# Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays

Authors: Ruiqing Shi; Junmei Qi; Sanyi Tang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We propose a three-dimensional stage-structured predatory-prey model with discrete and distributed delays. By use of a new variable, the original three-dimensional system transforms into an equivalent four-dimensional system. Firstly, we study the existence and local stability of positive equilibrium of the new system. And, by choosing the time delayτ as a bifurcation parameter, we show that Hopf bifurcation may occur as the time delay τ passes through some critical values. Secondly, by use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, some simple discussion is presented.


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## Body

## 1. Introduction

Since the pioneering theoretical works by Lotka [1] and Volterra [2], there were a lot of authors who studied all kinds of predator-prey models modeled by ordinary differential equations (ODEs) [3–5]. To reflect that the dynamical behavior of the models depends on the past history of the system, it is often necessary to incorporate time delays into the models. Therefore, a more realistic predator-prey model should be described by delayed differential equations (DDEs) [6–26]. Some of them investigated discrete delays [6–20]; others were about distributed delays [21–24]; and both discrete and distributed delays were studied in [25]. In general, delay differential equations exhibit more complicated dynamics on stability, periodic structure, bifurcation, and so on [26].In the natural world, many individuals have a life story that takes them through two stages, immature and mature. The predator only catches the mature prey, as the immature preys are protected by their eggshells or refuge. Some predator-prey models with stage structure were investigated in [27–33]. Motivated by [25, 27, 31] and the references cited therein, in the present paper, we will consider the following stage-structured predator-prey model with discrete and distributed delay:
(1)x1′(t)=rx2(t)-d1x1(t)-re-d1τx2(t-τ),x2′(t)=re-d1τx2(t-τ)-d2x2(t)-b1x2(t)y(t),y′(t)=y(t)[-d3+b2∫-∞t‍F(t-s)x2(s)ds-αy(t)],
where x1(t), x2(t), and y(t) can be interpreted as the population densities of the immature prey, mature prey, and predator at time t, respectively. r denotes the birth rate of the prey population; d1, d2, and d3 denote the death rate of the immature prey, mature prey, and the predator; α is the density-depended death rate of the predator; b1 denotes the per capita per unit time predation rate of the predator; the term b2∫-∞t‍F(t-s)x2(s)ds is the conversion rate from prey to predator, and the distributed delay may interpret as digest delay. The function F(s) is called the delayed kernel that is a nonnegative bounded function defined on [0,∞). Following the ideas of Cushing et al. [34], we define F(t) as the following weak kernel function:
(2)F(t)=βe-βt,β>0.Next, we define a new variable:(3)u(t)=∫-∞t‍βe-β(t-s)x2(s)ds.
Then by use of linear chain trick technique, system (1) can be transformed into the following equivalent system:
(4)x1′(t)=rx2(t)-d1x1(t)-re-d1τx2(t-τ),x2′(t)=re-d1τx2(t-τ)-d2x2(t)-b1x2(t)y(t),u′(t)=βx2(t)-βu(t),y′(t)=y(t)[-d3+b2u(t)-αy(t)].The organization of this paper is as follows: In Section2, we will get the conditions for the existence and stability of positive equilibrium of system (4). The occurring condition for Hopf bifurcation is also obtained. In Section 3, by use of normal form theory and central manifold argument, we illustrate the direction and stability of Hopf bifurcation. In Section 4, we give some brief discussion.


## 2. Stability of Positive Equilibrium and Existence of Hopf Bifurcation

In this section, we will firstly investigate the existence and stability of positive equilibrium of system (4) then study the effect of time delay on the system (4); that is, we will choose τ as bifurcating parameter to analyze Hopf bifurcation.Theorem 1.
There exists a unique positive equilibriumE* for system (4), if assumption(H1)
re-d2τ-d2>0 holds. And E*=(x1*,x2*,u*,y*), with
(5)x1*=r(1-e-d1τ)(αre-d1τ-αd2+b1b3)b1b2d1,x2*=u*=α(re-d1τ-d2)+b1b3b1b2,y*=re-d1τ-d2b1.Linearizing system (4) at E*, we get
(6)x1′(t)=-d1x1(t)+rx2(t)-re-d1τx2(t-τ),x2′(t)=(-d2-b1y*)x2(t)+re-d1τx2(t-τ)-b1x2*y(t),u′(t)=βx2(t)-βu(t),y′(t)=b2y*u(t)-αy*y(t),
and the characteristic equation for system (6) takes the form
(7)λ4+h1λ3+h2λ2+h3λ+h4λ4+[h5λ3+h6λ2+h7λ+h8]e-λτ=0,
where
(8)h1=(d1+d2+b1y*)+(β+αy*),h2=d1(d2+b1y*)+(β+αy*)(d1+d2+b1y*)+αβy*,h3=d1(d2+b1y*)(β+αy*)+αβy*(d1+d2+b1y*),h4=αβd1y*(d2+b1y*)+βb1b2x2*y*,h5=-re-d1τ,h6=-(β+αy*+d1)re-d1τ,h7=-(αβy*+d1β+αd1y*)re-d1τ,h8=-αβd1y*re-d1τ.Note that whenτ=0, (7) becomes
(9)λ4+(h1+h5)λ3+(h2+h6)λ2+(h3+h7)λ+h4+h8=0.
It is easy to confirm that
(10)h1+h5=d1+β+αy*>0,h2+h6=(β+αy*)d1+αβy*>0,h3+h7=αβy*d1>0,h4+h8=αβb1b2x2*y*>0,(h1+h5)(h2+h6)-(h3+h7)>0,(h1+h5)[(h2+h6)(h3+h7)-(h1+h5)(h4+h8)]-(h3+h7)2>0.
Thus, by the Routh-Hurwitz criterion we know that all the roots of (9) have negative real parts, which means that the positive equilibrium E* is locally asymptotically stable for τ=0.Next, we will consider the case forτ>0. Suppose that there is a pure imaginary root λ=iω, ω>0. Then we get
(11)ω4-h1iω3-h2ω2+h3iω+h4+(-h5iω3-h6ω2+h7iω+h8)(cosωτ-isinωτ)=0.Separating the real and imaginary parts, we have(12)(h6ω2-h8)cosωτ+(h5ω3-h7ω)sinωτ=ω4-h2ω2+h4,(h6ω2-h8)sinωτ+(-h5ω3+h7ω)cosωτ=h1ω3-h3ω.
Incorporating sin2ωτ+cos2ωτ=1, we have
(13)ω8+f1ω6+f2ω4+f3ω2+f4=0,
where
(14)f1=h12-2h2-h52,f2=2h4+h22-2h1h3-h62+2h5h7,f3=h32-2h2h4+2h6h8-h72,f4=h42-h82.Denotez=ω2. Then (13) becomes
(15)z4+f1z3+f2z2+f3z+f4=0.
Let
(16)G(z)=z4+f1z3+f2z2+f3z+f4.
Then the following assumption holds true.(H2)
Equation (15) has at least one positive real root.In fact, if all the parameters of system (4) are given, it is easy to calculate the root of (15). Since limz→∞G(z)=+∞, we conclude that if f4<0, then (15) has at least one positive real root. Without loss of generality, we assume that (15) has four positive root, defined by z1, z2, z3, z4, respectively. Then (13) has four positive roots as
(17)ω1=z1,ω2=z2,ω3=z3,ω4=z4.
From (12), we obtain
(18)sinωτ=(h5ω7+(h1h6-h7-h2h5)ω5+(h2h7+h4h5-h1h8-h3h6)ω3+(h3h8-h4h7)ω)×(h52ω6+(h62-2h5h7)ω4+(h72-2h6h8)ω2+h82)-1,(19)cosωτ=((h6-h1h5)ω6+(h1h7+h3h5-h2h6-h8)ω4+(h4h6+h2h8-h3h7)ω2+h4h8)×(h52ω6+(h62-2h5h7)ω4+(h72-2h6h8)ω2+h82)-1.Denote(20)e1=h52,e2=h62-2h5h7,e3=h72-2h6h8,e4=h82,e5=h5,e6=h1h6-h7-h2h5,e7=h2h7+h4h5-h1h8-h3h6,e8=h3h8-h4h7,e9=h6-h1h5,e10=h1h7+h3h5-h2h6-h8,e11=h4h6+h2h8-h3h7,e12=-h4h8.
Then cosωτ can be written as
(21)cosωτ=e9ω6+e10ω4+e11ω2+e12e1ω6+e2ω4+e3ω2+e4,
from which we can get
(22)τk(j)=1ωk{arccos(e9ωk6+e10ωk4+e11ωk2+e12e1ωk6+e2ωk4+e3ωk2+e4)+2jπ},hhhhhhhhhhhhhhhhhhhhhhhhk=1,2,3,4;j=0,1,2,….Thus,±iωk is a pair of purely imaginary root of (7). Define
(23)τ0=τk0(0)=mink∈{1,2,3,4}{τk(0)},ω0=ω,In order to obtain the main result, it is necessary to make the following assumption:(H3)
Re(dλ/dτ)|τ=τ0≠0.Taking the derivative ofλ with respect to τ in (7), it is easy to obtain
(24)(4λ3+3h1λ2+2h2λ+h3)dλdτ+(3h5λ2+2h6λ+h7)e-λτdλdτ-(h5λ3+h6λ2+h7λ+h8)e-λτ(λ+τdλdτ)=0,
and it may be rewritten as
(25)dλdτ=((h5λ4+h6λ3+h7λ2+h8λ)e-λτ)×(τe-λτ4λ3+3h1λ2+2h2λ+h3+(3h5λ2+2h6λ+h7)e-λτ-(h5λ3+h6λ2+h7λ+h8)τe-λτ)-1,
or equivalently, we have
(26)(dλdτ)-1=(4λ3+3h1λ2+2h2λ+h3+(3h5λ2+2h6λ+h7)e-λτ-(h5λ3+h6λ2+h7λ+h8)τe-λτ)×((h5λ4+h6λ3+h7λ2+h8λ)e-λτ)-1=(4λ3+3h1λ2+2h2λ+h3+(3h5λ2+2h6λ+h7)e-λτ)×(h5λ4+h6λ3+h7λ2+h8λ)-1-τλ.Takingλ=iω into the above equation, we get
(27)(dλdτ)-1=(-4iω3-3h1ω2+2h2iω+h3+(-3h5ω2+2h6iω+h7)(cosωτ-isinωτ))×(h5ω4-h6iω3-h7ω2+h8iω)-1-τiω=(-3h1ω2+h3-(3h5ω2-h7)cosωτ+2h6sinωτ)×(h5ω4-h7ω2+i(h8ω-h6ω3))-1+(i[-4ω3+2h2ω+2h6ωcosωτ+(3h5ω2-h7)sinωτ])×(h5ω4-h7ω2+i(h8ω-h6ω3))-1-τiω.Denote(28)Q=(h5ω4-h7ω2)2+(h8ω-h6ω3)2>0.
Then we have
(29)QRe(dλdτ)-1=[-3h1ω2+h3-(3h5ω2-h7)cosωτ+2h6sinωτ]×(h5ω4-h7ω2)+[-4ω3+2h2ω+2h6ωcosωτ+(3h5ω2-h7)sinωτ]×(h8ω-h6ω3).Note that(30)Sign{Re(dλdτ)∣τ=τ0}=Sign{Re(dλdτ)-1∣τ=τ0}.
Now, we can employ a result from [35] to analyze (7).Lemma 2 (see [35]).
Consider the exponential polynomial(31)P(λ,e-λτ1,e-λτ2,…,e-λτm)=λn+p1(0)λn-1+⋯+pn-1(0)λ+pn(0)+[p1(1)λn-1+⋯+pn-1(1)λ+pn(1)]e-λτ1+⋯+[p1(m)λn-1+⋯+pn-1(m)λ+pn(m)]e-λτm,
where τi≥0(i=1,2,…,m) and pj(i)(i=0,1,…,m;j=1,2,…,n) are constants. As (τ1,τ2,…,τm) vary, the sum of the order of the zeros of P(λ,e-λτ1,…,e-λτm) on the open right half-plane can change only if a zero appears on or crosses the imaginary axis.From Lemma2, it is easy to obtain the following theorem.Theorem 3.
Suppose that (H1), (H2), and (H3) hold. Then the following results hold true.(i)
The positive equilibriumE* of system (4) (x1*,x2*,u*,y*) is asymptotically stable for τ∈[0,τ0).(ii)
The positive equilibriumE* of system (4) undergoes a Hopf bifurcation when τ=τ0. That is system (4) has a periodic solution bifurcating from the positive equilibrium E* near τ=τ0.


## 3. Direction and Stability of the Hopf Bifurcation

In this section, we will derive the explicit formulae for determining the properties of the Hopf bifurcation at critical value ofτ0 by using the normal form and the center manifold theory [35]. Throughout this section, we always assume that system (4) undergoes Hopf bifurcation at the positive equilibrium E* for τ=τ0, and then ±ω0 is the corresponding purely imaginary roots of the characteristic equation at the positive equilibrium E*.Letu1=x1-x1*,  u2=x2-x2*,  u3=u-u*,  u4=y-y*,  ui¯(t)=ui(τt), and τ=τ0+μ, and dropping the bars for simplification of notations, then system (4) is transformed into FDE defined in C=C([-1,0],R4) as
(32)u′(t)=Lμ(ut)+g(μ,xt),
where ut=(u1(t),u2(t),u3(t),u4(t))T∈R4,  Lμ:C→R,g:R×C→R, and
(33)Lμ(ϕ)=(τ0+μ)(-d1r000-d2-b1y*0-b1x2*0β-β000b2y*-αy*)×(ϕ1(0)ϕ2(0)ϕ3(0)ϕ4(0))+(τ0+μ)(0-re-d1τ000re-d1τ0000000000)(ϕ1(-1)ϕ2(-1)ϕ3(-1)ϕ4(-1)),(34)g(μ,ϕ)=(τ0+μ)(0-b1ϕ2(0)ϕ4(0)0b2ϕ4(0)ϕ3(0)-αϕ4(0)2),
where ϕ(θ)=(ϕ1(θ),ϕ2(θ),ϕ3(θ),ϕ4(θ))T∈C. By the Riesz representation theorem, there exists a function η(θ,μ) of bounded variation for θ∈[-1,0], such that
(35)Lμϕ=∫-10‍dη(θ,μ)ϕ(θ),forϕ∈C.In fact, we can choose(36)η(θ,μ)=(τ0+μ)×(-d1r000-d2-b1y*0-b1x2*0β-β000b2y*-αy*)δ(θ)+(τ0+μ)(0-re-d1(τ0+μ)000re-d1(τ0+μ)0000000000)δ(θ+1),
where δ is the Dirac delta function. For ϕ∈C1([-1,0],R4), define
(37)A(μ)ϕ={dϕ(θ)dθ,θ∈[-1,0),∫-10dη(s,μ)ϕ(s),θ=0,R(μ)ϕ={0,θ∈[-1,0),g(μ,ϕ),θ=0.
Then system (32) is equivalent to
(38)u′(t)=A(μ)ut+R(μ)ut,
where ut=u(t+θ),θ∈[-1,0]. For ψ∈C1([0,1],v(R4)*), define
(39)A*ψ(s)={-dψ(s)ds,s∈(0,1],∫-10dηT(t,0)ψ(-t),s=0,
and a bilinear inner product
(40)〈ψ(s),ϕ(θ)〉=ψ¯(0)ϕ(0)-∫-10‍∫ξ=0θ‍ψ¯(ξ-θ)dη(θ)ϕ(ξ)dξ,
where η(θ)=η(θ,0). Then A(0) and A* are adjoint operators. By the discussion in Section 2, we know that ±iω0τ0 are eigenvalues of A(0). Thus, they are also eigenvalues of A*. We need to compute the eigenvector of A(0) and A* corresponding to iω0τ0 and -iω0τ0, respectively.Suppose thatq(θ)=(1,q1,q2,q3)Teiθω0τ0 is the eigenvectors of A(0) corresponding to iω0τ0. Then A(0)q(θ)=iω0τ0q(θ). It follows from the definition of A(0) and η(θ,μ) that
(41)τ0(-d1r000-d2-b1y*0-b1x2*0β-β000b2y*-αy*)q(0)+τ0(0-re-d1τ0000re-d1τ00000000000)q(-1)=iω0τ0q(0).
Because of q(-1)=q(0)e-iω0τ0, then we get
(42)(iω0+d1-r+re-d1τ0e-iω0τ0000iω0+d2+b1y*-re-d1τ0e-iω0τ00b1x2*0-βiω0+β000-b2y*iω0+αy*)×(1q1q2q3)=(0000),
from which we obtain
(43)q1=iω0-d1r(1-e-d1τ0e-iω0τ0),q2=β(d1-iω0)r(iω0+β)(1-e-d1τ0e-iω0τ0),q3=b2y*β(iω0+d1)r(iω0+β)(iω0+αy*)(1-e-d1τ0e-iω0τ0).Similarly, letq*(θ)=D(1,q1*,q2*,q3*)eiθω0τ0 be the eigenvectors of A* corresponding to -iω0τ0, and according to the definition of A* we have
(44)τ0(-d1000r-d2-b1y*β000-βb2y*0-b1x2*0-αy*)q*(0)+τ0(0000-re-d1τ0re-d1τ00000000000)q*(-1)=-iω0τ0q*(0).
Note that q*(-1)=q*(0)eiω0τ0. Then we get
(45)(-iω0+d1000re-d1τ0eiω0τ0-rd2-iω0+b1y*-re-d1τ0eiω0τ0-β000β-iω0-b2y*0b1x2*0αy*-iω0)×(1q1*q2*q3*)=(0000),
from which we can obtain
(46)q1*=((iω0-αy*)(iω0+β)(1-e-d1τ0eiω0τ0))×((iω0+β)(iω0-αy*)×(-iω0+d2+b1y*-re-d1τ0e-iω0τ0)-βb1x2*b2y*)-1,q2*=(rb1x2*b2y*(iω0+β)(1-e-d1τ0eiω0τ0))e×((ω02+β2)(iω0-αy*)e×(-iω0+d2+b1y*-re-d1τ0e-iω0τ0)e-(β-iω0)βb1x2*b2y*)-1,q3*=(rb1x2*(iω0+β)(1-e-d1τ0eiω0τ0))×((iω0+β)(iω0-αy*)×(-iω0+d2+b1y*-re-d1τ0e-iω0τ0)-βb1x2*b2y*)-1.
By (40), we get
(47)〈q*(s),q(θ)〉=D¯(1,q¯1*,q¯2*,q¯3*)(1,q1,q2,q3)T-∫-10‍∫ξ=0θ‍D¯(1,q¯1*,q¯2*,q¯3*)×e-iω0τ0(ξ-θ)dη(θ)(1,q1,q2,q3)Teiω0τ0ξdξ=D¯{-∫-10‍(1,q1*¯,q2*¯,q3*¯)θeiω0τ0θdη(θ)(1,q1,q2,q3)T}1+q1q¯1*+q2q¯2*+q3q¯3*-∫-10‍(1,q¯1*,q¯2*,q¯3*)θeiω0τ0θdη(θ)(1,q1,q2,q3)T}=D¯[(q¯1*-1)e-d2τ0e-iω0τ01+q1q¯1*+q2q¯2*+q3q¯3*+τ0rq1(q¯1*-1)e-d2τ0e-iω0τ0].
Then we can choose D¯ such that 〈q*(s),q(θ)〉=1, 〈q*(s),q¯(θ)〉=0.Nest, we will use the ideas in [35] to compute the coordinates describing center manifold C0 at μ=0. Define
(48)z(t)=〈q*,ut〉,W(t,θ)=ut(θ)-2Re{z(t),q(θ)}.On the center manifoldC0, we have W(t,θ)=W(z(t),z¯(t),θ), and
(49)W(z(t),z¯(t),θ)=W20(θ)z22+W11(θ)zz¯+W02z¯22+…,
where z and z¯ are local coordinates for center manifold C0 in the direction of q*(s) and q¯(θ). Note that W is real if ut is real, and we only consider real solutions. For the solution ut∈C0 of (38), since μ=0 and (38), we have
(50)z′(t)=iω0τ0z+q¯*(0)g(0,W(z,z¯,0))+2Re{zq(0)}=iω0τ0z+q¯*(0)g0(z,z¯).
Then, the above equation can be denoted as
(51)z′(t)=iω0τ0z(t)+f(z,z¯),
where
(52)f(z,z¯)=q¯*(0)g0(z,z¯)=f20z22+f11zz¯+f02z¯22+f21z2z¯2+⋯.
From (48) and (49), we have
(53)ut=(u1t(θ),u2t(θ),u3t(θ),u4t(θ))=W(t,θ)+2Re{z(t),q(θ)}=W(t,θ)+zq(θ)+z¯q(θ)¯,
and then we can obtain
(54)u2t(0)=W20(2)(0)z22+W11(2)(0)zz¯+W02(2)(0)z¯22+q1z+q1¯z¯+O(|z,z¯3|),u3t(0)=W20(3)(0)z22+W11(3)(0)zz¯+W02(3)(0)z¯22+q2z+q2¯z¯+O(|z,z¯|3),u4t(0)=W20(4)(0)z22+W11(4)(0)zz¯+W02(4)(0)z¯22+q3z+q3¯z¯+O(|(z,z¯)|3).From the definition ofg(μ,ut), we have
(55)f(z,z¯)=q¯*(0)g0(z,z¯)=τ0D¯(1,q¯1*,q¯2*,q¯3*)×(0-2b1u2t(0)u4t(0)0b2u3t(0)u4t(0)-αu4t2(0))(56)=τ0D¯{-αq3*(2W20(4)(0)q3¯+4W11(4)(0)q3)]z2[-2b1q¯1*q1q3+b2q¯3*q2q3-αq¯3*q32]eeeeeeeeeeeee+2zz¯[-2b1q¯1*Re{q1q3¯}+b2q¯3*Re{q2q3¯}eeeeeeeeeeeeeeeee-αq¯3*Re{q3q3¯}]eeeeeeeeeeeee+z2¯[-b1q¯1*q1q3¯+b2q¯3*q2¯q3¯-αq¯3*q3¯q3¯]eeeeeeeeeeeee+12z2z¯[-b1q1*(W20(2)(0)q3¯+2W11(2)(0)q3eeeeeeeeeeeeeeeeeeeee+2W11(4)(0)q1+W20(4)(0)q1¯)eeeeeeeeeeeeeeeeeeeee+b2q3*(W20(3)(0)q3¯+2W11(3)(0)q3eeeeeeeeeeeeeeeeeeeee+2W11(4)(0)q2+W20(4)(0)q2¯)eeeeeeeeeeeeeeeeeeeee-αq3*(2W20(4)(0)q3¯+4W11(4)(0)q3)]eeeeeeeeeeeee+⋯z2[-2b1q¯1*q1q3+b2q¯3*q2q3-αq¯3*q32]}.Comparing the coefficients with those of (52), we obtain
(57)f20=2τ0D¯[-b1q¯1*q1q3+b2q¯3*q2q3-αq¯3*q32],f11=2τ0D¯[-b1q¯1*Re{q1q3¯}e+b2q¯3*Re{q2q3¯}-αq¯3*Re{q3q3¯}],f02=2τ0D¯[-b1q¯1*q1¯q3¯+b2q¯3*q2¯q3¯-αq¯3*q32¯],f21=τ0D¯[-b1q1*(W20(2)(0)q3¯eeeeeeeeeeee+2W11(2)(0)q3+2W11(4)(0)q1eeeeeeeeeeee+W20(4)(0)q1¯)e+b2q3*(W20(3)(0)q3¯eeeeeeeeeeeee+2W11(3)(0)q3+2W11(4)(0)q2eeeeeeeeeeeee+W20(4)(0)q2¯)-αq3*(2W20(4)(0)q3¯+4W11(4)(0)q3)].In order to determinef21 we need to compute W20(θ) and W11(θ). From (38) and (48), we have
(58)W˙=u˙t-z˙q-zq-˙={A(0)W-2Re{q¯*(0)g0q(θ)},θ∈[-1,0),A(0)W-2Re{q¯*(0)g0q(θ)}+g0,θ=0,=A(0)W-H(z,z¯,θ),
where
(59)H(z,z¯,θ)=H20(θ)z22+H11(θ)zz¯+H02z2¯2+⋯.Note that on the center manifoldC0 near the origin
(60)W˙=Wzz˙+Wz¯z¯˙,
and thus we obtain
(61)(A(0)-2iω0τ0)W20(θ)=-H20(θ),A(0)W11(θ)=-H11(θ).By (58), we know that, for θ∈[-1,0),
(62)H(z,z¯,θ)=-q¯*(0)g0q(θ)-q0*g0¯q¯(θ)=-fq(θ)-f¯q(θ)¯.Comparing the coefficients with those in (59), we get
(63)H20(θ)=-f20q(θ)-f¯02q¯(θ),H11(θ)=-f11q(θ)-f¯11q¯(θ).From (61), (63), and the definition of A, we have
(64)W˙20(θ)=2iω0τ0W20(θ)+f20q(θ)+f¯02q¯(θ).Noting thatq(θ)=q(0)eiω0τ0θ, we get
(65)W20(θ)=if20ω0τ0q(0)eiω0τ0θ+if¯023ω0τ0q¯(0)e-iω0τ0θ+E1e2iω0τ0θ,
where E1=(E1(1),E1(2),E1(3),E1(4))∈R4 is a constant vector.Similarly, from (61) and (62), we can get
(66)W11(θ)=if¯11ω0τ0q¯(0)e-iω0τ0θ+E2,
where E2=(E2(1),E2(2),E2(3),E2(4))∈R4 is a constant vector.Next, we will find outE1 and E2. In fact, from the definition of A and (61), we can obtain
(67)∫-10‍dη(θ)W20(θ)=2iω0τ0W20(0)-H20(0),(68)∫-10‍dη(θ)W11(θ)=-H11(0),
where η(θ)=η(0,θ). From (59) and (61), when θ=0 we have
(69)H(z,z¯,0)=-2Re{q¯*(0)g0q(0)}+g0=-q¯*(0)g0q(0)-q*(0)g0¯q¯(0)+g0=-g(z,z¯)q(0)-g¯(z,z¯)q¯(0)+g0.
That is,
(70)H20(0)z22+H11(0)zz¯+H02(0)z2¯2=-q(0)(f20z22+f11zz¯+f02z2¯2+⋯)-q¯(0)(f¯20z¯22+f¯11zz¯+f02¯z22+⋯)+g0.By (34), we have
(71)g0=τ0(0-b1u2t(0)u4t(0)0b2u3t(0)u4t(0)-αu4t2(0)).By (48) and (49), we have
(72)ut(θ)=W(t,θ)+2Re{z(t)q(θ)}=W(t,θ)+z(t)q(θ)+z¯(t)q¯(θ)=W20(θ)z22+W11(θ)zz¯+z(t)q(θ)+z¯(t)q¯(θ)+⋯.Thus, we obtain(73)H20(0)=-f20q(0)-f¯02q¯(0)+2τ0(0-b1q1q30b2q2q3-αq32),(74)H11(0)=-f11q(0)-f¯11q¯(0)+2τ0×(0-b1Re{q1q3¯}0b2Re{q2q3¯}-αq¯3*|q3|2).Note that(75)(iω0τ0I-∫-10‍eiω0τ0θdη(θ))q(0)=0,(-iω0τ0I-∫-10‍e-iω0τ0θdη(θ))q¯(0)=0.
Then, by substituting (65) and (73) into (67), we will get
(76)(2iω0τ0I-∫-10‍e2iω0τ0θdη(θ))E1=2τ0(0-b1q1q30b2q2q3-αq32),
or equivalently,
(77)(2iω0+d1-r+re-d1τ0e-2iω0τ00002iω0+d2+b1y*-re-d1τ0e-2iω0τ00b1x2*0-β2iω0+β000-b2y*2iω0-αy*)E1=2(0-b1q1q30b2q2q3-αq32),
from which we can get
(78)E1=2(2iω0+d1-r+re-d1τ0e-2iω0τ00002iω0+d2+b1y*-re-d1τ0e-2iω0τ00b1x2*0-β2iω0+β000-b2y*2iω0-αy*)-1×(0-b1q1q30b2q2q3-αq32).Similarly, by substituting (66) and (74) into (68), we will get
(79)(d1-r+re-d1τ0000d2+b1y*-re-d1τ00b1x2*0-ββ000-b2y*-αy*)E2=2(0-b1Re{q1q3¯}0b2Re{q2q3¯}-αRe{q3q3¯}),E2=2(d1-r+re-d1τ0000d2+b1y*-re-d1τ00b1x2*0-ββ000-b2y*-αy*)-1×(0-b1Re{q1q3¯}0b2Re{q2q3¯}-αRe{q3q3¯})Thus, we can determineW20(θ) and W11(θ) from (65) and (66). Furthermore, f21 can be expressed by the parameters and delay. Thus, we can compute the following values:
(80)c1(0)=i2ω0τ0(f20f11-2|f11|2-|f02|23)+f212,μ2=-Re{c1(0)}Re{λ′(τ0)},β2=2Re{c1(0)},T2=-Im{c1(0)}+μ2Im{λ′(τ0)}ωτ0,
which determine the qualities of bifurcating periodic solution in the center manifold at critical value τ0; that is, μ2 determines the direction of the Hopf bifurcation: if μ2>0(μ2<0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solution exists for τ>τ0(τ<τ0);β2 determines the stability of the bifurcating periodic solution: the bifurcating periodic solution is stable (unstable) if β2<0(β2>0); and T2 determines period of the bifurcating periodic solution: the period increases (decreases) if T2>0(<0).


## 4. Discussion

In this paper, we propose a three-dimensional stage-structured predatory-prey model with discrete and distributed delay. Then, by introducing a new variable, the original system is transformed into an equivalent four-dimensional system (4). In Section 2, we analyze the existence and local stability of the positive equilibrium of the system (4). In fact, by use of the Routh-Hurwitz criterion, we know that the positive equilibrium E* is locally asymptotically stable for τ=0. Then, by some computation, we get a threshold τ=τ0 in Theorem 3. The results indicate that if τ<τ0, the positive equilibrium E* of system (4) is asymptotically stable; if τ=τ0, the positive equilibrium E* of system (4) will undergo a Hopf bifurcation. In Section 3, by use of normal form theory and central manifold argument, we establish the formulae for the direction and the stability of the Hopf bifurcation. Our theoretical results show that stage structure and delay play an important role in the dynamics of the system, and delay may lead to complicated dynamic behaviors, such as Hopf bifurcation.



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*Source: 201936-2013-11-03.xml*