Journal of Applied Mathematics
Volume 2012, Issue 1 384293
Research Article
Open Access

Stability and Bifurcation in a Delayed Holling-Tanner Predator-Prey System with Ratio-Dependent Functional Response

Juan Liu, 

Corresponding Author

Juan Liu

Department of Science, Bengbu College, Bengbu, Anhui 233030, China

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Zizhen Zhang, 

Zizhen Zhang

School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu, Anhui 233030, China aufe.edu.cn

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Ming Fu, 

Ming Fu

School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu, Anhui 233030, China aufe.edu.cn

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First published: 18 December 2012
https://doi.org/10.1155/2012/384293
Citations: 2
Academic Editor: C. Conca

Abstract

We analyze a delayed Holling-Tanner predator-prey system with ratio-dependent functional response. The local asymptotic stability and the existence of the Hopf bifurcation are investigated. Direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are studied by deriving the equation describing the flow on the center manifold. Finally, numerical simulations are presented for the support of our analytical findings.

1. Introduction

Predator-prey dynamics has long been and will continue to be of interest to both applied mathematicians and ecologists due to its universal existence and importance [1]. Although the early Lotka-Volterra model has given way to more sophisticated models from both a mathematical and biological point of view, it has been challenged by ecologists for its functional response suffers from paradox of enrichment and biological control paradox. The ratio-dependent models are discussed as a solution to these difficulties and found to be a more reasonable choice for many predator-prey interactions [2–4]. One type of the ratio-dependent models which plays a special role in view of the interesting dynamics it possesses is the ratio-dependent Holling-Tanner predator-prey system [5, 6]. A ratio-dependent Holling-Tanner predator-prey system takes the form of
()
where N(t) and P(t) represent the population of prey species and predator species at time t. It is assumed that in the absence of the predator, the prey grows logistically with carrying k and intrinsic growth rate r. The predator growth equation is of logistic type with a modification of the conventional one. The parameter m represents the maximal predator per capita consumption rate, and q is the half capturing saturation constant. The parameter s is the intrinsic growth rate of the predator and h is the number of prey required to support one predator at equilibrium, when y equals x/h. All the parameters are assumed to be positive.
Liang and Pan [6] established the sufficient conditions for the global stability of positive equilibrium of system (1.1) by constructing Lyapunov function. Considering the effect of time delays on the system, Saha and Chakrabarti [7] considered the following delayed system
()
where τ is the negative feedback delay of the prey. Saha and Chakrabarti [7] proved that the system (1.2) is permanent under certain conditions and obtained the conditions for the local and global stability of the positive equilibrium. It is well known that studies on dynamical systems not only involve a discussion of stability and persistence, but also involve many dynamical behaviors such as periodic phenomenon, bifurcation, and chaos [8–10]. In particular, the Hopf bifurcation has been studied by many authors [11–13]. Based on this consideration and since both species are growing logistically, we consider the Hopf bifurcation of the following system with two delays:
()
where τ1 and τ2 represent the negative feedbacks in prey and predator growth.
Before proceeding further we nondimensionalize our model system (1.3) with the following scaling rt → t, rτ1 → τ1, rτ2 → τ2, N(t)/k → x(t), mP(t)/rk → y(t). Then we get the nondimensional form of system (1.3):
()
where α = qr/m, β = sh/m, δ = m/hr.

This paper is organized as follows. In the next section, we will consider the local stability of the positive equilibrium and the existence of Hopf bifurcation of system (1.4). In Section 3, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. Some numerical simulations are also given to illustrate the theoretical prediction in Section 4.

2. Local Stability and Hopf Bifurcation

Considering the ecological significance of system (1.4), we are interested only in the positive equilibrium of system (1.4). It is not difficult to verify that system (1.4) has a unique positive equilibrium E(x0, y0), where x0 = (1 + αδ − δ)/(1 + αδ), y0 = δx0 if (H1): 1 + αδ > δ holds.

Let x(t) = u1(t) + x0, y(t) = u2(t) + y0, and we still denote u1(t) and u2(t) by x(t) and y(t), respectively, then system (1.4) can be rewritten as
()
where
()
Then the linearized system of (2.1) is
()
The characteristic equation of (2.3) is
()
where
()

Case 1 (τ1 = τ2 = 0). Equation (2.4) reduces to

()
If the condition (H2): A + B + C > 0 and D + E > 0 holds, it is clear that roots of (2.6) must have negative real parts.

Case 2 (τ1 > 0,  τ2 = 0). Equation (2.4) becomes

()
Let λ = iω1 (ω1 > 0) be a root of (2.7). Then, we have
()
From (2.8), we can get
()
If the condition (H3): D2 − E2 < 0 holds, then (2.9) has a unique positive root ω10
()
The corresponding critical value of time delay τ10 is
()
Next, differentiating (2.7) with respect to τ1 and substituting τ1 = τ10, then we get
()
From (2.10) and (2.12), we have
()
Therefore, if the condition (H3): D2 − E2 < 0 holds, then . Thus, we have the following results.

Theorem 2.1. For system (1.4), if the conditions (H2)-(H3) hold, then the positive equilibrium E(x0, y0) of system (1.4) is asymptotically stable for τ ∈ [0, τ10) and unstable when τ > τ10, system (1.4) undergoes a Hopf bifurcation at E(x0, y0) when τ1 = τ10.

Case 3 (τ1 = 0,  τ2 > 0). Equation (2.4) becomes

()
Let λ = iω2 (ω2 > 0) be a root of (2.14). Then, we get
()
It follows that
()
If the condition (H2) holds, then −(D + E) 2 < 0. Thus, (2.16) has a unique positive root ω20,
()
The corresponding critical value of time delay τ20 is
()
Similar as in Case 2, we know that if the condition (H2) holds, then we have
()
In conclusion, we have the following results.

Theorem 2.2. For system (1.4), if the condition (H2) holds, then the positive equilibrium E(x0, y0) of system (1.4) is asymptotically stable for τ ∈ [0, τ20) and unstable when τ > τ20, system (1.4) undergoes a Hopf bifurcation at E(x0, y0) when τ1 = τ20.

Case 4 (τ1 = τ2 = τ > 0). Equation (2.4) becomes

()
Multiplying eλτ on both sides of (2.20), we have
()
Let λ = iω (ω > 0) be a root of (2.21). Then, we get
()
Then, we can get
()
where
()
Thus, we can obtain
()
with
()
Let e = ω2, then (2.25) can be transformed into the following form
()
Next, we suppose that (H4): (2.27) has at least one positive root. Without loss of generality, we suppose that it has four positive roots which are denoted as e1, e2, e3, and e4. Thus, (2.25) has four positive roots , k = 1,2, 3,4. The corresponding critical value of time delay is
()
Let , k ∈ {1,2, 3,4}, .

Differentiating (2.21) regarding τ and substituting τ = τ0, we obtain

()
where
()
Obviously, if the condition (H5): MRNR + MINI ≠ 0 holds, then . Namely, the transversality condition is satisfied if H5 holds. From the above analysis, we have the following theorem.

Theorem 2.3. For system (1.4), if the conditions (H2), (H4), and (H5) hold, then the positive equilibrium E(x0, y0) of system (1.4) is asymptotically stable for τ ∈ [0, τ0) and unstable when τ > τ0, system (1.4) undergoes a Hopf bifurcation at E(x0, y0) when τ = τ0.

Case 5 (τ1 ≠ τ2 and τ1 > 0, τ2 > 0). We consider (2.4) with τ2 in its stable interval and τ1 is considered as a parameter. Let be the root of (2.4). Then we have

()
where
()
It follows that
()
where
()
Suppose that (H6): (2.33) has at least finite positive roots.

If the condition (H6) holds, we denote the roots of (2.33) as . Then, for every fixed , the corresponding critical value of time delay is
()
Let . The corresponding purely imaginary roots of (2.33) are denoted as ±iω*. Next, we give the following assumption. (H7): . Hence, we have the following theorem.

Theorem 2.4. Suppose that the conditions (H2), (H6), and (H7) hold and τ2 ∈ (0, τ20). The positive equilibrium E(x0, y0) of system (1.4) is asymptotically stable for τ1 ∈ [0, τ1*) and unstable when τ1 > τ1*, system (1.4) undergoes a Hopf bifurcation at E(x0, y0) when τ1 = τ0.

3. Direction and Stability of Bifurcated Periodic Solutions

In this section, we will employ the normal form method and center manifold theorem introduced by Hassard [14] to determine the direction of Hopf bifurcation and stability of bifurcating periodic solutions of system (1.4) at τ1 = τ2 = τ = τ0.

We denote τ as τ = τ0 + μ, μ ∈ R, t = sτ, Then μ = 0 is the Hopf bifurcation value of system (1.4). For convenience, we first rescale the time by s → t/τ, x(sτ) = x1(s), y(sτ) = x2(s) and still denote s = t, then system (1.4) can be transformed to the following form:
()
where
()
and Lμ, F are given by
()
where ϕ(θ) = (ϕ1(θ), ϕ2(θ)) T ∈ C([−1,0], R2), and
()
with
()
Hence, by the Riesz representation theorem, there exists a 2 × 2 matrix function η(θ, μ):[−1,0] → R2 whose elements are of bounded variation such that
()
In fact, we choose
()
where δ is the Dirac delta function, then (3.3) is satisfied.
For ϕ ∈ C([−1,0], R2), we define
()
Then system (3.1) can be transformed into the following operator equation
()
The adjoint operator A* of A(0) is defined by
()
associated with a bilinear form
()
where η(θ) = η(θ, 0).

From the above discussion, we know that that ±iω0τ0 are eigenvalues of A(0) and they are also eigenvalues of A*.

We assume that are the eigenvectors of A(0) belonging to the eigenvalue iτ0ω0 and are the eigenvectors of A* belonging to −iτ0ω0. Thus,
()
Then, we can obtain
()
Next, we get the coefficients used in determining the important quantities of the periodic solution by using a computation process similar to that in [15]:
()
with
()
where E1 and E2 can be computed as the following equations, respectively
()
with
()
Therefore, we can calculate the following values:
()
Based on the discussion above, we can obtain the following results.

Theorem 3.1. The direction of the Hopf bifurcation is determined by the sign of μ2: if μ2 > 0  (μ2 < 0), then the Hopf bifurcation is supercritical (subcritical). The stability of bifurcating periodic solutions is determined by the sign of β2: if β2 < 0  (β2 > 0), the bifurcating periodic solutions are stable (unstable).

4. Numerical Example

In this section, to illustrate the analytical results obtained in the previous sections, we present some numerical simulations. Let α = 1, β = 1, δ = 2, then we have the following particular case of system (1.4):
()
Obviously, 1 + αδ = 3 > δ = 2. Thus, the condition (H1) holds. Then we can get the unique positive equilibrium E(0.3333,0.6666) of system (4.1). By a simple computation, A + B + C = 2.1110 > 0, D + E = 0.5998 > 0. Namely, the condition (H2) holds.

Firstly, we can obtain that D2 − E2 = −0.4444 < 0. Namely, the condition (H3) is satisfied for τ1 > 0, τ2 = 0. Further, we have ω10 = 0.3734, τ10 = 3.7457. By Theorem 2.1, we can know that the positive equilibrium E(0.3333,0.6666) is asymptotically stable for τ1 ∈ [0, τ10) and unstable when τ1 > τ10. Let τ1 = 3.15 ∈ [0, τ10), then the positive equilibrium E(0.3333,0.6666) is asymptotically stable, which can be seen from Figure 1. When τ1 = 4.05 > τ10 = 3.7457, it can be seen from Figure 2 that the positive equilibrium E(0.3333,0.6666) is unstable and a Hopf bifurcation occurs. Similarly, we have ω20 = 2.0189, τ20 = 0.7311. For τ2 = 0.68 ∈ [0, τ20), the positive equilibrium E(0.3333,0.6666) is asymptotically stable from Theorem 2.2 and this property can be shown in Figure 3. If τ2 = 0.75 > τ20 = 0.7311, the positive equilibrium E(0.3333,0.6666) is unstable and a Hopf bifurcation occurs, and the corresponding waveform and phase plots are shown in Figure 4.

Details are in the caption following the image
Figure 1
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E(0.3333,0.6666) is asymptotically stable when τ1 = 3.15 < τ10 = 3.7457 with initial value 0.25, 1.15.
Details are in the caption following the image
Figure 2
Open in figure viewerPowerPoint
E(0.3333,0.6666) is unstable when τ1 = 4.05 > τ10 = 3.7457 with initial value 0.25, 1.15.
Details are in the caption following the image
Figure 3
Open in figure viewerPowerPoint
E(0.3333,0.6666) is asymptotically stable when τ2 = 0.68 < τ20 = 0.7311 with initial value 0.25, 1.15.
Details are in the caption following the image
Figure 4
Open in figure viewerPowerPoint
E(0.3333,0.6666) is unstable when τ2 = 0.75 > τ10 = 0.7311 with initial value 0.25, 1.15.

Secondly, we can get ω0 = 2.2937, τ0 = 0.7015, and λ′(τ0) = 0.9911 + 1.7808i for τ1 = τ2 = τ > 0. From Theorem 2.3, we know that E(0.3333,0.6666) is asymptotically stable for τ ∈ [0, τ0), which can be illustrated by Figure 5. As can be seen from Figure 5 that when τ = 0.64 ∈ [0, τ0) the positive equilibrium E(0.3333,0.6666) is asymptotically stable. However, if τ = 0.74 > τ0 = 0.7015, then the positive equilibrium E(0.3333,0.6666) becomes unstable and a family of bifurcated periodic solutions occur, which is illustrated by Figure 6. In addition, from (3.18), we get μ2 = 139.3 > 0, β2 = −276.12 < 0. Thus, by Theorem 3.1, we know that the Hopf bifurcation is supercritical and the bifurcated periodic solutions are stable.

Details are in the caption following the image
Figure 5
Open in figure viewerPowerPoint
E(0.3333,0.6666) is asymptotically stable when τ = 0.64 < τ0 = 0.7015 with initial value 0.25, 1.15.
Details are in the caption following the image
Figure 6
Open in figure viewerPowerPoint
E(0.3333,0.6666) is unstable when τ = 0.74 > τ0 = 0.7015 with initial value 0.25, 1.15.

Lastly, regard τ1 as a parameter and let τ2 = 0.5 ∈ (0, τ20), we can obtain that ω* = 0.3758. Further we have τ1* = 3.7688. Let τ1 = 3.25 ∈ [0, τ1*), we can know that the positive equilibrium E(0.3333,0.6666) is asymptotically stable from Theorem 2.4, which can be shown by Figure 7. When τ1 = 4.15 > τ1* = 3.7688 then the positive equilibrium E(0.3333,0.6666) becomes unstable and a Hopf bifurcation occurs, which can be illustrated in Figure 8.

Details are in the caption following the image
Figure 7
Open in figure viewerPowerPoint
E(0.3333,0.6666) is asymptotically stable when τ1 = 3.25 < τ1* = 3.7457 with τ2 = 0.5 and initial value 0.25, 1.15.
Details are in the caption following the image
Figure 8
Open in figure viewerPowerPoint
E(0.3333,0.6666) is unstable when τ1 = 4.15 > τ1* = 3.7457 with with τ2 = 0.5 and initial value 0.25, 1.15.

5. Conclusion

In the present paper, a Holling-Tanner predator-prey system with ratio-dependent functional response and two delays is investigated. We prove that the system is asymptotically stable under certain conditions. Compared with the system considered in [7], we not only consider the feedback delay of the prey but also the feedback delay of the predator. By choosing the delay as a bifurcation parameter, we show that the Hopf bifurcations can occur as the delay crosses some critical values. Furthermore, we get that the two species could also coexist with some available delays of the prey and the predator. This is valuable from the view of biology. In addition, Saha and Chakrabarti [7] only considered the stability of the system. It is well known that there are also some other behaviors for dynamical systems. Based on this consideration, we investigate the Hopf bifurcation and properties of the bifurcated periodic solutions of the system. The direction and the stability of the bifurcated periodic solutions are determined by applying the normal theory and the center manifold theorem. If the bifurcated periodic solutions are stable, then the two species may coexist in an oscillatory mode from the viewpoint of biology. Some numerical simulations supporting the theoretical results are also included.

Acknowledgments

The authors are grateful to the referees and the editor for their valuable comments and suggestions on the paper. This work is supported by Anhui Provincial Natural Science Foundation under Grant no. 1208085QA11.

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