two

A delayed three-species food chain system with two types of functional responses, Holling type and Beddington–DeAngelis type, is investigated. By analyzing the distribution of the roots of the associated characteristic equation, we get the sufficient conditions for the stability of the positive equilibrium and the existence of Hopf bifurcation. In particular, the properties of Hopf bifurcation such as direction and stability are determined by using the normal form theory and center manifold theorem. Finally, numerical simulations are


Introduction
In population dynamics, two-species predator-prey systems have been studied by many researchers [1,2,3,4,5,6].However, there is often the interaction among multiple species in nature, whose relationships are more complex than those in two species.Therefore, it is more realistic to consider the multiple-species predator-prey systems.Recently, Do et al. [7] proposed and studied the following three-species food chain system with Holling type II functional response and Beddington-DeAngelis functional response: , dy(t) dt = −d 1 y(t) + c 2 x(t)y(t) α 1 +x(t) − c 3 y(t)z(t) α 2 +y(t)+βz(t) , dz(t) dt = −d 2 z(t) + c 4 y(t)z(t) α 2 +y(t)+βz(t) , (1) where x(t), y(t) and z(t) denote the population densities of the prey, the mid-predator and the top predator, respectively.All the parameters in system (1) are positive constants.a is the birth rate of the prey.b is the intraspecific competition rate of the prey.c 1 and c 2 are the interspecific interaction coefficients between the prey and the mid-predator.c 3 and c 4 are the interspecific interaction coefficients between the mid-predator and the top predator.d 1 and d 2 are the death rates of the mid-predator and the top predator, respectively.α 1 and α 2 are the half-saturation constants and β scales the impact of the predator interference.In [7], Do et al.
proved that system (1) is dissipative and the conditions for the stability and the persistence of system (1) were obtained.
It is well-known that time delays have important effect on predator-prey systems.They could cause a stable equilibrium to become unstable and cause the population to fluctuate.And predator-prey systems with time delay have been investigated widely by many researchers [8,9,10,11,12,13].In [8], Xu investigated the stability and persistence of a predator-prey system with time delay and stage structure for the prey.In [12].Meng et al. investigated the stability and Hopf bifurcation of a delayed food web consisting of three species.Motivated by the work above, and considering that the consumption of prey by the predator throughout its past history governs the present birth rate of the predator, we incorporate time delay due to gestation of the mid-predator and the top predator into system (1) and get the following delayed predator-prey system: where the constant τ ≥ 0 is the time delay due to the gestation of the mid-predator and the top predator.In this paper, we shall investigate the effect of the time delay on the dynamics of system (2).The structure of this paper is arranged as follows.In Section 2, we will consider the local stability of the positive equilibrium and the existence of Hopf bifurcation at the positive equilibrium.In Section 3, we give the formula determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions.Finally, we give some simulations to support our theoretical predictions.

Properties of bifurcating periodic solutions
In Section 2, we have obtained the conditions for the existence of Hopf bifurcation when τ = τ 0 .In this section, we shall investigate the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions by using normal form theory and center manifold theorem in [15].
Then, we can get the following coefficients: Based on the discussion above, we can obtain the following results.

Theorem 2
The direction of the Hopf bifurcation is determined by the sign of µ 2 : if µ 2 > 0 (µ 2 < 0), the Hopf bifurcation is supercritical (subcritical).The stability of bifurcating periodic solutions is determined by the sign of β 2 : if β 2 < 0 (β 2 > 0), the bifurcating periodic solutions are stable (unstable).The period of the bifurcating periodic solutions is determined by the sign of T 2 : if T 2 > 0 (T 2 < 0), the period of the bifurcating periodic solutions increases (decreases).

Numerical simulation
In this section, we give a numerical example to support the theoretical analysis.We consider the following system: That is, the conditions (H 1 ) − (H 4 ) hold.Thus, from Theorem 1, the positive equilibrium E * (0.4097, 2.5617, 0.4232) is locally asymptotically stable when 0 ≤ τ < τ 0 , as is illustrated by Fig. 1.When the time delay τ passes through the critical value τ 0 , the positive equilibrium E * (0.4097, 2.5617, 0.4232) loses its stability and a Hopf bifurcation occurs and a family of periodic solutions bifurcate from the positive equilibrium E * (0.4097, 2.5617, 0.4232).This property can be seen from Fig. 2. In addition, from (13), we have C 1 (0) = −0.1317+ 0.2352i, µ 2 = −3.0557< 0, β 2 = −0.2634< 0, T 2 = 5.1947 > 0. Therefore, from Theorem 2, we know that the Hopf bifurcation is subcritical, the bifurcating periodic solutions are stable and the period of the bifurcating periodic solutions increases.

Conclusion
In the present paper, a three-species food chain system with time delay and the hybrid type of functional responses, Holling type and Beddington-DeAngelis type is studied.Compared with the system considered in [7], we mainly investigate the effect of the time delay due to gestation of the mid-predator and the top predator on the system.The sufficient conditions for the local stability of the positive equilibrium and the existence of periodic solutions via Hopf bifurcation at the positive equilibrium of system (2) are obtained.It is proved that when the conditions are satisfied, then there exists a critical value τ 0 of the time delay below which system (2) is stable and above which the system is unstable.Especially, system (2) undergoes a Hopf bifurcation at the positive equilibrium when τ = τ 0 .In reality, the occurrence of Hopf bifurcation means that the existence of the species in system (2) changes from the positive equilibrium to a limit cycle.For the further investigation, formulae are derived to determine the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem.If the bifurcating periodic solutions are stable, then we can conclude that the species in system (2) may coexist in an oscillatory mode.A numerical example verifying the theoretical results is also included.And from the numerical example, we can see that the species in system (2) may coexist in an oscillatory mode under some certain conditions.Do et al. [7] obtained that the species in system (2) without delay could coexist.However, we get that the species could also coexist with the time delay due to gestation of the mid-predator and the top predator.This is very valuable from the view of ecology.