HOPF BIFURCATION AND GLOBAL PERIODIC SOLUTION IN A DELAYED STAGE-STRUCTURED PREY-PREDATOR SYSTEM

A prey-predator model with stage-structured for predator and time delay is considered. The characteristic equations of the equilibrium points are analyzed, by applying the theorem of Hopf bifurcation, the conditions for the positive equilibrium occurring local Hopf bifurcation are given. The conditions for the existence of global Hopf bifurcation of the system are obtained. Finally, numerical simulation and brief conclusion are given. AMS Subject Classification: 34K13, 34K18, 34K20, 34K60, 92D25


Introduction
In population dynamics, time delays play an important role, which can make the system's equilibrium points loss of its stability and bifurcate various periodic solutions, even chaos.Recently, S.Y.Li and X.G.Xue [1,2] investigated the local Hopf bifurcation of a three-stage-structured prey-predator system, they took time delay τ as the bifurcation parameter, shown that, the positive equilibrium loses its stability and a local Hpof bifurcation occurring when the delay τ passes through the first critical value τ 0 .
In delayed population systems, are there existence of large-scale periodic solutions (global Hopf bifurcation) when τ far away from the first Hopf bifurcation critical values τ 0 ?C.J. Sun, M.A. Han and Y.P. Lin [3] studied the global Hopf bifurcation of a delayed logistic model by using a the result due to Wu [4].The global periodic solutions of a delayed predator-prey system have considered by X.P. Yan and W.T. Li [5].In this paper, we consider the Hopf bifurcation and global periodic solution of following stage-structured prey-predator system with hunting delay where γ = γ 1 + Ω, θ = θ 1 + a. x i (t)(i = 1, 2, 3) are the densities of immature preys, mature preys and old preys at time t, y(t) is the density of predator at time t, respectively.All of the parameters are positive, α is the birth rate of mature prey population, and γ 1 , θ 1 , b are the death rate of immature, mature and old prey population, respectively.Ω and a are the maturity rate and ageing rate of the prey population, respectively.η, c and f are the density dependent coefficients of immature and old prey population, predator population, respectively.k(0 < k < 1) is the rate of conversing prey into predator and E is the predation coefficient.τ is the hunting delay for predator population.The initial conditions for (1) are:
According to the Hopf bifurcation theorem for functional differential equations [7], we have the following result.

Global Hopf Bifurcation
In this section, we study the existence of global Hopf bifurcations.The method we used here is based on the global Hopf bifurcating theorem for general functional differential equations introduced by Wu [4].For convenience, we write system (1) as the following form where Let l (E 2 ,τ j ,2π/ω 0 ) be the connected component (E 2 , τ j , 2π/ω 0 ) in Σ, where τ j and ω 0 defined in (8).
Lemas 2. All solution of (1) are positive and uniformly bounded.
Proof.Assume (1) has a nontrivial periodic solution of period τ , then the following differential equations have periodic solutions.Let V (t) = 3 i=1 c i x i (t) + c 4 y(t), where c i (i = 1, 2, 3, 4) are positive constants to be determined, it follows that Applying Barbalat's lemma [8], we conclude that which contradicts the fact that (1) has periodic solutions.

Conclusion
In this paper, we studied a delayed stage-structured prey-predator system and analyzed the stability of the positive equilibrium, obtained the conditions of the positive equilibrium occurring local Hopf bifurcation and global Hopf bifurcation (global periodic solution).
Numerical examples by time-series plot, shown that the system considered local asymptotically stable and stable local Hopf bifurcation periodic solutions.And the populations can be coexistence with large periodic fluctuating under some conditions which caused by the large time delay far away from τ 0 , and the amplitudes of period oscillatory are increasing as time delays increased.In particular, we observe that the solutions of the delayed system could arbitrary close to zero when the delay tend to some critical values, which means that the prey or predator would tend to extinction.These are very interesting in mathematics and biology.