Journal of Applied MathematicsVolume 2012 (2012), Article ID 260798, 17 pagesdoi:10.1155/2012/260798

Research Article

Global Stability and Hopf Bifurcation for Gause-Type Predator-Prey System

Shuang Guo1,2 and Weihua Jiang¹

¹Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China²School of Mathematical Sciences, Daqing Normal University, Daqing 163712, China

Received 30 October 2011; Accepted 19 December 2011

Academic Editor: Junjie Wei

Copyright © 2012 Shuang Guo and Weihua Jiang. 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 class of three-dimensional Gause-type predator-prey model is considered. Firstly, local stability of equilibrium indicating the extinction of top-predator is obtained. Meanwhile, we construct a Lyapunov function, which is an extension of the Lyapunov functions constructed by Hsu for predator-prey system (2005), to give the global stability of the equilibrium. Secondly, we analyze the stability of coexisting equilibrium of predator-prey system with time delay when the predator catches the prey of pregnancy or with growth time. The delay can lead to periodic solutions, which is consistent with the law of growth for birds and some mammals. Further, an explicit formula is given which determines the stability of the bifurcating periodic solutions theoretically and the existence of periodic solutions is displayed by numerical simulations.

1. Introduction

The predator-prey systems have been extensively studied. The most popular one is Lotka-Volterra type model, which exhibits the well-known “paradox of enrichment” observed by [1, 2] and so on. However, there is often the interaction among multiple populations in nature, whose relationships are more complex than those in two populations. So the dynamics of three-dimensional model may become more complicated [3–8].

In general, the relationship among three groups may be competitive, a predator and two preys, two species catching the same prey, or a food chain. See [6, 9–11]. The Gause-type predator-prey food chain model was proposed by Freedman and Waltman in 1977 [5], which can be described as follows: where , , and are the population densities of prey, predator, and top predator at time , respectively. is the intrinsic growth rate of prey; and are the specific growth rates of predator and top predator; , are the death rates of and ; , are the conversion rates for prey and predator. For (1.1), there have been a lot of results, including the properties of equilibria and bifurcations. Freedman has argued that the unique interior equilibrium exists and it is locally asymptotically stable [5]. In [7, 12], the authors performed the normal forms for Hopf bifurcations and saddle node bifurcations near a degenerate equilibrium and showed that the model within a certain parameter range could take on chaos by numerical simulations. Many biologists believe that if the unique equilibrium of a predator-prey system is locally asymptotically stable, then it is globally asymptotically stable. For proving the global stability of the equilibrium, one can construct a Lyapunov function. If we are able to construct a Lyapunov function for the system, then the global stability is directly established from the modified LaSalle’s invariant principle [13]. Many authors presented some globally qualitative analysis of solutions of system (1.1) (see [8, 14–17]).

We assume that , , and satisfy the following conditions.(H1), there exists such that ( is called the carrying capacity of prey species), and for .(H2) and .(H3) and .

In general, there are three prototypes of monotone response functions which we often refer to as Holling I, II, and III, respectively. In [9, 15], Chiu and Hsu considered the three-dimensional food chain model that the response function is of Holling II; in [18], Hsu et al. obtained a complete classification about the asymptotic behavior of the solutions of Gause-type predator-prey system with Holling II function. In this paper, we discuss the general Gause-type food chain models, in particular, the case of Holling’s type I and II functional responses for predator and prey, respectively. So we let , , and . For simplicity, we nondimensionalize the system (1.1) with the following scaling: Then the system (1.1) takes the form which satisfies , , and , where

Through simple analysis, we know that (1.3) has four equilibriums: , , , and , where There is no obvious biological significance for and . The behavior of the solutions of (1.3) can be very complicated (see [12, 19–21]). We know that the time delay can be incorporated into (1.1) in three different ways: a time delay in the prey-specific growth term , a time delay in the predator response term , a time delay in the interaction term . The predators have a gestation period or reaction time such as grass-hare-fox food chain. Thus, a time delay can be incorporated into the predator response term , that is, In general, delay differential equations (DDEs) exhibit much more complicated dynamics than ordinary differential equations (ODEs) such as the existence of Bogdanov-Takens bifurcation and even chaos (see [21, 22], etc.). The reason is that a time delay could destabilize a stable equilibrium and induce bifurcations (see [23–26]). The cyclic phenomenon is very significant for the stability and the balance of ecosystems. Using the delay as a bifurcation parameter, we investigate the stability of the positive equilibrium and the existence of Hopf bifurcation of the model with delay. It is shown that Hopf bifurcation can occur as the delay crosses some critical values.

The rest of our paper is organized as follows: in Section 2, we construct a Lyapunov function to prove the global stability of the equilibrium indicating the extinction of top-predator. In Section 3, we first investigate the stability of coexisting equilibrium and the existence of Hopf bifurcation of (1.6). To determine the dynamics of the delay model, we study the characteristic equation of the linearized system at the equilibrium. In Section 4, we derive an explicit formula for determining the stability of bifurcating periodic solutions and the direction of Hopf bifurcation by the normal form method and the center manifold theory. In Section 5, we carry out some numerical simulations to illustrate the results obtained.

2. Global Stability of the Extinction of Top Predator

We consider the linearized system of (1.3) at . The characteristic equation at the equilibrium is given by where

When and , is locally asymptotically stable by the Routh-Hurwitz criterion, so we have the following.

Lemma 2.1. If and , is locally asymptotically stable.

To show that is globally stable, we construct a Lyapunov function:

It is obvious that , , and for . For simplicity, let , and ; then if and only if ; if and only if , where satisfies :

We have when , and it follows that Hence, we have the following theorem by Lasalle’s invariance principle [13].

Theorem 2.2. If and , is global asymptotically stable.

3. Stability and Hopf Bifurcation of Coexisting Equilibrium

In this part, we mainly study the stability of the coexisting equilibrium . If , then is a unique nontrivial equilibrium of (1.6). We consider the linearized system of (1.6) at . The equation of Jacobian determinant at the equilibrium is given by where The eigenvalue satisfies the characteristic equation where , , , , , and .

If , then all the eigenvalues of (3.3) have negative real parts when by the Routh-Hurwitz criterion. So we have the following lemma.

Lemma 3.1. If and , then the coexisting equilibrium of (1.3) is locally asymptotically stable.

It is known that is asymptotically stable if all roots of the corresponding characteristic equation (3.3) have negative real parts. We study the distribution of the roots of the transcendental equation (3.3) when . We assume that the equilibrium of the ODE model (1.3) is stable; then we derive some conditions to ensure that the steady state of the delay model is still stable.

Now we substitute into (3.3):

(1) when , ;

(2) when , . Separating the real and imaginary parts gives We get Let , , , and ; then (3.5) becomes From Ruan and Wei [27], we have the following results on the distribution of roots of (3.6).

Lemma 3.2. If one of the followings holds: (a),(b),then (3.6) has no positive root.

Therefore, the real parts of all the eigenvalues of (3.3) are negative for all delay , so we have the following.

Theorem 3.3. Suppose that (a) and ,(b)either or .Then the equilibrium of the delay model (1.6) is absolutely stable; that is, is asymptotically stable for all .

However, the stability of the steady state depends on the delay and the delay could even induce oscillations if the conditions in Lemma 3.2 are not satisfied.

Lemma 3.4. Denote If one of the followings holds: (a),(b),(c),then (3.6) has at least one positive root. This implies that the characteristic equation (3.3) has at least a pair of purely imaginary roots.

Suppose that the (3.6) has positive roots. Without loss of generality, we assume that it has three positive roots, denoted by , and , respectively. Then (3.6) has three positive roots . Let be the unique root of (3.4) such that . Also denote for .

So is the solution of (3.3). Clearly, We can define that is, are the purely imaginary roots of (3.3) for .

Denoting be the root of (3.3) satisfying , we have the following lemma.

Lemma 3.5. If , one has .

Proof. Assume that then Because we have If , then . There must be . This is because (3.3) has the positive real part roots as if . This contradicts to the fact when and is asymptotically stable.

By Lemma 3.5 we have the following theorem.

Theorem 3.6. Suppose that and . If Lemma 3.4 holds, then the equilibrium of the delay model (1.6) is asymptotically stable when , and unstable when , where is defined by (3.10). In addition, if , then Hopf bifurcation occurs when .

4. Direction and Stability of Hopf Bifurcation

Let , , , , , , and The system (1.6) is transformed into a functional differential equation (FDE) in [28], defining where . And the nonlinear term is Obviously is a Hopf bifurcation point. So the system (1.6) can transform into an abstract functional differential equation: where .

There exists a matrix (), whose elements are of bounded variation functions such that In fact, we can choose Then (4.5) is satisfied. For , we define So (4.5) is equivalent to the following abstract equation: where and for .

For , we define and a bilinear form: where . Then and are adjoint operators. We know that are eigenvalues of and therefore they are also eigenvalues of . The vectors and are the eigenvectors of and corresponding to the eigenvalue and , respectively, satisfying with , , , , and . Following the same algorithms as Hassard et al. [29], we can obtain the coefficients which will be used to determine the important quantities: Since there are and in , we still need to compute them. From [29], we have According to where we have , where with And similarly, According to where we have , where with

Consequently, can be expressed explicitly by the parameters and delay in the system (4.2). Thus, we can compute the following values: which determine the properties of bifurcating periodic solutions at the critical value . That is, determines the direction of Hopf bifurcation: if , then Hopf bifurcation at is forward (or backward); determines the stability of bifurcating periodic solutions: ; the bifurcating periodic solution is orbitally asymptotically stable (unstable); determines the period of the bifurcating periodic solutions: the period increases (decreases) if .

5. Numerical Simulations

In order to check our conclusions, we perform some numerical simulations. We choose the parameters as follows:(1), , , , , , and ;(2), , , , , , and .

Thus, all the conditions in Theorem 3.3 are satisfied. with initial value (0.48, 0.32, 0.25) for (1) and with initial value (0.08, 0.02, 0.25) for (2) are asymptotically stable for all (see Figure 1). From a biological sense, the prey, predator, and top predator will have a short-term shock in the initial stage as the effect of . But the population would tend to a steady level after a long period of time.

(a)

(b)

Figure 1: is asymptotically stable for all .

We choose another set of parameters which satisfies Theorem 3.6.

(3) , , , , , , , and .

By (3.10), we have , , and . Theorem 3.6 indicates that the equilibrium of the delay model (1.6) is asymptotically stable when (see Figure 2).

(a)

(b)

(c)

(d)

Figure 2: with initial value (0.985, 0.34, 0.4) is asymptotically stable when .

Hopf bifurcation occurs when , and the bifurcating periodic solution is orbitally asymptotically for (see Figure 3).

(a)

(b)

(c)

(d)

Figure 3: A stable periodic orbit of system (1.6) when initial value is (0.985, 0.34, 0.4) and .

In addition, the periodic solution of system (1.6) still exists when is large and its amplitude is larger compared with the solution in Figure 3 (see Figure 4). The numerical results of Figure 4 show that the global existence of periodic solutions is generated by the Hopf bifurcation. How to explain the phenomenon theoretically needs further researches.

(a)

(b)

Figure 4: The stable periodic orbits of system (1.6) when and with parameters given by (3).

6. Conclusion

In this paper, we analyze the dynamics of the equilibria of the extinction of top-predator and coexistence for a class of three-dimensional Gause-type predator-prey model. We obtain that the equilibrium of the extinction of top-predator is not only locally but also globally asymptotically stable for the certain parameters; introducing delay changes the stability of the coexisting equilibrium and Hopf bifurcation occurs with the increase of . The existence of periodic solutions for sufficiently large delay has been shown by numerical simulations. We know that the population outbreak may happen for the species with periodic fluctuation. The outbreaks of pests and mice are famous. For example, the number of locusts is estimated as many as and their weight is 5000 t, which appeared in a plague of locusts in Somalia in 1957 [30]. Thus, it is of great significance to research multiple periodic solutions of biological systems for controlling insect pests, preventing epidemics, and maintaining ecological balance.

Acknowledgments

The work is supported in part by the National Natural Science Foundation of China, by the Heilongjiang Provincial Natural Science Foundation (nos. A200806, A200813), and by the Heilongjiang Provincial Young Backbone Project (11251G0011).

References

H. I. Freedman, Deterministic Mathematical Models in Population Ecology, vol. 57, Marcel Dekker, New York, NY, USA, 1980.

N.G. Hairston, F. E. Smith, and L. B. Slobodkin, “Community structure, population control and competition,” The American Naturalist, vol. 94, pp. 421–425, 1960.

G. J. Butler, S. B. Hsu, and P. Waltman, “Coexistence of competing predators in a chemostat,” Journal of Mathematical Biology, vol. 17, no. 2, pp. 133–151, 1983.

K. S. Chêng, S. B. Hsu, and S. S. Lin, “Some results on global stability of a predator-prey system,” Journal of Mathematical Biology, vol. 12, no. 1, pp. 115–126, 1981.

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.

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

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

Z. Luo, “Optimal control for a predator-prey system with age-dependent,” International Journal of Biomathematics, vol. 2, no. 1, pp. 45–59, 2009.

C.-H. Chiu and S.-B. Hsu, “Extinction of top-predator in a three-level food-chain model,” Journal of Mathematical Biology, vol. 37, no. 4, pp. 372–380, 1998.

J.-M. Ginoux, B. Rossetto, and J.-L. Jamet, “Chaos in a three-dimensional Volterra-Gause model of predator-prey type,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 15, no. 5, pp. 1689–1708, 2005.

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.

S. Tang and L. Chen, “Global qualitative analysis for a ratio-dependent predator-prey model with delay,” Journal of Mathematical Analysis and Applications, vol. 266, no. 2, pp. 401–419, 2002.

Z. Q. Lu and G. S. K. Wolkowicz, “Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates,” SIAM Journal on Applied Mathematics, vol. 52, no. 1, pp. 222–233, 1992.

A. Ardito and P. Ricciardi, “Lyapunov functions for a generalized Gause-type model,” Journal of Mathematical Biology, vol. 33, no. 8, pp. 816–828, 1995.

S.-B. Hsu, “A survey of constructing Lyapunov functions for mathematical models in population biology,” Taiwanese Journal of Mathematics, vol. 9, no. 2, pp. 151–173, 2005.

S. B. Hsu and T. W. Huang, “Global stability for a class of predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 55, no. 3, pp. 763–783, 1995.

Y. Kuang, “Global stability of Gause-type predator-prey systems,” Journal of Mathematical Biology, vol. 28, no. 4, pp. 463–474, 1990.

S.-B. Hsu, T.-W. Hwang, and Y. Kuang, “Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system,” Journal of Mathematical Biology, vol. 42, no. 6, pp. 489–506, 2001.

L. Chen and F. Chen, “Global analysis of a harvested predator-prey model incorporating a constant prey refuge,” International Journal of Biomathematics, vol. 3, no. 2, pp. 205–223, 2010.

W. Ko and K. Ryu, “A qualitative study on general Gause-type predator-prey models with non-monotonic functional response,” Nonlinear Analysis. Real World Applications, vol. 10, no. 4, pp. 2558–2573, 2009.

D. Xiao and S. Ruan, “Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response,” Journal of Differential Equations, vol. 176, no. 2, pp. 494–510, 2001.

Yu. A. Kuznetsov and S. Rinaldi, “Remarks on food chain dynamics,” Mathematical Biosciences, vol. 134, no. 1, pp. 1–33, 1996.

W. Jiang and J. Wei, “Bifurcation analysis in a limit cycle oscillator with delayed feedback,” Chaos, Solitons and Fractals, vol. 23, no. 3, pp. 817–831, 2005.

H. Wang and W. Jiang, “Hopf-pitchfork bifurcation in van der Pol's oscillator with nonlinear delayed feedback,” Journal of Mathematical Analysis and Applications, vol. 368, no. 1, pp. 9–18, 2010.

J. Wei and W. Jiang, “Bifurcation analysis in van der Pol's oscillator with delayed feedback,” Journal of Computational and Applied Mathematics, vol. 213, no. 2, pp. 604–615, 2008.

J. J. Wei and S. G. Ruan, “Stability and global Hopf bifurcation for neutral differential equations,” Acta Mathematica Sinica, vol. 45, no. 1, pp. 93–104, 2002.

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.

J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 2nd edition, 1977.

B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41, Cambridge University Press, Cambridge, UK, 1981.

B. C. Birch, Hosea, Joel and Amos, Westminster John Knox, Kentucky, Ky, USA, 1997.