# On the Nature of Bifurcation in a Ratio-Dependent Predator-Prey Model with Delays

Authors: Changjin Xu; Yusen Wu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

A ratio-dependent predator-prey model with two delays is investigated. The conditions which ensure the local stability and the existence of Hopf bifurcation at the positive equilibrium of the system are obtained. It shows that the two different time delays have different effects on the dynamical behavior of the system. An example together with its numerical simulations shows the feasibility of the main results. Finally, main conclusions are included.


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## Body

## 1. Introduction

After the seminal models of Volterra and Lotka in the mid-1920s, understanding the dynamics of predator-prey models has been the focus of intense research in recent years. A great deal of excellent and interesting results have been reported. For example, Bhattacharyya and Mukhopadhyay [1] made a detailed discussion on the local and global dynamical behavior of an ecoepidemiological model, Kar and Ghorai [2] analyzed the local stability, global stability, influence of harvesting, and bifurcation of a delayed predator-prey model with harvesting, and Chakraborty et al. [3] studied the bifurcation and control of a bioeconomic model of a delayed prey-predator model. Bhattacharyya and Mukhopadhyay [4] focused on the spatial dynamics of nonlinear prey-predator models with prey migration and predator switching, and Chang and Wei [5] considered the bifurcation nature and optimal control of a diffusive predator-prey system with time delay and prey harvesting. For more related research, one can see [6–26].In 2011, Wang and Pei [27] investigated the stability and Hopf bifurcation of the following delayed ratio-dependent predator-prey system:
(1)x˙(t)=ax(t-τ1)-bx2-cxymy+x,y˙(t)=x(t-τ2)y(t-τ2)my(t-τ2)+x(t-τ2)-dy,
where x(t) and y(t) represent the densities of the prey and the predator population at time t, respectively. The parameters a, k, c, m, f, and d are positive constants that stand for prey intrinsic growth rate, carrying capacity, capturing rate, half capturing saturation constant, conversion rate, predator and death rate, respectively. The constants τ1≥0 and τ2≥0 denote the time delays due to gestation of the prey and predator, respectively. For more detailed biological meaning of the coefficients of system (1), one can see [27]. Applying the Nyquist criteria and the theory of Hopf bifurcation, Wang and Pei [27] considered the stability of the positive equilibrium and the existence of the local Hopf bifurcation of system (1). By means of the center manifold and normal form theories, they obtained explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solutions.We would like to point out that Wang and Pei [27] investigated the local stability of system (1) under the assumption τ1+τ2 in a certain range and focused on the local Hopf bifurcation by choosing the delay τ2 as bifurcation parameter. A natural problem arising from this is what effect the different delays τ1 and τ2 have on the dynamical behavior of system (1). In [27], Wang and Pei did not analyze this aspect. Thus, we think that it is important to deal with the effect of time delay on the dynamics of system (1). To the best of the authors knowledge, there are very few works’ which deal with this topic. In this paper, we will further investigate the stability and bifurcation of model (1) as a complementarity. It will be shown that the two different time delays τ1 and τ2 have different effects on the stability and Hopf bifurcation nature of system (1).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, numerical simulations are carried out to illustrate the validity of the main results. Some main conclusions are drawn in Section 4.


## 2. Stability and Local Hopf Bifurcations

In this section, we shall study the stability of the positive equilibrium and the existence of local Hopf bifurcations.From [27], we know that if the following conditions (H1)(2)d<f<dcc-ma,c>maorf>d,c≤mahold, then system (1) has a unique equilibrium point E(x*,y*), where
(3)x*=akl-b1(k+l)akl,y*=b1kl.Letx¯(t)=x(t)-x* and y¯(t)=y(t)-y* and still denote x¯(t),  y¯(t) by x(t), y(t), respectively. Then (1) reads as
(4)x˙(t)=p1x+p2y+p3x(t-τ1),y˙(t)=p4y+p5x(t-τ2)+p6y(t-τ2),
where
(5)p1=c(f2-d2)-2maf2mf2,p2=-d2cf2,p3=a,p4=-d,p5=(f-d)2mf,p6=d2f.The characteristic equation of (4) is given by
(6)det(λ-p1-p3e-λτ1-p2-p5e-λτ2λ-p4-p6e-λτ2)=0.That is,(7)λ2-(p1+p4)λ-p3λe-λτ1-p6λe-λτ2+p3p4e-λτ1+(p1p6-p2p5)e-λτ2+p3p6e-λ(τ1+τ2)+p1p4=0.The following lemma is important for us to analyze the distribution of roots of the transcendental equation (7).Lemma 1 (see [28]).
For the transcendental equation(8)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 five cases.Case 1.
τ
1
=
τ
2
=
0. Equation (7) becomes
(9)λ2-(p1+p3+p4+p6)λ2+(p3p4+p1p4+p1p6-p2p5+p3p6)=0.
All roots of (9) have a negative real part if the following condition holds:(H2)(10)p1+p3+p4+p6<0,p3p4+p1p4+p1p6-p2p5+p3p6>0.
Then the equilibrium pointE(x*,y*) is locally asymptotically stable when the conditions (H1) and (H2) are satisfied.Case 2.
τ
1
=
0, τ2>0. Equation (7) becomes
(11)λ2-(p1+p4)λ+p3p4+p1p4-(p6λ+p2p5-p1p6-p3p6)e-λτ2=0.
Forω>0, let iω be a root of (11). Then it follows that
(12)(p1p6-p2p5+p3p6)cosωτ2-p6ωsinωτ2=ω2-p3p4-p1p4,p6ωcosωτ2+(p1p6-p2p5+p3p6)sinωτ2=-(p1+p3+p4)ω
which is equivalent to
(13)ω4+r1ω2+r2=0,
where
(14)r1=(p1+p3+p4)2-2(p3p4-p1p4)-p62,r2=(p3p4-p1p4)2-(p1p6-p2p5+p3p6)2.
DefineΔ1=r12-4r2. Following Cao and Xiao [29] and Theorem  2.1 in Ge and Yan [30], we have the following result.Lemma 2.
If (H1) holds, then(i)
ifr1<0 and Δ1=0, then (11) with τ2=τ2n+ has a pair of pure imaginary roots ±iω+;(ii)
ifr1<0 and Δ1>0, then (11) with τ2=τ2n+ has two pairs of pure imaginary roots ±iω+ and ±iω-, where(15)τ2n±=1ω±arccos[(ω±2-p3p4-p1p4)(p1p6-p2p5+p3p6)-p6(p1+p3+p4)ω±2(p1p6-p2p5+p3p6)2+(p6ω±)2]+2nπω±and ω± satisfies(16)ω+2=-r1+Δ12,ω-2=-r1-Δ12;(iii)
ifr1>0 or Δ1<0, then all the roots of (11) have negative real parts for τ2≥0.   From Lemma 2, one has the following result.Theorem 3.
Letτ2n± be defined by (15). Under the condition (H1),(i)
ifr1<0 and Δ1=0, then the trivial solution (x*,y*)T of (11) is asymptotically stable for all τ2∈[0,τ20) and unstable for τ2>τ20. That is, Hopf bifurcation occurs when τ2=τ20;(ii)
ifr1>0 or Δ1<0, then there are Hopf bifurcations near the trivial solution (x*,y*)T of (11) when τ2=τ2n+ and τ2=τ2n-.Case 3.
τ
1
>
0, τ2=0. Equation (7) takes the form
(17)λ2-(p1+p3+p4+p6)λ+p1p6-p2p5+p1p4-(p3λ-p3p4-p3p6)e-λτ1=0.
Forη>0, let iη be a root of (17). Then it follows that
(18)(p3p4+p3p6)cosητ1-p3ηsinητ1=η2+p2p5-p1p6-p1p4,p3ηcosητ1+(p3p4+p3p6)sinητ1=-(p1+p4+p6)
which is equivalent to
(19)η4+s1η2+s2=0,
where
(20)s1=(p1+p4+p6)2+2(p2p5-p1p6-p1p4)-p32,s2=(p2p5-p1p6-p1p4)2-(p3p4+p3p6)2.
DefineΔ2=s12-4s2. Following the Theorem  2.1 in Ge and Yan [30], we have the following result.Lemma 4.
If (H1) holds, then(i)
ifs1<0 and Δ2=0, then (17) with τ1=τ1n+ has a pair of pure imaginary roots ±iη+;(ii)
ifs1<0 and Δ2>0, then (17) with τ1=τ1n+ has two pairs of pure imaginary roots ±iη+ and ±iη-, where(21)τ1n±=1η±arccos[(η±2+p2p5-p1p6-p1p4)(p3p4+p3p6)-p3(p1+p4+p6)η±(p3p4+p3p6)2+(p3η±)2]+2nπη±and η± satisfies
(22)η+2=-s1+Δ22,η-2=-s1-Δ22;(iii)
ifs1>0 or Δ2<0, then all the roots of (17) have negative real parts for τ1≥0.From Lemma4, we have the following result.Theorem 5.
Letτ1n± be defined by (21). Under the condition (H1),(i)
ifs1<0 and Δ2=0, then the trivial solution (x*,y*)T of (17) is asymptotically stable for all τ1∈[0,τ10) and unstable for τ1>τ10. That is, Hopf bifurcation occurs when τ1=τ10;(ii)
ifs1>0 or Δ2<0, then there are Hopf bifurcations near the trivial solution (x*,y*)T of (17) when τ1=τ1n+ and τ1=τ1n-.Case 4.
τ
1
>
0, τ2>0. We consider (7) with τ2 in its stable interval, by regarding τ1 as a parameter. Without loss of generality, we consider system (1) under the assumptions (H1) and (H2). Let iη*(η*>0) be a root of (7). Then we can obtain
(23)η*4+k1η*3+k2η*2+k3η*+k4=0,
where
(24)k1=2p6sinη*τ2,k2=p62sin2η*τ2-2(p1p6-p2p5)cosη*τ2k2=+(p-1+p4+p6cosη*τ2)2,k3=-2p6(p1p6-p2p5)sinη*τ2cosη*τ2,k4=p6(p1p6-p2p5)2+(p1p6-p2p5)2sin2η*τ2k2=+2(p1+p4+p6cosη*τ2)(p1p6-p2p5)sinη*τ2k2=-(p3+p4+p3p6cosη*τ2)2-(p3η+p3p6sinη*τ2)2.
Denote(25)H(η*)=η*4+k1η*3+k2η*2+k3η*+k4.
Assume that(H3)(26)k4<0.
It is easy to check thatH(0)<0 if (H5) holds and limη*→+∞H(η*)=+∞. We can obtain that (25) has finite positive roots η1*,η2*,…,ηn*. For every fixed ηi*, i=1,2,3,…,k, there exists a sequence {τ1ij∣j=1,2,3,…}, such that (25) holds. Let
(27)τ10=min{τ1ij∣i=1,2,…,k;j=1,2,…}.
Whenτ1=τ10, (7) has a pair of purely imaginary roots ±iη~* for τ2∈[0,τ20).
In the following, we assume that(H4)(28)[d(Reλ)dτ1]λ=iη~*≠0.
Thus, by the general Hopf bifurcation theorem for FDEs in Hale [31], we have the following result on the stability and Hopf bifurcation in system (1).Theorem 6.
For system (1), suppose that (H1), (H2), (H3), and (H4) are satisfied and τ2∈[0,τ20). Then the positive equilibrium E(x*,y*) is asymptotically stable when τ1∈[0,τ10), and system (1) undergoes a Hopf bifurcation at the positive equilibrium E(x*,y*) when τ1=τ10.Case 5.
τ
1
>
0, τ2>0. We consider (7) with τ1 in its stable interval, by regarding τ2 as a parameter. Without loss of generality, we consider system (1) under the assumptions (H1) and (H2). Let iω*(ω*>0) be a root of (7). Then we can obtain
(29)ω*4+t1ω*3+t2ω*2+t3ω*+t4=0,
where
(30)t1=2p3sinω*τ1,t2=p32sin2ω*τ1-2p3p4cosω*τ1t2=-(p1+p4+p3cosω*τ1)2,t3=2p3p4(p1+p4+p3sinω*τ1)sinω*τ1t3=-2p32p4sinω*τ1cosω*τ1,t4=(p3p4)2sin2ω*τ2-(p1p6-p2p5+p3p6cosω*τ1)2t3=-(p6ω*+p3p6sinω*τ1)2.
Denote(31)H*(ω*)=ω*4+t1ω*3+t2ω*2+t3ω*+t4.
Obviously,H(0)<0 if (H5) holds and limω*→+∞H*(ω*)=+∞. We can obtain that (31) has finite positive roots ω1*,ω2*,…,ωn*. For every fixed ωi*, i=1,2,3,…,k, there exists a sequence {τ2ij∣j=1,2,3,…}, such that (31) holds. Let
(32)τ20=min{τ2ij∣i=1,2,…,k;j=1,2,…}.
Whenτ2=τ20, (7) has a pair of purely imaginary roots ±iω*¯ for τ1∈[0,τ10).
In the following, we assume that(H5)(33)[d(Reλ)dτ2]λ=iω*¯≠0.
In view of the general Hopf bifurcation theorem for FDEs in Hale [31], we have the following result on the stability and Hopf bifurcation in system (1).Theorem 7.
For system (1), assume that (H1), (H2), (H3), and (H5) are satisfied and τ1∈[0,τ10). Then the positive equilibrium E(x*,y*) is asymptotically stable when τ2∈[0,τ20), and system (1) undergoes a Hopf bifurcation at the positive equilibrium E(x*,y*) when τ2=τ20.Case 6.
τ
1
=
τ
2
=
τ. Equation (7) becomes
(34)λ2-(p1+p4)λ+p1p4-[(p2+p6)λ-(p-3p4+p1p6-p2p5)]e-λτ+p3p6λe-2λτ=0,
which is equivalent to
(35)[λ2-(p1+p4)λ+p1p4]eλτ-[(p2+p6)λ-(p3p4+p1p6-p2p5)]+p3p6λe-λτ=0.
When τ=0, (35) becomes
(36)λ2-(p1+p3+p4+p6)λ2+(p1p4+p3p4+p1p6-p2p5+p3p6)=0.
It is easy to see that if the condition (H2) holds, then all roots of (36) have a negative real part. Then the equilibrium point E(x*,y*) is locally asymptotically stable when the conditions (H1) and (H2) are satisfied.
Forθ>0, let  iθ be a root of (35). Then it follows that
(37)(p1p4-θ2+p3p6)cosθτ2+(p3p6-p1p4+θ2)sinθτ=p2p5-p3p4-p1p6,(p1+p4)θcosθτ+(p3p6-p1p4+θ2)sinθτ=-(p3+p6)θ
which is equivalent to(38)sinθτ=(p2p5-p3p4-p1p6)(p1+p4)θ+(p3+p6)θ(p1p4-θ2+p3p6)[(p1+p4)θ]2-(p3p6)2+(p1p4-θ2)2,(39)cosθτ=(p3p4+p1p6-p2p5)(p3p6-p1p4+θ2)-(p3+p6)θ(p1+p4)θ[(p1+p4)θ]2-(p-3p6)2+(p1p4-θ2)2.
It follows fromsin2θτ+cos2θτ=1 that
(40)[xx(p2p5-p3p4-p1p6)(p1+p4)θ+(p3+p6)θ(p1p4-θ2+p3p6)xx]2+[(p3p4+p1p6-p2p5)(p3p6-p1p4+θ2)-(p3+p6)θ(p1+p4)θθ2]2={[(p1+p4)θ]2-(p3p6)2+(p1p4-θ2)2}2.
Then we have(41)θ8+u3θ6+u2θ4+u1θ2+u0=0,
where
(42)u0=[(p1p4)2-(p3p6)2]2u0=-[(p2p5-p3p4-p1p6)(p3p6-p1p4)]2,u1=2[(p1p4)2-(p3p6)2][(p1+p2)2-2p1p4]2u0=-[(p2p5-p3p4-p1p6)(p1+p4)u0=-+(p3+p6)(p1p4+p3p6)]u0=-2(p2p5-p3p4-p1p6)(p1+p4)u0=×[(p2p5-p3p4-p1p6)u0=×+(p3+p6)(p1+p4)],u2=2[(p1p4)2-(p3p6)2]+[(p1+p2)2-2p1p4]2u2=+2[(p2p5-p3p4-p1p6)(p1+p4)u2=+2+(p3+p6)(p1p4+p3p6)](p3+p6)u2=-[(p-2p5-p3p4-p1p6)+(p3+p6)(p1+p4)]2,u3=2[(p1+p2)2-2p1p4]-(p3+p6)2.
Letz=θ2. Then (41) becomes
(43)z4+u3z3+u2z2+u1z+u0=0.
Denote(44)h(z)=z4+u3z3+u2z2+u1z+u0.
Then(45)h′(z)=4z3+3u3z2+2u2z+u1.
Set(46)4z3+3u3z2+2u2z+u1=0.
Lety=z+u3/4. Then (46) becomes
(47)y3+p1y+q1=0,
where
(48)p1=u22-316u32,q1=u3332-u3u28+u14.
Define(49)D=(q12)2+(p13)3,σ=-1+32,y1=-q12+D3+-q12-D3,y2=-q12+Dσ3+-q12-Dσ23,y3=-q12+Dσ23+-q12-Dσ3,zi=yi-p14,i=1,2,3.
From [32, 33], we have the following result.Lemma 8.
Ifu0<0, then (43) has at least one positive root.Lemma 9.
Assume thatu0≥0. Then one has the following:(i)
ifD≥0, then (43) has positive roots if and only if z1>0 and h′(z1)<0;(ii)
ifD<0, then (43) has positive roots if and only if there exists at least one z*∈{z1,z2,z3} such that z*>0 and h′(z*)≤0.Without loss of generality, we assume that (43) has four positive roots, defined by z1, z2, z3, z4, respectively. Then (41) has four positive roots
(50)θ1=z1,θ2=z2,θ3=z3,θ4=z4.By (39), if we denote(51)τk(j)=1θk{arccos[(p3p4+p1p6-p2p5)(p3p6-p1p4+θ2)-(p3+p6)θ(p1+p4)θ[(p1+p4)θ]2-(p-3p6)2+(p1p4-θ2)2]+2jπ},where k=1,2,3,4; j=0,1,…, then ±iθk are a pair of purely imaginary roots of (35) with τk(j). Define
(52)τ0=τk0(0)=mink∈{1,2,3,4}{τk(0)},θ0=θk0.Based on above analysis, we have the following result.Lemma 10.
Forτ1=τ2=τ, if (H1) and (H2) hold, then all roots of (1) have a negative real part when τ∈[0,τ0), and (1) admits a pair of purely imaginary roots ±θki when τ=τk(j)(k=1,2,3,4,j=0,1,2,…).Letλ(τ)=α(τ)+iθ(τ) be a root of (35) near τ=τk(j), and let α(τk(j))=0 and θ(τk(j))=θk. Due to functional differential equation theory, for every τk(j), k=1,2,3,4, j=0,1,2,…, there exists ɛ>0 such that λ(τ) is continuously differentiable in τ for |τ-τk(j)|<ɛ. Substituting λ(τ) into the left-hand side of (35) and taking derivative with respect to τ, we have
(53)[dλdτ]-1=-[2λ-(p1+p4)]λeλτ-(p2+p6)+p3p6e-λτλ[λ2-(p1+p4)λ+p1p4]eλτ+p3p6λ2e-λτ-τλ.We can easily obtain(54)[d(Reλ(τ))dτ]τ=τk(j)-1=-Re{[2λ-(p1+p4)]λeλτ-(p2+p6)+p3p6e-λτλ[λ2-(p1+p4)λ+p1p4]eλτ+p3p6λ2e-λτ}τ=τk(j)=-Re{A1-A2iB1+B2i}=-A1B1-A2B2B12+B22,
where
(55)A1=(p3p6-2θk2)cosθkτk(j)+(p1+p4)θksinθkτk(j)-(p2+p6),A2=(p1+p4)θkτk(j)cosθkτk(j)+(2θk2+p3p6)sinθkτk(j),B1=(p1+p4)θk2cosθkτk(j)-(p1p4-θk2)θksinθkτk(j)-p3p6θk2,B2=(p1+p4)θk2sinθkτk(j)+(p1p4-θk2)θkcosθkτk(j)+p3p6θk2sinθkτk(j).Now we assume that(H6)(56)A1B1≠A2B2.In view of the above analysis and the results of Kuang [33] and Hale [31], we have  the following.Theorem 11.
Forτ=0, if (H1) and (H2) hold, then the positive equilibrium E(x*,y*) of system (1) is asymptotically stable for τ∈[0,τ0). In addition to the conditions (H1) and (H2), one further assumes that (H6) holds. Then system (1) undergoes a Hopf bifurcation at the positive equilibrium E(x*,y*) when τ=τk(j), k=1,2,3,4, j=0,1,2,….


## 3. Computer Simulations

In this section, we present some numerical results of system (1) to verify the analytical predictions obtained in the previous section. Let us consider the following system:
(57)x˙(t)=2x(t-τ1)-0.12x2-2xy0.77y+x,y˙(t)=0.205x(t-τ2)y(t-τ2)0.77y(t-τ2)+x(t-τ2)-0.1y,
which has a positive equilibrium E(5.61,7.02). We can easily obtain that (H1)–(H6) hold true. When τ1=0, applying MATLAB 7.0, we can get ω0≈0.5524,τ20≈2.3345. The positive equilibrium E(5.61,7.02) is asymptotically stable for τ2<τ20≈2.3345 and unstable for τ2>τ20≈2.3345 which is shown in Figure 1. When τ2=τ20≈2.3345, (57) undergoes a Hopf bifurcation around the positive equilibrium E(5.61,7.02). That is, a small amplitude periodic solution occurs near E(5.61,7.02) when τ1=0 and τ2 is close to τ20=2.3345 which is shown in Figure 2.  The bifurcation diagram of the case is shown in Figure 3.Trajectory portrait and phase portrait of system (57) with τ1=0, τ2=2.2<τ20≈2.3345. The positive equilibrium E(5.61,7.02) is asymptotically stable. The initial value is (5.6, 7.45).
(a)
(b)
(c)
(d)Trajectory portrait and phase portrait of system (57) with τ1=0, τ2=2.5>τ20≈2.3345. Hopf bifurcation occurs from the positive equilibrium E(5.61,7.02). The initial value is (5.6, 7.45).
(a)
(b)
(c)
(d)Figure 3
Bifurcation diagram with respect to the time delayτ2 for system (57) with τ1=0.Letτ2=0.25∈(0,1.8), and choose τ1 as a parameter. We have τ10≈0.8013. Then the positive equilibrium is asymptotically  stable when τ1∈[0,τ10). The Hopf bifurcation value of (57) is τ10≈0.8013 (see Figures 4 and 5) The bifurcation diagram of the case is shown in Figure 6.Trajectory portrait and phase portrait of system (57) with τ2=1.8, τ1=0.75<τ10≈0.8013. The positive equilibrium E(5.61, 7.02) is asymptotically stable. The initial value is (5.6, 7.45).
(a)
(b)
(c)
(d)Trajectory portrait and phase portrait of system (57) with τ2=1.8, τ1=0.98>τ10≈0.8013. Hopf bifurcation occurs from the positive equilibrium E(5.61,7.02). The initial value is (5.6, 7.45).
(a)
(b)
(c)
(d)Figure 6
Bifurcation diagram with respect to the time delayτ1 for system (57) with τ2=1.8.Whenτ2=0, using MATLAB 7.0, we obtain η0≈0.9056, τ10≈0.9122. The positive equilibrium E(5.61,7.02) is asymptotically stable for τ1<τ10≈0.9122 and unstable for τ1>τ10≈0.9122 which is shown in Figure 7. When τ1=τ10≈0.9122, (57) undergoes a Hopf bifurcation at the positive equilibrium E0(5.61,7.02). That is, a small amplitude periodic solution occurs around E(5.61,7.02) when τ2=0 and τ1 is close to τ10=0.9122 which is illustrated in Figure 8. The bifurcation diagram of the case is shown in Figure 9.Trajectory portrait and phase portrait of system (57) with τ2=0, τ1=0.90<τ10≈0.9122. The positive equilibrium E(5.61,7.02) is asymptotically stable. The initial value is (5.6, 8).
(a)
(b)
(c)
(d)Trajectory portrait and phase portrait of system (57) with τ2=0, τ1=1.0>τ10≈0.9122. Hopf bifurcation occurs from the positive equilibrium E0(5.61,7.02). The initial value is (5.6, 7.45).
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(d)Figure 9
Bifurcation diagram with respect to the time delayτ1 for system (57) with τ2=0.Letτ1=0.5∈(0,0.9122), and choose τ2 as a parameter. We have τ20≈0.7723. Then the positive equilibrium is asymptotically stable when τ2∈[0,τ20). The Hopf bifurcation value of (57) is τ20≈0.7723(see Figures 10 and 11). The bifurcation diagram of the case is shown in Figure 12.Trajectory portrait and phase portrait of system (57) with τ1=0.5, τ2=0.5<τ20≈0.7723. The positive equilibrium E(5.61,7.02) is asymptotically stable. The initial value is (5.6, 7.45).
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(b)
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(d)Trajectory portrait and phase portrait of system (57) with τ1=0.5, τ2=1.6>τ20≈0.7723. Hopf bifurcation occurs from the positive equilibrium E(5.61,7.02). The initial value is (5.6, 7.45).
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(d)Figure 12
Bifurcation diagram  with respect to the time delayτ2 for system (57) with τ1=0.5.Whenτ1=τ2=τ, using MATLAB 7.0, we obtain θ0≈0.8725, τ0≈1.2032. The positive equilibrium E(5.61,7.02) is asymptotically stable for τ<τ0≈1.2032 and unstable for τ>τ0≈1.2032 which is shown in Figure 13.  When τ=τ0≈1.2032, (57) undergoes a Hopf bifurcation at the positive equilibrium E0(5.61,7.02). That is, a small amplitude periodic solution occurs around E(5.61,7.02) when τ is close to τ0≈1.2032 which is illustrated in Figure 14.   The bifurcation diagram of the case is shown in Figure 15.Trajectory portrait and phase portrait of system (57) with τ1=0τ2=τ=1.0<τ0≈1.2032. The positive equilibrium E(5.61,7.02) is asymptotically stable. The initial value is (5.6, 7.45).
(a)
(b)
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(d)Trajectory portrait and phase portrait of system (57) with τ1=τ2=τ=1.25>τ0≈1.2032. Hopf bifurcation occurs from the positive equilibrium E(5.61,7.02). The initial value is (5.6, 7.45).
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(d)Figure 15
Bifurcation diagram with respect to the time delayτ for system (57) with τ1=τ2=τ.


## 4. Conclusions

In this paper, we have investigated local stability of the positive equilibriumE(x*,y*) and local Hopf bifurcation of a ratio-dependent predator-prey model with two delays. It is shown that if some conditions hold true and τ2∈[0,τ20), then the positive equilibrium E(x*,y*) is asymptotically stable when τ1∈(0,τ10). When the delay τ1 increases, the positive equilibrium E(x*,y*) loses its stability and a sequence of Hopf bifurcations occur at the positive equilibrium E(x*,y*). That is, a family of periodic orbits bifurcates from the the positive equilibrium E(x*,y*). We also showed that if a certain condition is satisfied and τ1∈[0,τ10), then the positive equilibrium E(x*,y*) is asymptotically stable when τ2∈(0,τ20), when the delay τ2 increases, the positive equilibrium E(x*,y*) loses its stability and a sequence of Hopf bifurcations occur at the positive equilibrium E(x*,y*). In case τ1=τ2=τ, we have shown that if some conditions are satisfied, and τ∈[0,τ0), then the positive equilibrium E(x*,y*) is asymptotically stable when τ∈(0,τ0). When the delay τ increases, the positive equilibrium E(x*,y*) loses its stability and a sequence of Hopf bifurcations occur at the positive equilibrium E(x*,y*) which means a family of periodic orbits bifurcates from the the positive equilibrium E(x*,y*). Some numerical simulations verifying our theoretical results are carried out. In addition, we must point out that although Ko and Ryu [6] have also investigated the the existence of Hopf bifurcation for system (1) with respect to positive equilibrium E(x*,y*), it is assumed that τ1+τ2 in a certain range and choose the delay τ2 as bifurcation parameter to consider the Hopf bifurcation nature. But what effect different time delays have on the dynamics of system (1)? Ko and Ryu [6] did not deal with this issue. It is important for us to consider what effect the two different time delays has on the dynamical behavior of system (1). Thus we think that our work generalizes the known results of Ko and Ryu [6]. In addition, we can study the Hopf bifurcation nature of system (1) by regarding the delay τ2 as bifurcation parameter. We will further focus on the topic elsewhere in the near future.



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*Source: 679602-2013-06-20.xml*