinary differential equations:

dX dT rX

1− X

K

− mXY aX, dY

dT Y

s

1−hY X

,

1.1

whereXandY denote the population densities of prey species and predator species at time T, respectively. The first equation in system1.1shows that the prey grows logistically with the carrying capacityK and the intrinsic growth rater in the absence of the predator. And the growth of the prey is hampered by the predator at a rate proportional to the functional responsemX/aXin the presence of the predator. The second equation shows that the predator consumes the prey according to the functional responsemX/aX and grows logistically with the intrinsic growth rates and carrying capacityX/hproportional to the number of the prey. The parametermdenotes the maximal predator per capita consumption rate.ais a saturation value; it corresponds to the number of prey necessary to achieve one half the maximum ratem. The parameterhdenotes the number of prey required to support one predator at equilibrium whenyequalsX/h.

Recently, there has been considerable interest in predator-prey systems with the Beddington-DeAngelis functional response. And it has been shown that the predator-prey systems with the Beddington-DeAngelis functional response have rich but biologically reasonable dynamics. For more details about this functional response one can refer to17–

21. Zhang16, Lu and Liu22considered the following modified Holling-Tanner delayed predator-prey system:

dX dT rX

1− X

K

− αXY abXcY, dY

dT Y

s

1−hYt−τ Xt−τ

,

1.2

whereτis incorporated in the negative feedback of the predator density.αXY/abXcYis the Beddington-DeAngelis functional response. The parametersα,a,b, andcare assumed to be positive.αis the maximum value at which per capita reduction rate of the prey can attain.

a measures the extent to which environment provides protection to the prey. bdescribes the effect of handling time on the feeding rate. c describes the magnitude of interference among predators. Zhang16investigated the local Hopf bifurcation of system1.2. Lu and Liu 22 proved that system1.2 is permanent under some conditions and obtained the sufficient conditions of local and global stability of system1.2. Since both the species are growing logistically, it is reasonable to assume delay in prey species as well. Based on this consideration, we incorporate the negative feedback of the prey density into system1.2and obtain the following system:

dX dT rX

1−XT−T1 K

− αXY abXcY, dY

dT Y

s

1−hYT−T₂ XT−T2

,

1.3

whereT1 andT2 are the feedback time delays of the prey density and the predator density respectively. Let X Kx, Y rK/αy,t rT,τ₁ rT₁,τ₂ rT₂, system1.3 can be

transformed into the following nondimensional form:

dx

dt x1−xt−τ1− xy a1bxc1y, dy

dt y

δ−βyt−τ₂ xt−τ2

,

1.4

wherea1 a/K, c1 cr/α, δs/r, βsh/αare the non-dimensional parameters and they are positive.

The main purpose of this paper is to consider the effect of multiple delays on system 1.4. The local stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. By employing normal form and center manifold theory, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined. Finally, some numerical simulations are also included to illustrate the theoretical analysis.

2. Local Stability and the Existence of Hopf Bifurcation

In this section, we study the local stability of each of feasible equilibria and the existence of Hopf bifurcation at the positive equilibrium. Obviously, system1.4has a unique boundary equilibriumE₁1,0and a unique positive equilibriumE_∗x∗, y_∗, where

x_∗ −

a1−bβ 1−c₁δ

a1−bβ 1−c₁δ24a₁β bβc₁δ

2 bβc₁δ , y_∗ δ

βx_∗. 2.1

The Jacobian matrix of system1.4atE₁takes the form

JE1

⎛

⎜⎝−e^−λτ1 − 1 a1b

0 δ

⎞

⎟⎠. 2.2

The characteristic equation of system1.4atE₁is of the form

λ−δ

λe^−λτ1

0. 2.3

Clearly, the boundary equilibriumE₁1,0is unable.

Next, we discuss the existence of Hopf bifurcation at the positive equilibriumEx∗, y_∗. Let xt z₁t x_∗, yt z₂t y_∗, and still denotez₁tand z₂tby xt and yt, respectively, then system1.4becomes

dx

dt a₁₁xt a₁₂yt b₁₁xt−τ₁

ijk≥2

f₁^ijkxⁱy^jx^kt−τ₁, dy

dt c21xt−τ2 c22yt−τ2

ijk≥2

f₂^ijkyⁱx^jt−τ2y^kt−τ2,

2.4

where

a₁₁ bx_∗y_∗

a1bx_∗c1y_∗2, a₁₂ − a1bx_∗x_∗ a1bx_∗c1y_∗2, b11−x∗, c21 δ²

β , c22−δ, f₁x1−xt−τ₁− xy

a1bxc1y, f₂y

δ−βyt−τ2 xt−τ2

,

f₁^ijkxⁱy^jx^kt−τ₁ 1 i!j!k!

∂^ijkf₁

∂xⁱ∂y^j∂x^kt−τ₁|_x∗,y∗, f₂^ijkyⁱx^jt−τ₂y^kt−τ₂ 1

i!j!k!

∂^ijkf₂

∂yⁱ∂x^jt−τ₂∂y^kt−τ₂|_x∗,y∗.

2.5

Then we can obtain the linearized system of system1.4 dx

dt a₁₁xt a₁₂yt b₁₁xt−τ₁, dy

dt c21xt−τ2 c22yt−τ2.

2.6

The characteristic equation of system2.6is

λ²−a₁₁λ−b₁₁λe^−λτ1 a₁₁c₂₂−a₁₂c₂₁−c₂₂λe^−λτ2b₁₁c₂₂e^−λτ1τ20. 2.7 Case 1. τ₁τ₂τ0.

The characteristic equation of system1.4becomes

λ² ABDλCE0, 2.8

where

A−a11, B−b11, Ca11c22−a12c21, D−c22, Eb11c22. 2.9

It is easy to verify that

CEδx_∗ a1δ²x_∗

β a₁bx_∗c₁y_∗2 >0. 2.10 Therefore, ifH1:ABD >0, the roots of2.8must have negative real parts. Then, we know that the positive equilibriumE_∗x∗, y_∗of system1.4is locally stable in the absence of delay, ifH1holds.

Case 2. τ₁τ₂τ >0.

The associated characteristic equation of the system is

λ²A1λ B1C1λe^−λτD1e^−2λτ0, 2.11 where

A₁−a11, B₁a₁₁c₂₂−a₁₂c₂₁, C₁−b11c₂₂, D₁ b₁₁c₂₂. 2.12 Multiplyinge^λτon both sides of2.11, we can obtain

λ²A1λ

e^λτ B1C1λ D1e^−λτ 0. 2.13 Now, forτ >0, ifλiωω >0be a root of2.13. Then, we have

D1−ω²

cosτω−A1ωsinτω−B1,

D1ω²

sinτω−A1ωcosτωC1ω.

2.14

It follows from2.14that

sinτω C₁ω² A₁B₁−C₁D₁ω

ω⁴A²₁ω²−D₁² , cosτω B1−A₁C₁ω²B₁D₁

ω⁴A²₁ω²−D₁² . 2.15 Then we have

ω⁸e₃ω⁶e₂ω⁴e₁ω²e₀0, 2.16

where

e₃2A²₁−C²₁, e₂A⁴₁2C²₁D₁−A²₁C₁²−B²₁−2D²₁,

e14A1B1C1D1−A²₁B²₁−C²₁D²₁−2A²₁D²₁−2B²₁D1, e0D⁴₁−B₁²D₁².

2.17

Letvω², then2.16becomes

v⁴e₃v³e₂v²e₁ve₀0. 2.18 Next, we give the following assumption.H2:2.18has at least one positive real root.

Suppose thatH2holds. Without loss of generality, we assume that2.18has four real positive roots, which are defined byv₁,v₂,v₃, andv₄, respectively. Then2.16has four positive rootsω_k√

v_k, k 1,2,3,4. Therefore,

τ_k^j 1 ωk

arccos

B1−A₁C₁ω_k²B₁D₁ ω⁴_kA²₁ω²_k−D²₁ 2jπ

, k1,2,3,4;j0,1,2. . . 2.19

Then we can know that ±iωk are a pair of purely imaginary roots of 2.11with τ τ_k^j. Define

τ₀ τ_k⁰min τ_k⁰

, ω₀ω_k0, k1,2,3,4. 2.20 Letλτ ατ iωτbe the root of2.11nearττ0which satisfiesατ0 0, ωτ0 ω0. Taking the derivative ofλwith respect toτin2.13, we obtain

C1dλ

dτ 2λA1e^λτdλ dτ

λ²A1λ e^λτ

λτdλ dτ

−D1e^−λτ

λτdλ dτ

0. 2.21

it follows that

dλ

dτ λ D₁e^−λτ− λ²A₁λ e^λτ

2λA₁e^λτC₁−τ D₁e^−λτ−λ²A₁λe^λτ. 2.22 Thus

dλ dτ

₋₁

2λA₁e^λτC₁

D1λe^−λτ−λ³A1λ²e^λτ −τ

λ. 2.23

Let Λ1

D₁ω₀−ω³₀

sinτ₀ω₀A₁ω²₀cosτ₀ω₀, Λ2

D₁ω₀ω₀³

cosτ₀ω₀A₁ω²₀sinτ₀ω₀, Λ3A1cosτ0ω0−2ω0sinτ0ω0C1, Λ4A1sinτ0ω0−2ω0cosτ0ω0.

2.24

Substituteλiω₀ω0>0into2.23, we can get

Re

dλτ0 dτ

₋₁ Re

2λA1e^λτC1

D1λe^−λτ−λ³A1λ²e^λτ

λiω0

Λ1×Λ3 Λ2×Λ4

Λ²₁ Λ²₂ . 2.25

Noting that

signRe

dλτ0 dτ

sign

dReλτ0 dτ

₋₁

. 2.26

Therefore, we make the following assumption in order to give the main results:H3 :Λ1× Λ3 Λ2×Λ4/0. Then, by Corollary 2.4 in23,24, we have the following theorem.

Theorem 2.1. For system1.4, if the conditionsH1–H3hold, then the equilibriumE_∗x_∗, y_∗of system1.4is asymptotically stable forτ ∈0, τ0and unstable whenτ > τ₀. And system1.4has a branch of periodic solution bifurcation from the zero solution nearττ0.

Case 3. τ₁/τ₂,τ₁>0 andτ₂>0.

The associated characteristic equation of the system is

λ²A₂λB₂λe^−λτ1 C₂D₂λe^−λτ2E₂e^−λτ1τ2, 2.27 where

A2−a11, B2−b11, C2a11c22−a12c21, D2−c22, E2b11c22. 2.28 We consider2.27withτ₂in its stable interval, regardingτ₁as a parameter. Without loss of generality, we consider system1.4under the case considered in16, andτ2∈0, τ20.τ20is defined as in16and can be obtained by

τ₂₀ 1 ωarccos

a11c₂₂b₁₁c₂₂−a₁₂c₂₁ω²−a11b₁₁c22ω² a11c22b11c22−a12c21² c22ω²

, 2.29

with

ω −

a11b₁₁²−c²₂₂

a11b₁₁²−c²₂₂2

4a11c₂₂b₁₁c₂₂−a₁₂c₂₁²

2 . 2.30

Letλiωω >0be a root of2.27. Then we obtain

B2ω−E2sinτ2ωsinτ1ωE2cosτ2ωcosτ1ωω²−C2cosτ2ω−D2ωsinτ2ω,

B2ω−E₂sinτ₂ωcosτ₁ω−E₂cosτ₂ωsinτ₁ωC₂sinτ₂ω−D₂ωcosτ₂ω−A₂ω. 2.31 It follows from2.31that

sinτ₁ω M₁N₁−M₂N₂

M²₁M²₂ , cosτ₁ω M₁N₂M₂N₁

M²₁M²₂ , 2.32

With

M₁B₂ω−E₂sinτ₂ω, M₂E₂cosτ₂ω,

N1ω²−C2cosτ2ω−D2ωsinτ2ω, N2−A2ωC2sinτ2ω−D2ωcosτ2ω.

2.33

Then we have

P₁ω P₂ωsinτ₂ωP₃ωcosτ₂ω0, 2.34

where

P₁ω ω⁴

A²₂D²₂−B₂²

ω²C²₂−E²₂,

P2ω −2D2ω³−2A2C2ω2B2E2ω, P3ω 2A2D2−C2ω².

2.35

Suppose thatH4:2.34has at least finite positive roots. IfH4holds, we define the roots of 2.34as ω1, ω2, . . . , ωk. Then, for every fixedωii 1,2, . . . , k, there exists a sequence {τ₁^j_i |j1,2, . . .}which satisfies2.34. Let

τ_1∗min τ₁^j

i |i1,2, . . . , k, j0,1, . . .

. 2.36

Whenτ1τ_1∗,2.27has a pair of purely imaginary roots±iω∗forτ2∈0, τ20.

To verify the transversality condition of Hopf bifurcation, we take the derivative ofλ with respect toτ₁in2.27, we can obtain

dλ

dτ₁ λe^−λτ1 B2λE2e^−λτ2 2λA₂B₂e^−λτ1D₂e^−λτ2

−τ₁e^−λτ1 B₂λE₂e^−λτ2

−τ₂e^−λτ2 C₂D₂λE₂e^−λτ1. 2.37

Thus dλ

dτ_{1 −1}

2λA₂B₂e^−λτ1D₂e^−λτ2−τ₂e^−λτ2 C₂D₂λE₂e^−λτ1 λe^−λτ1 B₂λE₂e^−λτ2 −τ1

λ. 2.38

Substituteλiω_∗ω∗>0into2.38, we can get

Re

dλτ_1∗

dτ1

₋₁

Δ1×Δ3 Δ2×Δ4

Δ²₁ Δ²₂ , 2.39

where

Δ1E2ω_∗cosτ2ω_∗sinτ_1∗ω_∗−ω_∗cosτ_1∗ω_∗B2ω_∗−E2ω_∗sinτ2ω_∗, Δ2E₂ω_∗cosτ₂ω_∗cosτ_1∗ω_∗ω_∗sinτ_1∗ω_∗B2ω_∗−E₂ω_∗sinτ₂ω_∗,

Δ3 A₂ D₂−τ₂C₂cosτ₂ω_∗−τ₂D₂ω_∗sinτ₂ω_∗

τ₂E₂sinτ₂ω_∗sinτ_1∗ω_∗ B₂−τ₂E₂cosτ₂ω_∗cosτ_1∗ω_∗, Δ4 2ω_∗ τ₂C₂τ₂D₂ω_∗−D₂sinτ₂ω_∗

τ2E2sinτ2ω_∗cosτ_1∗ω_∗−B2−τ2E2cosτ2ω_∗sinτ_1∗ω_∗.

2.40

Next, we make the following assumption:H5:Δ1×Δ3 Δ2×Δ4/0.

Thus, by the discussion above and by the general Hopf bifurcation theorem for FDEs in Hale25, we have the following results.

Theorem 2.2. Forτ₂ ∈0, τ20,τ₂₀is defined by2.29. If the conditionsH4-H5hold, then the equilibriumE_∗x∗, y_∗of system 1.4is asymptotically stable forτ1 ∈ 0, τ1∗and unstable when τ > τ_1∗. System1.4has a branch of periodic solution bifurcation from the zero solution nearτ τ_1∗.

3. Direction and Stability of Bifurcated Periodic Solutions

In this section, we shall investigate the direction of the Hopf bifurcation and the stability of bifurcating periodic solution of system1.4w.r. toτ₁ forτ₂ ∈0, τ20, andτ₂₀ is defined by 2.29. The idea employed here is the normal form and center manifold theory described in Hassard et al.26. Throughout this section, it is considered that system1.4undergoes the Hopf bifurcation at τ₁ τ_1∗,τ₂ ∈ 0, τ20at E_∗x∗, y_∗. Letτ₁ τ_1∗ μ, μ ∈ R so that the Hopf bifurcation occurs atμ0. Without loss of generality, we assume thatτ_2∗ < τ_1∗, where τ_2∗∈0, τ20.

Letu₁t xt−x_∗,u₂t yt−y_∗, and rescaling the time delayt → t/τ1, Then system1.4can be transformed into an FDE inCC−1,0, R²as:

ut ˙ L_μu_tF μ, u_t

, 3.1

whereut u1t, u2t^T∈R²andL_μ:C → R²,F:R×C → R²are given, respectively, by

Lμφ τ_1∗μ

Aφ0 Cφ

−τ_2∗

τ_1∗

Bφ−1

, F μ, φ

τ_1∗μ f1, f2

_T ,

3.2

with

A

a₁₁ a₁₂

0 0

, B

b₁₁ 0 0 0

, C

0 0 c21 c22

, f₁g₁φ²₁0 g₂φ₁0φ20 g₃φ₂²0 g₄φ₁0φ1−1

h1φ₁³0 h2φ²₁0φ20 h3φ10φ₂²0 h4φ³₂0 · · ·, f₂g₁φ²₁

−τ_2∗

τ_1∗

g₂φ₁

−τ_2∗

τ_1∗

φ₂0 g₃φ₁

−τ_2∗

τ_1∗

φ₂

−τ_2∗

τ_1∗

, g₄φ₂0φ2

−τ_2∗

τ_1∗

h₁φ³₁

−τ_2∗

τ_1∗

h₂φ²₁

−τ_2∗

τ_1∗

φ₂0, h₃φ²₁

−τ_2∗

τ_1∗

φ2

−τ_2∗

τ_1∗

h4φ³₂0 · · ·,

3.3

where

g₁ by_∗ a1c1y_∗

a1bx_∗c1y_∗3, g₂−a²₁a1bx_∗a1c1y_∗2bc1x_∗y_∗ a1bx_∗c1y_∗3 , g₃ c₁x_∗a1bx_∗

a1bx_∗c1y_∗3, g₄−1, h1− b²y_∗ a₁c₁y_∗

a₁bx_∗c₁y_∗4, h2 a²₁ba1b²x_∗2b²c1x_∗y_∗−bc²₁y²_∗ a₁bx_∗c₁y_∗4 , h₃ a²₁c₁a₁c₁²y_∗2bc²₁x_∗y_∗−b²c₁x_∗²

a1bx_∗c1y_∗4 , h₄− c₁²x_∗a1bx_∗ a1bx_∗c1y_∗4, g₁ −βy_∗²

x_∗³ , g₂ βy_∗

x²_∗ , g₃ βy_∗

x²_∗ , g₄ −β x_∗, h₁ βy²_∗

x⁴_∗ , h₂−βy_∗

x³_∗ , h₃−βy_∗ x_∗³ .

3.4

Using Riesz representation theorem, there exists a 2×2 matrix functionηθ, μ, θ ∈ −1,0 whose elements are of bounded variation, such that

L_μφ ₀

−1dη θ, μ

φθ, φ∈C−1,0, R2. 3.5

In fact, choosing

η θ, μ

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩ τ_1∗μ

ACB, θ0, τ_1∗μ

CB, θ∈

−τ_2∗

τ1

,0

, τ_1∗μ

B, θ∈

−1,−τ_2∗

τ1

,

0, θ−1.

3.6

Forφ∈C−1,0, we define

A μ φ

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ dφθ

dθ , −1≤θ <0, ₀

−1dη θ, μ

φθ, θ0,

R μ φ

#0, −1≤θ <0, F μ, φ

, θ0.

3.7

Then system3.1can be transformed into the following operator equation

ut ˙ A μ

u_tR μ

u_t, 3.8

whereu_tutθ u1tθ, u2tθ.

Forϕ ∈ C¹0,1,R^2∗, where R^2∗ is the 2-dimensional space of row vectors, we further define the adjoint operatorA^∗of A0:

A^∗ ϕ

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−dϕs

ds , 0< s≤1, ₀

−1ϕ−ξdηξ,0, s0,

3.9

and a bilinear inner product:

$ϕs, φθ%

ϕ^T0φ0− ₀

θ−1

_θ

ξ0ϕ^Tξ−θdηθφξdξ, 3.10

whereηθ ηθ,0.

Since ±iω∗τ_1∗ are eigenvalues of A0, they are also eigenvalues of A^∗. Let qθ 1, q2^Te^{iω∗τ1∗θ} be the eigenvectors of A0 corresponding to iω_∗τ_1∗ and q^∗s 1/ρ1, q^∗₂^Te^{iω∗τ1∗s} be the eigenvectors of A^∗ corresponding to −iω∗τ_1∗. By a simple computation, we can get

q2 iω_∗−a11−b11e^iω∗τ1∗

a₁₂ , q₂^∗−iω_∗a11b11e^iω∗τ1∗

c21e^iω∗τ2∗ , ρ1q2q^∗₂b11τ_1∗e^{−iω∗τ1∗}c21τ_2∗q^∗₂e^{−iω∗τ2∗}c22τ_2∗q2q^∗₂e^{−iω∗τ2∗}.

3.11

Then q^∗, q1, q^∗, q0.

In the remainder of this section, Following the algorithms given in 26 and using similar computation process to that in 16, we can get that the coefficients which will be used to determine the important qualities of the bifurcating periodic solutions,

g20 2τ_1∗

ρ

#

g1g2q²0 g3

q²02

g4q¹−1

q^∗₂

g₁

q¹

−τ_2∗

τ_1∗

₂

g₂q¹

−τ_2∗

τ1^∗

q²0 g₃q¹

−τ_2∗

τ_1∗

q²

−τ_2∗

τ_1∗

g₄q²0q²

−τ_2∗

τ_1∗

&

,

g₁₁ τ_1∗

ρ '

2g₁g₂

q²0 q²0

2g₃q²0q²0 g₄

q¹−1 q¹−1 q^∗₂

2g₁q¹

−τ_2∗

τ_1∗

q¹

−τ_2∗

τ_1∗

g₂

q¹

−τ_2∗

τ_1∗

q²0 q¹

−τ_2∗

τ_1∗

q²0

g₃

q¹

−τ_2∗

τ_1∗

q²

−τ_2∗

τ_1∗

q¹

−τ_2∗

τ_1∗

q²

−τ_2∗

τ_1∗

g₄

q²0q²

−τ_2∗

τ_1∗

q²0q²

−τ_2∗

τ_1∗

( ,

g₀₂ 2τ_1∗

ρ

#

g₁g₂q²0 g₃

q²0₂

g₄q¹−1

q^∗₂

g₁

q¹

−τ_2∗

τ_1∗

₂

g₂q¹

−τ_2∗

τ_1∗

q²0 g₃q¹

−τ_2∗

τ_1∗

q²

−τ_2∗

τ_1∗

g₄q²0q²

−τ_2∗

τ_1∗

( ,

g21 2τ_1∗

ρ

# g1

W₂₀¹0 2W₁₁¹0 g2

1

2W₂₀²0 W₁₁²0 1

2W₂₀¹0q²0 W₁₁¹0q²0

g₃

W₂₀²0q²0 2W₁₁²0q²0 g₄

1

2W₂₀¹−1 W₁₁¹−1 1

2W₂₀¹0q¹−1 W₁₁¹0q¹−1

3h₁

h₂

2q²0 q²0 h₃

2q²0q²0

q²0₂ 3h₄

q²0₂ q²0 q^∗₂

g₁

W₂₀¹

−τ_2∗

τ_1∗

q¹

−τ_2∗

τ_1∗

2W₁₁¹

−τ_2∗

τ_1∗

q¹

−τ_2∗

τ_1∗

g₂ 1

2W₂₀¹

−τ_2∗

τ_1∗

q²0 W₁₁¹

−τ_2∗

τ_1∗

q²0 1

2W₂₀²0q¹

−τ_2∗

τ_1∗

W₁₁²0q¹

−τ_2∗

τ_1∗

g₃ 1

2W₂₀¹

−τ_2∗

τ_1∗

q²

−τ_2∗

τ_1∗

W₁₁¹

−τ_2∗

τ_1∗

q²

−τ_2∗

τ_1∗

1 2W₂₀²

−τ_2∗

τ_1∗

q¹

−τ_2∗

τ_1∗

g₄ 1

2W₂₀²0q²

−τ_2∗

τ_1∗

W₁₁²0q²

−τ_2∗

τ_1∗

1

2W₂₀²

−τ_2∗

τ_1∗

q²0 W₁₁²

−τ_2∗

τ_1∗

q²0

3h₁

q¹

−τ_2∗

τ_1∗

2

q¹

−τ_2∗

τ_1∗

h₂

q¹

−τ_2∗

τ_1∗

₂

q²0 2q¹

−τ_2∗

τ_1∗

q¹

−τ_2∗

τ_1∗

q²0

h₃ q¹

−τ_2∗

τ_1∗

₂ q²

−τ_2∗

τ_1∗

2q¹

−τ_2∗

τ_1∗

q¹

−τ_2∗

τ_1∗

q²

−τ_2∗

τ_1∗

&

, 3.12

with

W₂₀θ ig₂₀q0

τ_1∗ω_∗ e^{iτ1∗ω∗θ}ig₀₂q0

3τ_1∗ω_∗ e^{−iτ1∗ω∗θ}E₂₀e^{2iτ1∗ω∗θ}, W₁₁θ −ig₁₁q0

τ_1∗ω_∗ e^{iτ1∗ω∗θ}ig₁₁q0

τ_1∗ω_∗ e^{−iτ1∗ω∗θ}E₁₁,

3.13