Abstract and Applied Analysis

Volume 2013 (2013), Article ID 795358, 12 pages

http://dx.doi.org/10.1155/2013/795358

Research Article

Stability and Hopf Bifurcation Analysis for a Gause-Type Predator-Prey System with Multiple Delays

Juan Liu,¹ Changwei Sun,² and Yimin Li³

¹Department of Science, Bengbu College, Bengbu 233030, China

²Department of Mechanical and Electronic Engineering, Bengbu College, Bengbu 233030, China

³Faculty of Science, Jiangsu University, Zhenjiang 212013, China

Received 5 May 2013; Accepted 13 May 2013

Academic Editor: Luca Guerrini

Copyright © 2013 Juan Liu et al. 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.

This paper is concerned with a Gause-type predator-prey system with two delays. Firstly, we study the stability and the existence of Hopf bifurcation at the coexistence equilibrium by analyzing the distribution of the roots of the associated characteristic equation. A group of sufficient conditions for the existence of Hopf bifurcation is obtained. Secondly, an explicit formula for determining the stability and the direction of periodic solutions that bifurcate from Hopf bifurcation is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the main theoretical results.

1. Introduction

Multispecies predator-prey models have been studied by many scholars [1–7]. Guo and Jiang [7] studied the following three-species food-chain system:

where , , and are the population densities of the prey, the predator and the top predator at time . The prey grows with intrinsic growth rate and carrying capacity in the absence of predation. The predator captures the prey with capture rate and Holling type II functional response . The top predator captures its prey (the predator) with capture rate and Holling type I functional response . The predator and the top predator contribute to their growth with the conversion rates and , respectively. The parameters and are the death rates of the predator and the top predator, respectively. All the parameters , , , , , , , , and in system (1) are assumed to be positive. The constant represents the time delay due to the gestation of the prey. Guo and Jiang [7] investigated the bifurcation phenomenon and the properties of periodic solutions of system (1).

Predator-prey systems with single delay as system (1) have been investigated extensively [8–12]. However, there are some papers on the bifurcations of a population dynamics with multiple delays [13–16]. Gakkhar and Singh [15] studied the effects of two delays on a delayed predator-prey system with modified Leslie-Gower and Holling type II functional response and established the existence of periodic solutions via Hopf bifurcation with respect to both delays. Motivated by the work of Guo and Jiang [7] and Gakkhar and Singh [15], we consider the following predator-prey system with two delays:

where denotes the time delay due to the gestation of the predator and denotes the time delay due to the gestation of the top predator.

This paper is organized as follows. In the next section, we will consider the stability of the positive equilibrium of system (2) and the existence of local Hopf bifurcation at the positive equilibrium. In Section 3, we can determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions from the Hopf bifurcation. Some numerical simulations are also given to illustrate the theoretical prediction in Section 4.

2. Local Stability and Hopf Bifurcation

Because we are only interested in the case in which the species can coexist, then we only consider the positive equilibrium of system (2). It is not difficult to know that if conditions and hold, then system (2) has a unique positive equilibrium , where

where

Let , , and and still denote , , and by , , and respectively. Then system (2) can be transformed to the following form:

where

The linearized system of system (5) is

Then the associated characteristic equation of system (7) at the origin is of the form

where

Case 1. One has .

Equation (8) becomes

Obviously, if conditions and hold, then all the roots of (10) must have negative real parts. Then, we can conclude that the positive equilibrium is locally asymptotically stable in the absence of delay.

Case 2. One has .

Equation (8) becomes

Letting be a root of (11), then we have

It follows that

where

Letting , then (13) becomes

Obviously, . Thus, we assume that (15) has at least one positive solution. Without loss of generality, we assume that it has three positive roots, which are denoted as , , and . Then (13) has three positive roots , .

From (12), we can get

Then, we denote

Next, we verify the transversality condition. Differentiating the two sides of (11) with respect to and noticing that is a function of , we can get

Therefore

From (13), we have

Therefore

with

Obviously, if , then . Thus, if condition holds, the transversality condition is satisfied. In conclusion, we have the following results.

Theorem 1. Suppose that conditions hold. The positive equilibrium of system (2) is asymptotically stable for and unstable when . And system (2) undergoes a Hopf bifurcation at when .

Case 3. One has .

Equation (8) becomes

Let be a root of (23), then we have

which follows that

with

Let , then (24) becomes

Similar as in Case 2, we assume that (27) has at least one positive solution. Without loss of generality, we assume that it has three positive roots, which are denoted by , and . Then (25) has three positive roots , .

From (24), we get

Then, we denote

Similar as in Case 2, we know that if condition holds, where

then, . Namely, if condition holds, the transversality condition is satisfied. Therefore, we have the following results. Therefore, we have the following theorem.

Theorem 2. Suppose that conditions hold. The positive equilibrium of system (2) is asymptotically stable for and unstable when . And system (2) undergoes a Hopf bifurcation at when .

Case 4. One has .

It is considered that with (8), in its stable interval and is considered as a parameter.

Let be the root of (8). Separating real and imaginary parts leads to

Eliminating leads to

where

Suppose that (32) has finite positive roots. If condition holds, we denote the roots of (32) by . For every fixed , there exists a sequence satisfying (32).

From (31), we can get

Let

when (8) has a pair of purely imaginary roots for .

To verify the transversality condition of Hopf bifurcation, differentiating (8) with respect to and substituting , we can get

where

Clearly, if condition holds, then . Namely, if condition holds, the transversality condition is satisfied. Therefore, we have the following results. Thus, we have the following theorem.

Theorem 3. Suppose that conditions hold and . The positive equilibrium of system (2) is asymptotically stable for and unstable when . And system (2) undergoes a Hopf bifurcation at when .

Case 5. One has .

We consider (8) with in its stable interval, regarding as a parameter.

Let be a root of (8). Then we get

It follows that

where

Similar as in Case 4, we suppose that (39) has finite positive roots. And we denote the roots of (39) by . The corresponding critical value of is

Let

when (8) has a pair of purely imaginary roots for .

Similar as in Case 4, we give the following assumption , where

Therefore, if condition holds, then we can get . That is, the transversality condition is satisfied. Hence, we have the following theorem.

Theorem 4. Suppose that conditions hold and . The positive equilibrium of system (2) is asymptotically stable for and unstable when . And system (2) undergoes a Hopf bifurcation at when .

3. Direction and Stability of the Hopf Bifurcation

In this section, we will employ the normal form method and center manifold theorem introduced by Hassard et al. [17] to determine the direction of Hopf bifurcation and stability of the bifurcated periodic solutions of system (2) with respect to for . Without loss of generality, we assume that , where .

Let . Then is the Hopf bifurcation value of system (2). Rescaling the time delay , then system (2) can be rewritten as

where

With

By Riesz representation theorem, there exists a matrix function whose elements are of bounded variation, such that

In fact, we can choose

For , we define

Then system (44) can be transformed into the following operator equation:

The adjoint operator of is defined by

associated with a bilinear form

where .

From the above discussion, we know that are the eigenvalues of and they are also eigenvalues of . We assume that is the eigenvector belonging to the eigenvalue and is the eigenvector belonging to the eigenvalue . Then, by a simple computation, we can obtain

Then we have .

Next, we get the coefficients used to determine the important quantities of the periodic solution by using a computation process similar to that in [18]:

with

where and can be computed as the following equations, respectively

with

Thus, we can calculate the following values:

Based on the discussion above, we can obtain the following results.

Theorem 5. If , then the Hopf bifurcation is supercritical (subcritical); if , the bifurcating periodic solutions are stable (unstable); if , the period of the bifurcating periodic solutions increases (decreases).

4. Numerical Simulation and Discussion

In this section, we present some numerical simulations to illustrate the analytical results obtained in the previous sections. Let , , , , , , , , and . Then we have the following particular case of system (2):

which has a positive equilibrium .

For , we have , . From Theorem 1, we know that the positive equilibrium is asymptotically stable for . As can be seen from Figure 1, if , is asymptotically stable. However, if , then is unstable and system (59) undergoes a Hopf bifurcation at , and a family of periodic solutions bifurcate from the positive equilibrium . This property can be illustrated by Figure 2. For , by a simple computation, we can easily get , . The corresponding waveform and the phase plots are shown in Figures 3 and 4.

Figure 1: is asymptotically stable for with initial values 0.975, 1.025, and 0.625.

Figure 2: is unstable for with initial values 0.975, 1.025, and 0.625.

Figure 3: is asymptotically stable for with initial values 0.975, 1.025, and 0.625.

Figure 4: is unstable for with initial values 0.975, 1.025, and 0.625.

For and , we get , . That is, when increases from zero to the critical value , the positive equilibrium is asymptotically stable; then it will lose stability, and a Hopf bifurcation occurs once . This property can be illustrated by Figures 5 and 6. Further, we get , . Then we have , , . Therefore, from Theorem 5, we can know that the Hopf bifurcation is supercritical and the bifurcating periodic solutions are stable.

Figure 5: is asymptotically stable for and with initial values 0.975, 1.025, and 0.625.

Figure 6: is unstable for and with initial values 0.975, 1.025, and 0.625.

At last, for and , we obtain , . The corresponding waveform and the phase plots are shown in Figures 7 and 8.

Figure 7: is asymptotically stable for and with initial values 0.975, 1.025, and 0.625.

Figure 8: is unstable for and with initial values 0.975, 1.025, and 0.625.

Guo and Jiang [7] have obtained that the three species in system (2) with only one time delay can coexist, however, we get that the species could also coexist with some available time delays of the predator and the top predator. This is valuable from the view of ecology. As the future work, we shall consider the following more general and more complicated system with multiple delays:

where is feedback delay of the prey and , are the time delays due to the gestation of the predator and the top predator, respectively.

Acknowledgments

This work was supported by National Natural Science Foundation of China (11072090), Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ2013B137) and Anhui Provincial Natural Science Foundation under Grant no. 1208085QA11.

References

A. Klebanoff and A. Hastings, “Chaos in three-species food chains,” Journal of Mathematical Biology, vol. 32, no. 5, pp. 427–451, 1994.

M. C. Varriale and A. A. Gomes, “A study of a three species food chain,” Ecological Modelling, vol. 110, no. 2, pp. 119–133, 1998.

H. I. Freedman and P. Waltman, “Mathematical analysis of some three-species food-chain models,” Mathematical Biosciences, vol. 33, no. 3-4, pp. 257–276, 1977.

K. McCann and P. Yodzis, “Bifurcation structure of a 3-species food chain model,” Theoretical Population Biology, vol. 48, no. 2, pp. 93–125, 1995.

A. Hastings and T. Powell, “Chaos in three-species food chain,” Ecology, vol. 72, no. 3, pp. 896–903, 1991.

S. Guo and W. Jiang, “Global stability and Hopf bifurcation for Gause-type predator-prey system,” Journal of Applied Mathematics, vol. 2012, Article ID 260798, 17 pages, 2012.

S. Guo and W. Jiang, “Hopf bifurcation analysis on general Gause-type predator-prey models with delay,” Abstract and Applied Analysis, vol. 2012, Article ID 363051, 17 pages, 2012.

E. Beretta and Y. Kuang, “Geometric stability switch criteria in delay differential systems with delay dependent parameters,” SIAM Journal on Mathematical Analysis, vol. 33, no. 5, pp. 1144–1165, 2002.

N. H. Gazi and M. Bandyopadhyay, “Effect of time delay on a detritus-based ecosystem,” International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 25619, 28 pages, 2006.

Y. Xue and X. Wang, “Stability and local Hopf bifurcation for a predator-prey model with delay,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 252437, 17 pages, 2012.

Y. Bai and X. Zhang, “Stability and Hopf bifurcation in a diffusive predator-prey system with Beddington-DeAngelis functional response and time delay,” Abstract and Applied Analysis, vol. 2011, Article ID 463721, 22 pages, 2011.

J.-F. Zhang, “Bifurcation analysis of a modified Holling-Tanner predator-prey model with time delay,” Applied Mathematical Modelling, vol. 36, no. 3, pp. 1219–1231, 2012.

X. Li, S. Ruan, and J. Wei, “Stability and bifurcation in delay-differential equations with two delays,” Journal of Mathematical Analysis and Applications, vol. 236, no. 2, pp. 254–280, 1999.

Y. Song, M. Han, and Y. Peng, “Stability and Hopf bifurcations in a competitive Lotka-Volterra system with two delays,” Chaos, Solitons & Fractals, vol. 22, no. 5, pp. 1139–1148, 2004.

S. Gakkhar and A. Singh, “Complex dynamics in a prey predator system with multiple delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 914–929, 2012.

X.-Y. Meng, H.-F. Huo, X.-B. Zhang, and H. Xiang, “Stability and Hopf bifurcation in a three-species system with feedback delays,” Nonlinear Dynamics, vol. 64, no. 4, pp. 349–364, 2011.

B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, Mass, USA, 1981.

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.