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

and Applied Analysis 3 For E1 = (K, 0), the corresponding characteristic equation is (λ + rθ) (λ + a − Kβεe −mτ 1 + Kεe e −λτ ) = 0. (14)


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 predatorprey interactions has been an active topic [1][2][3][4][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][7][8][9][10][11][12].
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; 0 <  < 1 and 1 −  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 predatorprey 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 .

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

Positivity of Solutions
Theorem 1. Solutions of system (2) with initial conditions (3) are positive for all  ≥ 0.
Next, we will prove the boundedness of solutions.

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 lim sup Hence, for  sufficiently small, there is a Calculating the derivative of  along solutions of system (2), we obtain where  0 =  − ( + ).Then there exists an  > 0, depending only on the parameters of system (2), such that () ≤  for all  large enough.Then (), () have an ultimately above bound.
For  0 = (0, 0), the corresponding characteristic equation is and the roots are which implies that the equilibrium  0 is always unstable.
For  1 = (, 0), the corresponding characteristic equation is It follows that or Denote and then we have for any  ≥ 0. Hence () = 0 has no positive root for  0 ≤ 1, and at least one positive for  0 > 1.Therefore, for all  ≥ 0, the equilibrium  1 is stable when  0 ≤ 1 and unstable when  0 > 1.
Summarizing the discussion above, we obtain the following conclusion.

Theorem 3. (i)
The equilibrium  0 is always unstable for all  ≥ 0.

The Hopf Bifurcation of DDEs.
In the following, we investigate the existence of purely imaginary roots  =  ( > 0) 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  ∈ [0,  max ), where  max is the maximum value in which  * exists.(e) each positive root () of (, ) = 0 is continuous and differentiable in  whenever it exists.
Let  be defined as in (d).From we have where It is obvious that property (d) is satisfied.Let ( 0 ,  0 ) be a point of its domain of definition such that ( 0 ,  0 ) = 0. We know that the partial derivatives   and   exist and are continuous in a certain neighborhood of ( 0 ,  0 ), and   ( 0 ,  0 ) ̸ = 0.By implicit function theorem, (e) is also satisfied.
Now let  =  ( > 0) be a root of (19).Substituting it into (19) and separating the real and imaginary parts yields (36) 0 100 200 300 400 500 600 700 800 900 1000 1 From (36), it follows that By the definitions of (, ), (, ) as in (20), and applying the property (a), (20) can be written as which yields Assume that  ∈  +0 is the set where () is a positive root of and for  ∉ , () is not defined.Then for all  in , () is satisfying Let  2 = ℎ; we have that We set Then, when Δ() ≥ 0, (ℎ, ) = 0 has real roots given by Note that and summarizing the discussion above, we have the following conclusion.
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].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.
(iii) If  1 (0) < 0 and the function   () has no positive zeros in , then the equilibrium  * is always unstable for all  ∈ [0,  max ).

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  * = ( * ,  * ).
(88) 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 Hence we have the following theorem by the result of Hassard et al. [17].

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.For the parameters (i),  max ≈ 1.45 and  = [0, 1.45).The function  0 () 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  * ≈ 0.58.(1) When  = 0, Figure 1(b) shows that the positive equilibrium of system (1) is unstable.
(4) If  = 0, that is the system (2) has no term  − , then the positive equilibrium if it exists is always unstable for  ≥ 0 (see Figures 1(b) and 5).
For the parameters (ii), the positive equilibrium of the DDEs is always stable for all  ∈  (see Figures 1(a) and 6).
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 [6][7][8][9][10][11].
For the DDEs, the time delay could be looked as a bifurcation parameter.Under certain condition, there is a critical delay  * and a  max , if the delay  ∈ [0,  * ), the system is unstable, and if  ∈ ( * ,  max ), the system is stable.But under this condition, the DDEs without the term  − will always be unstable.

Figure 4 :
Figure 4: The infected steady state  * of system (2) is unstable when  = 0.55 (a) (b) and stable when  = 0.75 (c).The rest parameter values are given in (i), and the initial values are  0 = 8 and  0 = 5.