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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 679602, 17 pages
http://dx.doi.org/10.1155/2013/679602
Research Article

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

1Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang 550004, China
2Department of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471003, China

Received 1 April 2013; Revised 29 May 2013; Accepted 2 June 2013

Academic Editor: Jinde Cao

Copyright © 2013 Changjin Xu and Yusen Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [626].

In 2011, Wang and Pei [27] investigated the stability and Hopf bifurcation of the following delayed ratio-dependent predator-prey system: where and represent the densities of the prey and the predator population at time , respectively. The parameters , , , , , and 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 and 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 in a certain range and focused on the local Hopf bifurcation by choosing the delay as bifurcation parameter. A natural problem arising from this is what effect the different delays and 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 and have different effects on the stability and Hopf bifurcation nature of system (1).

The remainder of the paper is organized as follows. In Section 2, 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) hold, then system (1) has a unique equilibrium point , where

Let and and still denote ,   by , , respectively. Then (1) reads as where

The characteristic equation of (4) is given by

That is,

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 as vary, the sum of orders of the zeros of 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. . Equation (7) becomes
All roots of (9) have a negative real part if the following condition holds:(H2)
Then the equilibrium point is locally asymptotically stable when the conditions (H1) and (H2) are satisfied.

Case 2. , . Equation (7) becomes
For , let be a root of (11). Then it follows that which is equivalent to where
Define . 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)if and , then (11) with has a pair of pure imaginary roots ;(ii)if and , then (11) with has two pairs of pure imaginary roots and , whereand satisfies(iii)if or , then all the roots of (11) have negative real parts for . From Lemma 2, one has the following result.

Theorem 3. Let be defined by (15). Under the condition (H1),(i)if and , then the trivial solution of (11) is asymptotically stable for all and unstable for . That is, Hopf bifurcation occurs when ;(ii)if or , then there are Hopf bifurcations near the trivial solution of (11) when and .

Case 3. , . Equation (7) takes the form
For , let be a root of (17). Then it follows that which is equivalent to where
Define . Following the Theorem  2.1 in Ge and Yan [30], we have the following result.

Lemma 4. If (H1) holds, then(i)if and , then (17) with has a pair of pure imaginary roots ;(ii)if and , then (17) with has two pairs of pure imaginary roots and , whereand satisfies (iii)if or , then all the roots of (17) have negative real parts for .

From Lemma 4, we have the following result.

Theorem 5. Let be defined by (21). Under the condition (H1),(i)if and , then the trivial solution of (17) is asymptotically stable for all and unstable for . That is, Hopf bifurcation occurs when ;(ii)if or , then there are Hopf bifurcations near the trivial solution of (17) when and .

Case 4. , . We consider (7) with in its stable interval, by regarding as a parameter. Without loss of generality, we consider system (1) under the assumptions and . Let be a root of (7). Then we can obtain where
Denote
Assume that (H3)
It is easy to check that if (H5) holds and . We can obtain that (25) has finite positive roots . For every fixed , , there exists a sequence , such that (25) holds. Let
When , (7) has a pair of purely imaginary roots for .
In the following, we assume that (H4)
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 . Then the positive equilibrium is asymptotically stable when , and system (1) undergoes a Hopf bifurcation at the positive equilibrium when .

Case 5. , . We consider (7) with in its stable interval, by regarding as a parameter. Without loss of generality, we consider system (1) under the assumptions (H1) and (H2). Let be a root of (7). Then we can obtain where
Denote
Obviously, if (H5) holds and . We can obtain that (31) has finite positive roots . For every fixed , , there exists a sequence , such that (31) holds. Let
When , (7) has a pair of purely imaginary roots for .
In the following, we assume that (H5)
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 , , , and are satisfied and . Then the positive equilibrium is asymptotically stable when , and system (1) undergoes a Hopf bifurcation at the positive equilibrium when .

Case 6. . Equation (7) becomes which is equivalent to When , (35) becomes
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 is locally asymptotically stable when the conditions (H1) and (H2) are satisfied.
For , let be a root of (35). Then it follows that which is equivalent to
It follows from that
Then we have where
Let . Then (41) becomes
Denote
Then
Set
Let . Then (46) becomes where
Define
From [32, 33], we have the following result.

Lemma 8. If , then (43) has at least one positive root.

Lemma 9. Assume that . Then one has the following:(i)if , then (43) has positive roots if and only if and ;(ii)if , then (43) has positive roots if and only if there exists at least one such that and .

Without loss of generality, we assume that (43) has four positive roots, defined by , , , , respectively. Then (41) has four positive roots

By (39), if we denotewhere ; , then are a pair of purely imaginary roots of (35) with . Define

Based on above analysis, we have the following result.

Lemma 10. For , if (H1) and (H2) hold, then all roots of (1) have a negative real part when , and (1) admits a pair of purely imaginary roots when .

Let be a root of (35) near , and let and . Due to functional differential equation theory, for every , , , there exists such that is continuously differentiable in for . Substituting into the left-hand side of (35) and taking derivative with respect to , we have

We can easily obtain where

Now we assume that (H6)

In view of the above analysis and the results of Kuang [33] and Hale [31], we have the following.

Theorem 11. For , if and (H2) hold, then the positive equilibrium of system (1) is asymptotically stable for . 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 when , , .

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: which has a positive equilibrium . We can easily obtain that (H1)–(H6) hold true. When , applying MATLAB 7.0, we can get . The positive equilibrium is asymptotically stable for and unstable for which is shown in Figure 1. When , (57) undergoes a Hopf bifurcation around the positive equilibrium . That is, a small amplitude periodic solution occurs near when and is close to which is shown in Figure 2. The bifurcation diagram of the case is shown in Figure 3.

fig1
Figure 1: Trajectory portrait and phase portrait of system (57) with , . The positive equilibrium is asymptotically stable. The initial value is (5.6, 7.45).
fig2
Figure 2: Trajectory portrait and phase portrait of system (57) with , . Hopf bifurcation occurs from the positive equilibrium . The initial value is (5.6, 7.45).
679602.fig.003
Figure 3: Bifurcation diagram with respect to the time delay for system (57) with .

Let , and choose as a parameter. We have . Then the positive equilibrium is asymptotically stable when . The Hopf bifurcation value of (57) is (see Figures 4 and 5) The bifurcation diagram of the case is shown in Figure 6.

fig4
Figure 4: Trajectory portrait and phase portrait of system (57) with , . The positive equilibrium (5.61, 7.02) is asymptotically stable. The initial value is (5.6, 7.45).
fig5
Figure 5: Trajectory portrait and phase portrait of system (57) with , . Hopf bifurcation occurs from the positive equilibrium . The initial value is (5.6, 7.45).
679602.fig.006
Figure 6: Bifurcation diagram with respect to the time delay for system (57) with .

When , using MATLAB 7.0, we obtain , . The positive equilibrium is asymptotically stable for and unstable for which is shown in Figure 7. When , (57) undergoes a Hopf bifurcation at the positive equilibrium . That is, a small amplitude periodic solution occurs around when and is close to which is illustrated in Figure 8. The bifurcation diagram of the case is shown in Figure 9.

fig7
Figure 7: Trajectory portrait and phase portrait of system (57) with , . The positive equilibrium is asymptotically stable. The initial value is (5.6, 8).
fig8
Figure 8: Trajectory portrait and phase portrait of system (57) with , . Hopf bifurcation occurs from the positive equilibrium . The initial value is (5.6, 7.45).
679602.fig.009
Figure 9: Bifurcation diagram with respect to the time delay for system (57) with .

Let , and choose as a parameter. We have . Then the positive equilibrium is asymptotically stable when . The Hopf bifurcation value of (57) is (see Figures 10 and 11). The bifurcation diagram of the case is shown in Figure 12.

fig10
Figure 10: Trajectory portrait and phase portrait of system (57) with , . The positive equilibrium is asymptotically stable. The initial value is (5.6, 7.45).
fig11
Figure 11: Trajectory portrait and phase portrait of system (57) with , . Hopf bifurcation occurs from the positive equilibrium . The initial value is (5.6, 7.45).
679602.fig.0012
Figure 12: Bifurcation diagram with respect to the time delay for system (57) with .

When , using MATLAB 7.0, we obtain , . The positive equilibrium is asymptotically stable for and unstable for which is shown in Figure 13. When , (57) undergoes a Hopf bifurcation at the positive equilibrium . That is, a small amplitude periodic solution occurs around when is close to which is illustrated in Figure 14. The bifurcation diagram of the case is shown in Figure 15.

fig13
Figure 13: Trajectory portrait and phase portrait of system (57) with . The positive equilibrium is asymptotically stable. The initial value is (5.6, 7.45).
fig14
Figure 14: Trajectory portrait and phase portrait of system (57) with . Hopf bifurcation occurs from the positive equilibrium . The initial value is (5.6, 7.45).
679602.fig.0015
Figure 15: Bifurcation diagram with respect to the time delay for system (57) with .

4. Conclusions

In this paper, we have investigated local stability of the positive equilibrium and local Hopf bifurcation of a ratio-dependent predator-prey model with two delays. It is shown that if some conditions hold true and , then the positive equilibrium is asymptotically stable when . When the delay increases, the positive equilibrium loses its stability and a sequence of Hopf bifurcations occur at the positive equilibrium . That is, a family of periodic orbits bifurcates from the the positive equilibrium . We also showed that if a certain condition is satisfied and , then the positive equilibrium is asymptotically stable when , when the delay increases, the positive equilibrium loses its stability and a sequence of Hopf bifurcations occur at the positive equilibrium . In case , we have shown that if some conditions are satisfied, and , then the positive equilibrium is asymptotically stable when . When the delay increases, the positive equilibrium loses its stability and a sequence of Hopf bifurcations occur at the positive equilibrium which means a family of periodic orbits bifurcates from the the positive equilibrium . 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 , it is assumed that in a certain range and choose the delay 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 as bifurcation parameter. We will further focus on the topic elsewhere in the near future.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 11261010 and no. 11101126), the Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), Natural Science and Technology Foundation of Guizhou Province (J20122100), Governor Foundation of Guizhou Province (201253) and Doctoral Foundation of Guizhou University of Finance and Economics (2010).

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