Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay

and Applied Analysis 3 When τ = 0, characteristic equation (10) or (12) becomes λ 3 + (d 1 + d 2 ) λ 2 + (d 3 + d 5 ) λ + d 4 + d 6 = 0. (13) It is easy to confirm that d 1 + d 2 > 0, d 4 + d 6 > 0 and (d 1 + d 2 )(d 3 + d 5 ) > d 4 + d 6 . By the Routh-Hurwitz criterion we know that all the roots of (13) have negative real parts. Thus, the positive equilibrium E 2 is locally asymptotically stable for τ = 0. Next, we will consider the eigenvalues of (12) for τ > 0. Suppose that there is a pure imaginary root λ = iω, ω > 0, then we get (−iω 3 − d 1 ω 2 ) (cosωτ + i sinωτ) − d 2 ω 2 + d 3 iω + d 4 + (d 5 iω + d 6 ) (cosωτ − i sinωτ) = 0. (14) Separating the real and imaginary parts, we have (−d 1 ω 2 + d 6 ) cosωτ + (ω3 + d 5 ω) sinωτ = d 2 ω 2 − d 4 , (d 5 ω − ω 3 ) cosωτ − (d 1 ω 2 + d 6 ) sinωτ = −d 3 ω. (15) By simple calculation, we can obtain the following equations: sinωτ = d 2 ω 5 + (d 1 d 3 − d 4 − d 2 d 5 ) ω 3 + (d 4 d 5 − d 3 d 6 ) ω ω 6 + d 2 1 ω 4 − d 2 5 ω 2 − d 2 6 , cosωτ = (d 3 − d 1 d 2 ) ω 4 + (d 1 d 4 − d 2 d 6 + d 3 d 5 ) ω 2 + d 4 d 6 ω 6 + d 2 1 ω 4 − d 2 5 ω 2 − d 2 6 .


Introduction
Since the pioneering theoretical works by Lotka [1] and Volterra [2], there were a lot of authors who studied all kinds of predator-prey models modeled by ordinary differential equations (ODEs).To reflect that the dynamical behavior of the models depends on the past history of the system, it is often necessary to incorporate time delays into the models.Therefore, a more realistic predator-prey model should be described by delayed differential equations (DDEs) [3][4][5][6][7][8][9][10][11].In general, delay differential equations exhibit more complicated dynamics on stability, periodic structure, bifurcation, and so on [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].In [27,28], the authors investigated the effect of the discrete delay on the stability of the model.In [29], the effect of the distributed delay on the stability of the model was investigated.In [11], the authors proposed a Logistic model with discrete and distributed delays:

𝑓 (𝑡 − 𝑠) 𝑥 (𝑠) d𝑠] ,
(1) where the parameters , ,  1 ,  2 are positive constants.The function  in (1) is called the delayed kernel, which is the weight given to the population  time units ago.And it was assumed that () ≥ 0 for all  ≥ 0, together with the normalization condition which ensures that the steady states of the model (1) are unaffected by the delay.They studied the stability of the positive equilibrium and existence of Hopf bifurcations, and direction and stability of the Hopf bifurcation were also analyzed.In [7], the authors proposed and investigated the following predator-prey model with time delay: where () and () can be interpreted as the population densities of the prey and the predator at time , respectively. 1 > 0 denotes the intrinsic growth rate of the prey, and  2 > 0 denotes the death rate of the predator.For the convenience of computation, they chose the same  > 0 as delays; the delay  represents the feedback time delay of the prey species to the growth of itself in term  11 ( − ), represents the feedback time delay of the predator species to the growth of itself in term  22 ( − ), represents the hunting delay in term  12 ( − ), and represents the time of the predator maturation in term  21 (−).The parameters   (,  = 1, 2) are all positive constants.They studied the stability of the positive equilibrium and existence of Hopf bifurcations.Motivated by [7,11,[27][28][29] and the references cited therein, in the present paper, we will consider the following predator-prey model with discrete and distributed delay: where () and () can be interpreted as the population densities of the prey and the predator at time , respectively. 1 > 0 denotes the intrinsic growth rate of the prey and  2 > 0 denotes the death rate of the predator;  > 0 represents the feedback time delay of the prey species to the growth of itself in term  11 ( − ), represents the feedback time delay of the predator species to the growth of itself in term  22 ( − ), and represents the hunting delay in term  12 ( − );   > 0 (,  = 1, 2).The function () is the same as the function () in system (1).Following the ideas of Cushing [30], we define () as the following weak kernel function: Next, we define a new variable: then using the linear chain trick technique, system (4) can be transformed into the following equivalent system: The organization of this paper is as follows.In Section 2, we will consider the existence and stability of equilibria of system (7).The existence of Hopf bifurcation is also discussed.In Section 3, by use of normal form theory and central manifold argument, we illustrate the direction and stability of Hopf bifurcation.In Section 4, we provide an example with some numerical simulations to verify the theoretical results, and we also give some brief discussion.

Local Stability of Equilibria and the Existence of Hopf Bifurcations
In this section, we will finish two tasks: (a) investigating the existence and stability of equilibriums of system (7) and (b) studying the effect of time delay on the system (7); that is, we will choose  as bifurcating parameter to analyze Hopf bifurcation.
Let the right equations of system (7) equal zero; we get the following algebraic equations: By simple computation, we know that the trivial equilibrium and boundary equilibrium of system (7) always exist with values  0 = (0, 0, 0) and  1 = ( 1 / 11 , 0,  1 / 11 ), respectively.In addition, we have the following results.
(ii) The eigenvalues of characteristic equations at the boundary equilibrium  1 are  1 = 0 and  2 = − < 0, and other eigenvalues are determined by  +  1  − = 0.When  = 0, it is easy to see that  3 = − 1 < 0, which means that this equilibrium is locally stable; whereas when  > 0, the sign of the real part of the eigenvalues can not be determined, which means that this equilibrium may be locally stable or unstable.
Next, we will consider the eigenvalues of (12) for  > 0. Suppose that there is a pure imaginary root  = ,  > 0, then we get Separating the real and imaginary parts, we have By simple calculation, we can obtain the following equations: Let then sin , cos  can be written as By adding the square of ( 18) and ( 19), we obtain where Denote  =  2 , then (20) becomes Let then the following assumption holds true.
In order to obtain the main result, it is necessary to make the following assumption.
Taking the derivative of  with respect to  in (12), it is easy to obtain which is equivalent to By (12), we have Take  =  into the above equation, we get Let Then we get Now, we can use the following lemma to get our result.
(ii) The positive equilibrium  2 of system (7) or system (4) undergoes a Hopf bifurcation when  =  0 .That is, system (7) has a branch of periodic of solutions bifurcating from the positive equilibrium  2 near  =  0 .

Direction and Stability of the Hopf Bifurcation
In this section, following the ideas of [33], we derive the explicit formulae for determining the properties of the Hopf bifurcation at critical value of  0 by using the normal form and the center manifold theory.Throughout this section, we always assume that system (7) undergoes Hopf bifurcation at the positive equilibrium  2 for  =  0 , and then ± 0 is the corresponding purely imaginary roots of the characteristic equation at the positive equilibrium  2 .
Next, we will use the ideas in [33] to compute the coordinates describing center manifold  0 at  = 0. Define On the center manifold  0 , we have where  and  are local coordinates for center manifold  0 in the direction of  * () and ().Note that  is real if   is real.We consider only real solutions.For the solution   ∈  0 of (43), since  = 0, we have where From ( 53) and ( 54), we have then we can obtain From the definition of (,   ), we have In order to determine  21 we need to compute  20 () and  11 ().From (43) and (53), we have where Note that on the center manifold  0 near to the origin, From ( 64), (66), and the definition of , we have Noting () = (0)  0  0  , hence where  1 = ( (1) 1 ,  (2)  1 ,  (3)  1 )  ∈  3 is a constant vector.Similarly, we have where 2 )  ∈  3 is a constant vector.In the following, we will find out  1 and  2 .From the definition of  and (64), we can obtain where () = (, 0).From ( 61) and ( 62 That is, By ( 39) and (53) we have Thus, Since  0  0 is the eigenvalues of (0) and (0) is the corresponding eigenvector, we obtain (77) Thus, substituting (68) and ( 75) into (70), we have or From which we can get Similarly, substituting (69) and ( 76) into (71), we can get or Thus, we can determine  20 () and  11 () from (68) and (69).Furthermore,  21 can be expressed by the parameters and delay.Thus, we can compute the following values: which determine the qualities of bifurcating periodic solution in the center manifold at critical value  0 ; that is,  2 determine the direction of the Hopf bifurcation: if  2 > 0 (  2 < 0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solution exists for  >  0 ( <  0 );  2 determines the stability of the bifurcating periodic solution: the bifurcating periodic solution is stable (unstable) if  2 < 0 ( 2 > 0); and  2 determines the period of the bifurcating periodic solution: the period increases (decreases) if  2 > 0 (< 0).

Numerical Investigations and Discussion
In this paper, we propose a two-dimensional predatory-prey model with discrete and distributed delay.Then, by introducing a new variable, the original system is transformed into an equivalent three-dimensional system.In Section 2, we analyze the existence and local stability of the equilibria of the three-dimensional system.The condition for the existence of a Hopf bifurcation is also obtained.In Section 3, by the use of normal form theory and central manifold argument, we establish the formulae for the direction and the stability of the Hopf bifurcation.
In order to confirm our main results obtained in this work, we consider the following special system:  By simple calculation, it is easy to see that model (84) exists a unique positive equilibrium  2 and  2 = (50/37, 6/37, 50/37).Note that the parameter set provided in model (84) satisfies the conditions of Theorem 2. When  = 1.1, the positive equilibrium  2 is asymptotically stable, as shown in Figures 1(a) and 1(c).It follows from the discussion in Section 2 that  0 ≈ 0.7246,  0 ≈ 1.1746, and   ( 0 ) = 0.09345 − 0.07497.Thus,  2 is stable when 0 ≤  <  0 , as indicated in Figures 1(a) and 1(c).
When  passes through the critical value  0 ,  2 loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcate from  2 , as shown in Figures 1(b) and 1(d).Since  2 > 0 and  2 < 0, the Hopf bifurcation is supercritical and the direction of the bifurcation is  >  0 and these bifurcating periodic solutions from  2 are stable; please see Figures 1(b) and 1(d) and Figure 2(a) for  = 1.5.Note that the model (84) may have very complex dynamics if we choose the time delay  as a bifurcation parameter.It follows from Figure 2 that the period of periodic solution is doubled as  increase, and around  = 1.6 (Figure 2 Both our theoretical and numerical results show that the positive equilibrium is asymptotically stable if  <  0 , which indicates that the dynamical behavior is simple for the considered system.However, if  >  0 , bifurcation and chaos may occur, which means that the considered system can take on very complex dynamics, and this may explain some complex phenomenon in the natural world.

Figure 1 :
Figure 1: The stability of unique positive equilibrium  2 .(a), (c) The equilibrium  2 is stable for  = 1.1.(b), (d) The equilibrium  2 is unstable and a stable periodic solution appears for  = 1.4.
(b)).If the time delay  is increasing further, a periodic solution with 4-time period appears around  = 1.75 (Figure 2(c)).Finally, the chaotic solution exists once the time delay reaches around  = 1.8 (Figure 2(d)).