A delayed predator-prey system with stage structure for the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of equilibria of the system is discussed. The existence of Hopf bifurcation at the positive equilibrium is established. By using an iteration technique and comparison argument, respectively, sufficient conditions are derived for the global stability of the positive equilibrium and two boundary equilibria of the system. Numerical simulations are carried out to illustrate the theoretical results.

Stage-structure is a natural phenomenon and represents, for example, the division of a population into immature and mature individuals. As is common, the dynamics-eating habits, susceptibility to predators, and so forth. are often quite different in these two subpopulations. Hence, it is of ecological importance to investigate the effects of such a subdivision on the interaction of species. In [

In [

We note that most of the predator-prey models with time delays studied in the literature are all of the Kolmogorov-type. In [

Motivated by the work of Gao et al. [

The initial conditions for system (

This paper is organized as follows. In the next section, we introduce some notations and state several lemmas which will be essential to our proofs. In Section

In this section, we introduce some notations and state several results which will be useful in next section. Let

We now consider

To proceed further, we need the following results from [

Fix

We make the following assumptions for (

If

For all

The matrix

For each

If

The following result was established by Wang et al. [

Let (h1)–(h4) hold. Then the hypothesis (h0) is valid and the following.

If

If

Lemma

The following definitions and results are useful in proving our lemma.

System (

A square matrix

If (

We now consider the following delay differential system:

System (

The characteristic equation of system (

If

For system (

If

If

We represent the right-hand side of (

The matrix

From the definition of

Thus, the conditions of Lemma

By a similar argument one can show that all solutions of system (

In this section, we discuss the local stability of each equilibria and the existence of Hopf bifurcation of system (

It is easy to show that system (

then system (

then system (

We now study the local stability of each of the nonnegative equilibrium of system (

Let

The characteristic equation of system (

The characteristic equation of system (

The characteristic equation of system (

If

Noting that if

Let

We therefore obtain the following results.

For system (

The positive equilibrium

If

Let

Let (H2) hold. If

In this section, we are concerned with the global stability of the equilibria

Let (H2) hold. Then the positive equilibrium

Let

We derive from the first equation of system (

We derive from the second and the third equations of system (

For

For

For

Again, we derive from the second and the third equations of system (

For

For

If

Let

For

By Theorem

Let

Let

For

For

In this section, we give some examples to illustrate the main results.

In system (

The temporal solution found by numerical integration of system (

In system (

The temporal solution found by numerical integration of system (

In system (

The temporal solution found by numerical integration of system (

In system (

When

When

In this paper, we considered a delayed predator-prey model with stage structure for the predator. By using the iteration technique and comparison argument, respectively, sufficient conditions were established for the global stability of the positive equilibrium and two boundary equilibria of system (

The authors wish to thank the reviewers for their valuable comments and suggestions that greatly improved the presentation of this work. This work was supported by the National Natural Science Foundation of China (no. 10671209) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.