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A delayed predator-prey system with Holling type II functional response and stage structure for both the predator and the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By means of persistence theory on infinite dimensional systems, it is proved that the system is permanent. By using Lyapunov functions and the LaSalle invariant principle, the global stability of each of the feasible equilibria of the model is discussed. Numerical simulations are carried out to illustrate the main theoretical results.

The predator-prey system is very important in population modelling and has been studied by many authors (see, e.g., [

It is generally recognized that some kinds of time delays are inevitable in population interactions and tend to be destabilizing in the sense that longer delays may destroy the stability of positive equilibria (see [

Motivated by the work of [

The initial conditions for system (

It is well known by the fundamental theory of functional differential equations [

The organization of this paper is as follows. In the next section, we investigate the local stability of each of the feasible equilibria of system (

In this section, we discuss the local stability of each equilibrium of system (

The characteristic equation of system (

The characteristic equation of system (

The characteristic equation of system (

When

If

If the inequality in

We now claim that

In conclusion, we have the following results.

For system (

If

If

if

Let

We now give an example to illustrate the main results in Theorem

In (

The temporal solution found by numerical integration of system (

In this section, we are concerned with the permanence of system (

System (

There are positive constants

Let

In order to study the permanence of system (

Let

Suppose that

there is a

T(t) is point dissipative in

Then

We are now able to state and prove the result on the permanence of system (

If

We need only to show that the boundaries of

In the following, we show that the conditions in Lemma

We now verify that

Since

In this section, we are concerned with the global stability of each of the feasible equilibria of system (

If

Let

The predator-extinction equilibrium

Assume that

Define

Define

The coexistence equilibrium

Let

System (

Define

Define

We now look for the invariant subset

We give an example to illustrate the result in Theorem

In (

The temporal solution found by numerical integration of system (

In this paper, we have incorporated stage structure for both the predators and the prey into a predator-prey model with time delay due to the gestation of the predator and Holling type II functional response. By using Lyapunov functionals and the LaSalle invariant principle, we have established sufficient conditions for the globally stability of each of the feasible equilibria of the system. As a result, we have shown the threshold for the permanence and extinction of the system. By Theorems

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (no. 11101117).