Journal of Applied Mathematics

Volume 2012 (2012), Article ID 286961, 18 pages

doi:10.1155/2012/286961

Research Article

The Effect of Time Delay on Dynamical Behavior in an Ecoepidemiological Model

Changjin Xu¹ and Yusen Wu²

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 26 October 2012; Revised 12 November 2012; Accepted 26 November 2012

Academic Editor: Shiping Lu

Copyright © 2012 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 delayed predator-prey model with disease in the prey is investigated. The conditions for the local stability and the existence of Hopf bifurcation at the positive equilibrium of the system are derived. The effect of the two different time delays on the dynamical behavior has been given. Numerical simulations are performed to illustrate the theoretical analysis. Finally, the main conclusions are drawn.

1. Introduction

During the past decades, epidemiological models have received considerate attention since the seminal SIR model of Kermack and McKendrich [1]. Great attention has been paid to the dynamics properties of the predator-prey models which have significant biological background. Numerous excellent and interesting results have been reported. For example, Bhattacharyya and Mukhopadhyay [2] studied the spatial dynamics of nonlinear prey-predator models with prey migration and predator switching, Bhattacharyya and Mukhopadhyay [3] analyzed the local and global dynamical behavior of an ecoepidemiological model, Kar and Ghorai [4] made a detailed discussion on the local stability, global stability, influence of harvesting and bifurcation of a delayed predator-prey model with harvesting, Chakraborty et al. [5] focused on the bifurcation and control of a bio-economic model of a delayed prey-predator model. For more related research, one can see [6–19].

In 2005, Song et al. [20] investigated the stability and Hopf bifurcation of a delayed ecoepidemiological model as follows:

where , , represent the susceptible prey, infected prey and predator population, respectively. can be interpreted as the prey carrying capacity with an intrinsic birth rate constant . is called the transmission coefficient. The predator has a death rate constant and the predation coefficient . The death rate of infected prey is positive constant . The coefficient in conversing prey into predator is . and are the time required for mature of predator and the time required for the gestation of predator, respectively. The more detail biological meaning of the coefficients of system (1.1), one can see [20].

For the sake of simplicity, Song et al. [20] rescales time , then system (1.1) can be transformed into the following form:

where , , , , , , , , .

We would like to point out that although Song et al. [20] investigated the local stability and Hopf bifurcation of system (1.2) under the assumption and obtained some good results, but they did not discuss what the different time delay and have effect on the stability and Hopf bifurcation behavior of system (1.2). Thus it is important for us to deal with the effect of time delay on the dynamics of system (1.2). There are some work which deal with this topic [21–24]. In this paper, we will further investigate the stability and bifurcation of model (1.2) as a complementarity. We will show that the two different time delay and have different effect on the stability and Hopf bifurcation behavior of system (1.2).

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 will study the stability of the positive equilibrium and the existence of local Hopf bifurcations.

If the following condition:

holds, then system (1.2) has a unique equilibrium point , where

Let , , and still denote , , by , , , respectively, then (1.2) reads as

The characteristic equation of (2.3) is given by

That is

where

where

The following lemma is important for us to analyze the distribution of roots of the transcendental equation (2.5).

Lemma 2.1 (see [13]). 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 four cases.

Case 1. , (2.5) becomes

All roots of (2.9) have a negative real part if the following condition holds:

Then the equilibrium point is locally asymptotically stable when the conditions and are satisfied.

Case 2. , , (2.5) becomes

For , be a root of (2.11), then it follows that

which is equivalent to

Let , then (2.13) takes the form

where , , . Denote

Let , where , . There are three cases for the solutions of (2.15).

(i)If , (2.15) has a real root and a pair of conjugate complex roots. The real root is positive and is given by

(ii)If , (2.15) has three real roots, of which two are equal. In particular, if , there exists only one positive root, ; If , there exists only one positive root, for , and there exist three positive roots for , , .(iii)If , there are three distinct real roots, , , , where . Furthermore, if , there exists only one positive root. Otherwise, if , there may exist either one or three positive real roots. If there is only one positive real root, it is equal to .

Obviously, the number of positive real roots of (2.15) depends on the sign of . If , (2.15) has only one positive real root. Otherwise, there may exist three positive roots.

Without loss of generality, we assume that (2.14) has three positive roots, defined by , , , respectively. Then (2.13) has three positive roots,

By (2.12), we have

Thus, if we denote

where ; , then are a pair of purely imaginary roots of (2.11) with . Define

Based on above analysis, we have the following result.

Lemma 2.2. If and hold, then all roots of (1.2) have a negative real part when and (1.2) admits a pair of purely imaginary roots when   ; .

Let be a root of (2.11) near , and , 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 (2.11) and taking derivative with respect to , we have

We can easily obtain

where . Thus, we have

Since , , we can conclude that the sign of is determined by that of .

The analysis above leads to the following result.

Theorem 2.3. Suppose that and , where is defined by (2.15). Then

and the of is consistent with that of .

In the sequel, we assume that

According to above analysis and the results of Kuang [25] and Hale [26], we have the following.

Theorem 2.4. For , if and hold, then the positive equilibrium of system (1.2) is asymptotically stable for . In addition to the conditions and , we further assume that holds, then system (1.2) undergoes a Hopf bifurcation at the positive equilibrium when , ; .

Case 3. , , (2.5) takes the form

For , be a root of (2.26), then it follows that

which is equivalent to

Let , then (2.13) takes the form

where , , . Denote

Let , where , . For (2.15), Similar analysis on the solutions of system (2.30) as that in Case 2. Here we omit it.

Without loss of generality, we assume that (2.30) has three positive roots, defined by , , , respectively. Then (2.29) has three positive roots

By (2.27), we have

Thus, if we denote

where ; , then are a pair of purely imaginary roots of (2.26) with . Define

The above analysis leads to the following result.

Lemma 2.5. If and hold, then all roots of (1.2) have a negative real part when and (1.2) admits a pair of purely imaginary roots when ; .

Let be a root of (2.26) near , and , 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 (2.26) and taking derivative with respect to , we have

We can easily obtain

where . Thus, we have

Since , we can conclude that the sign of is determined by that of .

From the analysis above, we obtain the following result.

Theorem 2.6. Suppose that and , where is defined by (2.30). Then

and the of is consistent with that of .

In the sequel, we assume that

Based on above analysis and in view of Kuang [25] and Hale [26], we get the following result.

Theorem 2.7. For , if and hold, then the positive equilibrium of system (1.2) is asymptotically stable for . In addition to the condition and , one further assumes that holds, then system (1.2) undergoes a Hopf bifurcation at the positive equilibrium when , ; .

Case 4. , . We consider (2.5) with in its stable interval. Regarding as a parameter. Without loss of generality, we consider system (1.2) under the assumptions and . Let be a root of (2.5), then we can obtain

where

Denote

Assume that

It is easy to check that if holds and . We can obtain that (2.42) has finite positive roots . For every fixed , , there exists a sequence , such that (2.42) holds. Let

When , (2.5) has a pair of purely imaginary roots for .

In the following, we assume that

Thus, by the general Hopf bifurcation theorem for FDEs in Hale [26], we have the following result on the stability and Hopf bifurcation in system (1.2).

Theorem 2.8. For system (1.2), suppose , , , , and are satisfied, and , then the positive equilibrium is asymptotically stable when , and system (1.2) undergoes a Hopf bifurcation at the positive equilibrium when .

Case 5. , . We consider (2.5) with in its stable interval. Regarding as a parameter. Without loss of generality, we consider system (1.2) under the assumptions and . Let be a root of (2.5), then we can obtain

where , , , and are defined by (2.41). Denote

Obviously, if holds and . We can obtain that (2.47) has finite positive roots . For every fixed , , there exists a sequence , such that (2.47) holds. Let

When , (2.5) has a pair of purely imaginary roots for .

In the following, we assume that

In view of the general Hopf bifurcation theorem for FDEs in Hale [26], we have the following result on the stability and Hopf bifurcation in system (1.2).

Theorem 2.9. For system (1.2), assume that , , , , and are satisfied and , then the positive equilibrium is asymptotically stable when , and system (1.2) undergoes a Hopf bifurcation at the positive equilibrium when .

3. Computer Simulations

In this section, we present some numerical results of system (1.2) 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 are satisfied. When , using Matlab 7.0, we obtain , . The positive equilibrium is asymptotically stable for and unstable for which is shown in Figure 1. When , (3.1) undergoes a Hopf bifurcation at the positive equilibrium , that is, a small amplitude periodic solution occurs around when and is close to which is shown in Figure 2.

(a)

(b)

(c)

(d)

Figure 1: Trajectory portrait and phase portrait of system (3.1) with , . The positive equilibrium is asymptotically stable. The initial value is .

(a)

(b)

(c)

(d)

Figure 2: Trajectory portrait and phase portrait of system (3.1) with , . Hopf bifurcation occurs from the positive equilibrium . The initial value is .

Let and choose as a parameter. We have , Then the positive equilibrium is asymptotically when . The Hopf bifurcation value of (3.1) is (see Figures 3 and 4).

(a)

(b)

(c)

(d)

Figure 3: Trajectory portrait and phase portrait of system (3.1) with , . The positive equilibrium is asymptotically stable. The initial value is .

(a)

(b)

(c)

(d)

Figure 4: Trajectory portrait and phase portrait of system (3.1) with , . Hopf bifurcation occurs from the positive equilibrium . The initial value is .

When , using Matlab 7.0, we obtain , . The positive equilibrium is asymptotically stable for and unstable for which is shown in Figure 5. When , (3.1) 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 6.

(a)

(b)

(c)

(d)

Figure 5: Trajectory portrait and phase portrait of system (3.1) with , . The positive equilibrium is asymptotically stable. The initial value is .

(a)

(b)

(c)

(d)

Figure 6: Trajectory portrait and phase portrait of system (3.1) with , . Hopf bifurcation occurs from the positive equilibrium . The initial value is .

Let and choose as a parameter. We have . Then the positive equilibrium is asymptotically stable when . The Hopf bifurcation value of (3.1) is (see Figures 7 and 8).

(a)

(b)

(c)

(d)

Figure 7: Trajectory portrait and phase portrait of system (3.1) with , . The positive equilibrium is asymptotically stable. The initial value is .

(a)

(b)

(c)

(d)

Figure 8: Trajectory portrait and phase portrait of system (3.1) with , . Hopf bifurcation occurs from the positive equilibrium . The initial value is .

4. Conclusions

In this paper, we have investigated local stability of the positive equilibrium and local Hopf bifurcation of an ecoepidemiological 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 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 . Some numerical simulations verifying our theoretical results is performed. In addition, we must point out that although Song et al. [20] have also investigated the the existence of Hopf bifurcation for system (1.2) with respect to positive equilibrium , it is assumed that . But what effect different time delay has on the dynamical behavior of system (1.2)? Song et al. [20] did not consider this issue. Thus we think that our work generalizes the known results of Song et al. [20]. In addition, we can investigate the Hopf bifurcation nature of system (1.2) by choosing the delay or as bifurcation parameter. We will further investigate the topic elsewhere in the near future.

Acknowledgments

This work is supported by National Natural Science Foundation of China (no. 11261010 and no. 11101126), Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), Natural Science and Technology Foundation of Guizhou Province (J2100), Governor Foundation of Guizhou Province (53), and Doctoral Foundation of Guizhou University of Finance and Economics (2010).

References

W. Kermack and A. McKendrich, “Contributions to the mathematical theory of epidemic,” Proceedings of the Royal Society A, vol. 138, no. 834, pp. 55–83, 1932.

R. Bhattacharyya and B. Mukhopadhyay, “Spatial dynamics of nonlinear prey-predator models with prey migration and predator switching,” Ecological Complexity, vol. 3, no. 2, pp. 160–169, 2006.

R. Bhattacharyya and B. Mukhopadhyay, “On an eco-epidemiological model with prey harvesting and predator switching: local and global perspectives,” Nonlinear Analysis. Real World Applications, vol. 11, no. 5, pp. 3824–3833, 2010.

T. K. Kar and A. Ghorai, “Dynamic behaviour of a delayed predator-prey model with harvesting,” Applied Mathematics and Computation, vol. 217, no. 22, pp. 9085–9104, 2011.

K. Chakraborty, M. Chakraborty, and T. K. Kar, “Bifurcation and control of a bioeconomic model of a prey-predator system with a time delay,” Nonlinear Analysis. Hybrid Systems, vol. 5, no. 4, pp. 613–625, 2011.

S. Gao, L. Chen, and Z. Teng, “Hopf bifurcation and global stability for a delayed predator-prey system with stage structure for predator,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 721–729, 2008.

T. K. Kar and U. K. Pahari, “Modelling and analysis of a prey-predator system with stage-structure and harvesting,” Nonlinear Analysis. Real World Applications, vol. 8, no. 2, pp. 601–609, 2007.

Y. Kuang and Y. Takeuchi, “Predator-prey dynamics in models of prey dispersal in two-patch environments,” Mathematical Biosciences, vol. 120, no. 1, pp. 77–98, 1994.

K. Li and J. Wei, “Stability and Hopf bifurcation analysis of a prey-predator system with two delays,” Chaos, Solitons & Fractals, vol. 42, no. 5, pp. 2606–2613, 2009.

R. M. vcMay, “Time delay versus stability in population models with two and three trophic levels,” Ecology, vol. 54, no. 2, pp. 315–325, 1973.

H. P. Prajneshu, “A prey-predator model with switching effect,” Journal of Theoretical Biology, vol. 125, no. 1, pp. 61–66, 1987.

S. Ruan, “Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays,” Quarterly of Applied Mathematics, vol. 59, no. 1, pp. 159–173, 2001.

S. Ruan and J. Wei, “On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 10, no. 6, pp. 863–874, 2003.

Y. Song and J. Wei, “Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system,” Journal of Mathematical Analysis and Applications, vol. 301, no. 1, pp. 1–21, 2005.

E. Teramoto, K. Kawasaki, and N. Shigesada, “Switching effect of predation on competitive prey species,” Journal of Theoretical Biology, vol. 79, no. 3, pp. 303–315, 1979.

R. Xu, M. A. J. Chaplain, and F. A. Davidson, “Periodic solutions for a delayed predator-prey model of prey dispersal in two-patch environments,” Nonlinear Analysis. Real World Applications, vol. 5, no. 1, pp. 183–206, 2004.

R. Xu and Z. Ma, “Stability and Hopf bifurcation in a ratio-dependent predator-prey system with stage structure,” Chaos, Solitons & Fractals, vol. 38, no. 3, pp. 669–684, 2008.

T. Zhao, Y. Kuang, and H. L. Smith, “Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 28, no. 8, pp. 1373–1394, 1997.

X. Zhou, X. Shi, and X. Song, “Analysis of nonautonomous predator-prey model with nonlinear diffusion and time delay,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 129–136, 2008.

X. Y. Song, Y. N. Xiao, and L. S. Chen, “Stability and Hopf bifurcation of an eco-epidemiological model with delays,” Acta Mathematica Scientia A, vol. 25, no. 1, pp. 57–66, 2005 (Chinese).

M. De la Sen, “Sufficiency-type stability and stabilization criteria for linear time-invariant systems with constant point delays,” Acta Applicandae Mathematicae, vol. 83, no. 3, pp. 235–256, 2004.

J. Cao, D. W. C. Ho, and X. Huang, “LMI-based criteria for global robust stability of bidirectional associative memory networks with time delay,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 66, no. 7, pp. 1558–1572, 2007.

M. De la Sen, R. P. Agarwal, A. Ibeas, and S. Alonso-Quesada, “On a generalized time-varying SEIR epidemic model with mixed point and distributed time-varying delays and combined regular and impulsive vaccination controls,” Advances in Difference Equations, vol. 2010, Article ID 281612, 42 pages, 2010.

Z. Liu, S. Lü, S. Zhong, and M. Ye, “Improved robust stability criteria of uncertain neutral systems with mixed delays,” Abstract and Applied Analysis, vol. 2009, Article ID 294845, 18 pages, 2009.

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993.

J. Hale, Theory of Functional Differential Equations, Springer, Berlin, Germany, 1977.