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Shaoli Wang, Zhihao Ge, "The Hopf Bifurcation for a Predator-Prey System with -Logistic Growth and Prey Refuge", Abstract and Applied Analysis, vol. 2013, Article ID 168340, 13 pages, 2013. https://doi.org/10.1155/2013/168340
The Hopf Bifurcation for a Predator-Prey System with -Logistic Growth and Prey Refuge
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.
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 [1–5]. 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 [6–12].
Motivated by the work of Ko and Ryu  and Tsoularis and Wallace , 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 .
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
2.1. Positivity of Solutions
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
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
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
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)).
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  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 , 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).
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.
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 .
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
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. . 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.  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