# Dynamical Analysis in a Delayed Predator-Prey Model with Two Delays

Authors: Changjin Xu; Peiluan Li
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

A class of Beddington-DeAngelis functional response predator-prey model is considered. The conditions for the local stability and the existence of Hopf bifurcation at the positive
equilibrium of the system are derived. Some explicit formulae for determining the stability and
the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are
obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, main conclusions are given.


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## Body

## 1. Introduction

In recent years, population dynamics (including stable, unstable, persistent, and oscillatory behavior) has become very popular since Vito Volterra and James Lotka proposed the seminal models of predator-prey models in the mid-1920s. Great attention has been paid to the dynamics properties of the predator-prey models which have significant biological background. Many excellent and interesting results have been obtained [1–21]. In 2009, Gakkhar et al. [4] investigated the local stability and Hopf bifurcation of the autonomous delayed predator-prey system with Beddington-DeAngelis functional response:(1.1)u̇1(t)=u1(t)[1-u1(t-τ1)]-a1u1(t)u2(t)a+u1(t)+bu2(t),u̇2(t)=d1u2(t)[-d+a1u1(t-τ2)a+u1(t-τ2)+bu2(t-τ2)],
where u1(t),u2(t) represent the prey density and the predator density, respectively. The delay terms occur in growth as well as interaction terms. For this, it means that the prey takes time τ1 to convert the food into its growth [11], whereas the predator takes time τ2 for the same [22]. All the parameters in the model take positive values, that is, a1>0,d>0,d1>0,a>0,b>0. The more detail biological meaning of the coefficients of system (1.1), one can see [11] or [22].We would like to point out that Gakkhar et al. [4] studied the local stability and Hopf bifurcation of system (1.1) under the assumption: τ1=τ2=τ and obtained some excellent results. While in most cases, τ1≠τ2. Considering the factor, we further investigate the model (1.1) with τ1≠τ2 as a complementarity.In this paper, we go on to study the stability, the local Hopf bifurcation for system (1.1). To the best of our knowledge, it is the first time to deal with the research of Hopf bifurcation for model (1.1) under the assumption τ1≠τ2.The remainder of the paper is organized as follows. In Section2, we investigate the stability of the positive equilibrium and the occurrence of local Hopf bifurcations. In Section 3, the direction and stability of the local Hopf bifurcation are established. In Section 4, numerical simulations are carried out to illustrate the validity of the main results. Some main conclusions are drawn in Section 5.


## 2. Stability of the Positive Equilibrium and Local Hopf Bifurcations

In this section, we shall study the stability of the positive equilibrium and the existence of local Hopf bifurcations.Since time delay does not change the equilibrium of system and according to [4], we know that the delayed prey predator model (1.1) has three equilibrium points: two boundary equilibrium E1(0,0) and E2(1,0), and a nontrivial equilibrium point E0(u1*,u2*), where u1*,u2* are the positive solutions of the following quadratic equations:(2.1)u12+α1u1+β1=0,u22+α2u2+β2=0,
where(2.2)α1=a1-da1-bb,β1=-ada1d,α2=a1(d-1)2+b(2ad+d-1)b2d,β2=-a(ad+d-1)b2d.
Since β1<0, (2.1) in u1 admits a unique positive solution. If one of the following conditions:(2.3)(a)α2>0,β2<0,(b)α2<0,β2=α224,(c)α2<0,0<β2<α224,
holds, then system (2.1) has at least one positive equilibrium point E0(u1*,u2*).Letu¯1(t)=u1(t)-u1*,u¯2(t)=u2(t)-u2* and still denote u¯i(t)(i=1,2) by ui(t)(i=1,2), respectively, then (1.1) becomes(2.4)u̇1(t)=m1u1(t)+m2u2(t)+m3u1(t-τ1)+F1,u̇2(t)=n1u2(t)+n2u1(t-τ2)+n3u2(t-τ2)+F2,
where mi,ni(i=1,2,3) and Fj(j=1,2) are defined by Appendix A.The linearization of (2.4) at (0,0) is(2.5)u̇1(t)=m1u1(t)+m2u2(t)+m3u1(t-τ1),u̇2(t)=n1u2(t)+n2u1(t-τ2)+n3u2(t-τ2),
whose characteristic equation is(2.6)λ2-(m1+n1)λ+m1n1-(n3λ-m1n3+m2n2)e-λτ2-(m3λ+m3n1)e-λτ1+m3n3e-λ(τ1+τ2)=0.
In order to investigate the distribution of roots of the transcendental equation (2.6), the following Lemma is useful.Lemma 2.1 (see [23]).
For the transcendental equation(2.7)P(λ,e-λτ1,…,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=0,
as (τ1,τ2,τ3,…,τm) vary, the sum of orders of the zeros of P(λ,e-λτ1,…,e-λτm) in the open right half plane can change, and only a zero appears on or crosses the imaginary axis.In the sequel, we consider three cases.Case a.
τ1=τ2=0, (2.6) becomes
(2.8)λ2-(m1+m3+n1+n3)λ+m1n1+m1n3+m3n3-m2n2-m3n1=0.
A set of necessary and sufficient conditions for all roots of (2.8) to have a negative real part are given in the following form:
(2.9)(H1)(m1+m3+n1+n3)<0,m1n1+m1n3+m3n3-m2n2-m3n1>0.
Then, the equilibrium point E0(u1*,u2*) is locally asymptotically stable when the condition (H1) holds.Case b.
τ1=0,τ2>0, (2.6) becomes
(2.10)λ2+pλ+r+(sλ+q)e-λτ2=0,
where
(2.11)p=-(m1+m3+n1),r=m1n1-m3n1,s=-n3,q=m1n3+m3n3-m2n2.
For ω>0,iω be a root of (2.10), then it follows that
(2.12)qcos⁡ωτ2+sωsin⁡ωτ2=ω2-r,sωcos⁡ωτ2-qsin⁡ωτ2=-pω,
which leads to
(2.13)ω4+(p2-s2-2r)ω2+r2-q2=0.
It is easy to see that if the condition
(2.14)(H2)p2-s2-2r>0,r2-q2>0
holds, then (2.13) has no positive roots. Hence, all roots of (2.10) have negative real parts when τ2∈[0,+∞) under the conditions (H1) and (H2).
If(H1) and
(2.15)(H3)r2-q2<0
hold, then (2.13) has a unique positive root ω02. Substituting ω02 into (2.12), we obtain
(2.16)τ2n=1ω0{arccosq(ω02-r)-psω02s2ω02+q2+2nπ},n=0,1,2,….
If (H1) and
(2.17)(H4)p2-s2-2r<0,r2-q2>0,(p2-s2-2r)2>4(r2-q2)
hold, then (2.10) has two positive roots ω+2 and ω-2. Substituting ω±2 into (2.12), we obtain
(2.18)τ2k±=1ω±{arccosq(ω±2-r)-psω±2s2ω±2+q2+2kπ},k=0,1,2,….
Let λ(τ2)=α(τ2)+iω(τ2) be a root of (2.10) near τ2=τ2n and α(τ2n)=0,ω(τ2n)=ω0. Due to functional differential equation theory, for every τ2n,n=0,1,2,…, there exists ɛ>0 such that λ(τ2) is continuously differentiable in τ2 for |τ2-τ2n|<ɛ. Substituting λ(τ2) into the left-hand side of (2.10) and taking derivative with respect to τ2, we have
(2.19)[dλdτ2]-1=(2λ+p)eλτ2λ(sλ+q)+sλ(sλ+q)-τ2λ,
which leads to
(2.20)[d(Reλ(τ))dτ2]τ2=τ2n-1=Re{(2λ+p)eλτ2λ(sλ+q)}|τ2=τ2n+Re{sλ(sλ+q)}|τ2=τ2n=Re{pcos⁡ω0τ2n-2ω0sin⁡ω0τ2n+i(2ω0cos⁡ω0τ2n+psin⁡ω0τ2n)-sω02+iqω0}+Re{s-sω02+iqω0}=1Λ{-sω02(pcos⁡ω0τ2n-2ω0sin⁡ω0τ2n)+qω0(2ω0cos⁡ω0τ2n+psin⁡ω0τ2n)-s2ω02}=1Λ{pω0(qsin⁡ω0τ2n-sω0cos⁡ω0τ2n)+2ω02(qcos⁡ω0τ2n+sω0sin⁡ω0τ2n)-s2ω02}=1Λ{p2ω02+2ω04-2rω02-s2ω02}=ω02Λ{2ω02+p2-s2-2r}=1Λ{-p2+s2+2r+Δ*+p2-s2-2r}=ω02ΛΔ*>0,
where
(2.21)Λ=s2ω04+q2ω02>0,Δ*=(s2-p2+2r)2-4(r2-q2).
Noting that
(2.22)sign⁡{d(Reλ)dτ2}|τ2=τ2n=sign⁡{Re[dλdτ2]}|τ2=τ2n=1,
we have
(2.23)d(Reλ)dτ2|τ2=τ2n>0.
Similarly, we can obtain
(2.24)d(Reλ)dτ2|τ2=τ2k+>0,d(Reλ)dτ2|τ2=τ2k-<0.
According to above analysis the Corollary  2.4 in Ruan and Wei [23], we have the following results.Lemma 2.2.
Forτ1=0,  assume that one of the conditions (a), (b), (c), and (d) holds and (H1) is satisfied. Then, the following conclusions hold.(i)
If(H2) holds, then the positive equilibrium E0(u1*,u2*) of system (1.1) is asymptotically stable for all τ2≥0.(ii)
If(H3) holds, then the positive equilibrium E0(u1*,u2*) of system (1.1) is asymptotically stable for τ2<τ20, and unstable for τ2<τ20. Furthermore, system (1.1) undergoes a Hopf bifurcation at the positive equilibrium E0(u1*,u2*) when τ2=τ20.(iii)
If(H4) holds, then there is a positive integer m such that the positive equilibrium E0(u1*,u2*) is stable when τ2∈[0,τ20+)∪(τ20-,τ21+)∪⋯∪(τ2m-1-,τ2m+), and unstable when τ2∈[τ20+,τ20-)∪(τ21+,τ21-)∪⋯∪(τ2m+,τ2m-)∪(τ2m+,∞). Furthermore, system (1.1) undergoes a Hopf bifurcation at the positive equilibrium E0(u1*,u2*) when τ2=τ2k±,k=0,1,2,….Case c (τ1>0,τ2>0)
We consider (2.6) with τ2 in its stable interval. Regarding τ1 as a parameter, without loss of generality, we consider system (1.1) under the assumptions (H1) and (H3). Let iω(ω>0) be a root of (2.6), then we can obtain
(2.25)ω4+k1ω3+k2ω2+k3ω+k4=0,
where
(2.26)k1=2n3sin⁡ωτ2,k2=2[(m2n2-m1n3)cos⁡ωτ2-m1n1]+(m1+n1+n3cos⁡ωτ2)2-m32,k3=2[(m2n2-m1n3)cos⁡ωτ2]n3sin⁡ωτ2-2m32n3sin⁡ωτ2+2(m1+n1+n3cos⁡ωτ2)(m2n2-m1n3)sin⁡ωτ2,k4=[(m2n2-m1n3)cos⁡ωτ2-m1n1]2+[(m2n2-m1n3)sin⁡ωτ2]2-(m3n1-m3n3cos⁡ωτ2)2-(m3n3sin⁡ωτ2)2.
Denote
(2.27)H(ω)=ω4+k1ω3+k2ω2+k3ω+k4.
Assume that
(2.28)(H5)(m2n2-m1n3-m1n1)2<(m3n1-m3n3)2.
It is easy to check that H(0)<0 if (H5) holds and limω→+∞H(ω)=+∞. We can obtain that (2.25) has finite positive roots ω1,ω2,…,ωn. For every fixed ωi,i=1,2,3,…,k, there exists a sequence {τ1ij∣j=1,2,3,…}, such that (2.25) holds. Let
(2.29)τ10=min⁡{τ1ij∣i=1,2,…,k;j=1,2,…}.
When τ1=τ10, (2.6) has a pair of purely imaginary roots ±iω* for τ2∈[0,τ20).
In the following, we assume that(2.30)(H6)[d(Reλ)dτ1]λ=iω*≠0.
Thus, by the general Hopf bifurcation theorem for FDEs in Hale [24], we have the following result on the stability and Hopf bifurcation in system (1.1).Theorem 2.3.
For system (1.1), assume that one of the conditions (a), (b), (c), and (d) holds and suppose (H1),(H3), and (H5) are satisfied, and τ2∈[0,τ20),  then the positive equilibrium E0(u1*,u2*) is asymptotically stable when τ1∈(0,τ10),  and system (1.1) undergoes a Hopf bifurcation at the positive equilibrium E0(u1*,u2*) when τ1=τ10.


## 3. Direction and Stability of the Hopf Bifurcation

In the previous section, we obtained conditions for Hopf bifurcation to occur whenτ1=τ10. In this section, we shall derive the explicit formulae determining the direction, stability, and period of these periodic solutions bifurcating from the positive equilibrium E0(u1*,u2*) at this critical value of τ1, by using techniques from normal form and center manifold theory [7]. Throughout this section, we always assume that system (1.1) undergoes Hopf bifurcation at the positive equilibrium E0(u1*,u2*) for τ1=τ10, and then ±iω* is corresponding purely imaginary roots of the characteristic equation at the positive equilibrium E0(u1*,u2*).Without loss of generality, we assume thatτ2*<τ10, where τ2*∈(0,τ20). For convenience, let u¯i(t)=ui(τt)(i=1,2) and τ1=τ10+μ, where τ10 is defined by (2.28) and μ∈R, drop the bar for the simplification of notations, then system (1.1) can be written as an FDE in C=C([-1,0],R2) as(3.1)u̇(t)=Lμ(ut)+F(μ,ut),
where u(t)=(u1(t),u2(t))T∈C and ut(θ)=u(t+θ)=(u1(t+θ),u2(t+θ))T∈C, and Lμ:C→R,F:R×C→R are given by(3.2)Lμϕ=(τ10+μ)B(ϕ1(0)ϕ2(0))+(τ10+μ)C(ϕ1(-τ2*τ10)ϕ1(-τ2*τ10))+(τ10+μ)D(ϕ1(-1)ϕ2(-1)),(3.3)F(μ,ϕ)=(τ10+μ)(f1,f2)T,
respectively, where ϕ(θ)=(ϕ1(θ),ϕ2(θ))T∈C,(3.4)B=(m1m20n1),C=(00n2n3),D=(m3000),f1=-ϕ1(0)ϕ1(-1)+l1ϕ12(0)+l2ϕ22(0)+l3ϕ1(0)ϕ2(0)+l4ϕ13(0)+l5ϕ23(0)+l6ϕ12(0)ϕ2(0)+l7ϕ1(0)ϕ22(0)+h.o.t.,f2=k1ϕ1(-τ2*τ10)ϕ2(0)+k2ϕ2(0)ϕ2(-τ2*τ10)+k3ϕ12(-τ2*τ10)+k4ϕ1(-τ2*τ10)ϕ2(-τ2*τ10)+e1ϕ12(-τ2*τ10)ϕ2(0)+e2ϕ1(-τ2*τ10)φ2(-τ2*τ10)ϕ2(0)+e3ϕ13(-τ2*τ10)+e4ϕ1(-τ2*τ10)ϕ22(-τ2*τ10)+e5ϕ23(-τ2*τ10)+h.o.t.
From the discussion in Section 2, we know that, if μ=0, then system (3.1) undergoes a Hopf bifurcation at the positive equilibrium E0(u1*,u2*) and the associated characteristic equation of system (3.1) has a pair of simple imaginary roots ±ω*τ10.By the representation theorem, there is a matrix function with bounded variation componentsη(θ,μ),θ∈[-1,0] such that(3.5)Lμϕ=∫-10dη(θ,μ)ϕ(θ),forϕ∈C.
In fact, we can choose(3.6)η(θ,μ)={(τ10+μ)(B+C+D),θ=0,(τ10+μ)(C+D),θ∈[-τ2*τ10,0),(τ10+μ)D,θ∈(-1,-τ2*τ10),0,θ=-1.Forϕ∈C([-1,0],R2), define
(3.7)A(μ)ϕ={dϕ(θ)dθ,-1≤θ<0,∫-10dη(s,μ)ϕ(s),θ=0,Rϕ={0,-1≤θ<0,F(μ,ϕ),θ=0.
Then, (3.1) is equivalent to the abstract differential equation(3.8)u̇t=A(μ)ut+R(μ)ut,
where ut(θ)=u(t+θ),θ∈[-1,0].Forψ∈C([0,1],(R2)*), define(3.9)A*ψ(s)={-dψ(s)ds,s∈(0,1],∫-10dηT(t,0)ψ(-t),s=0.
For ϕ∈C([-1,0],R2) and ψ∈C([0,1],(R2)*), define the bilinear form(3.10)〈ψ,ϕ〉=ψ¯(0)ϕ(0)-∫-10∫ξ=0θψT(ξ-θ)dη(θ)ϕ(ξ)dξ,
where η(θ)=η(θ,0), the A=A(0) and A* are adjoint operators. By the discussions in Section 2, we know that ±iω*τ10 are eigenvalues of A(0), and they are also eigenvalues of A* corresponding to iω*τ10 and -iω*τ10, respectively. By direct computation, we can obtain(3.11)q(θ)=(1,α)Teiω*τ10θ,q*(s)=M(1,α*)eiω*τ10s,M=1K,
where(3.12)α=iω*-m1-m3e-iω*τ10m2,α*=-iω*+m1+m3e-iω*τ10n2e-iω*τ2*/τ10,K=1+α¯α*+τ10[m3eiω*τ10+τ2*τ10n2α¯eiω*(τ2*/τ10)+α¯α*n3τ2*τ10eiω*(τ2*/τ10)].
Furthermore, 〈q*(s),q(θ)〉=1 and 〈q*(s),q¯(θ)〉=0.Next, we use the same notations as those in Hassard [7] and we first compute the coordinates to describe the center manifold C0 at μ=0. Let ut be the solution of (3.1) when μ=0.Define(3.13)z(t)=〈q*,ut〉,W(t,θ)=ut(θ)-2Re{z(t)q(θ)},
on the center manifold C0, and we have(3.14)W(t,θ)=W(z(t),z¯(t),θ),
where(3.15)W(z(t),z¯(t),θ)=W(z,z¯)=W20z22+W11zz¯+W02z¯22+⋯,
and z and z¯ are local coordinates for center manifold C0 in the direction of q* and q¯*. Noting that W is also real if ut is real, we consider only real solutions. For solutions ut∈C0 of (3.1),(3.16)ż(t)=iω*τ10z+q¯*(θ)F(0,W(z,z¯,θ))+2Re{zq(θ)}=defiω*τ10z+q¯*(0)F0.
That is(3.17)ż(t)=iω*τ10z+g(z,z¯),
where(3.18)g(z,z¯)=g20z22+g11zz¯+g02z¯22+⋯.
Hence, we have to obtain the expression of g(z,z¯) (see Appendix B). Then, it is easy to obtain the expression of g20,g11,g02,g21 (see Appendix B).For unknownW20(i)(θ),W11(i)(θ),(i=1,2) in g21, we still need to compute them.From (3.8), (3.13), we have(3.19)W′={AW-2Re{q¯*(0)f¯q(θ)},-1≤θ<0,AW-2Re{q¯*(0)f¯q(θ)}+F¯,θ=0,=defAW+H(z,z¯,θ),
where(3.20)H(z,z¯,θ)=H20(θ)z22+H11(θ)zz¯+H02(θ)z¯22+⋯.
Comparing the coefficients, we obtain(3.21)(A-2iτ10ω*)W20=-H20(θ),(3.22)AW11(θ)=-H11(θ),
and we know that, for θ∈[-1,0),(3.23)H(z,z¯,θ)=-q¯*(0)f0q(θ)-q*(0)f¯0q¯(θ)=-g(z,z¯)q(θ)-g¯(z,z¯)q¯(θ).
Comparing the coefficients of (3.23) with (3.20) gives that(3.24)H20(θ)=-g20q(θ)-g¯02q¯(θ),(3.25)H11(θ)=-g11q(θ)-g¯11q¯(θ).
From (3.21), (3.24), and the definition of A, we get(3.26)Ẇ20(θ)=2iω*τ10W20(θ)+g20q(θ)+g¯02q¯(θ).
Noting that q(θ)=q(0)eiω*τ10θ, we have(3.27)W20(θ)=ig20ω*τ10q(0)eiω*τ10θ+ig¯023ω*τ10q¯(0)e-iω*τ10θ+E1e2iω*τ10θ,
where E1=(E1(1),E1(2))T∈R2 is a constant vector.Similarly, from (3.22), (3.25), and the definition of A, we have(3.28)Ẇ11(θ)=g11q(θ)+g¯11q¯(θ),(3.29)W11(θ)=-ig11ω*τ10q(0)eiω*τ10θ+ig¯11ω*τ10q¯(0)e-iω*τ10θ+E2,
where E2=(E2(1),E2(2))T∈R2 is a constant vectorIn what follows, we shall seek appropriateE1,E2 in (3.27), (3.29), respectively. It follows from the definition of A and (3.24), (3.25) that(3.30)∫-10dη(θ)W20(θ)=2iω*τ10W20(0)-H20(0),(3.31)∫-10dη(θ)W11(θ)=-H11(0),
where η(θ)=η(0,θ).From (3.21), we have(3.32)H20(0)=-g20q(0)-g¯02q¯(0)+2τ10(H1,H2)T,(3.33)H11(0)=-g11q(0)-g¯11(0)q¯(0)+2τ10(P1,P2)T,
where(3.34)H1=-e-iω*τ10+l1+l2α2+l3α,H2=(k1α+k2α2)e-iω*τ2*+(k3+k4α)e-2iω*τ2*,P1=-12(e-iω*τ10+eiω*τ10)+l1+l2|α|2+2Re{α},P2=k1Re{αeiω*τ2*}+k2|α|2(e-iω*τ2*+eiω*τ2*)+k3+k4Re{α}.
Noting that(3.35)(iω*τ10I-∫-10eiω*τ10θdη(θ))q(0)=0,(-iω*τ10I-∫-10e-iω*τ10θdη(θ))q¯(0)=0,
and substituting (3.27) and (3.32) into (3.30), we have(3.36)(2iω*τ10I-∫-10e2iω*τ10θdη(θ))E1=2τ10(H1,H2)T.
That is(3.37)(2iω*-m3e-2iω*τ10-m2-n2e-2iω*τ2*2iω*-n1e-2iω*τ10-n3e-2iω*τ2*)E1=2(H1,H2)T.
It follows that(3.38)E1(1)=Δ11Δ1,E1(2)=Δ12Δ1,
where(3.39)Δ1=(2iω*-m3e-2iω*τ10)(2iω*-n1e-2iω*τ10-n3e-2iω*τ2*)-m2n2e-2iω*τ2*,Δ11=2H1(2iω*-n1e-2iω*τ10-n3e-2iω*τ2*)+2H2m2,Δ12=2H2(2iω*-m3e-2iω*τ10)+2H1n2e-2iω*τ2*.
Similarly, substituting (3.28) and (3.33) into (3.31), we have(3.40)(∫-10dη(θ))E2=2τ10(P1,P2)T.
That is(3.41)(m1+m3m2n2n1+n3)E2=2(-P1,-P2)T.
It follows that(3.42)E2(1)=Δ21Δ2,E2(2)=Δ22Δ2,
where(3.43)Δ2=(m1+m3)(n1+n3)-m2n2,Δ21=2m2P2-2P1(n1+n3),Δ22=2n2P1-2P2(m1+m3).
From (3.27), (3.29), (3.38), (3.42), we can calculate g21 and derive the following values:(3.44)c1(0)=i2ω*τ10(g20g11-2|g11|2-|g02|23)+g212,μ2=-Re{c1(0)}Re{λ′(τ10)},β2=2Re(c1(0)),T2=-Im⁡{c1(0)}+μ2Im⁡{λ′(τ10)}ω*τ10.
These formulas give a description of the Hopf bifurcation periodic solutions of (3.1) at τ=τ10 on the center manifold. From the discussion above, we have the following result.Theorem 3.1.
The periodic solution is forward (backward) ifμ2>0(μ2<0);  the bifurcating periodic solutions are orbitally asymptotically stable with asymptotical phase (unstable) if β2<0(β2>0); the periods of the bifurcating periodic solutions increase (decrease) if T2>0(T2<0).


## 4. Numerical Examples

In this section, we present some numerical results of system (1.1) to verify the analytical predictions obtained in the previous section. From Section 3, we may determine the direction of a Hopf bifurcation and the stability of the bifurcation periodic solutions. Let us consider the following system:(4.1)u̇1(t)=u1(t)[1-u1(t-τ1)]-u1(t)u2(t)0.05+u1(t)+0.6u2(t),u̇2(t)=u2(t)[-0.5+u1(t-τ2)0.05+u1(t-τ2)+0.6u2(t-τ2)],
which has a positive equilibrium E0(u1*,u2*)≈(0.3038,0.4230). When τ1=0, then we can easily obtain that (H1) and (H3) are satisfied. Take n=0, for example, by some computation by means of Matlab 7.0, we get ω0≈0.1342,τ20≈1.52. From Lemma 2.2, we know that the transversal condition is satisfied. Thus, the positive equilibrium E0≈(0.3038,0.4230) is asymptotically stable for τ2<τ20≈1.52 and unstable for τ2>τ20≈1.52 which is shown in Figure 1. When τ2=τ20≈1.52, (4.1) undergoes a Hopf bifurcation at the positive equilibrium E0≈(0.3038,0.4230), that is, a small amplitude periodic solution occurs around E0≈(0.3038,0.4230) when τ1=0 and τ2 is close to τ20=1.52 which is shown in Figure 2.Trajectory portrait and phase portrait of system (4.1) with τ1=0,τ2=1.5<τ20≈1.52. The positive equilibrium E0(0.3038,0.4230) is asymptotically stable. The initial value is (0.2,0.2).
(a)(b)(c)(d)Trajectory portrait and phase portrait of system (4.1) with τ1=0,τ2=1.6>τ20≈1.52. Hopf bifurcation occurs from the positive equilibrium E0(0.3038,0.4230). The initial value is (0.2,0.2).
(a)(b)(c)(d)Letτ2=1.5∈(0,1.52) and choose τ1 as a parameter. We have τ10≈1.2930. Then, the positive equilibrium is asymptotically when τ1∈[0,τ10). The Hopf bifurcation value of (4.1) is τ10≈1.2930. By the algorithm derived in Section 3, we can obtain(4.2)λ′(τ10)≈0.5018-7.2021i,c1(0)≈-2.0231-3.3225i,μ2≈0.2646,β2≈-4.2342,T2≈8.3701.
Furthermore, it follows that μ2>0 and β2<0. Thus, the positive equilibrium E0≈(0.3038,0.4230) is stable when τ1<τ10 as is illustrated by the computer simulations (see Figure 3). When τ1 passes through the critical value τ10, the positive equilibrium E0≈(0.3038,0.4230) loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcations from the positive equilibrium E0≈(0.3038,0.4230). Since μ2>0 and β2<0, the direction of the Hopf bifurcation is τ1>τ10, and these bifurcating periodic solutions from E0≈(0.3038,0.4230) at τ10 are stable, which are depicted in Figure 4.Trajectory portrait and phase portrait of system (4.1) with τ2=0.5,τ1=1.2<τ10≈1.2930. The positive equilibrium E0(0.3038,0.4230) is asymptotically stable. The initial value is (0.2,0.2).
(a)(b)(c)(d)Trajectory portrait and phase portrait of system (4.1) with τ2=0.5,τ1=1.35>τ10≈1.2930. Hopf bifurcation occurs from the positive equilibrium E*=(0.32,0.96). The initial value is (0.2,0.2).
(a)(b)(c)(d)


## 5. Conclusions

In this paper, we have investigated local stability of the positive equilibriumE0(u1*,u2*) and local Hopf bifurcation of a Beddington-DeAngelis functional response predator-prey model with two delays. We have showed that, if one of the conditions (a), (b), (c), and (d) holds and (H1),(H3), and (H5) are satisfied, and τ2∈[0,τ20), then the positive equilibrium E0(u1*,u2*) is asymptotically stable when τ1∈(0,τ10), as the delay τ1 increases, the positive equilibrium E0(u1*,u2*) loses its stability and a sequence of Hopf bifurcations occur at the positive equilibrium E0(u1*,u2*), that is, a family of periodic orbits bifurcates from the the positive equilibrium E0(u1*,u2*). At last, the direction of Hopf bifurcation and the stability of the bifurcating periodic orbits are discussed by applying the normal form theory and the center manifold theorem. A numerical example verifying our theoretical results is carried out. In addition, we must point out that, although Gakkhar et al. [4] have also investigated the existence of Hopf bifurcation for system (1.1) with respect to positive equilibrium E0(u1*,u2*), it is assumed that τ1=τ2. For τ1≠τ2, only numerical simulations are carried out to discuss the existence of Hopf bifurcation. In this paper, under the case τ1≠τ2, we investigate the existence of Hopf bifurcation quantitatively. Our work generalizes the known results of Sunita Gakkhar et al. [4]. Similarly, we can investigate the Hopf bifurcation of system (1.2) by choosing the delay τ2 as bifurcation parameter. We will further investigate the topic elsewhere in the near future.



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*Source: 652947-2012-05-08.xml*