Bifurcation Analysis of a Stage-structured Prey-predator System with Discrete and Continuous Delays

A three-stage-structured prey-predator model with discrete and continuous time delays is studied. The characteristic equations and the stability of the boundary and positive equilibrium are analyzed. The conditions for the positive equilibrium occurring Hopf bifurcation are given, by applying the theorem of Hopf bifurcation. Finally, numerical simulation and brief conclusion are given.


Introduction
In the natural world, there are many species whose individual members have a life history that takes them through two stages: immature and mature.In 1990, Aiello and Freedman [1] introduced single-species stagestructured model with time delay, and the stability of the system was studied.In 1997, Wang and Chen [2] introduced single-species stage-structured model without time delay and found that an orbitally asymptotically stable periodic orbit existence.In these papers [3], the authors assume that the life history of each population is divided into distinctive stages: the immature and mature members of the population, where only the mature member can reproduce themselves.However, in the nature many species go through three life stages: immature, mature and old.For example, many female animals lose reproductive ability when they are old.
A single species with three life history stage and cannibalism model have considered by S. J. Gao [4], and shown that the stability of the positive equilibrium can change a finite number of times at most as time delay is increased when the model under some parameters values.Recently, a nonautonomous three-stage-structured predator-prey system with time delay have studied by S. J. Yang and B. Shi [5], by using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution is obtained.And the local Hopf bifurcation and global periodic solutions for a delayed threestage-structured predator-prey considered by Li et al. [6,7].

Formulation of the Model
In this paper, we consider following three-stage-structured prey-predator model with discrete and continuous time delays , , x t x t x t are the densities of immature preys, mature preys and old preys population at time   , t y t is the density of predator population at time t, respectively.
All of the parameters are positive,  is the birth rate of mature prey population, and The initial conditions for (2.2) are 0, 0, 0, 0, , 0 .

Local Stability Analysis
Obviously, system (2.2) has two boundary equilibrium , holds), and an unique positive equilib- , , , , x y Q E x  be any arbitrary equilibrium.The linearized equations are where and the characteristic equation about E is given by Proof. 1) From (3.2), the characteristic equation about is given by  2) From (3.2), the characteristic equation about is given by Then, 1 2 ,

Existence of Local Hopf Bifurcation
The characteristic equation about the positive equilibrium is given by where and Note that if condition holds.By Routh-Hurwits criterion, all ro ve negative real parts.Then, the equilibri Suppose

  
) have at least one positive root.Without we assume distinct roots ve four pair of aginary And the direction of  jn  pass through the imaginar axis [8] when According to the Hopf bifurcation theorem for functional differential equations [9], (2.2) can undergoes a Hopf bifurcation at the positive equilibrium when
2) If condition 2) is asymptotically stable and unstable when  .To determine the stability, directio her properties of bifurcating periodic solutions, the normal form theory and ce er manifold argument should be considered [10].

Numerical Simulation n and ot nt
We consider following stage-structured delay system   .That is to say, time delay can make the positive equilibrium lose stability.It is shown that populations can be coexistence with periodic fluctuating under some conditions and such fluctuation is caused by the time delay.The bifurcating periodic solution (limit cycle) is stable when  from 5 to 40 and the amplitudes of pe- riod oscillatory are increasing as time delays increased.But, too large time delay would make the population to     be extinct, because the population arbitrary close to zero as time delay increase to some critical value.These are very interesting in mathematics and biology.
equations [9], (2.2) can a Hopf bifurcation at the positive equilibrium ing to the Hopf bifurcation theorem

Remark 1 .
is defined by(3.12).It must be pointe eorem 2 can not determine the stability and the direction of bifurcating periodic solutions, that is, the periodic solutions may exists either delays increased.But, too large time delay would make amplitudes of period oscillatory are the population to be die out, because the population very close to zero as time delay increase to some critical value.