# Stability and Global Hopf Bifurcation Analysis on a Ratio-Dependent Predator-Prey Model with Two Time Delays

Authors: Huitao Zhao; Yiping Lin; Yunxian Dai
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A ratio-dependent predator-prey model with two time delays is studied. By means of an iteration technique, sufficient conditions are obtained for the global attractiveness of the positive equilibrium. By comparison arguments, the global stability of the semitrivial equilibrium is addressed. By using the theory of functional equation and Hopf bifurcation, the conditions on which positive equilibrium exists and the quality of Hopf bifurcation are given. Using a global Hopf bifurcation result of Wu (1998) for functional differential equations, the global existence of the periodic solutions is obtained. Finally, an example for numerical simulations is also included.


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## Body

## 1. Introduction

The main purpose of this paper is to investigate the bifurcation phenomena from the delays for the following predator-prey system:(1)x˙(t)=x(t)[r1-a11x(t)-a12x(t)y(t-τ2)my2(t-τ2)+x2(t)],y˙(t)=a21x2(t-τ1)y(t)my2(t)+x2(t-τ1)-r2y(t),
where x(t) and y(t) stand for the population (or density) of the prey and the predator at time t, respectively. From the biological sense, we assume that x2+y2≠0. r1,r2,a11,a12,a21, and m are positive constants, in which r1 denotes the intrinsic growth rate of the prey, a11 is the intraspecific competition rate of the prey, a12 is the capturing rate of the predator, a21/a12 describes the efficiency of the predator in converting consumed prey into predator offspring, m is the interference coefficient of the predators, and r2 is the predator mortality rate. The delay τ1≥0 denotes the gestation period of the predator; τ2≥0 is the hunting delay of the predator to prey.This model is labeled “ratio-dependent,” which means that the functional and numerical responses depend on the densities of both prey and predators, especially when predator has to search for food. Such a functional response is called a ratio-dependent response function (see [1] for more details). In system (1), the ratio-dependent response function is of the form (x/y)=c(x/y)2/(m+(x/y)2)=cx2/(my2+x2).The ratio-dependent predator-prey model has been studied by several researchers recently and very rich dynamics have been observed [2–5]. For example, Xu et al. [4] studied a delayed ratio-dependent predator-prey model with the same ratio-dependent response function of system (1). By means of an iteration technique, they obtained the sufficient conditions for the global attractiveness of the positive equilibrium. By comparison arguments, they proved the global stability of the semitrivial equilibrium. Finally using the theory of functional equation and Hopf bifurcation, they gave the condition on which positive equilibrium exists and the formulae to determine the quality of Hopf bifurcation. But in their work, the global continuation of local Hopf bifurcation was not mentioned.In general, periodic solutions through the Hopf bifurcation in delay differential equations are local for the values of parameters which are only in a small neighborhood of the critical values (see, e.g., [6, 7]). Therefore we would like to know if these nonconstant periodic solutions obtained through local bifurcation can continue for a large range of parameter values. Recently, a great deal of research has been devoted to the topics [8–12]. One of the methods used in them is the global Hopf bifurcation theorem by Wu [13]. For example, Song et al. [12] studied a predator-prey system with two delays, and using the methods in [13], they get the global existence of periodic solutions.Motivated by [12], we will study the system (1); special attention is paid to the global continuation of local Hopf bifurcation. We suppose that the initial condition for system (1) takes the form
(2)x(θ)=ϕ(θ),y(θ)=ψ(θ),ϕ(θ)≥0,ψ(θ)≥0,θ∈[-τ,0](τ=τ1+τ2),ϕ(0)>0,ψ(0)>0,
where (ϕ(θ),ψ(θ))∈𝒞([-τ,0],R+02), which is the Banach space of continuous functions mapping the interval [-τ,0] into R+02, where R+02={(x,y)∣x≥0,y≥0}.By the fundamental theory of functional differential equations [14], system (1) has a unique solution (x(t),y(t)) satisfying initial condition (2).The rest of the paper is organized as follows. In Section2, we show the positivity and the boundedness of solutions of system (1) with initial condition (2). In Section 3, we study the existence of Hopf bifurcation for system (1) at the positive equilibrium. In Section 4, using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcation and the stability and other properties of bifurcating periodic solutions. In Section 5, by means of an iteration technique, sufficient conditions are obtained for the global attractiveness of the positive equilibrium. In Section 6, we consider the global existence of bifurcating periodic solutions and give some numerical simulations. In Section 7, a brief discussion is given.


## 2. Positivity and Boundedness

In this section, we study the positivity and boundedness of solutions of system (1) with initial conditions (2).Theorem 1.
Solutions of system (1) with initial condition (2) are positive for all t≥0.Proof.
Assume(x(t),y(t)) to be a solution of system (1) with initial condition (2). Let us consider y(t) for t≥0. It follows from the second equation of system (1) that
(3)y(t)=y(0)e∫0t‍((a21x2(s-τ1)/my2(s)+x2(s-τ1))-r2)ds;
then, from initial condition (2), we have y(t)>0, for t≥0. We derive from the first equation of system (1) that
(4)x(t)=x(0)e∫0t‍(r1-a11x(s)-(a12x(s)y(s-τ2)/my2(s-τ2)+x2(s)))ds;
that is, x(t)>0 for t≥0. This ends the proof.For the following discussion of boundedness, we first consider the following ordinary differential equation:(5)u˙=a21A12u(t)mu2(t)+A12-r2u(t),u(0)>0,
where a21,r2,A1, and m are positive constants. From Lemma 2.1 in [5], it is easy to verify the following result.Lemma 2.
Ifa21<r2, the trivial equilibrium u0=0 of (5) is globally stable. If a21>r2, then (5) admits a unique positive equilibrium u*=(a21-r2)/mr2A1 which is globally asymptotically stable in Λ={u∣u≥0}.Theorem 3.
Positive solutions of system (1) with initial condition (2) are ultimately bounded.Proof.
Let(x(t),y(t)) be a positive solution of system (1) with initial condition (2). From the first equation of system (1), we have
(6)x˙(t)≤x(t)[r1-a11x(t)],
which yields
(7)limsupt→+∞x(t)≤r1a11;
hence, for ϵ>0 sufficiently small, there is a T1>0 such that if t>T1, x(t)<(r1/a11)+ϵ.
We now consider the boundedness ofy(t). If a21≤r2, we derive from the second equation of system (1) that
(8)y˙(t)≤(a21-r2)y(t)≤0;
from monotone bounded theorem, it is easy to show that limt→+∞y(t)≤y(0).
Therefore, we assume below thata21>r2. We derive from the second equation of system (1) that, for t>T1+τ,
(9)y˙(t)≤a21(r1/a11+ϵ)2y(t)my2(t)+(r1/a11+ϵ)2-r2y(t);
noting that a21>r2, by Lemma 2, a comparison argument shows that
(10)limsupt→+∞y(t)≤a21-r2mr2(r1a11+ϵ).
This completes the proof.


## 3. Local Stability and Hopf Bifurcation

In this section, we discuss the local stability of the positive equilibrium and the semitrivial equilibrium of system (1) and establish the existence of Hopf bifurcation at the positive equilibrium.It is easy to show that system (1) always has a semitrivial equilibrium E1(r1/a11,0). Further, if the following condition holds:(H1)r12a212m>a122r2(a21-r2)>0,then system (1) has a unique positive equilibrium E*(x*,y*), where
(11)x*=r1a21-r2a12ha11a21,y*=hx*,
where
(12)h=a21-r2mr2.For convenience, let us introduce new variablesX(t)=x(t-τ1),Y(t)=y(t),τ=τ1+τ2, rewriting X(t),Y(t) as x(t),y(t), so that system (1) can be written as the following system with a single delay:
(13)x˙(t)=x(t)[r1-a11x(t)-a12x(t)y(t-τ)my2(t-τ)+x2(t)],y˙(t)=a21x2(t)y(t)my2(t)+x2(t)-r2y(t).
Clearly, system (13) has the same equilibrium as system (1).The characteristic equation of system (13) at the semitrivial equilibrium E1(r1/a11,0) is of the form
(14)(λ+r1)(λ+r2-a21)=0.
Clearly, (14) always has a root λ=-r1, and if a21<r2, the other root of (14) is negative; if a21>r2, the other root of (14) is positive. Hence the semitrivial equilibrium E1(r1/a11,0) is locally asymptotically stable (unstable) if a21<r2 (a21>r2).The characteristic equation of system (13) at the positive equilibrium E*(x*,y*) is of the form
(15)λ2+p0λ+p1+p2e-λτ=0,
where
(16)p0=r1-2a12r22ha212+2r2(a21-r2)a21,p1=(r1-2a12r22ha212)2r2(a21-r2)a21,p2=2a12r22h(2r2-a21)(a21-r2)a213,
where h is defined as (12).Whenτ=0, (15) becomes
(17)λ2+p0λ+p1+p2=0.
It is easy to show that
(18)p1+p2=2r2(a21-r2)(r1a21-a12r2h)a212.
Obviously, if (H1) holds, then p1+p2>0. Hence, the positive equilibrium E*(x*,y*) of system (13) is locally stable when τ=0 if
(19)r1>2a12r22ha212-2r2(a21-r2)a21,
and it is unstable when τ=0 if
(20)r1<2a12r22ha212-2r2(a21-r2)a21.We assume thatλ=iω(ω>0) is a root of (15); this is the case if and only if ω satisfies the following equation:
(21)-ω2+p0ωi+p1+p2e-iωτ=0.
Separating the real and imaginary parts, we obtain the following system for ω:
(22)p2cosωτ=ω2-p1,p2sinωτ=p0ω.
It follows that
(23)ω4+(p02-2p1)ω2+p12-p22=0.Lettingz=ω2, (42) becomes
(24)z2+(p02-2p1)z+p12-p22=0.
By a direct calculation, it follows that
(25)p02-2p1=(r1-2a12r22ha212)2+(2r2(a21-r2)a21)2>0,p1-p2=2r2(a21-r2)a21(r1-4a12r22h+a12a21r2ha212).
Note that if (H1) holds, then p1+p2>0. Hence if (H1) and p1-p2>0 hold, (24) has no positive roots. Accordingly, if (H1) and p1-p2>0 hold, the positive equilibrium E* of system (13) exists and is locally asymptotically stable for all τ≥0. If (H1) and p1-p2<0 hold, then (24) has a unique positive root ω0, where
(26)ω02=12(2p1-p02+p04-4p02p1+4p22).
Then, we can get
(27)τn=1ω0arccosω02-p1p2+2nπω0,n=0,1,2,…,
at which (15) admits a pair of purely imaginary roots of the form ±ω0.Letp1-p2<0 and τ0 be defined above. Denote
(28)λ(τ)=α(τ)+iω(τ)
the root of (15) satisfying
(29)α(τn)=0,ω(τn)=ω0.It is not difficult to verify that the following result holds.Lemma 4.
If (H1) andp1-p2<0 hold, the transversal condition (d(Reλ)/dτ)∣τ=τn>0 holds.Proof.
Differentiating (15) with respect τ, we obtain that
(30)2λdλdτ+p0dλdτ-p2τe-λτdλdτ=p2λe-λτ;
it follows that
(31)(dλdτ)-1=2λ+p0-λp2e-λτ-τλ;
from (15) and (31), we have
(32)(dλdτ)-1=2λ+p0-λ(λ2+p0λ+p1)-τλ.
We therefore derive that(33)sign{d(Reλ)dτ|τ=τn}=sign{Re(dλdτ)-1|τ=τn}=sign{Re[2λ+p0-λ(λ2+p0λ+p1)]τ=τn}=sign{ω02(p02-2p1+ω02)ω04p02+(ω0p1-ω03)2}.
Noting that p02-2p1>0, hence, if (H1) and p1-p2<0 hold, we have (d(Reλ)/dτ)∣τ=τn>0. Accordingly, the transversal condition holds and a Hopf bifurcation occurs at τ=τn.By Lemma B in [5], we have the following results.Theorem 5.
Suppose (H1) holds and leth be defined in (12), for system (13), one has the following.(i)
Ifr1>(2a12r22h/a212)-(2r2(a21-r2)/a21) and r1>(4a12r22h+a12a21r2h)/a212, then the positive equilibrium E* is locally asymptotically stable for all τ≥0.(ii)
Ifr1>(2a12r22h/a212)-(2r2(a21-r2)/a21) and r1<(4a12r22h+a12a21r2h)/a212, then there exists a positive number τ0 such that the positive equilibrium E* is locally asymptotically stable if τ∈[0,τ0) and is unstable if τ>τ0. Further, system (13) undergoes a Hopf bifurcation at E* when τ=τ0.


## 4. Direction and Stability of Hopf Bifurcations

In Section3, we have shown that system (13) admits a periodic solution bifurcated from the positive equilibrium E* at the critical value τ0. In this section, we derive explicit formulae to determine the direction of Hopf bifurcations and stability of periodic solutions bifurcated from the positive equilibrium E* at critical value τ0 by using the normal form theory and the center manifold reduction (see, e.g., [15, 16]).Setτ=τ0+μ; then μ=0 is a Hopf bifurcation value of system (13). Thus we can consider the problem above in the phase space 𝒞=𝒞([-τ,0],R2).Letu1(t)=x(t)-x*,u2(t)=y(t)-y* . System (13) is transformed into
(34)u˙1(t)=c1u1(t)+c4u2(t-τ)+∑i+j≥21i!j!fij(1)u1i(t)u2j(t-τ),u˙2(t)=c2u1(t)+c3u2(t)+∑i+j≥2‍1i!j!fij(2)u1i(t)u2j(t),
where
(35)c1=-r1+2a12r22ha212,c2=2r2h(a21-r2)a21,c3=-r2+r2(2r2-a21)a21,c4=-a12r2(2r2-a21)a212,f(1)=x(t)[r1-a11x(t)-a12x(t)y(t-τ)my2(t-τ)+x2(t)],f(2)=a21x2(t)y(t)my2(t)+x2(t)-r2y(t),fij(1)=∂i+jf(1)∂xi∂y(t-τ)j|(x*,y*),fij(2)=∂i+jf(2)∂xi∂yj|(x*,y*),i,j≥0.For the simplicity of notations, we rewrite (34) as
(36)u˙(t)=Lμut+f(μ,ut),
where u(t)=(u1(t),u1(t))T∈R2, ut(θ)∈𝒞 is defined by ut(θ)=u(t+θ), and Lμ:𝒞→R,f:R×𝒞→R are given, respectively, by
(37)Lμϕ=[c10c2c3]ϕ(0)+[0c400]ϕ(-τ),(38)f(μ,ϕ)=[∑i+j≥2‍1i!j!fij(1)ϕ1i(t)ϕ2j(t-τ)∑i+j≥2‍1i!j!fij(2)ϕ1i(t)ϕ2j(t)].
By the Riesz representation theorem, there exists a function η(θ,μ) of bounded variation for θ∈[-τ,0] such that
(39)Lμϕ=∫-τ0‍dη(θ,μ)ϕ(θ),forϕ∈𝒞.
In fact, we can choose
(40)η(θ,μ)=[c10c2c3]δ(θ)+[0c400]δ(θ+τ),
where δ is the Dirac delta function. For ϕ∈𝒞1([-τ,0],R2), define
(41)A(μ)ϕ={dϕ(θ)dθ,θ∈[-τ,0),∫-τ0‍dη(s,μ)ϕ(s),θ=0,R(μ)ϕ={0,θ∈[-τ,0),f(μ,ϕ),θ=0.Then whenθ=0, system (36) is equivalent to
(42)u˙t=A(μ)ut+R(μ)ut,
where ut(θ)=u(t+θ) for θ∈[-τ,0].Forψ∈𝒞1([0,τ],(R2)*), define
(43)A*ψ(s)={-dψ(s)ds,s∈(0,τ],∫-τ0‍dηT(t,0)ψ(-t),s=0,
and a bilinear inner product,
(44)〈ψ(s),ϕ(θ)〉=ψ-(0)ϕ(0)-∫-τ0‍∫ξ=0θ‍ψ-(ξ-θ)dη(θ)ϕ(ξ)dξ,
where η(θ)=η(θ,0) and (·)- denotes the conjugate complex of (·). Then A(0) and A* are adjoint operators. By the discussion in Section 3, we know that ±iω0 are eigenvalues of A(0). Thus, they are also eigenvalues of A*. We first need to compute the eigenvector of A(0) and A* corresponding to iω0 and -iω0, respectively.Suppose thatq(θ)=(1,ρ)Teiω0θ is the eigenvector of A(0) corresponding to iω0. Then (0)q(θ)=iω0q(θ). From the definition of A(0), it is easy to get ρ=(iω0-c3)/c2.Similarly, letq*(s)=D(1,ρ*)e-iω0s be the eigenvector of A* corresponding to -iω0. By the definition of A*, we can compute ρ*=(-iω0-c1)/c2.In order to assure〈q*(s),q(θ)〉=1, we need to determine the value of D. From (44) and the definitions of q and q*, we have D=1/(1+ρ-*ρ+c4ρτ0eiτ0ω0) such that 〈q*(s),q(θ)〉=1 and 〈q*(s),q-(θ)〉=0.In the following, we first compute the coordinates to describe the center manifoldC0 at μ=0. Define
(45)z(t)=〈q*,ut〉,W(t,θ)=ut(θ)-2Re{z(t)q(θ)}.
On the center manifold C0, we have
(46)W(t,θ)=W(z(t),z-(t),θ)=W20(θ)z22+W11(θ)zz-+W02(θ)z-22+W30(θ)z36+⋯,
where z and z- are local coordinates for center manifold C0 in the directions of q and q-. Note that W is real if ut is real. We consider only real solutions. For the solution ut∈C0, since μ=0, we have
(47)z˙=ω0z+i〈q*(θ),f({z(t)q(θ)}0,W(z(t),z-(t),θ)hhhhhhhhhhhhh+2Re{z(t)q(θ)})〉=iω0z+q-*(0)f({z(t)q(0)}0,W(z(t),z-(t),0)ffffffffffffffffffff+2Re{z(t)q(0)})≜iω0z+q-*(0)f0(z,z-)=iω0z+g(z,z-),
where
(48)g(z,z-)=q-*(0)f0(z,z-)=g20z22+g11zz-+g02z-22+g21z2z-2+⋯.
By (45), we have
(49)ut(θ)=(u1t(θ),u2t(θ))T=W(t,θ)+zq(θ)+z-q-(θ).
It follows from (38) and (48) that
(50)g20=2D-[12f20(1)ρ2+f11(1)ρe-iτ0ω0+12f02(1)e-2iτ0ω0hhh+ρ-*(12f20(2)ρ2+f11(2)ρ+12f02(2))],g11=D-[f20(1)ρρ-+f11(1)(ρeiτ0ω0+ρ-e-iτ0ω0)hh+f02(1)+ρ-*(f20(2)ρρ-+f11(2)(ρ+ρ-)+f02(2))],g02=2D-[12f20(1)ρ-2+f11(1)ρeiτ0ω0DD+12f02(1)e2iτ0ω0+ρ-*(12f20(2)ρ-2hhhhhhhhhhhhhhhhh+f11(2)ρ-+12f02(2))],g21=2D-[12f20(1)(2ρW11(1)(0)+ρ-W20(1)(0))ffffff+f11(1)(ρW11(2)(-τ0)+12ρ-W20(2)(-τ0)fffffhhh+12W20(1)(0)eiτ0ω0+W11(1)(0)e-iτ0ω0)ffffff+12f02(1)(2W11(2)(-τ0)e-iτ0ω0+W20(2)(-τ0)eiτ0ω0)ffffff+12f21(1)(ρ2eiτ0ω0+2ρρ-e-iτ0ω0)ffffff+12f12(1)(ρ-e-2iτ0ω0+2ρ)fffffffff+12f30(1)ρ2ρ-+12f03(1)e-iτ0ω0]+2D-ρ-*[12f20(2)(2ρW11(1)(0)+ρ-W20(1)(0))fffffh+f11(2)(ρW11(2)(0)+12ρ-W20(2)(0)fffffffhhhhh+12W20(1)(0)+W11(1)(0))fffffh+12f02(2)(2W11(2)(0)+W20(2)(0))fffffh+12f21(2)(ρ2+2ρρ-)fffffh+12f12(2)(ρ-+2ρ)+12f30(2)ρ2ρ-+12f03(2)].In order to assure the value ofg21, we need to compute W20(θ) and W11(θ). By (42) and (45), we have
(51)W˙=u˙t-z˙q-z-˙q-={AW-2Re{q-*(0)f0q(θ)},θ∈[-τ0,0),AW-2Re{q-*(0)f0q(θ)}+f0,θ=0,hhhhhhhhhhhhhhhhhhhhhhh≜AW+H(z,z-,θ),
where
(52)H(z,z-,θ)=H20(θ)z22+H11(θ)zz-+H02(θ)z-22+⋯.
Notice that near the origin on the center manifold C0, we have
(53)W˙=Wzz˙+Wz-z-˙;
thus, we have
(54)(A-2iωkτkI)W20(θ)=-H20(θ),AW11(θ)=-H11(θ).
By (51), for θ∈[-τ0,0), we have
(55)H(z,z-,θ)=-q-*(0)f0q(θ)-q*(0)f-0q-(θ)=-gq(θ)-g-q-(θ).
Comparing the coefficients with (51) gives that
(56)H20(θ)=-g20q(θ)-g-02q-(θ),H11(θ)=-g11q(θ)-g-11q-(θ).
From (56), (54), and the definition of A(0), we can get
(57)W˙20(θ)=2iω0W20(θ)+g20q(θ)+g-02q-(θ).Notice thatq(θ)=q(0)eiω0θ; we have
(58)W20(θ)=ig20ω0q(0)eiω0θ+ig-023ω0q-(0)e-iω0θ+E1e2iω0θ,
where E1=(E1(1),E1(2))∈R2 is a constant vector. In the same way, we can also obtain
(59)W11(θ)=-ig11ω0q(0)eiω0θ+ig-11ω0q-(0)e-iω0θ+E2,
where E2=(E2(1),E2(2))∈R2 is also a constant vector. In what follows, we will compute E1 and E2. From the definition of A(0) and (54), we have
(60)∫-τ00‍dη(θ)W20(θ)=2iω0W20(0)-H20(0),(61)∫-τ00‍dη(θ)W11(θ)=-H11(0),
where η(θ)=η(0,θ).From (51), (58), and (60) and noting that
(62)[iω0I-∫-τ00‍eiω0θdη(θ)]q(0)=0,
we have
(63)E1(1)=1A1|e1-c4e-2iω0τ0e22iω0-c3|,E1(2)=1A1|2iω0-c1e1-c2e2|,
where
(64)A1=(2iω0-c1)(2iω0-c3)-c2c4e-2iω0τ0,e1=f20(1)ρ2+2f11(1)ρe-iτ0ω0+f02(1)e-2iτ0ω0,e2=f20(2)ρ2+2f11(2)ρ+f02(2).From (52), (59), and (61) and noting that
(65)[-iω0I-∫-τ00‍e-iω0θdη(θ)]q-(0)=0,
we have
(66)E2(1)=1A2|e3-c4e4-c3|,E2(2)=1A2|-c1e3-c2e4|,
where
(67)A2=c1c3-c2c4,e3=f20(1)ρρ-+f11(1)(ρeiτ0ω0+ρ-e-iτ0ω0)+f02(1),e4=f20(2)ρρ-+f11(2)(ρ+ρ-)+f02(2).Thus, we can determineW20(θ) and W11(θ) from (58) and (59). Furthermore, we can determine each gij. Therefore, each gij is determined by the parameters and delay in (13). Thus, we can compute the following values [15]:
(68)c1(0)=i2ω0τ0(g20g11-2|g11|2-13|g02|2)+g212,μ2=-Re{c1(0)}Re{λ′(τ0)},T2=-Im{c1(0)}+μ2Im{λ′(τ0)}ω0τ0,β2=2Re{c1(0)},
which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value τk; that is, μ2 determines the directions of the Hopf bifurcation: if μ2>0(<0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcation exists for τ>τ0(<τ0); β2 determines the stability of the bifurcation periodic solutions: the bifurcating periodic solutions are stable (unstable) if β2<0(>0); and T2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if T2>0(<0).


## 5. Global Attractiveness

In this section, following Chaplygin [17], taking into account the upper and lower solution technique and using monotone iterative methods [18, 19], we discuss the global attractiveness of the positive equilibrium E*(x*,y*) and the global stability of the semitrivial equilibrium E1(r1/a11,0) of system (1), respectively.Theorem 6.
Suppose (H1) holds and leth be defined above, then the positive equilibrium E*(x*,y*) of system (1) is globally attractive provided that the following holds:(H2)r1>max{a12/2m,(3a12/m)+(2a12r2/a21)h},Proof.
Let(x(t),y(t)) be any positive solution of system (1) with initial conditions (2).
Let(69)U1=limsupt→+∞x(t),V1=liminft→+∞x(t),U2=limsupt→+∞y(t),V2=liminft→+∞y(t).
Using iteration method, we will proof that U1=V1=x*,U2=V2=y*.
From the first equation of system (1), we have
(70)x˙(t)≤x(t)[r1-a11x(t)];
by comparison, it follows that
(71)U1=limsupt→+∞x(t)≤r1a11:=M1x;
hence, for ϵ>0 sufficiently small, there exists a T1>0 such that if t>T1,x(t)≤M1x+ϵ.
From the second equation of system (1), we have, for t>T1+τ,
(72)y˙(t)≤a21(M1x+ϵ)2y(t)my2(t)+(M1x+ϵ)2-r2y(t).
Consider the following auxiliary equation:
(73)u˙(t)=a21(M1x+ϵ)2u(t)mu2(t)+(M1x+ϵ)2-r2u(t).
Since (H1) holds, by Lemma 2, it follows from (73) that
(74)limt→+∞u(t)=(M1x+ϵ)h,
where h is defined in (12). By comparison, we obtain that
(75)U2=limsupt→+∞y(t)≤(M1x+ϵ)h;
since this inequality holds true for arbitrary ϵ>0 sufficiently small, it follows that U2≤M1y, where
(76)M1y=M1xh.
Hence, for ϵ>0 sufficiently small, there is a T2>T1+τ such that if t>T2,y(t)≤M1y+ϵ.
Forϵ>0 sufficiently small, noting that my2(t-τ2)+x2≥2mxy(t-τ2), we derive from the first equation of system (1) that, for t>T2,
(77)x˙(t)≥x(t)[r1-a11x(t)-a122m];
by comparison, it follows that
(78)V1=liminft→+∞x(t)≥1a11(r1-a122m):=N1x;
hence, for ϵ>0 sufficiently small, there is a T3>T2+τ, such that if t>T3, x(t)≥N1x-ε.
Forϵ>0 sufficiently small, we derive from the second equation of system (1) that, for t>T3+τ,
(79)y˙(t)≥a21(N1x-ϵ)2y(t)my2(t)+(N1x-ϵ)2-r2y(t).
Consider the following auxiliary equation:
(80)u˙(t)=a21(N1x-ϵ)2u(t)mu2(t)+(N1x-ϵ)2-r2u(t).
Since (H1) holds, by Lemma (5), it follows from (80) that
(81)limt→+∞u(t)=(N1x-ϵ)h;
by comparison we derive that
(82)V2=liminft→+∞y(t)≥(N1x-ϵ)h.
Since this inequality holds true for arbitrary ϵ>0 sufficiently small, we conclude that V2≥N1y, where
(83)N1y=N1xh.
Therefore, for ϵ>0 sufficiently small, there is a T4>T3+τ such that if t>T4, y(t)≥N1y-ϵ.
Again, forϵ>0 sufficiently small, it follows from the first equation of system (1) that, for t>T4,
(84)x˙(t)≤x(t)[r1-a11x(t)-a12(N1x-ϵ)(N1y-ϵ)m(M1y+ϵ)2+(M1x+ϵ)2];
by comparison we derive that
(85)U1=limsupt→+∞x(t)≤1a11(r1-a12(N1x-ϵ)(N1y-ϵ)m(M1y+ϵ)2+(M1x+ϵ)2).
Since the above inequality holds true for arbitrary ϵ>0 sufficiently small, it follows that U≤M2x, where
(86)M2x=1a11(r1-a12N1xN1ym(M1y)2+(M1x)2);
hence, for ϵ>0 sufficiently small, there is a T5>T4+τ such that if t>T5, x(t)≤M2x+ϵ.
It follows from the second equation of system (1) that, for t>T5,
(87)y˙(t)≤a21(M2x+ϵ)2y(t)my2(t)+(M2x+ϵ)2-r2y(t).
By Lemma 2 and a comparison argument we derive from (87) that
(88)U2=limsupt→+∞y(t)≤(M2x+ϵ)h;
since this inequality holds true for ϵ>0 sufficiently small, we get U2≤M2y, where
(89)M2y=M2xh;
hence, for ϵ>0 sufficiently small, there is a T6>T5+τ such that if t>T6, y(t)≤M2y+ϵ.
Forϵ>0 sufficiently small, it follows from the first equation of system (1) that, for t>T6,
(90)x˙(t)≥x(t)[r1-a11x(t)-a12(M2x+ϵ)(M2y+ϵ)m(N1y-ϵ)2+(N1x-ϵ)2];
by comparison, we can obtain that
(91)V1=liminft→+∞x(t)≥1a11(r1-a12(M2x+ϵ)(M2y+ϵ)m(N1y-ϵ)2+(N1x-ϵ)2).
Since the above inequality holds true for arbitrary ϵ>0 sufficiently small, it follows that V≥N2x, where
(92)N2x=1a11(r1-a12M2xM2ym(N1y)2+(N1x)2);
therefore, for ϵ>0 sufficiently small, there is a T7>T6+τ such that if t>T7, x(t)≥N2x-ϵ.
Forϵ>0 sufficiently small, we derive from the second equation of system (1) that, for t>T7+τ,
(93)y˙(t)≥a21(N2x-ϵ)2y(t)my2(t)+(N2x-ϵ)2-r2y(t).
Since (H1) holds, by Lemma 2 and a comparison argument, it follows (93) that
(94)V2=liminft→+∞y(t)≥(N2x-ϵ)h;
since, for arbitrary ϵ>0 sufficiently small, this inequality holds true, we conclude that V2≥N2y, where
(95)N2y=N2xh.
Continuing this process, we obtain four sequences Mnx,Mny,Vnx,andVny(n=1,2,…) such that, for n≥2,
(96)Mnx=1a11(r1-a12Nn-1xNn-1ym(Mn-1y)2+(Mn-1x)2),Nnx=1a11(r1-a12MnxMnym(Nn-1y)2+(Nn-1x)2),Mny=Mnxh,Nny=Nnxh,
where h is defined in (12). It is readily seen that
(97)Nnx≤V1≤U1≤Mnx,Nny≤V2≤U2≤Mny.
It is easy to know that the sequences Mnx,Mny are not increasing and the sequences Nnx,Nny are not decreasing; from accumulation point theorem, the limit of each sequence in Mnx,Mny,Nnx, and Nny exists, Denote
(98)x-=limt→+∞Mnx,x_=limt→+∞Nnx,y-=limt→+∞Mny,y_=limt→+∞Nny.
We therefore obtain from (96) and (98) that
(99)x-=1a11(r1-a12x_y_my-2+x-2),x_=1a11(r1-a12x-y-my_2+x_2),y-=x-h,y_=x_h.
To complete the proof, it is sufficient to prove that x-=x_,y-=y_. It follows from (99) that
(100)a11(1+mh2)x-3=r1(1+mh2)x-2-a12hx_2,(101)a11(1+mh2)x_3=r1(1+mh2)x_2-a12hx-2.
Letting (100) minus (101), we have
(102)a11(1+mh2)(x--x_)(x-2+x-x_+x_2)=[r1(1+mh2)+a12h](x--x_)(x-+x_).
If x-≠x_, we derive from (102) that
(103)a11(1+mh2)(x-2+x-x_+x_2)=[r1(1+mh2)+a12h](x-+x_).
Letting A=a11(1+mh2),B=r1(1+mh2)+a12h, we derive from (103) that
(104)x-x_=(x-+x_)2-BA(x-+x_).
It follows from (104) that
(105)(x-+x_)2-4x-x_=(x-+x_)2-4[(x-+x_)2-BA(x-+x_)]=(x-+x_)[4BA-3(x-+x_)];
noting that x-≥N1x,x_≥N1x, we derive from (105) that
(106)(x-+x_)2-4x-x_≤2(x-+x_)[2BA-3N1x].
Substituting (78) into (106), it follows that
(107)(x-+x_)2-4x-x_≤-2(x-+x_)a11[r1-3a12m-2a12h1+mh2].
Hence, if (H2) holds, we have (x-+x_)2-4x-x_<0; this is a contradiction. Accordingly, we have x-=x_. Therefore, from (99), we have y-=y_. Hence, the positive equilibrium E* is globally attractive. The proof is complete.Using the same methods in [4, 20], we can also get a similar result.Theorem 7.
Ifr1>a12/2m and a21<r2, the semitrivial equilibrium E1(r1/a11,0) of system (1) is globally asymptotically stable.


## 6. Global Continuation of Local Hopf Bifurcations

In this section, we study the global continuation of periodic solutions bifurcating from the positive equilibriumE* of system (13). Throughout this section, we follow closely the notations in [13]. For simplification of notations, setting z(t)=(z1(t),z2(t))T=(x(t),y(t))T, we may rewrite system (13) as the following functional differential equation:
(108)z˙(t)=ℱ(zt,τ,p),
where zt(θ)=(z1t(θ),z2t(θ))T=(z1(t+θ),andz2(t+θ))T∈𝒞([-τ,0],R2). It is obvious that if (H1) holds, then system (13) has a semitrivial equilibrium E1(r1/a11,0) and a positive equilibrium E*(x*,y*). Following the work of [13], we need to define
(109)X=𝒞([-τ,0],R2),Γ=Cl{(z,τ,p)∈X×R×R+;zisanonconstanthhhhhperiodicsolutionof(108)},𝒩={(z-,τ-,p-);ℱ(z-,τ-,p-)=0}.
Let ℓ(E*,τj,2π/ω0) denote the connected component passing through (E*,τj,2π/ω0) in Γ, where τj is defined by (26). From Theorem 5, we know that ℓ(E*,τj,2π/ω0) is nonempty.We first state the global Hopf bifurcation theory due to Wu [13] for functional differential equations.Lemma 8.
Assume that(z*,τ,p) is an isolated center satisfying the hypotheses (A1)–(A4) in [13]. Denote by ℓ(z*,τ,p) the connected component of (z*,τ,p) in Γ. Then either(i)
ℓ(z*,τ,p) is unbounded or(ii)
ℓ(z*,τ,p) is bounded; ℓ(z*,τ,p)∩Γ is finite and
(110)∑(z,τ,p)∈ℓ(z*,τ,p)∩𝒩γm(z*,τ,p)=0,for all m=1,2,…, where γm(z*,τ,p) is the mth crossing number of (z*,τ,p) if m∈J(z*,τ,p) or it is zero if otherwise.Clearly, if (ii) in Lemma8 is not true, then ℓ(z*,τ,p) is unbounded. Thus, if the projections of ℓ(z*,τ,p) onto z-space and onto p-space are bounded, then the projection onto τ-space is unbounded. Further, if we can show that the projection of ℓ(z*,τ,p) onto τ-space is away from zero, then the projection of ℓ(z*,τ,p) onto τ-space must include interval [τ,+∞). Following this ideal, we can prove our results on the global continuation of local Hopf bifurcation.Lemma 9.
If condition (H1) holds, then all nonconstant periodic solutions of (13) with initial conditions,
(111)x(θ)=ϕ(θ),y(θ)=ψ(θ),ϕ(θ)≥0,ψ(θ)≥0,θ∈[-τ,0](τ=τ1+τ2),ϕ(0)>0,ψ(0)>0,
are uniformly bounded.Proof.
Suppose thatx=x(t),y=y(t) are nonconstant periodic solutions of system (13) and define
(112)x(ξ1)=min{x(t)},x(η1)=max{x(t)},y(ξ2)=min{y(t)},y(η2)=max{y(t)}.
It follows from system (13) that
(113)x(t)=x(0)exp{∫0t‍(a12x(s)y(s-τ)my2(s-τ)+x2(s)r1-a11x(s)hhhhhhhhhhhhh-a12x(s)y(s-τ)my2(s-τ)+x2(s))ds∫0t‍},y(t)=y(0)exp{∫0t‍(-r2+a21x2(s)my2(s)+x2(s))ds},
which implies that the solutions of system (13) cannot cross the x-axis and y-axis. Thus the nonconstant periodic orbits must be located in the interior of each quadrant. It follows from initial conditions of system (13) that (t)>0,y(t)>0. From system (13), we can get
(114)0=r1-a11x(η1)-a12x(η1)y(η1-τ)my2(η1-τ)+x2(η1),0=-r2+a21x2(η2)my2(η2)+x2(η2).
Since x(t)>0,y(t)>0, it follows from the first equation of (114) that
(115)0<x(η1)≤r1a11;
on the other hand, by the second equation of (114) and (115), we have
(116)0<y(η2)≤hr1a11,
where h is defined in (12). From the discussion above, the lemma follows immediately.Lemma 10.
If conditions (H1) and (H2) hold, then system (13) has no nonconstant periodic solution with period τ.Proof.
Suppose for a contradiction that system (13) has nonconstant periodic solution with period τ. Then the following system (117) of ordinary differential equations has nonconstant periodic solution:
(117)x˙(t)=x(t)[r1-a11x(t)-a12x(t)y(t)my2(t)+x2(t)],y˙(t)=a21x2(t)y(t)my2(t)+x2(t)-r2y(t),
which has the same equilibria as system (13), that is, E1(r1/a11,0) and a positive equilibrium E*(x*,y*). Note that x-axis and y-axis are the invariable manifold of system (13) and the orbits of system (13) do not intersect each other. Thus, there is no solution crossing the coordinate axis. On the other hand, note the fact that if system (117) has a periodic solution, then there must be the equilibrium in its interior and E1 are located on the coordinate axis. Thus, we conclude that the periodic orbit of system (117) must lie in the first quadrant. From the proof of Theorem 6, we known that if (H1) and (H2) hold, the positive equilibrium is asymptotically stable and globally attractive; thus, there is no periodic orbit in the first quadrant. This ends the proof.Theorem 11.
Suppose the conditions (H1) and (H2) hold; letω0 and τj(j=0,1,…) be defined in (26). If (2a12r22h/a212)-(2r2(a21-r2)/a21)<r1<((4a12r22h+a12a21r2h)/a212), then system (13) has at least j-1 periodic solutions for every τ>τj,(j=1,2,…).Proof.
It is sufficient to prove that the projection ofℓ(E*,τj,2π/ω0) onto τ-space is [τ-,+∞) for each j>0, where τ-≤τj.
The characteristic matrix of (108) at an equilibrium z-=(z-(1),z-(2))∈R2 takes the following form:
(118)Δ(z-,τ,p)(λ)=λId-Dℱ(z-,τ-,p-)(eλId).(z-,τ-,p-) is called a center if ℱ(z-,τ-,p-)=0 and det(Δ(z-,τ-,p-)((2π/p)i))=0. A center is said to be isolated if it is the only center in some neighborhood of (z-,τ-,p-). It follows from (118) that
(119)det(Δ(E1,τ,p)(λ))=(λ+r1)(λ+r2-a21)=0,(120)det(Δ(E*,τ,p)(λ))=λ2+p0λ+p1+p2e-λτ=0,
where p0, p1, and p2 are defined as in Section 3. From the discussion in Section 3, each of (119) and (120) has no purely imaginary root provided that r1>(4a12r22h+a12a21r2h)/a212. Thus, we conclude that (108) has no the center of the form as (E1,τ,p) and (E*,τ,p). On the other hand, from the discussion in Section 3 about the local Hopf bifurcation, it is easy to verify that (E*,τj,2π/ω0) is an isolated center, and there exist  ϵ>0, δ>0, and a smooth curve λ:(τj-δ,τj+δ)→𝒞 such that det(Δ(λ(τ)))=0,|λ(τ)-ω0|<ϵ for all τ∈[τj-δ,τj+δ] and
(121)λ(τj)=ω0i,dReλ(τ)dτ|τ=τj>0.
Let
(122)Ωϵ,(2π/ω0)={(η,p);0<η<ϵ,|p-2πω0|<ϵ}.
It is easy to verify that, on [τj-δ,τj+δ]×∂Ωϵ,2π/ω0,
(123)det(Δ(E*,τ,p)(η+2πpi))=0if  and  only  ifη=0,τ=τj,p=2πω0.
Therefore, the hypotheses (A1)–(A4) in [13] are satisfied. Moreover, if we define
(124)H±(E*,τj,2πω0)(η,p)=det(Δ(E*,τj±δ,p)(η+2πpi)),
then we have the crossing number of isolated center (E*,τj,(2π/ω0)) as follows:
(125)γ(E*,τj,2πω0)=degB(H-(E*,τj,2πω0),Ωϵ,2π/ω0)-degB(H+(E*,τj,2πω0),Ωϵ,2π/ω0)=-1.
Thus, we have
(126)∑(z-,τ-,p-)∈𝒞(E*,τj,2π/ω0)γ(z-,τ-,p-)<0,
where (z-,τ-,p-) has all or parts of the form (E*,τk,2π/ω0)(k=0,1,…). It follows from Lemma 8 that the connected component ℓ(E*,τj,2π/ω0) through (E*,τj,2π/ω0) in Γ is unbounded. From (26), we can know that if (H1) holds, for j≥1,
(127)τj=1ω0arccosω02-p1p2+2jπω0>2πω0.
Now we prove that the projection ofℓ(E*,τj,2π/ω0) onto τ-space is [τ-,+∞), where τ-≤τj. Clearly, it follows from the proof of Lemma 10 that system (13) with τ=0 has no nontrivial periodic solution. Hence, the projection of ℓ(E*,τj,(2π/ω0)) onto τ-space is away from zero.
For a contradiction, we suppose that the projection ofℓ(E*,τj,(2π/ω0)) onto τ-space is bounded; this means that the projection of ℓ(E*,τj,(2π/ω0)) onto τ-space is included in an interval (0,τ*). Noticing (2π/ω0)<τj and applying Lemma 10 we have 0<p<τ* for (z(t),τ,p) belonging to ℓ(E*,τj,(2π/ω0)). Applying Lemma 9, we know that the projection of ℓ(E*,τj,(2π/ω0)) onto z-space is bounded. So the component of ℓ(E*,τj,(2π/ω0)) is bounded. This contradicts our conclusion that ℓ(E*,τj,(2π/ω0)) is unbounded. The contradiction implies that the projection of ℓ(E*τj,(2π/ω0)) onto τ-space is unbounded above.
Hence, system (13) has at least j-1 periodic solution for every τ>τj,(j=1,2,…). This completes the proof.Example 12.
In system (1), we first choose a11=0.1,a12=1,a21=3/2, and m=2. As depicted in Figure 1, a bifurcation diagram is given for system (1) with respect to the parameters r1 and r2. By the discussion in Section 3, system (1) always has a semitrivial equilibrium E1, and if r2>a21, E1 is asymptotically stable; otherwise, E1 is unstable. So if we choose 0<r2<a21=3/2, as depicted in Figure 1, E1 is always unstable. In domains II, V, and VI, the positive equilibrium is not feasible. In domains I, III, and IV, system (1) has a unique positive equilibrium; it is locally asymptotically stable in domain I and is unstable in domain IV. In domain III, system (1) undergoes a Hopf bifurcation at the positive equilibrium at some τ0. Further, we choose r1=5/12,r2=1, a11=0.1,a12=1,a21=3/2,andm=2. In this case, system (1) has a positive equilibrium E*=(5/6,5/12). By computation, we have ω0≈0.1063,τ0≈10.8795, and τ1≈69.9876. From Theorem 5, E* is stable when τ<τ0 as illustrated by numerical simulations (see Figure 2). When τ passes through the critical value τ0, the equilibrium E* loses its stability and a Hopf bifurcation occurs; that is, a family of periodic solution bifurcates from E*. By the algorithm derived in Section 3 and Section 4, we have λ′(τ0)=0.0053-0.0058i,c1(0)=-0.4357+0.0265i, which implies that μ2>0,β2<0, and T2>0. Thus, by the discussion in Section 4, the Hopf bifurcation is supercritical for τ>τ0, the bifurcating periodic solutions from E* at τ0 are asymptotically stable, and the period of these periodic solutions is increasing with the increasing of τ, which are depicted in Figures 3, 4, and 5. Furthermore, Figure 5 shows that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of τ1=69.9876.Figure 1
The bifurcation diagram of system (1) with a11=0.1,a12=1,a21=3/2, and m=2, where L1:r1=-(2/9)r22+r2/3,L2:r1=(8/9)r22((3/2)-r2)/(2r2)-(4/3)r2((3/2)-r2), and L3:r1=((16/9)r2+(1/6))((3/2)-r2)/(2r2).The trajectories and phase graphs of system (1) with τ=τ1+τ2=6+4=10.
(a)
(b)
(c)The trajectories and phase graphs of system (1) with τ=τ1+τ2=6+6=12.
(a)
(b)
(c)The trajectories and phase graphs of system (1) with τ=τ1+τ2=10+8=18.
(a)
(b)
(c)The trajectories and phase graphs of system (1) with τ=τ1+τ2=10+60=70.
(a)
(b)
(c)


## 7. Discussion

In this paper, we have studied a ratio-dependent predator-prey model with two time delays. By analyzing the corresponding characteristic equation, the local stability of the positive equilibrium and the semitrivial equilibrium of system (1) was discussed. We have obtained the estimated length of gestation delay which would not affect the stable coexistence of both prey and predator species at their equilibrium values. The existence of Hopf bifurcation for system (1) at the positive equilibrium was also established. From theoretical analysis it was shown that the larger values of gestation time delay cause fluctuation in individual population density and hence the system becomes unstable. As the estimated length of delay to preserve stability and the critical length of time delay for Hopf bifurcation are dependent upon the parameters of system, it is possible to impose some control, which will prevent the possible abnormal oscillation in population density. The global attractiveness result in Theorem 6 implied that system (1) is permanent if the intrinsic growth rate of the prey and the conversion rate and the interference rate of the predator are high, and the death rate of the predator is low. From Theorem 7 we see that if the death rate of the predator is greater than the conversion rate of the predator, the predator population become extinct for any gestation delay. In particular, the results about boundedness and attractiveness are similar to the results of [4]. From the discussion in Sections 3 and 4, we see that if the values of r1,r2,a11,a12,a21, and m are given, we can get the Hopf bifurcation value of τ, and further we may determine the direction of Hopf bifurcation and the stability of periodic solutions bifurcating from the positive equilibrium E* at the critical point τ0. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay.



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*Source: 321930-2013-12-18.xml*