# The Hopf Bifurcation for a Predator-Prey System withθ-Logistic Growth and Prey Refuge

Authors: Shaoli Wang; Zhihao Ge
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The Hopf bifurcation for a predator-prey system withθ-logistic growth and prey refuge is studied. It is shown that the ODEs undergo a Hopf bifurcation at the positive equilibrium when the prey refuge rate or the index-θ passed through some critical values. Time delay could be considered as a bifurcation parameter for DDEs, and using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results.


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## Body

## 1. Introduction

The construction and study of models for the population dynamics of predator-prey systems have long been and will continue to be one of the dominant themes in both ecology and mathematical ecology since the famous Lotka-Volterra equations. In recent years, the study of the consequences of hiding behavior of prey on the dynamics of predator-prey interactions has been an active topic [1–5]. Some of the empirical and theoretical work have investigated the effects of prey refuges and drawn a conclusion that the refuges used by prey have a stabilizing effect on the considered interactions and prey extinction can be prevented by the addition of refuges [6–12].Motivated by the work of Ko and Ryu [13] and Tsoularis and Wallace [14], we construct the following θ-logistic growth predator-prey system with Holling type-II functional response and prey refuge:
(1)x˙=rx[1-(xK)θ]-βɛxy1+ɛx,y˙=βɛxy1+ɛx-ay,
where x and y represent the densities of prey and predator, respectively, and r, K, θ, β, ɛ, and a are all positive constants and have their biological meanings accordingly. r is the θ-logistic intrinsic growth rate of the prey in the absence of the predator; K is the carrying capacity; θ is the logistic index; β is the predation rate of predator; 0<ɛ<1 and 1-ɛ is the prey refuge rate; a is the death rate of the predator.By assuming that the reproduction of predator after predating the prey will not be instantaneous but mediated by some discrete time lag required for gestation of predator, we incorporate a delay in system (1) to make the model more realistic. We aim to discuss the effect of time delay due to gestation of the predator on the global dynamics of system (1). To this end, we consider the following delayed predator-prey system with θ-logistic growth and prey refuge:
(2)x˙=rx[1-(xK)θ]-βɛxy1+ɛx,y˙=e-mτβɛx(t-τ)y(t-τ)1+ɛx(t-τ)-ay,
where m is positive constant. The constant τ denotes a time delay due to the gestation of the predator and the term e-mτ denotes the probability of the predators, which capture the prey at time t-τ and still alive at times t.The initial conditions for system (2) take the form
(3)x(η)=ϕ(η)≥0,y(η)=ψ(η)≥0,η∈[-τ,0],ϕ(0)>0,ψ(0)>0,
where (ϕ(η),ψ(η))∈C([-τ,0],R+02), the Banach space of continuous functions mapping the interval [-τ,0] into R+02, where R+02={(x,y):x≥0,y≥0}.It is well known by the fundamental theory of functional differential equations [15] that system (2) has a unique solution x(t) and y(t) satisfying initial conditions (3).The organization of this paper is as follows. In Section2, we show the positivity and the boundedness of solutions of system (2) with initial conditions (3). In Section 3, we discuss the stability of boundary equilibria of system (2). In Section 4, we study the existence of Hopf bifurcations for system (1) and (2) at the positive equilibrium. In Section 5, using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Finally, numerical simulations are carried out to illustrate the main results, and a brief discussion is given to conclude this work in Section 6.


## 2. Positivity and Boundedness

In this section, we show the positivity and boundedness of solutions of system (2) with initial conditions (3).


### 2.1. Positivity of Solutions

Theorem 1.
Solutions of system (2) with initial conditions (3) are positive for all t≥0.Proof.
Let(x(t),y(t)) be a solution of system (2) with initial conditions (3). From the first equation of system (2), we have
(4)x(t)=x(0)e∫0t‍(r(1-(x(ς)/K)θ)-βɛy(ς)/(1+ɛx(ς)))dς.
Since ϕ(η)≥0. Hence, x(t) is positive.
To show thaty(t) is positive on [0,+∞), suppose that there exists t1 such that y(t1)=0, and y(t)>0 for t∈[0,t1). Then y˙(t1)≤0. From the second equation of (2), we have
(5)y˙(t1)=e-mτβɛx(t1-τ)y(t1-τ)1+ɛx(t1-τ)-ay(t1)=e-mτβɛx(t1-τ)y(t1-τ)1+ɛx(t1-τ)>0,
which is a contradiction.Next, we will prove the boundedness of solutions.


### 2.2. Boundedness of Solutions

Theorem 2.
Positive solutions of system (2) with initial conditions (3) are ultimately bounded.Proof.
Let(x(t),y(t)) be a solution of system (2) with initial conditions (3). From the first equation of (2), we have
(6)x˙≤rx[1-(xK)θ],
which yields
(7)x(t)≤x0K[x0β+(Kβ-x0β)e-rθt]1/θ,
and therefore
(8)limsupt→+∞x(t)≤K.
Hence, for ρ sufficiently small, there is a T1>0 such that if t>T1, x(t)<K+ρ. Set
(9)V(t)=x(t-τ)e-mτ+y(t).
Calculating the derivative of V along solutions of system (2), we obtain
(10)V˙(t)=e-mτrx(t-τ)[1-(x(t-τ)K)θ]-ay(t)≤e-mτrx(t-τ)-ay(t)≤M0-aV(t),
where M0=e-mτr(K+ρ). Then there exists an M>0, depending only on the parameters of system (2), such that V(t)≤M for all t large enough. Then x(t), y(t) have an ultimately above bound.


## 2.1. Positivity of Solutions

Theorem 1.
Solutions of system (2) with initial conditions (3) are positive for all t≥0.Proof.
Let(x(t),y(t)) be a solution of system (2) with initial conditions (3). From the first equation of system (2), we have
(4)x(t)=x(0)e∫0t‍(r(1-(x(ς)/K)θ)-βɛy(ς)/(1+ɛx(ς)))dς.
Since ϕ(η)≥0. Hence, x(t) is positive.
To show thaty(t) is positive on [0,+∞), suppose that there exists t1 such that y(t1)=0, and y(t)>0 for t∈[0,t1). Then y˙(t1)≤0. From the second equation of (2), we have
(5)y˙(t1)=e-mτβɛx(t1-τ)y(t1-τ)1+ɛx(t1-τ)-ay(t1)=e-mτβɛx(t1-τ)y(t1-τ)1+ɛx(t1-τ)>0,
which is a contradiction.Next, we will prove the boundedness of solutions.


## 2.2. Boundedness of Solutions

Theorem 2.
Positive solutions of system (2) with initial conditions (3) are ultimately bounded.Proof.
Let(x(t),y(t)) be a solution of system (2) with initial conditions (3). From the first equation of (2), we have
(6)x˙≤rx[1-(xK)θ],
which yields
(7)x(t)≤x0K[x0β+(Kβ-x0β)e-rθt]1/θ,
and therefore
(8)limsupt→+∞x(t)≤K.
Hence, for ρ sufficiently small, there is a T1>0 such that if t>T1, x(t)<K+ρ. Set
(9)V(t)=x(t-τ)e-mτ+y(t).
Calculating the derivative of V along solutions of system (2), we obtain
(10)V˙(t)=e-mτrx(t-τ)[1-(x(t-τ)K)θ]-ay(t)≤e-mτrx(t-τ)-ay(t)≤M0-aV(t),
where M0=e-mτr(K+ρ). Then there exists an M>0, depending only on the parameters of system (2), such that V(t)≤M for all t large enough. Then x(t), y(t) have an ultimately above bound.


## 3. Stability of the Boundary Equilibria

In this section, we discuss the stability of the boundary equilibria of system (2) with initial conditions (3).We denoteR0=Kɛ(βe-mτ-a)/a and always assume β>a. The system (2) always has two boundary equilibria E0=(0,0) and E1=(K,0); if R0>1, it also has a positive equilibrium E*=(x*,y*), where
(11)x*=aɛ(βe-mτ-a),y*=r[1-(x*/K)θ](1+ɛx*)βɛ.Now we consider the stability of boundary equilibria.ForE0=(0,0), the corresponding characteristic equation is
(12)(λ-r)(λ+a)=0,
and the roots are
(13)λ1=r>0,λ2=-a<0,
which implies that the equilibrium E0 is always unstable.ForE1=(K,0), the corresponding characteristic equation is
(14)(λ+rθ)(λ+a-Kβɛe-mτ1+Kɛe-mτe-λτ)=0.
It follows that
(15)λ1=-rθ
or
(16)λ+a-Kβɛe-mτ1+Kɛe-λτ=0.
Denote
(17)f(λ)=λ+a-Kβɛe-mτ1+Kɛe-λτ,
and then we have
(18)f(0)=a-Kβɛe-mτ1+Kɛ=11+Kɛ(a+Kaɛ-Kβɛe-mτ)=a1+Kɛ(1-R0),limλ→∞f(λ)=∞
for any τ≥0. Hence f(λ)=0 has no positive root for R0≤1, and at least one positive for R0>1. Therefore, for all τ≥0, the equilibrium E1 is stable when R0≤1 and unstable when R0>1.Summarizing the discussion above, we obtain the following conclusion.Theorem 3.
(i) The equilibriumE0 is always unstable for all τ≥0.
(ii) The equilibriumE1 is stable when R0≤1 and unstable when R0>1 for all τ≥0.


## 4. The Hopf Bifurcation

### 4.1. The Hopf Bifurcation of ODEs

WhenR0>1, the system (2) also has a positive equilibrium E*. The characteristic of the linearized system of (2) near the infected equilibrium E* is given by
(19)P(λ,τ)+Q(λ,τ)e-λτ=0,
where
(20)P(λ,τ)=λ2+b1(τ)λ+b2(τ),Q(λ,τ)=b3(τ)λ+b4(τ),(21)b1(τ)=a+rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2,b2(τ)=a[rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2],b3(τ)=-βɛx*e-mτ1+ɛx*,b4(τ)=β2ɛ2x*y*e-mτ(1+ɛx*)3-βɛx*e-mτ1+ɛx*[rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2].
When τ=0, R0 becomes R0*=Kɛ(β-a)/a and (19) becomes
(22)λ2+a1(0)λ+a2(0)=0,
where
(23)a1(0)=b1(0)+b3(0)=a+rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2-βɛx*1+ɛx*=rθ(x*K)θ-aɛy*1+ɛx*=r(θ+aβ)(aKɛ(β-a))θ-arβ,(24)a2(0)=b2(0)+b4(0)=a[rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2]+β2ɛ2x*y*(1+ɛx*)3-βɛx*1+ɛx*[rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2]=β2ɛ2x*y*(1+ɛx*)3>0.Theorem 4.
(i) IfR0*>1 and a1(0)>0, then the positive equilibrium E* of system (1) is asymptotically stable.
(ii) IfR0*>1 and a1(0)<0, then system (1) is unstable.
(iii) Suppose thatR0*>1. Then system (1) undergoes a Hopf bifurcation when ɛ (or θ) passes through ɛc (or θc).Proof.
Obviously (i) and (ii) hold.
(iii) From (23), we know that the root of the characteristic equation (22) satisfies
(25)Reλ=-b1(0)+b3(0)2=r2[aβ-(β+θ)(aKɛ(β-a))θ].
Solving Reλ=0, we have
(26)ɛc=a(1+βθ/a)1/θK(β-a)
or
(27)a=(a+βθc)[aKɛ(β-a)]θc.
Calculating the derivative, we obtain
(28)dRe(λ(ɛ))dɛ|ɛ=ɛc=-arθ(a+βθ)2Kβ(β-a)[aKɛ(β-a)]θ-1|ɛ=ɛc<0
or
(29)dRe(λ(θ))dθ|θ=θc=-r2[aKɛ(β-a)]θ-rθ(a+βθ)2β[aKɛ(β-a)]θ-1|θ=θc<0.
Hence the positive equilibrium is stable when ɛ<ɛc (θ<θc) and unstable when ɛ>ɛc (θ>θc). Thus Hopf bifurcation occurs at ɛ=ɛc (θ=θc).Example 5.
We choose a series of parameterr=0.1, K=10, β=0.2, a=0.13, and θ=3, then ɛc≈0.33. If ɛ=0.3<ɛc, then R0*≈1.62>1 and a1(0)≈0.022>0, thus the system (1) is stable (see Figure 1(a)); if ɛ=0.42>ɛc, then R0*≈2.26>1 and a1(0)≈-0.033<0, and thus the system (1) is unstable (see Figure 1(b)).
If we choose a series of parameterr=0.1, K=10, β=0.2, a=0.13, and ɛ=0.3, then R0*≈1.62>1 and θc≈4.18 (θc≈2.08×10-22 is unserviceable). If θ=4<θc, then a1(0)≈0.003>0, and thus the system (1) is stable (see Figure 2(a)); if θ=5>θc, then a1(0)≈-0.014<0, and thus the system (1) is unstable (see Figure 2(b)).The infected steady stateE* of system (1) is asymptotically stable when ɛ=0.3 (a) and unstable when ɛ=0.42 (b). The rest parameter values are r=0.1, K=10, β=0.2, a=0.13, and θ=3, and the initial values are x0=8 and y0=5.
(a)
(b)The infected steady stateE* of system (1) is asymptotically stable when θ=4 (a), and unstable when θ=5 (b). The rest parameter values are r=0.1, K=10, β=0.2, a=0.13, and ɛ=0.3, and the initial values are x0=8 and y0=5.
(a)
(b)


### 4.2. The Hopf Bifurcation of DDEs

In the following, we investigate the existence of purely imaginary rootsλ=iω(ω>0) to (19). Equation (19) takes the form of a second-degree exponential polynomial in λ, with all the coefficients of P and Q depending on τ. Beretta and Kuang [16] established a geometrical criterion which gives the existence of purely imaginary root of a characteristic equation with delay dependent coefficients.In order to apply the criterion due to Beretta and Kuang [16], we need to verify the following properties for all τ∈[0,τmax), where τmax is the maximum value in which E* exists.(a)
P(0,τ)+Q(0,τ)≠0;(b)
P(iω,τ)+Q(iω,τ)≠0;(c)
limsup{|P(λ,τ)/Q(λ,τ)|:|λ|→∞,Reλ≥0}<1;(d)
F(ω,τ)=|P(iω,τ)|2-|Q(iω,τ)|2 has a finite number of zeros;(e)
each positive rootω(τ) of F(ω,τ)=0 is continuous and differentiable in τ whenever it exists.Here,P(λ,τ) and Q(λ,τ) are defined as in (20).Letτ∈[0,τmax), and using (20) and (21), we have
(30)P(0,τ)+Q(0,τ)=b2(τ)+b4(τ)=β2ɛ2x*y*e-mτ(1+ɛx*)3>0,
and then
(31)P(iω,τ)+Q(iω,τ)=-ω2+b2(τ)+b4(τ)+iω[b1(τ)+b3(τ)]≠0.
Therefore, (a) and (b) are satisfied.From (20), we know that
(32)lim|λ|→+∞|Q(λ,τ)P(λ,τ)|=lim|λ|→+∞|b3(τ)λ+b4(τ)λ2+b1(τ)λ+b2(τ)|=0.
Therefore, (c) follows.LetF be defined as in (d). From
(33)|P(iω,τ)|2=ω4+[b1(τ)2-2b2(τ)]ω2+b2(τ)2,|Q(iω,τ)|2=b3(τ)2ω2+b4(τ)2,
we have
(34)F(ω,τ)=ω4+a1(τ)ω2+a2(τ),
where
(35)a1(τ)=b12(τ)-2b2(τ)-b32(τ),a2(τ)=b22(τ)-b42(τ).It is obvious that property (d) is satisfied. Let(ω0,τ0) be a point of its domain of definition such that F(ω0,τ0)=0. We know that the partial derivatives Fω and Fτ exist and are continuous in a certain neighborhood of (ω0,τ0), and Fω(ω0,τ0)≠0. By implicit function theorem, (e) is also satisfied.Now letλ=iω(ω>0) be a root of (19). Substituting it into (19) and separating the real and imaginary parts yields
(36)ω2-b2(τ)=b3(τ)ωsinωτ+b4(τ)cosωτ,b1(τ)ω=b4(τ)sinωτ-b3(τ)ωcosωτ.
From (36), it follows that(37a)sinωτ=ω[b3(τ)(ω2-b2(τ))+b1(τ)b4(τ)]b3(τ)ω2+b42(τ),(37b)cosωτ=(b4(τ)-b1(τ)b3(τ))ω2-b2(τ)b4(τ)b3(τ)ω2+b42(τ).By the definitions ofP(λ,τ), Q(λ,τ) as in (20), and applying the property (a), (20) can be written as(38a)sinωτ=ImP(iω,τ)Q(iω,τ)(38b)cosωτ=-ReP(iω,τ)Q(iω,τ),which yields
(39)|P(iω,τ)|2=|Q(iω,τ)|2.Assume thatI∈R+0 is the set where ω(τ) is a positive root of
(40)F(ω,τ)=|P(iω,τ)|2-|Q(iω,τ)|2
and for τ∉I,ω(τ) is not defined. Then for all τ in I, ω(τ) is satisfying
(41)F(ω,τ)=0.
Let ω2=h; we have that
(42)F(h,τ)=h2+a1(τ)h+a2(τ)=0.
We set
(43)Δ(τ)=a1(τ)2-4a2(τ)=[b12(τ)-2b2(τ)-b32(τ)]2-4[b22(τ)-b42(τ)].
Then, when Δ(τ)≥0, F(h,τ)=0 has real roots given by
(44)h+(τ)=-a1(τ)+Δ(τ)2,h-(τ)=-a1(τ)-Δ(τ)2.
Note that
(45)a1(τ)=b12(τ)-2b2(τ)-b32(τ)=[a+rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2]2-2a[rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2]-(βɛx*e-mτ1+ɛx*)2=[rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2]2>0,
and summarizing the discussion above, we have the following conclusion.Proposition 6.
IfΔ(τ)≥0 and a2(τ)<0, the F(h,τ)=0 has only one positive root denoted by h+. Furthermore, F(ω,τ)=0 has a unique positive root given by ω=h+.
Defineα(τ)∈[0,2π) such that sinα(τ) and cosα(τ) are given by the right hand sides of (37a) and (37b), respectively, with α(τ) given by (38a) and (38b).
And the relation between the argumentα and ωτ in (37a) and (37b) for τ>0 must be
(46)ωτ=α+2nπ,n=0,1,2,….
Hence we can define the maps τn:I→R+0 given by
(47)τn(τ):=α(τ)+2nπω(τ),τn>0,n=0,1,2,…,
where a positive root ω(τ) of (42) exists in I.
Let one introduces the functionsSn(τ):I→R,
(48)Sn(τ)=τ-α(τ)+2nπω(τ),n=0,1,2,…,
that are continuous and differentiable in τ. Thus, we give the following theorem which is due to Beretta and Kuang [16].Theorem 7.
Assume thatω(τ) is a positive root of (19) defined for τ∈I,I⊆R+0, and at some τ*∈I, Sn(τ*)=0 for some n∈N0, then a pair of simple conjugate pure imaginary roots λ=±iω exists at τ=τ* which crosses the imaginary axis from left to right if δ(τ*)>0 and crosses the imaginary axis from right to left if δ(τ*)<0, where
(49)δ(τ*)=sign{Fω′(ωτ*,τ*)}sign{dSn(τ)dτ|τ=τ*}.Applying Theorem3 and the Hopf bifurcation theorem for functional differential equation [12], we can conclude the existence of a Hopf bifurcation as stated in the following theorem.Theorem 8.
For system (2), the following conclusions are hold.(i)
Ifa1(0)>0 and the function S0(τ) has no positive zero in I, then the equilibrium E* is asymptotically stable for all τ∈[0,τmax).(ii)
Ifa1(0)>0 and the function S0(τ) has positive zero in I, then there exists τ*∈I, such that the equilibrium E* is asymptotically stable for τ∈[0,τ*) and becomes unstable for τ∈(τ*,τmax), with a Hopf bifurcation occurring when τ=τ*.(iii)
Ifa1(0)<0 and the function Sn(τ) has no positive zeros in I, then the equilibrium E* is always unstable for all τ∈[0,τmax).(iv)
Ifa1(0)<0 and the function Sn(τ) has positive zeros in I, for some n∈N, then there exists τ*∈I, such that the equilibrium E* is unstable for 0≤τ<τ* and becomes asymptotically stable for τ∈(τ*,τmax), with a Hopf bifurcation occurring when τ=τ*.Remark 9.
Ifτ≥(lnβ-lna(1+1/Kɛ))/m:=τmax, then R0≤1, y*≤0 and the equilibrium E* converges to E1=(K,0).


## 4.1. The Hopf Bifurcation of ODEs

WhenR0>1, the system (2) also has a positive equilibrium E*. The characteristic of the linearized system of (2) near the infected equilibrium E* is given by
(19)P(λ,τ)+Q(λ,τ)e-λτ=0,
where
(20)P(λ,τ)=λ2+b1(τ)λ+b2(τ),Q(λ,τ)=b3(τ)λ+b4(τ),(21)b1(τ)=a+rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2,b2(τ)=a[rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2],b3(τ)=-βɛx*e-mτ1+ɛx*,b4(τ)=β2ɛ2x*y*e-mτ(1+ɛx*)3-βɛx*e-mτ1+ɛx*[rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2].
When τ=0, R0 becomes R0*=Kɛ(β-a)/a and (19) becomes
(22)λ2+a1(0)λ+a2(0)=0,
where
(23)a1(0)=b1(0)+b3(0)=a+rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2-βɛx*1+ɛx*=rθ(x*K)θ-aɛy*1+ɛx*=r(θ+aβ)(aKɛ(β-a))θ-arβ,(24)a2(0)=b2(0)+b4(0)=a[rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2]+β2ɛ2x*y*(1+ɛx*)3-βɛx*1+ɛx*[rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2]=β2ɛ2x*y*(1+ɛx*)3>0.Theorem 4.
(i) IfR0*>1 and a1(0)>0, then the positive equilibrium E* of system (1) is asymptotically stable.
(ii) IfR0*>1 and a1(0)<0, then system (1) is unstable.
(iii) Suppose thatR0*>1. Then system (1) undergoes a Hopf bifurcation when ɛ (or θ) passes through ɛc (or θc).Proof.
Obviously (i) and (ii) hold.
(iii) From (23), we know that the root of the characteristic equation (22) satisfies
(25)Reλ=-b1(0)+b3(0)2=r2[aβ-(β+θ)(aKɛ(β-a))θ].
Solving Reλ=0, we have
(26)ɛc=a(1+βθ/a)1/θK(β-a)
or
(27)a=(a+βθc)[aKɛ(β-a)]θc.
Calculating the derivative, we obtain
(28)dRe(λ(ɛ))dɛ|ɛ=ɛc=-arθ(a+βθ)2Kβ(β-a)[aKɛ(β-a)]θ-1|ɛ=ɛc<0
or
(29)dRe(λ(θ))dθ|θ=θc=-r2[aKɛ(β-a)]θ-rθ(a+βθ)2β[aKɛ(β-a)]θ-1|θ=θc<0.
Hence the positive equilibrium is stable when ɛ<ɛc (θ<θc) and unstable when ɛ>ɛc (θ>θc). Thus Hopf bifurcation occurs at ɛ=ɛc (θ=θc).Example 5.
We choose a series of parameterr=0.1, K=10, β=0.2, a=0.13, and θ=3, then ɛc≈0.33. If ɛ=0.3<ɛc, then R0*≈1.62>1 and a1(0)≈0.022>0, thus the system (1) is stable (see Figure 1(a)); if ɛ=0.42>ɛc, then R0*≈2.26>1 and a1(0)≈-0.033<0, and thus the system (1) is unstable (see Figure 1(b)).
If we choose a series of parameterr=0.1, K=10, β=0.2, a=0.13, and ɛ=0.3, then R0*≈1.62>1 and θc≈4.18 (θc≈2.08×10-22 is unserviceable). If θ=4<θc, then a1(0)≈0.003>0, and thus the system (1) is stable (see Figure 2(a)); if θ=5>θc, then a1(0)≈-0.014<0, and thus the system (1) is unstable (see Figure 2(b)).The infected steady stateE* of system (1) is asymptotically stable when ɛ=0.3 (a) and unstable when ɛ=0.42 (b). The rest parameter values are r=0.1, K=10, β=0.2, a=0.13, and θ=3, and the initial values are x0=8 and y0=5.
(a)
(b)The infected steady stateE* of system (1) is asymptotically stable when θ=4 (a), and unstable when θ=5 (b). The rest parameter values are r=0.1, K=10, β=0.2, a=0.13, and ɛ=0.3, and the initial values are x0=8 and y0=5.
(a)
(b)


## 4.2. The Hopf Bifurcation of DDEs

In the following, we investigate the existence of purely imaginary rootsλ=iω(ω>0) to (19). Equation (19) takes the form of a second-degree exponential polynomial in λ, with all the coefficients of P and Q depending on τ. Beretta and Kuang [16] established a geometrical criterion which gives the existence of purely imaginary root of a characteristic equation with delay dependent coefficients.In order to apply the criterion due to Beretta and Kuang [16], we need to verify the following properties for all τ∈[0,τmax), where τmax is the maximum value in which E* exists.(a)
P(0,τ)+Q(0,τ)≠0;(b)
P(iω,τ)+Q(iω,τ)≠0;(c)
limsup{|P(λ,τ)/Q(λ,τ)|:|λ|→∞,Reλ≥0}<1;(d)
F(ω,τ)=|P(iω,τ)|2-|Q(iω,τ)|2 has a finite number of zeros;(e)
each positive rootω(τ) of F(ω,τ)=0 is continuous and differentiable in τ whenever it exists.Here,P(λ,τ) and Q(λ,τ) are defined as in (20).Letτ∈[0,τmax), and using (20) and (21), we have
(30)P(0,τ)+Q(0,τ)=b2(τ)+b4(τ)=β2ɛ2x*y*e-mτ(1+ɛx*)3>0,
and then
(31)P(iω,τ)+Q(iω,τ)=-ω2+b2(τ)+b4(τ)+iω[b1(τ)+b3(τ)]≠0.
Therefore, (a) and (b) are satisfied.From (20), we know that
(32)lim|λ|→+∞|Q(λ,τ)P(λ,τ)|=lim|λ|→+∞|b3(τ)λ+b4(τ)λ2+b1(τ)λ+b2(τ)|=0.
Therefore, (c) follows.LetF be defined as in (d). From
(33)|P(iω,τ)|2=ω4+[b1(τ)2-2b2(τ)]ω2+b2(τ)2,|Q(iω,τ)|2=b3(τ)2ω2+b4(τ)2,
we have
(34)F(ω,τ)=ω4+a1(τ)ω2+a2(τ),
where
(35)a1(τ)=b12(τ)-2b2(τ)-b32(τ),a2(τ)=b22(τ)-b42(τ).It is obvious that property (d) is satisfied. Let(ω0,τ0) be a point of its domain of definition such that F(ω0,τ0)=0. We know that the partial derivatives Fω and Fτ exist and are continuous in a certain neighborhood of (ω0,τ0), and Fω(ω0,τ0)≠0. By implicit function theorem, (e) is also satisfied.Now letλ=iω(ω>0) be a root of (19). Substituting it into (19) and separating the real and imaginary parts yields
(36)ω2-b2(τ)=b3(τ)ωsinωτ+b4(τ)cosωτ,b1(τ)ω=b4(τ)sinωτ-b3(τ)ωcosωτ.
From (36), it follows that(37a)sinωτ=ω[b3(τ)(ω2-b2(τ))+b1(τ)b4(τ)]b3(τ)ω2+b42(τ),(37b)cosωτ=(b4(τ)-b1(τ)b3(τ))ω2-b2(τ)b4(τ)b3(τ)ω2+b42(τ).By the definitions ofP(λ,τ), Q(λ,τ) as in (20), and applying the property (a), (20) can be written as(38a)sinωτ=ImP(iω,τ)Q(iω,τ)(38b)cosωτ=-ReP(iω,τ)Q(iω,τ),which yields
(39)|P(iω,τ)|2=|Q(iω,τ)|2.Assume thatI∈R+0 is the set where ω(τ) is a positive root of
(40)F(ω,τ)=|P(iω,τ)|2-|Q(iω,τ)|2
and for τ∉I,ω(τ) is not defined. Then for all τ in I, ω(τ) is satisfying
(41)F(ω,τ)=0.
Let ω2=h; we have that
(42)F(h,τ)=h2+a1(τ)h+a2(τ)=0.
We set
(43)Δ(τ)=a1(τ)2-4a2(τ)=[b12(τ)-2b2(τ)-b32(τ)]2-4[b22(τ)-b42(τ)].
Then, when Δ(τ)≥0, F(h,τ)=0 has real roots given by
(44)h+(τ)=-a1(τ)+Δ(τ)2,h-(τ)=-a1(τ)-Δ(τ)2.
Note that
(45)a1(τ)=b12(τ)-2b2(τ)-b32(τ)=[a+rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2]2-2a[rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2]-(βɛx*e-mτ1+ɛx*)2=[rθ(x*K)θ-βɛ2x*y*(1+ɛx*)2]2>0,
and summarizing the discussion above, we have the following conclusion.Proposition 6.
IfΔ(τ)≥0 and a2(τ)<0, the F(h,τ)=0 has only one positive root denoted by h+. Furthermore, F(ω,τ)=0 has a unique positive root given by ω=h+.
Defineα(τ)∈[0,2π) such that sinα(τ) and cosα(τ) are given by the right hand sides of (37a) and (37b), respectively, with α(τ) given by (38a) and (38b).
And the relation between the argumentα and ωτ in (37a) and (37b) for τ>0 must be
(46)ωτ=α+2nπ,n=0,1,2,….
Hence we can define the maps τn:I→R+0 given by
(47)τn(τ):=α(τ)+2nπω(τ),τn>0,n=0,1,2,…,
where a positive root ω(τ) of (42) exists in I.
Let one introduces the functionsSn(τ):I→R,
(48)Sn(τ)=τ-α(τ)+2nπω(τ),n=0,1,2,…,
that are continuous and differentiable in τ. Thus, we give the following theorem which is due to Beretta and Kuang [16].Theorem 7.
Assume thatω(τ) is a positive root of (19) defined for τ∈I,I⊆R+0, and at some τ*∈I, Sn(τ*)=0 for some n∈N0, then a pair of simple conjugate pure imaginary roots λ=±iω exists at τ=τ* which crosses the imaginary axis from left to right if δ(τ*)>0 and crosses the imaginary axis from right to left if δ(τ*)<0, where
(49)δ(τ*)=sign{Fω′(ωτ*,τ*)}sign{dSn(τ)dτ|τ=τ*}.Applying Theorem3 and the Hopf bifurcation theorem for functional differential equation [12], we can conclude the existence of a Hopf bifurcation as stated in the following theorem.Theorem 8.
For system (2), the following conclusions are hold.(i)
Ifa1(0)>0 and the function S0(τ) has no positive zero in I, then the equilibrium E* is asymptotically stable for all τ∈[0,τmax).(ii)
Ifa1(0)>0 and the function S0(τ) has positive zero in I, then there exists τ*∈I, such that the equilibrium E* is asymptotically stable for τ∈[0,τ*) and becomes unstable for τ∈(τ*,τmax), with a Hopf bifurcation occurring when τ=τ*.(iii)
Ifa1(0)<0 and the function Sn(τ) has no positive zeros in I, then the equilibrium E* is always unstable for all τ∈[0,τmax).(iv)
Ifa1(0)<0 and the function Sn(τ) has positive zeros in I, for some n∈N, then there exists τ*∈I, such that the equilibrium E* is unstable for 0≤τ<τ* and becomes asymptotically stable for τ∈(τ*,τmax), with a Hopf bifurcation occurring when τ=τ*.Remark 9.
Ifτ≥(lnβ-lna(1+1/Kɛ))/m:=τmax, then R0≤1, y*≤0 and the equilibrium E* converges to E1=(K,0).


## 5. Direction and Stability of the Hopf Bifurcation of the DDEs

In the above section, we have obtained some conditions which guarantee that the delayed predator-prey system withθ-logistic growth and prey refuge undergoes the Hopf bifurcation at some value of τ=τ*. In this section, we will study the direction, stability, and the period of the bifurcating periodic solutions. The approach we used here is based on the normal form approach and the center manifold theory introduced by Hassard et al. [17]. Throughout this section, we always assume that system (2) undergoes Hopf bifurcation at the positive equilibrium E*=(x*,y*) for τ=τ*, and then ±iω is corresponding purely imaginary roots of the characteristic equation at the positive equilibrium E*=(x*,y*).Letu1(t)=x(t)-x*, u2(t)=y(t)-y*.System (2) is transformed into
(50)u˙1(t)=α1u1(t)+α2u2(t)+Σi+j≥21i!j!fij(1)u1i(t)u2j(t),u˙2(t)=β1u2(t)+β2u1(t-τ)+β3u2(t-τ)u˙2(t)=+Σi+j+l≥21i!j!l!fijl(2)u2i(t)u1j(t-τ)u2l(t-τ),
where
(51)α1=-rθ(x*K)θ+βɛ2y*(1+ɛx*)2,α2=-βɛx*1+ɛx*,β1=-a,β2=βɛy*e-mτ(1+ɛx*)2,β3=βɛx*e-mτ1+ɛx*,f(1)=rx[1-(xK)θ]-βɛxy1+ɛx,f(2)=e-mτβɛx(t-τ)y(t-τ)1+ɛx(t-τ)-ay,fij(1)=∂i+jf(1)∂xi∂yj|(x*,y*),fijl(2)=∂i+j+lf(2)∂yi∂x(t-τ)j∂y(t-τ)l|(y*,x*,y*),i,j,l≥0.
For the simplicity of notations, we rewrite (50) as
(52)u˙(t)=Lυ(ut)+f(υ,ut),
where u(t)=(u1(t),u2(t))T∈R2, ut(ζ)∈C is defined by ut(ζ)=u(t+ζ), and Lυ:C→R2, f:R×C→R2 are given, respectively, by
(53)Lυ(ϕ)=(α1α20β1)ϕ(0)+(00β2β3)ϕ(-τ),(54)f(υ,ϕ)=(Σi+j≥21i!j!fij(1)ϕ1i(t)ϕ2j(t)Σi+j+l≥21i!j!l!fijl(2)ϕ2i(t)ϕ1j(t-τ)ϕ2l(t-τ)).
By the Riesz representation theorem, there exists a function η(ζ,υ) of bounded variation for ζ∈[-τ,0], such that
(55)Lυ(ϕ)=∫-τ0‍dη(ζ,0)ϕ(ζ)forϕ∈C.
In fact, we can choose
(56)η(ζ,υ)=(α1α20β1)δ(ζ)-(00β2β3)δ(ζ+τ),
where δ denotes the Dirac delta function. For ϕ∈C1([-τ,0],R2), define
(57)A(υ)ϕ={dϕ(ζ)dζ,ζ∈[-τ,0),∫-τ0‍{dϕ(ζ)dζ}dη(s,υ)ϕ(s),ζ=0,R(υ)ϕ={0,ζ∈[-τ,0),f(υ,ϕ),ζ=0.
Then system (52) is equivalent to
(58)u˙(t)=A(υ)ut+R(υ)ut,
where ut(ζ)=u(t+ζ) for ζ∈[-τ,0].Forψ∈C1([0,τ],(R2)*), define
(59)A*ψ(s)={-dψ(s)ds,s∈(0,τ],∫-τ0‍{dψ(s)ds}ψ(-t)dη(t,0),s=0,
and a bilinear inner product
(60)〈ψ(s),ϕ(ζ)〉=ψ¯(0)ϕ(0)-∫-τ0‍∫ξ=0ζ‍ψ¯(ξ-ζ)dη(ζ)ϕ(ξ)dξ,
where η(ζ)=η(ζ,0). Then A(0) and A* are adjoint operators. By the discussion in Section 4, we know that ±iω*τ* 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 iω*τ* and -iω*τ*, respectively.Suppose thatq(ζ)=(1,q)Teiω*τ*ζ is the eigenvector of A(0) corresponding to iω*τ*; then A(0)q(ζ)=iω*τ*q(ζ). It follows from (55) and (56) and the definition of A(0) that
(61)(α1-iω*α2β2e-iω*τ*β1+β3e-iω*τ*-iω*)q(0)=(00).
Solving the equations above, we derive that
(62)q(0)=(1,ρ)T=(1,iω*-α1α2)T.
On the other hand, suppose that q*(s)=D(1,σ)eiω*τ*s is the eigenvector of A* corresponding to -iω*τ*. It follows from (55) and (56) and the definition of A* that
(63)(α1+iω*β2eiω*τ*α2β1+β3eiω*τ*+iω*)(q*(0))T=(00),
which yields
(64)q*(0)=D(1,σ)=D(1,-α1+iω*β2e-iω*τ*).In order to assure〈q*(s),q(ζ)〉=1, we need to determine the value of D. By (58), we have
(65)〈q*(s),q(ζ)〉=D¯(1,σ¯)(1,ρ)T-∫-τ*0‍∫ξ=0ζ‍D¯(1,σ¯)e-iω*τ*(ξ-ζ)dη(ζ)(1,ρ)Teiω*τ*ξdξ=D¯{1+ρσ¯-∫-τ*0‍(1,σ¯)eiω*τ*ζdη(ζ)(1,ρ)T}=D¯{1+ρσ¯+(β2+ρβ3)σ¯τ*eiω*τ*}.
Thus, we can choose D as
(66)D=11+ρσ¯+(β2+ρβ3)σ¯τ*eiω*τ*
such that
(67)〈q*(s),q(ζ)〉=1,〈q*(s),q¯(ζ)〉=0.In the following, we apply the ideas in Hassard et al. [17] to describe the center manifold C0 at μ=0, similar to that in [18, 19]. Let ut be the solution of (52) when μ=0. Define
(68)z(t)=〈q*,ut〉,W(t,ζ)=ut(ζ)-2Re{z(t)q(ζ)}.
On the center manifold C0, we have
(69)W(t,ζ)=W(z(t),z¯(t),ζ),
where
(70)W(z(t),z¯(t),ζ)=W20(ζ)z22+W11(ζ)zz¯+W02(ζ)z¯22+···,
and z and z¯ are local coordinates for center manifold C0 in the direction of q* and q¯*. Note that W is real if ut is real. We only consider real solutions. For solution ut∈C0 of (52), since μ=0, we have
(71)z˙(t)=iω*τ*z+q¯*(0)f(0,W(z,z¯,0)+2Re{zq(ζ)})=iω*τ*z+q¯*(0)f0(z,z¯).We rewrite this equation as(72)z˙(t)=iω*τ*z+g(z,z¯),
where
(73)g(z,z¯)=q¯*(0)f0(z,z¯)=g20z22+g11zz¯+g02z¯22+g21z2z¯2+⋯.
It follows from (68) that ut(ζ)=W(t,ζ)+zq(ζ)+z¯q¯(ζ) and q(ζ)=(1,ρ)Teiω*τ*ζ, and then
(74)u1t(0)=z+z¯+W20(1)(0)z22+W11(1)(0)zz¯u1t(0)=+W02(1)(0)z¯22+···,u1t(-τ*)=ze-iω*τ*+z¯eiω*τ*+W20(1)(-τ*)z22u1t(-τ*)=+W11(1)(-τ*)zz¯+W02(1)(-τ*)z¯22+···,u2t(0)=zρ+z¯ρ¯+W20(2)(0)z22+W11(2)(0)zz¯u2t(0)=+W02(2)(0)z¯22+···,u2t(-τ*)=zρe-iω*τ*+z¯ρ¯eiω*τ*+W20(2)(-τ*)z22u2t(-τ*)=+W11(2)(-τ*)zz¯+W02(2)(-τ*)z¯22+···.
It follows from (54) and (73) that
(75)g20=D¯[+f002(2)ρ2e-2iω*τ*)f20(1)+f02(1)ρ2+2f11(1)ρ=D¯+σ¯(2f011(2)ρe-2iω*τ*+f020(2)e-2iω*τ*=D¯+f002(2)ρ2e-2iω*τ*)],g11=D¯[f20(1)+f02(1)ρρ¯+f11(1)(ρ+ρ¯)=D¯+σ¯(f011(2)(ρ+ρ¯)+f020(2)+f002(2)ρρ¯)],g02=D¯[=D¯+f002(2)ρ¯2e2iω*τ*)f20(1)+f02(1)ρ¯2+2f11(1)ρ¯=D¯+σ¯(2f011(2)ρ¯e2iω*τ*+f020(2)e2iω*τ*=D¯+f002(2)ρ¯2e2iω*τ*)],g21=2D¯×{(W11(2)(0)+12W20(2)(0)+ρW11(1)(0)+12ρ¯W20(1)(0))σ¯×[(W11(2)(0)+12W20(2)(0)+ρW11(1)(0)+12ρ¯W20(1)(0))f011(2)×(12eiω*τ*W20(2)(-τ*)+e-iω*τ*W11(2)(-τ*)+ρe-iω*τ*W11(1)(-τ*)+12ρ¯eiω*τ*W20(1)(-τ*))+f020(2)(e-iω*τ*W11(1)(-τ*)+12eiω*τ*W20(1)(-τ*))+f002(2)(ρe-iω*τ*W11(2)(-τ*)+12eiω*τ*W20(2)(-τ*))+12f030(2)e-iω*τ*+12f003(2)ρ2ρ¯e-iω*τ*+12f012(2)ρ(ρ+2ρ¯)e-iω*τ*+12f021(2)(ρ+2ρ¯)e-iω*τ*]+[f20(1)(W11(1)(0)+12W20(1)(0))+f02(1)(ρW11(2)(0)+12ρ¯W20(2)(0))+f11(1)(W11(2)(0)+12W20(2)(0)+f11(1)+ρW11(1)(0)+12ρ¯W20(1)(0))]}.
We now calculate W20(ζ) and W11(ζ). It follows from (58) and (68) that
(76)W˙=u˙t-z˙q-z¯˙q¯={A(0)W-2Re{q¯*f0q(ζ)},s∈[-τ*,0),A(0)W-2Re{q¯*f0q(0)}+f0,ζ=0,≔AW+H(z,z¯,ζ),
where
(77)H(z,z¯,ζ)=H20(ζ)z22+H11(ζ)zz¯+H02(ζ)z¯22+···.
On the other hand, on C0 near the origin
(78)W˙=Wzz˙+Wz¯z¯˙.
We derive from (76)–(78) that
(79)(A-2iω0)W20(ζ)=-H20(ζ),AW11(ζ)=-H11(ζ),….
It follows from (73) and (76) that for ζ∈[-τ*,0),
(80)H(z,z¯,ζ)=-q¯*f0q(ζ)-q*f¯0q¯(ζ)=-gq(ζ)-g¯q¯(ζ).
Comparing the coefficients in (77) gives that for ζ∈[-τ*,0),
(81)H20(ζ)=-g20q(ζ)-g¯02q¯(ζ),(82)H11(ζ)=-g11q(ζ)-g¯11q¯(ζ).
From (79), (81), and following the definition of A, we get
(83)W˙20=2iω*τ*W20(ζ)+g20q(ζ)+g¯20q¯(ζ).
Notice that q(ζ)=q(0)eiω*τ*ζ, and hence
(84)W20=ig20ω*τ*q(0)eiω*τ*ζ+ig¯02ω*τ*q¯(0)e-iω*τ*ζ+E1e2iω*τ*ζ,
where E1=(E11,E12)∈R2 is a constant vector. Similarly, from (79) and (82), we obtain
(85)W11=-ig11ω*τ*q(0)eiω*τ*ζ+ig¯11ω*τ*q¯(0)e-iω*τ*ζ+E2,
where E2=(E21,E22)∈R2 is also a constant vector.In what follows, we will seek appropriateE1 and E2. From the definition of A and (79), we obtain that
(86)∫-τ*0‍dη(ζ)W20(ζ)=2iω0-H20(0),(87)∫-τ*0‍dη(ζ)W11(ζ)=-H11(0),
where η(ζ)=η(0,ζ).Set(88)d1=f20(1)+f02(1)ρ2+2f11(1)ρ,d2=2f011(2)ρe-2iω*τ*+f020(2)e-2iω*τ*+f002(2)ρ2e-2iω*τ*,d3=f20(1)+f02(1)ρρ¯+f11(1)(ρ+ρ¯),d4=f011(2)(ρ+ρ¯)+f020(2)+f002(2)ρρ¯.
By (76), we have
(89)H20(ζ)=-g20q(0)-g¯20q¯(0)+[d1d2],(90)H11(ζ)=-g11q(0)-g¯11q¯(0)+[d3d4].
Substituting (84) and (89) into (86), we obtain
(91)(2iω*τ*I-∫-τ*0‍e2iω*τ*ζdη(ζ))E1=[d1d2],
which leads to
(92)AE1=[d1d2],
where
(93)A=[2iω*τ*-α1-α2-β2e-2iω*τ*2iω*τ*-β1-β3e-2iω*τ*].
It follows that
(94)E1(1)=1|A||d1-α2d22iω*τ*-β1-β3e-2iω*τ*|,E1(2)=1|A||2iω*τ*-α1d1-β2e-2iω*τ*d2|.
Similarly, substituting (85) and (90) into (87), we can get
(95)BE2=[d3d4],
where
(96)B=[-α1-α2-β2-β1-β3],
and hence
(97)E2(1)=1|B||d3-α2d4-β1-β3|,E2(2)=1|B||-α1d3-β2d4|.
Thus, we can determine W20 and W11 from (84) and (85). Furthermore, g21 in (75) can be expressed by the parameters and delay given in system (50). Based on the above analysis, we can see that each gij can be determined by the parameters. Thus we can compute the following quantities:
(98)c1(0)=i2ω*τ*(g11g20-2|g11|2-|g02|23)+g212,μ2=-Re{c1(0)}Re{λ′(τ*)},β2=2Re{c1(0)},T2=-Im{c1(0)}+μ2Im{λ′(τ*)}ω*τ*.Hence we have the following theorem by the result of Hassard et al. [17].Theorem 10.
In (98), μ2 determines the directions of the Hopf bifurcation: if μ2>0(μ2<0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for τ>τ*(τ<τ*); β2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if β2<0(β2>0); T2 determines the period of the bifurcating periodic solutions: the period increases (decrease) if T2>0(<0).


## 6. Numerical Simulations and Discussion

In this section, we will carry out some numerical simulations for supporting our theoretical analysis.In the following, we choose two sets of parameters.(i)
r=0.1, K=10, β=0.2, a=0.13, m=0.15, θ=3, ɛ=0.42,(ii)
r=0.1, K=10, β=0.2, a=0.13, m=0.15, θ=3, ɛ=0.3.For the parameters (i),τmax≈1.45 and I=[0,1.45). The function S0(τ) is drawn for τ∈I in Figure 3, from which we can see that there is only one positive critical value of the delay τ, denoted by τ*, and τ*≈0.58.(1)
Whenτ=0, Figure 1(b) shows that the positive equilibrium of system (1) is unstable.(2)
Whenτ=0.55<τ*, we can see the positive equilibrium of system (2) is unstable (see Figures 4(a) and 4(b)).(3)
Whenτ=0.75∈(τ*,τmax), the positive equilibrium of system (2) is stable and there is a Hopf bifurcation that occurs when τ=τ*. Furthermore, we can obtain Re(c1(0))<0 and μ2<0. Therefore, the Hopf bifurcation of system (2) at the infected equilibrium is subcritical and the bifurcating periodic solutions are orbitally asymptotically stable (see Figure 4(b)).(4)
Ifm=0, that is the system (2) has no term e-mτ, then the positive equilibrium if it exists is always unstable for τ≥0 (see Figures 1(b) and 5).Figure 3
Graph of the functionS0(τ).The infected steady stateE* of system (2) is unstable when τ=0.55 (a) (b) and stable when τ=0.75 (c). The rest parameter values are given in (i), and the initial values are x0=8 and y0=5.
(a)
(b)
(c)The infected steady stateE* of system (2) without term e-mτ is unstable when τ=0.75 (a) and τ=10 (b). The rest parameter values are given in (i), and the initial values are x0=8 and y0=5.
(a)
(b)For the parameters (ii), the positive equilibrium of the DDEs is always stable for allτ∈I (see Figures 1(a) and 6).The infected steady stateE* of system (2) is stable when τ=0.55 (a) and τ=0.75 (b). The rest parameter values are given in (ii), and the initial values are x0=8 and y0=5.
(a)
(b)In this paper, the Hopf bifurcation for a predator-prey system withθ-logistic growth and prey refuge is studied. It is shown that the decreasing prey refuge rate or the increasing logistic index will derive the stable ODEs unstable. Therefore, the refuges used by prey have a stabilizing effect on the considered interactions [6–11].For the DDEs, the time delay could be looked as a bifurcation parameter. Under certain condition, there is a critical delayτ* and a τmax, if the delay τ∈[0,τ*), the system is unstable, and if τ∈(τ*,τmax), the system is stable. But under this condition, the DDEs without the term e-mτ will always be unstable.



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