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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 168340, 13 pages
http://dx.doi.org/10.1155/2013/168340
Research Article

The Hopf Bifurcation for a Predator-Prey System with -Logistic Growth and Prey Refuge

School of Mathematics and Information Sciences, Henan University, Kaifeng 475001, China

Received 12 April 2013; Accepted 6 June 2013

Academic Editor: Luca Guerrini

Copyright © 2013 Shaoli Wang and Zhihao Ge. 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

The Hopf bifurcation for a predator-prey system with -logistic growth and prey refuge is studied. It is shown that the ODEs undergo a Hopf bifurcation at the positive equilibrium when the prey refuge rate or the index- passed through some critical values. Time delay could be considered as a bifurcation parameter for DDEs, and using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results.

1. Introduction

The construction and study of models for the population dynamics of predator-prey systems have long been and will continue to be one of the dominant themes in both ecology and mathematical ecology since the famous Lotka-Volterra equations. In recent years, the study of the consequences of hiding behavior of prey on the dynamics of predator-prey interactions has been an active topic [15]. Some of the empirical and theoretical work have investigated the effects of prey refuges and drawn a conclusion that the refuges used by prey have a stabilizing effect on the considered interactions and prey extinction can be prevented by the addition of refuges [612].

Motivated by the work of Ko and Ryu [13] and Tsoularis and Wallace [14], we construct the following -logistic growth predator-prey system with Holling type-II functional response and prey refuge: where and represent the densities of prey and predator, respectively, and , , , , , and are all positive constants and have their biological meanings accordingly. is the -logistic intrinsic growth rate of the prey in the absence of the predator; is the carrying capacity; is the logistic index; is the predation rate of predator; and is the prey refuge rate; is the death rate of the predator.

By assuming that the reproduction of predator after predating the prey will not be instantaneous but mediated by some discrete time lag required for gestation of predator, we incorporate a delay in system (1) to make the model more realistic. We aim to discuss the effect of time delay due to gestation of the predator on the global dynamics of system (1). To this end, we consider the following delayed predator-prey system with -logistic growth and prey refuge: where is positive constant. The constant denotes a time delay due to the gestation of the predator and the term denotes the probability of the predators, which capture the prey at time and still alive at times .

The initial conditions for system (2) take the form where , the Banach space of continuous functions mapping the interval into , where .

It is well known by the fundamental theory of functional differential equations [15] that system (2) has a unique solution and satisfying initial conditions (3).

The organization of this paper is as follows. In Section 2, we show the positivity and the boundedness of solutions of system (2) with initial conditions (3). In Section 3, we discuss the stability of boundary equilibria of system (2). In Section 4, we study the existence of Hopf bifurcations for system (1) and (2) at the positive equilibrium. In Section 5, using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Finally, numerical simulations are carried out to illustrate the main results, and a brief discussion is given to conclude this work in Section 6.

2. Positivity and Boundedness

In this section, we show the positivity and boundedness of solutions of system (2) with initial conditions (3).

2.1. Positivity of Solutions

Theorem 1. Solutions of system (2) with initial conditions (3) are positive for all .

Proof. Let be a solution of system (2) with initial conditions (3). From the first equation of system (2), we have Since . Hence, is positive.
To show that is positive on , suppose that there exists such that , and for . Then . From the second equation of (2), we have which is a contradiction.

Next, we will prove the boundedness of solutions.

2.2. Boundedness of Solutions

Theorem 2. Positive solutions of system (2) with initial conditions (3) are ultimately bounded.

Proof. Let be a solution of system (2) with initial conditions (3). From the first equation of (2), we have which yields and therefore Hence, for sufficiently small, there is a such that if , . Set Calculating the derivative of along solutions of system (2), we obtain where . Then there exists an , depending only on the parameters of system (2), such that for all large enough. Then , have an ultimately above bound.

3. Stability of the Boundary Equilibria

In this section, we discuss the stability of the boundary equilibria of system (2) with initial conditions (3).

We denote and always assume . The system (2) always has two boundary equilibria and ; if , it also has a positive equilibrium , where

Now we consider the stability of boundary equilibria.

For , the corresponding characteristic equation is and the roots are which implies that the equilibrium is always unstable.

For , the corresponding characteristic equation is It follows that or Denote and then we have for any . Hence has no positive root for , and at least one positive for . Therefore, for all , the equilibrium is stable when and unstable when .

Summarizing the discussion above, we obtain the following conclusion.

Theorem 3. (i) The equilibrium is always unstable for all .
(ii) The equilibrium is stable when and unstable when for all .

4. The Hopf Bifurcation

4.1. The Hopf Bifurcation of ODEs

When , the system (2) also has a positive equilibrium . The characteristic of the linearized system of (2) near the infected equilibrium is given by where When , becomes and (19) becomes where

Theorem 4. (i) If and , then the positive equilibrium of system (1) is asymptotically stable.
(ii) If and , then system (1) is unstable.
(iii) Suppose that . Then system (1) undergoes a Hopf bifurcation when (or ) passes through (or ).

Proof. Obviously (i) and (ii) hold.
(iii) From (23), we know that the root of the characteristic equation (22) satisfies Solving , we have or Calculating the derivative, we obtain or Hence the positive equilibrium is stable when () and unstable when (). Thus Hopf bifurcation occurs at ().

Example 5. We choose a series of parameter , , , , and , then . If , then and , thus the system (1) is stable (see Figure 1(a)); if , then and , and thus the system (1) is unstable (see Figure 1(b)).
If we choose a series of parameter , , , , and , then and ( is unserviceable). If , then , and thus the system (1) is stable (see Figure 2(a)); if , then , and thus the system (1) is unstable (see Figure 2(b)).

fig1
Figure 1: The infected steady state of system (1) is asymptotically stable when (a) and unstable when (b). The rest parameter values are , , , , and , and the initial values are and .
fig2
Figure 2: The infected steady state of system (1) is asymptotically stable when (a), and unstable when (b). The rest parameter values are , , , , and , and the initial values are and .
4.2. The Hopf Bifurcation of DDEs

In the following, we investigate the existence of purely imaginary roots to (19). Equation (19) takes the form of a second-degree exponential polynomial in , with all the coefficients of and depending on . Beretta and Kuang [16] established a geometrical criterion which gives the existence of purely imaginary root of a characteristic equation with delay dependent coefficients.

In order to apply the criterion due to Beretta and Kuang [16], we need to verify the following properties for all , where is the maximum value in which exists.(a);(b);(c);(d) has a finite number of zeros;(e)each positive root of is continuous and differentiable in whenever it exists.

Here, and are defined as in (20).

Let , and using (20) and (21), we have and then Therefore, (a) and (b) are satisfied.

From (20), we know that Therefore, (c) follows.

Let be defined as in (d). From we have where

It is obvious that property (d) is satisfied. Let be a point of its domain of definition such that . We know that the partial derivatives and exist and are continuous in a certain neighborhood of , and . By implicit function theorem, (e) is also satisfied.

Now let be a root of (19). Substituting it into (19) and separating the real and imaginary parts yields From (36), it follows that

By the definitions of , as in (20), and applying the property (a), (20) can be written aswhich yields

Assume that is the set where is a positive root of and for is not defined. Then for all in , is satisfying Let ; we have that We set Then, when , has real roots given by Note that and summarizing the discussion above, we have the following conclusion.

Proposition 6. If and , the has only one positive root denoted by . Furthermore, has a unique positive root given by .
Define such that and are given by the right hand sides of (37a) and (37b), respectively, with given by (38a) and (38b).
And the relation between the argument and in (37a) and (37b) for must be Hence we can define the maps given by where a positive root of (42) exists in .
Let one introduces the functions , that are continuous and differentiable in . Thus, we give the following theorem which is due to Beretta and Kuang [16].

Theorem 7. Assume that is a positive root of (19) defined for , and at some , for some , then a pair of simple conjugate pure imaginary roots exists at which crosses the imaginary axis from left to right if and crosses the imaginary axis from right to left if , where

Applying Theorem 3 and the Hopf bifurcation theorem for functional differential equation [12], we can conclude the existence of a Hopf bifurcation as stated in the following theorem.

Theorem 8. For system (2), the following conclusions are hold.(i)If and the function has no positive zero in , then the equilibrium is asymptotically stable for all .(ii)If and the function has positive zero in , then there exists , such that the equilibrium is asymptotically stable for and becomes unstable for , with a Hopf bifurcation occurring when .(iii)If and the function has no positive zeros in , then the equilibrium is always unstable for all .(iv)If and the function has positive zeros in , for some , then there exists , such that the equilibrium is unstable for and becomes asymptotically stable for , with a Hopf bifurcation occurring when .

Remark 9. If , then , and the equilibrium converges to .

5. Direction and Stability of the Hopf Bifurcation of the DDEs

In the above section, we have obtained some conditions which guarantee that the delayed predator-prey system with -logistic growth and prey refuge undergoes the Hopf bifurcation at some value of . In this section, we will study the direction, stability, and the period of the bifurcating periodic solutions. The approach we used here is based on the normal form approach and the center manifold theory introduced by Hassard et al. [17]. Throughout this section, we always assume that system (2) undergoes Hopf bifurcation at the positive equilibrium for , and then is corresponding purely imaginary roots of the characteristic equation at the positive equilibrium .

Let , .

System (2) is transformed into where For the simplicity of notations, we rewrite (50) as where , is defined by , and , are given, respectively, by By the Riesz representation theorem, there exists a function of bounded variation for , such that In fact, we can choose where denotes the Dirac delta function. For , define Then system (52) is equivalent to where for .

For , define and a bilinear inner product where . Then and are adjoint operators. By the discussion in Section 4, we know that are eigenvalues of . Thus, they are also eigenvalues of . We first need to compute the eigenvector of and corresponding to and , respectively.

Suppose that is the eigenvector of corresponding to ; then . It follows from (55) and (56) and the definition of that Solving the equations above, we derive that On the other hand, suppose that is the eigenvector of corresponding to . It follows from (55) and (56) and the definition of that which yields

In order to assure , we need to determine the value of . By (58), we have Thus, we can choose as such that

In the following, we apply the ideas in Hassard et al. [17] to describe the center manifold at , similar to that in [18, 19]. Let be the solution of (52) when . Define On the center manifold , we have where and and are local coordinates for center manifold in the direction of and . Note that is real if is real. We only consider real solutions. For solution of (52), since , we have

We rewrite this equation as where It follows from (68) that and , and then It follows from (54) and (73) that We now calculate and . It follows from (58) and (68) that where On the other hand, on near the origin We derive from (76)–(78) that It follows from (73) and (76) that for , Comparing the coefficients in (77) gives that for , From (79), (81), and following the definition of , we get Notice that , and hence where is a constant vector. Similarly, from (79) and (82), we obtain where is also a constant vector.

In what follows, we will seek appropriate and . From the definition of and (79), we obtain that where .

Set By (76), we have Substituting (84) and (89) into (86), we obtain which leads to where It follows that Similarly, substituting (85) and (90) into (87), we can get where and hence Thus, we can determine and from (84) and (85). Furthermore, in (75) can be expressed by the parameters and delay given in system (50). Based on the above analysis, we can see that each can be determined by the parameters. Thus we can compute the following quantities:

Hence we have the following theorem by the result of Hassard et al. [17].

Theorem 10. In (98), determines the directions of the Hopf bifurcation: if , then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for ; determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if ; determines the period of the bifurcating periodic solutions: the period increases (decrease) if .

6. Numerical Simulations and Discussion

In this section, we will carry out some numerical simulations for supporting our theoretical analysis.

In the following, we choose two sets of parameters.(i), , , , , , ,(ii), , , , , , .

For the parameters (i), and . The function is drawn for in Figure 3, from which we can see that there is only one positive critical value of the delay , denoted by , and .(1)When , Figure 1(b) shows that the positive equilibrium of system (1) is unstable.(2)When , we can see the positive equilibrium of system (2) is unstable (see Figures 4(a) and 4(b)).(3)When ), the positive equilibrium of system (2) is stable and there is a Hopf bifurcation that occurs when . Furthermore, we can obtain and . Therefore, the Hopf bifurcation of system (2) at the infected equilibrium is subcritical and the bifurcating periodic solutions are orbitally asymptotically stable (see Figure 4(b)).(4)If , that is the system (2) has no term , then the positive equilibrium if it exists is always unstable for (see Figures 1(b) and 5).

168340.fig.003
Figure 3: Graph of the function .
fig4
Figure 4: The infected steady state of system (2) is unstable when (a) (b) and stable when (c). The rest parameter values are given in (i), and the initial values are and .
fig5
Figure 5: The infected steady state of system (2) without term is unstable when (a) and (b). The rest parameter values are given in (i), and the initial values are and .

For the parameters (ii), the positive equilibrium of the DDEs is always stable for all (see Figures 1(a) and 6).

fig6
Figure 6: The infected steady state of system (2) is stable when (a) and (b). The rest parameter values are given in (ii), and the initial values are and .

In this paper, the Hopf bifurcation for a predator-prey system with -logistic growth and prey refuge is studied. It is shown that the decreasing prey refuge rate or the increasing logistic index will derive the stable ODEs unstable. Therefore, the refuges used by prey have a stabilizing effect on the considered interactions [611].

For the DDEs, the time delay could be looked as a bifurcation parameter. Under certain condition, there is a critical delay and a , if the delay , the system is unstable, and if , the system is stable. But under this condition, the DDEs without the term will always be unstable.

Acknowledgments

The authors would like to thank the referees for their helpful suggestions, which improved the quality of this paper greatly. This work is supported by the National Natural Science Foundation of China (no. 11271298).

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